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Statistical Dynamics of Classical Systems
Abstract
The statistical dynamics of a classical random variable that satisfies a nonlinear equation of motion is recast in terms of closed self-consistent equations in which only the observable correlations at pairs of points and the exact response to infinitesimal disturbances appear. The self-consistent equations are developed by introducing a second field that does not commute with the random variable. Techniques used in the study of the interacting quantum fields can then be employed, and systematic approximations can be obtained. It is also possible to carry out a "charge normalization" eliminating the nonlinear coupling in favor of a dimensionless parameter which measures the deviation from Gaussian behavior. No assumptions of spatial or time homogeneity or of small deviation from equilibrium enter. It is shown that previously inferred renormalization schemes for homogeneous systems were incomplete or erroneous. The application of the method to classical microscopic systems, where it leads from first principles to a coupled-mode description is briefly indicated.
... In order to apply renormalization group techniques, one can cast the stochastic equation into a path integral formulation which encompasses all the trajectories emanating from different noise realizations, following the Martin-Siggia-Rose-Janssen-de Dominicis formalism [24][25][26]. For the one-dimensional Burgers equation, this is straightforward and one obtains [21] ...
... Thus, one can choose a regulator matrix R κ with only three non-zero elements R κ,uū = R κ,ūu ≡ M κ and R κ,ūū ≡ N κ . Moreover, since the auxiliary fields do not enter any n > 2 vertex function, they completely decouple from the flow equations (24). One can thus focus on the sole velocity and response velocity sector and consider the reduced vertex functions (10), and the reduced propagator matrixḠ κ given by (see Appendix B) ...
... The reduced propagator matrix is purely longitudinal for the Burgers-KPZ equation as a consequence of irrotationality, whereas it is purely transverse for the Navier-Stokes equation as a consequence of incompressibility [23,27]. The exact flow equations (24) can be projected onto the reduced (u,ū) sector. The different terms involved in these flow equations can be written as diagrams, representing the field u by a line carrying an ingoing-into-avertex arrow and the response fieldū by a line carrying an outgoing-of-a-vertex arrow. ...
A new scaling regime characterized by a $z=1$ dynamical critical exponent has been reported in several numerical simulations of the one-dimensional Kardar-Parisi-Zhang and noisy Burgers equations. This scaling was found to emerge in the tensionless limit for the interface and in the inviscid limit for the fluid. Based on functional renormalization group, the origin of this scaling has been elucidated. It was shown to be controlled by a yet unpredicted fixed point of the one-dimensional Burgers-KPZ equation, termed inviscid Burgers fixed point. The associated universal properties, including the scaling function, were calculated. Here, we generalize this analysis to the multi-dimensional Burgers-KPZ equation. We show that the inviscid-Burgers fixed point exists in all dimensions $d$, and that it controls the large momentum behavior of the correlation functions in the inviscid limit. It turns out that it yields in all $d$ the same super-universal value $z=1$ for the dynamical exponent.
... To compute the nonlinear response of diffusive systems, we use the EFT of diffusion developed by Crossley, Glorioso and Liu [4], see [28] for a review and [29,30] for related work. It differs from previous approaches to fluctuating hydrodynamics [31,32] and macroscopic fluctuation theory [26] in that it provides a systematic controlled expansion in fluctuations. In particular, it captures general nonlinearities in the noise field that are not visible at the level of constitutive relations, and are missed in other approaches. ...
... , d is a spacetime index. This is a minimal way of producing a gauge-invariant partition function; the number of degrees of freedom (two per symmetry) matches those of previous approaches to fluctuating hydrodynamics [31,32] which contain one density and a noise field for each continuous symmetry. One then proceeds as usual in EFT by including in L all possible operators allowed by symmetries, in an expansion in derivatives and fields. ...
... The only Wilsonian coefficients involved are therefore the Taylor expansion coefficients of the diffusivity and conductivity (or susceptibility) around the density of interest (2). This agrees with other existing approaches to hydrodynamics [31,32], including macroscopic fluctuation theory [26] -we therefore expect that the results in this paper on the leading late time density three and four-point functions could be obtained from these other methods (to the best of our knowledge, they have not so far). However, we emphasize that the EFT approach of [4] allows to systematically compute corrections to this action (and therefore to observables such as the ones studied here), including terms that are missed in previous approaches. ...
Nonintegrable systems thermalize, leading to the emergence of fluctuating hydrodynamics. Typically, this hydrodynamics is diffusive. We use the effective field theory (EFT) of diffusion to compute higher-point functions of conserved densities. We uncover a simple scaling behavior of correlators at late times, and, focusing on three and four-point functions, derive the asymptotically exact universal scaling functions that characterize nonlinear response in diffusive systems. This allows for precision tests of thermalization beyond linear response in quantum and classical many-body systems. We confirm our predictions in a classical lattice gas.
... Like every theoretical model in physics, the validity of the assertions above crucially depends on the type and precision of the questions being asked. Even with a clear separation of scales, interactions between the "slow" and the ignored "fast" degrees of freedom can add up over macroscopic scales and lead to significant and qualitative deviations from the naive hydrodynamic predictions [1][2][3][4][5][6]. As an example, n-point retarded correlation functions or response functions of hydrodynamic observables (i.e. ...
... The state-of-the-art solution to these problems is presented by the Martin-Siggia-Rose (MSR) formalism [2], where the hydrodynamic constitutive relations are supplemented with random stochastic noise and physical observables are computed by averaging over all possible noise configurations sampled from a Gaussian distribution. The ambiguity in the spread of the Gaussian distribution is fixed by invoking the fluctuation-dissipation theorem (FDT) from thermal field theory [9]. ...
... As shown in [44], this ensures that stochastic loop corrections do not alter the analyticity properties of n-point correlation functions. 2 In particular, provided that retarded 2-point correlation functions have no unstable or acausal poles at tree-level (i.e. in the classical linear theory), an FDT-compatible UV-regularisation prescription ensures that stability and causality remains intact perturbatively at arbitrary loop orders in the SK-EFT. ...
A bstract
We construct stable and causal effective field theories (EFTs) for describing statistical fluctuations in relativistic diffusion and relativistic hydrodynamics. These EFTs are fully non-linear, including couplings to background sources, and enable us to compute n -point time-ordered correlation functions including the effects of statistical fluctuations. The EFTs we construct are inspired by the Maxwell-Cattaneo model of relativistic diffusion and Müller-Israel-Stewart model of relativistic hydrodynamics respectively, and have been derived using both the Martin-Siggia-Rose and Schwinger-Keldysh formalisms. The EFTs non-linearly realise the dynamical Kubo-Martin-Schwinger (KMS) symmetry, which ensures that n -point correlation functions and interactions in the theory satisfy the appropriate fluctuation-dissipation theorems. Since these EFTs typically admit ultraviolet sectors that are not fixed by the low-energy infrared symmetries, we find that they simultaneously admit multiple realisations of the dynamical KMS symmetry. We also comment on certain obstructions to including statistical fluctuations in the recently-proposed stable and causal Bemfica-Disconzi-Noronha-Kovtun model of relativistic hydrodynamics.
... where ρ is the Martin-Siggia-Rose-Janssen-De Dominicis (MSRJD) response field [85][86][87][88] and the effective Hamiltonian ...
... The slow coupling makes fluctuations at the boundary comparable to bulk fluctuations which modifies the fluctuating hydrodynamics (see (12a)) for the SSEP and introduces additional terms (see (14)) in the corresponding MSRJD-action [85][86][87][88]. In the MFT formulation for large deviation, these additional terms significantly modify the boundary condition of the hydrodynamic fields which makes solution of the corresponding Euler-Lagrange equation challenging. ...
... The action in (14a) is the Martin-Siggia-Rose-Janssen-De Dominicis action [85][86][87][88] of the stochastic differential equation (12) with ρ being the response field. One simple way to see this is by writing the transition probability between density profiles ρ(x, t i ) and ρ(x, t f ) governed by the dynamics (12) as ...
We revisit the one-dimensional model of the symmetric simple exclusion process slowly coupled with two unequal reservoirs at the boundaries. In its non-equilibrium stationary state, the large deviations functions of density and current have been recently derived using exact microscopic analysis in B Derrida, O Hirschberg and T Sadhu, J Stat Phys 182, 15 (2021). We present an independent derivation using the hydrodynamic approach of the macroscopic fluctuation theory (MFT). The slow coupling introduces additional boundary terms in the MFT-action which modifies the spatial boundary conditions for the associated variational problem. For the density large deviations, we explicitly solve the corresponding Euler-Lagrange equations using a simple local transformation of the optimal fields. For the current large deviations, our solution is obtained using the addi-tivity principle. In addition to recovering the expression of the large deviation functions, our solution describes the most probable path for these rare fluctuations.
... Schwartz and Katzav have shown how to apply SCE to stochastic nonlinear field theory using the Fokker-Planck equation [52,61]. Here, we provide a field-theoretic framework for the deterministic dynamical model of Barenblatt's using the Martin-Siggia-Rose (MSR) formalism [64] and apply SCE to this alternative form as well. We remark that the MSR technique has previously been used to formulate an action for Barenblatt's equation, enabling the use the exact RG to obtain the known asymptotic result perturbatively [65]. ...
... In order to make progress, we use the fact that it is possible to express the solution of a differential equation -a deterministic function -as the limit of a probability distribution. Thus, our strategy is to convert the differential equation to a field theory using the so-called Martin-Siggia-Rose (MSR) formalism [64,[79][80][81], where the solution to differential equations becomes an expectation over a fluctuating field. For Barenblatt's equation, this strategy was first implemented by Yoshida [65], who used the so-called exact or functional renormalization group to calculate the anomalous dimension to second order in ϵ. ...
The method of self-consistent expansions is a powerful tool for handling strong coupling problems that might otherwise be beyond the reach of perturbation theory, providing surprisingly accurate approximations even at low order. First applied in its embryonic form to fully-developed turbulence, it has subsequently been successfully applied to a variety of problems that include polymer statistics, interface dynamics and high order perturbation theory for the anharmonic oscillator. Here we show that the self-consistent expansion can be applied to singular perturbation problems arising in the theory of partial differential equations. We demonstrate its application to Barenblatt's nonlinear diffusion equation for porous media filtration, where the long-time asymptotics exhibits anomalous dimensions which can be systematically calculated using the perturbative renormalization group. We find that even the first order self-consistent expansion improves the approximation of the anomalous dimension obtained by the first order perturbative renormalization group, especially in the strong coupling regime. We also develop a field-theoretic framework for deterministic partial differential equations to facilitate the application of self-consistent expansions to other dynamic systems, and illustrate its application using the example of Barenblatt's equation. The scope of our results on the combination of renormalization group and self-consistent expansions is limited to partial differential equations whose long-time asymptotics is controlled by incomplete similarity. However, our work suggests that these methods could be applied to a broader suite of singular perturbation problems such as boundary layer theory, multiple scales analysis and matched asymptotic expansions, for which excellent approximations using renormalization group methods alone are already available.
... Following the recent literature on effective theories for thermal systems [30][31][32][33][34], we write down a dissipative Lagrangian for the dynamics of an RLC circuit with Gaussian white noise. The approach is closely related to the much older Martin-Siggia-Rose (MSR) Lagrangian [35] for stochastic dynamics. Indeed, a crucial aspect of our construction is that it necessarily incorporates both dissipation along with stochastic fluctuations, which are mandated by the fluctuation-dissipation theorem. ...
... This approach is inspired by the desire to calculate the transition probability P (x α (t) = a α |x α (0) = b α ) -namely, the probability to go from microstate b to microstate a in time t. One popular way of trying to evaluate this transition probability (density function) is by using the Martin-Siggia-Rose (MSR) path integral: [35] ...
We revisit the theory of dissipative mechanics in RLC circuits, allowing for circuit elements to have nonlinear constitutive relations, and for the circuit to have arbitrary topology. We systematically generalize the dissipationless Hamiltonian mechanics of an LC circuit to account for resistors and incorporate the physical postulate that the resulting RLC circuit thermalizes with its environment at a constant positive temperature. Our theory explains stochastic fluctuations, or Johnson noise, which are mandated by the fluctuation-dissipation theorem. Assuming Gaussian Markovian noise, we obtain exact expressions for multiplicative Johnson noise through nonlinear resistors in circuits with convenient (parasitic) capacitors and/or inductors. With linear resistors, our formalism is describable using a Kubo-Martin-Schwinger-invariant Lagrangian formalism for dissipative thermal systems. Generalizing our technique to quantum circuits could lead to an alternative way to study decoherence in nonlinear superconducting circuits without the Caldeira-Leggett formalism.
... It has long been recognized that the equations governing the population level behavior of Markov processes based on stochastic individual-based dynamics are often very similar to the equations describing the behavior of many interacting particles in statistical physics Black and McKane, 2012) and quantum mechanics (Martin et al., 1973;Hochberg et al., 1999;Baez and Biamonte, 2018). As a consequence, powerful heuristic tools originally developed in physics can be leveraged, under many situations, to study the behavior of systems in which a large number of individuals interact in a stochastic manner (Martin et al., 1973;Doi, 1976;Peliti, 1985;Hochberg et al., 1999;Thomas et al., 2014;Chow and Buice, 2015;Weber and Frey, 2017;Baez and Biamonte, 2018). ...
... It has long been recognized that the equations governing the population level behavior of Markov processes based on stochastic individual-based dynamics are often very similar to the equations describing the behavior of many interacting particles in statistical physics Black and McKane, 2012) and quantum mechanics (Martin et al., 1973;Hochberg et al., 1999;Baez and Biamonte, 2018). As a consequence, powerful heuristic tools originally developed in physics can be leveraged, under many situations, to study the behavior of systems in which a large number of individuals interact in a stochastic manner (Martin et al., 1973;Doi, 1976;Peliti, 1985;Hochberg et al., 1999;Thomas et al., 2014;Chow and Buice, 2015;Weber and Frey, 2017;Baez and Biamonte, 2018). Indeed, various tools from statistical and quantum mechanics have already been successfully applied to study stochasticity in biological populations (O'Dwyer et al., 2009;de Vladar and Barton, 2011;Black and McKane, 2012;Schraiber, 2014). ...
Infinitely many distinct trait values may arise in populations bearing quantitative traits, and modelling their population dynamics is thus a formidable task. While classical models assume fixed or infinite population size, models in which the total population size fluctuates due to demographic noise in births and deaths can behave qualitatively differently from constant or infinite population models due to density-dependent dynamics. In this paper, I present a stochastic field theory for the eco-evolutionary dynamics of finite populations bearing one-dimensional quantitative traits. I derive stochastic field equations that describe the evolution of population densities, trait frequencies, and the mean value of any trait in the population. These equations recover well-known results such as the replicator-mutator equation, Price equation, and gradient dynamics in the infinite population limit. For finite populations, the equations describe the intricate interplay between natural selection, noise-induced selection, eco-evolutionary feedback, and neutral genetic drift in determining evolutionary trajectories. My methods use ideas from statistical physics and present an alternative to some recently proposed measure-theoretic frameworks.
... Model f (x) g(x, x ′ ) Ornstein-Uhlenbeck (OU) process [18] x x ′ SIS model of epidemic spreading [9] x (1 − x)x ′ Lotka-Volterra (LV) model [7,8] x(x−1) xx ′ Neural network (NN) model [1,4] x tanh(x ′ ) Kuramoto model [14] 0 sin(x ′ −x) The foremost problem in the study of complex systems is how to reduce the dynamics of many interacting elements to the dynamics of a few variables [21]. Dynamical mean-field theory (DMFT) [22][23][24] is a powerful method to tackle this problem in the limit N → ∞, yielding a solution in terms of the path-probability for the effective dynamics of a single degree of freedom. The application of DMFT to models described by Eq. (1) has been attracting an enormous interest [2-5, 7, 8, 15, 19, 20, 25-31], especially in the context of neural networks and ecosystems. ...
... Solution through DMFT. We solve the coupled dynamics of Eq. (1) on directed networks by using dynamical mean-field theory (DMFT) [22][23][24]. We consider the sparse regime, where the mean degree c is finite, independent of N . ...
Although real-world complex systems typically interact through sparse and heterogeneous networks , analytic solutions of their dynamics are limited to models with all-to-all interactions. Here, we solve the dynamics of a broad range of nonlinear models of complex systems on sparse directed networks with a random structure. By generalizing dynamical mean-field theory to sparse systems, we derive an exact equation for the path-probability describing the effective dynamics of a single degree of freedom. Our general solution applies to key models in the study of neural networks, ecosystems, epidemic spreading, and synchronization. Using the population dynamics algorithm, we solve the path-probability equation to determine the phase diagram of a seminal neural network model in the sparse regime, showing that this model undergoes a transition from a fixed-point phase to chaos as a function of the network topology.
... Keeping these questions in mind, in the following, we construct the EFT including fluctuations for a selfinteracting system with a single conserved density together with a single gapped mode; the longest-lived gapped mode, acting as a UV-regulator for the EFT. Our effective action is consistent with the general framework of [1], however, derived through the traditional Martin-Siggia-Rose (MSR) formalism [22]. Explicitly calculating the one-loop retarded density-density Green's function, we discuss how the renormalization of the diffusion constant at τ ¼ 0, performed in [6,7], is affected by the gapped mode. ...
... For this, we construct an effective action for n whose equation of motion is (5). We do this in the framework of MSR formalism [22]. The idea is to put a noise term on the right side of (5) and then impose the fluctuation-dissipation theorem to fix its strength. ...
In a system with one conserved charge the charge diffusion is modified by nonlinear self-interactions within an effective field theory (EFT) of diffusive fluctuations. We include the slowest ultraviolet (UV) mode, constructing a UV-regulated EFT. The relaxation time of this UV mode is protected from renormalization, as supported by experimental data in a bad metal system. Furthermore, the retarded density-density Green’s function acquires four branch points, eventually increasing the range of applicability. We discuss the fate of long time tails as well as implications for the quark gluon plasma.
... where the subscript 0 denotes the linear approximation. Substituting expression (16) for in terms of φ into Eq. (19) and averaging the product (t, x) (0, y) in accordance with Eq. (10), we obtain ...
... The nonlinear interaction of flow fluctuations can be consistently examined in the framework of the Wyld diagrammatic technique [15]. The diagrammatic technique can be derived from the representation of correlation functions as functional integrals over the observed variables and auxiliary fields [16]. The integration is performed like in the quantum field theory [17]. ...
We analytically examine fluctuations of vorticity excited by an external random force in two-dimensional fluid. We develop the perturbation theory enabling one to calculate nonlinear corrections to correlation functions of the flow fluctuations found in the linear approximation. We calculate the correction to the pair correlation function and the triple correlation function. It enables us to establish the criterion of validity of the perturbation theory for different ratios of viscosity and bottom friction. We find that the corrections to the second moment are anomalously weak in the cases of small bottom friction and small viscosity and relate the weakness to the energy and enstrophy balances. We demonstrate that at small bottom friction the triple correlation function is characterized by universal scaling behavior in some region of lengths. The developed perturbation method was verified and confirmed by direct numerical simulations.
... A stochastic dynamics problem is equivalent to a quantum field theory of the doubled set of fields, E and E ′ (each basic field acquires an auxiliary field) [13,14,[54][55][56], in the sense that the statistical averages ⟨. . .⟩ of the dynamic quantities over the ensemble of configurations in the stochastic problem can be identified with the functional averages taken with weight exp S(E, E ′ ) for some action functional S(E, E ′ ).The generating functional of full Green functions for the stochastic problem (5) can be represented by the following functional integral [14]: ...
... In Figure 1, we drew the one-loop order diagram expansions for the pairwise correlation function ⟨EE⟩ (as can be seen in Figure 1a) and the simplest response function ⟨δE/δ f ⟩ (see Figure 1b). The diagram expansions shown in Figure 1 are typical for the stochastic nonlinear dynamics problems [12][13][14][54][55][56]. In particular, the first graphs of the diagram expansions (presented in Figure 1) coincide with the Wyld diagrams obtained for the stochastic Navier-Stokes equation in the theory of fully developed turbulence [57], although they differ in the orders with two or more loops [14]. ...
The long-term, large-scale behavior in a problem of stochastic nonlinear dynamics corresponding to the Abelian sandpile model is studied with the use of the quantum-field theory renormalization group approach. We prove the multiplicative renormalization of the model including an infinite number of coupling parameters, calculate an infinite number of renormalization constants, identify a plane of fixed points in the infinite dimensional space of coupling parameters, discuss their stability and critical scaling in the model, and formulate a simple law relating the asymptotic size of an avalanche to a model exponent quantifying the time-scale separation between the slow energy injection and fast avalanche relaxation processes.
... Further developments of Wiener's path integration include the works by Onsager and Machlup on irreversible processes [7,8]. Moreover, using the operator formulation proposed by Martin, Siggia, and Rose [9], Janssen and De Dominics [10][11][12] developed an alternative formalism for path integration (known as the MSRJD integral), which opened the possibility of investigating diverse phenomena associated with non-equilibrium thermodynamics. Moreover, the MSRJD path integration can be understood as a functional representation of classical differential equations, such as the Fokker-Planck and Langevin equations. ...
We show that every operator in $L^{2}$ has an associated measure on a space of functions and prove that it can be used to find solutions to abstract Cauchy problems, including partial differential equations. We find explicit formulas to compute the integral of functions with respect to this measure and develop approximate formulas in terms of a perturbative expansion. We show that this method can be used to represent solutions of classical equations, such as the diffusion and Fokker-Plank equations, as Wiener and Martin-Siggia-Rose-Jansen-de Dominics integrals, and propose an extension to paths in infinite dimensional spaces.
... The most promising direction for studying the stochastic problem (1)-(5) is considered to be its reduction to a quantum field theory model. According to the fundamental theorem [6], the stochastic equation (1)-(5) is equivalent to a quantum field theory with a double set of transverse fields with the action given as ...
Within the framework of the renormalization group approach in the stochastic model of fully developed turbulence, the $\beta$-function has been calculated in the fourth order of perturbation theory for high-dimensional spaces $d \rightarrow \infty$. The position of the fixed point of the renormalization group in the fourth order of the $\varepsilon$-expansion has been determined, and the index $\omega$, which defines the infrared stability of this point, has been calculated. We demonstrate the possibility of significantly reducing the number of Feynman diagrams through mutual cancellation. The results obtained allow us to find the four terms of the $\varepsilon$-expansion of the index $\omega$: $$ \omega = 2\,\varepsilon + \frac{2}{3}\,\varepsilon^2 + \frac{10}{9}\,\varepsilon^3 + \frac{56}{27}\,\varepsilon^4.
... The present work generalizes these results to the study of linearly-coupled SDEs on sparse graphs introducing a message passing algorithm for local cavity moments, such as one-time averages, and two-time response functions and correlation functions. This is done first applying the dynamic cavity method to a graphical model representation obtained from the path-integral formulation of coupled SDEs (Martin-Siggia-Rose-Janssen-DeDominicis formalism [48][49][50]) and then employing a second-order expansion in the interaction strength to the obtained action, in the spirit of [27]. Notably, our expansion aligns with a Gaussian ansatz for the cavity messages, which is exact for linear systems of SDEs with additive noise and in the presence of global constraints, like in the case of spherical 2-spin models. ...
Stochastic dynamics on sparse graphs and disordered systems often lead to complex behaviors characterized by heterogeneity in time and spatial scales, slow relaxation, localization, and aging phenomena. The mathematical tools and approximation techniques required to analyze these complex systems are still under development, posing significant technical challenges and resulting in a reliance on numerical simulations. We introduce a novel computational framework for investigating the dynamics of sparse disordered systems with continuous degrees of freedom. Starting with a graphical model representation of the dynamic partition function for a system of linearly-coupled stochastic differential equations, we use dynamic cavity equations on locally tree-like factor graphs to approximate the stochastic measure. Here, cavity marginals are identified with local functionals of single-site trajectories. Our primary approximation involves a second-order truncation of a small-coupling expansion, leading to a Gaussian form for the cavity marginals. For linear dynamics with additive noise, this method yields a closed set of causal integro-differential equations for cavity versions of one-time and two-time averages. These equations provide an exact dynamical description within the local tree-like approximation, retrieving classical results for the spectral density of sparse random matrices. Global constraints, non-linear forces, and state-dependent noise terms can be addressed using a self-consistent perturbative closure technique. The resulting equations resemble those of dynamical mean-field theory in the mode-coupling approximation used for fully-connected models. However, due to their cavity formulation, the present method can also be applied to ensembles of sparse random graphs and employed as a message-passing algorithm on specific graph instances.
... To show this, we use a stochastic field theory formalization [33,[51][52][53][54][55][56] of a biologically realistic model of stochastic leaky and fire (sLIF) neurons with nonlinear intensity functions. We study its dynamics to compare the impact of threshold power-law and exponential intensity functions on population activity. ...
Neurons in the brain continuously process the barrage of sensory inputs they receive from the environment. A wide array of experimental work has shown that the collective activity of neural populations encodes and processes this constant bombardment of information. How these collective patterns of activity depend on single neuron properties is often unclear. Single-neuron recordings have shown that individual neural responses to inputs are nonlinear, which prevents a straightforward extrapolation from single neuron features to emergent collective states. In this work, we use a field theoretic formulation of a stochastic leaky integrate-and-fire model to study the impact of nonlinear intensity functions on macroscopic network activity. We show that the interplay between nonlinear spike emission and membrane potential resets can i) give rise to metastable transitions between active firing rate states, and ii) can enhance or suppress mean firing rates and membrane potentials in opposite directions.
... In this work, we follow and alternative road. By means of the Martin-Siggia-Rose-Jenssen-De Dominicis (MSRJD) formalism [9][10][11] , we built a generating functional for correlation functions, subsequently considering the transition to the continuum limit. In this limit, we show a hidden symmetry generated by area preserving diffeomorphisms transformations. ...
We investigate a system of two-dimensional interacting Brownian particles at finite density. In the continuum limit, we uncover a hidden symmetry under area-preserving diffeomorphism transformations. This symmetry leads to the conservation of local vorticity. By calculating the generating functional within the saddle-point plus Gaussian fluctuations approximation, we reveal the emergence of a $U(1)$ gauge symmetry. This emergent symmetry enables us to describe the dynamics of density fluctuations as a gauge theory. We solve the corresponding equations of motion for various local and non-local two-body potentials, demonstrating the presence of multiple dynamical regimes and associated dynamical phase transitions.
... At the ultraviolet cutoff k = Λ, the effective action Γ Λ [Φ] coincides with the classical action S[Φ] and at k = 0 the effective action is the quantum action as fluctuations of different momentum modes are integrated in successively. To formulate the effective action in the closed time path, starting from the Langevin equation in Eq. (1) and (2), we perform the path integral about the noise variables that result in the Martin-Siggia-Rose response field [63]. The renormalization group (RG) scale k dependent effective action of Model H in the Schwinger-Keldysh formalism is ...
The critical dynamics of Model H with a conserved order parameter coupled to a transverse momentum density which describes the gas-liquid or binary-fluid transitions is investigated within the functional renormalization group approach formulated on the closed time path. According to the dynamic scaling analysis, Model H and QCD critical end point belong to the same dynamic universality class in the critical region. The higher-order correction of the transport coefficient $\bar\lambda$ and shear viscosity $\bar\eta$ arising from mode-couplings are obtained by calculating the two-point correlation functions. The flow equation of a dimensionless coupling constant for nondissipative interactions is derived to look for the fixed-point solution of the system. The scaling relation between the critical exponent of the transport coefficient and that of the shear viscosity is estimated. Finally, the dynamic critical exponent $z$ is obtained as a function of the spatial dimension $d$.
... In order to apply the RG method, it is advantageous to recast the Langevin-like formulation of the model in terms of path integrals. Following the well-known Janssen-De Dominicis procedure [29,53], stochastic equations (9) and (10) are fully equivalent to the field-theoretic model of a double set of (unrenormalized) fieldsΦ = {v,b,v ′ ,b ′ }, wherev ′ andb ′ are Martin-Siggia-Rose (MSR) response fields [63]. For the ensuing application of the RG method, it is necessary to distinguish between bare (unrenormalized) and renormalized parameters [29,40]. ...
We investigate the stochastic version of the paradigmatic model of magnetohydrodynamic turbulence. The model can be interpreted as an active vector admixture subject to advective processes governed by turbulent flow. The back influence on fluid dynamics is explicitly taken into account. The velocity field is generated through a fully developed turbulent flow taking into account the violation of spatial parity, which is introduced through the helicity parameter ρ. We consider a generalized setup in which parameter A is introduced in model formulation, which is associated with the interaction part of the model, and its actual value represents different physical systems. The model is analyzed by means of the field-theoretic renormalization group. The calculation is performed using ε-expansion, where ε is the deviation from the Kolmogorov scaling. Two-loop numerical calculations of the renormalization constant associated with the renormalization of the magnetic field are presented.
... The same hydrodynamics extends for multi-lane generalizations with the number of lanes ≪ ℓ d . Our detailed derivation for (1), presented in the Supplemental Materials (SM) [42], is based on an evaluation of the Martin-Siggia-Rose-Janssen-de Dominicis Action [43][44][45][46] of the stochastic differential equation (1) following a coarsegraining of the microscopic generator [47]. This method is generalizable for variations of the dynamics and dimensionality. ...
We present a fluctuating hydrodynamic description of a non-polar active lattice gas model with excluded volume interactions that exhibits motility-induced phase separation under appropriate conditions. For quasi-one dimension and higher, stability analysis of the noiseless hydrodynamics gives quantitative bounds on the phase boundary of the motility-induced phase separation in terms of spinodal and binodal. Inclusion of the multiplicative noise in the fluctuating hydrodynamics describes the exponentially decaying two-point correlations in the stationary-state homogeneous phase. Our hydrodynamic description and theoretical predictions based on it are in excellent agreement with our Monte-Carlo simulations and pseudo-spectral iteration of the hydrodynamics equations. Our construction of hydrodynamics for this model is not suitable in strictly one-dimension with single-file constraints, and we argue that this breakdown is associated with micro-phase separation.
... In this work, we would like to study the critical dynamics of model A within the functional renormalization group formulated on the Schwinger-Keldysh closed time path. The real-time FRG with the Schwinger-Keldysh path integral and the relevant techniques thereof have been discussed in detail in our former work [21]; see also [41,42]. Following the approach there, one is able to arrive at the renormalization group (RG) scale dependent effective action corresponding to Eq. (1) with Eq. (2), that is, ...
The Schwinger-Keldysh functional renormalization group developed by Y.-y. Tan [Real-time dynamics of the O ( 4 ) scalar theory within the fRG approach, ] is employed to investigate critical dynamics related to a second-order phase transition. The effective action of model A is expanded to the order of O ( ∂ 2 ) in the derivative expansion for the O ( N ) symmetry. By solving the fixed-point equations of effective potential and wave function, we obtain static and dynamic critical exponents for different values of the spatial dimension d and the field component number N . It is found that one has z ≥ 2 in the whole range of 2 ≤ d ≤ 4 for the case of N = 1 , while in the case of N = 4 , the dynamic critical exponent turns to z < 2 when the dimension approach towards d = 2 .
Published by the American Physical Society 2024
... We derive this result using a Martin-Siggia-Rose path integral formalism [29] for DMFT in Appendix D. Full DMFT equations can be found in Appendix D.2. Following prior works on DMFT for infinite width feature learning, the large-N limit can be straightforwardly obtained from a saddle point of the DMFT action [15,11,30,17]. ...
In this work, we analyze various scaling limits of the training dynamics of transformer models in the feature learning regime. We identify the set of parameterizations that admit well-defined infinite width and depth limits, allowing the attention layers to update throughout training--a relevant notion of feature learning in these models. We then use tools from dynamical mean field theory (DMFT) to analyze various infinite limits (infinite key/query dimension, infinite heads, and infinite depth) which have different statistical descriptions depending on which infinite limit is taken and how attention layers are scaled. We provide numerical evidence of convergence to the limits and discuss how the parameterization qualitatively influences learned features.
... Derivation: In the following, we outline our derivation of (2b) within the fluctuating hydrodynamics framework of MFT. The crucial idea behind MFT is to recognize the relevant hydrodynamic modes for a coarse-grained description of the dynamics and characterize the probability of their evolution in terms of an Action, which is analogous to the Martin-Siggia-Rose-Janssen-De Dominicis (MSRJD) Action [58][59][60][61] of the associated fluctuating hydrodynamics equation. For SSEP, the relevant hydrodynamic mode is the locally conserved density ρ(x, t) evolving by [48,52] ...
... Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. example [2,3]. Their application to the modern study of classical stochastic processes can be traced back to the work of Onsager and Machlup [4], Stratonovich [5], and Graham [6]. ...
Transition probabilities for stochastic systems can be expressed in terms of a functional integral over paths taken by the system. Approximately evaluating this integral by the saddle point method in the weak-noise limit leads to a remarkable mapping between dominant stochastic paths through the potential V and conservative, Hamiltonian mechanics with an effective potential −|∇V|2 . The conserved ‘energy’ in this effective system has dimensions of power. We show that this power, H, can be identified with the Laplace parameter of the time-transformed dynamics. As H → 0, corresponding to the long-time limit, the equilibrium Boltzmann density is recovered. However, keeping H finite leads to insights into the non-equilibrium behaviour of the system. Moreover, it facilitates the explicit summation over families of trajectories, which is far harder in the time domain, and turns out to be essential for making contact with the long-time limit in some cases. We illustrate the validity of these results using simple examples that can be explicitly solved by other means.
... To show that our results hold also in this case, at least if the asymmetry is not too strong, we have analyzed the case of small asymmetry in perturbation theory. The analysis of the Martin-Siggia-Rose-De Dominicis-Janssen action [67][68][69][70] allows us to conclude that a small degree of asymmetry (γ = 1 − , 1) does not affect qualitatively the results we shall present in the next section, therefore establishing that our findings for the symmetric case also holds for small asymmetry (see Appendix D for more details). We have also confirmed this result by numerical simulations for γ < 1. ...
How diversity is maintained in natural ecosystems is a long-standing question in Theoretical Ecology. By studying a system that combines ecological dynamics, heterogeneous interactions, and spatial structure, we uncover a new mechanism for the survival of diversity-rich ecosystems in the presence of demographic fluctuations. For a single species, one finds a continuous phase transition between an extinction and a survival state, that falls into the universality class of Directed Percolation. Here we show that the case of many species with heterogeneous interactions is different and richer. By merging theory and simulations, we demonstrate that with sufficiently strong demographic noise, the system exhibits behavior akin to the single-species case, undergoing a continuous transition. Conversely, at low demographic noise, we observe unique features indicative of the ecosystem's complexity. The combined effects of the heterogeneity in the interaction network and migration enable the community to thrive, even in situations where demographic noise would lead to the extinction of isolated species. The emergence of mutualism induces the development of global bistability, accompanied by sudden tipping points. We present a way to predict the catastrophic shift from high diversity to extinction by probing responses to perturbations as an early warning signal.
... However, fluctuations should always appear on par with dissipative effects, according to the fluctuation-dissipation theorem [77]. This can be amended via a bottom-up construction by including fluctuations as noise in the equations, thus turning hydrodynamics into a proper EFT, following the Martin-Siggia-Rose formalism [42,73,[78][79][80]. More recently, also a top-down view on dissipative fluctuating hydrodynamics has been developed [43,44,81,82], based on the closed time-path formalism of Schwinger and Keldysh [83,84], which has the advantage of working at the full non-linear level, providing access to the n-point correlators. ...
At its core, hydrodynamics is a many-body low-energy effective theory for the
long-wavelength, long-timescale dynamics of conserved charges in systems
close to thermodynamic equilibrium. It has a wide range of applications,
that span from nuclear physics, astrophysics, cosmology, and more recently
strongly-interacting electronic phases of matter. In condensed matter, however,
symmetries are often only approximate, and softly broken by the
presence of the lattice, impurities and defects, or because the symmetry
is accidental. Therefore, the hydrodynamic regime must be expanded to
include weak non-conservation effects, which lead to a theory known as
quasihydrodynamics.
In this thesis we make progress in understanding the theory of (quasi)
hydrodynamics, with a specific focus on applications to condensed matter
systems and their holographic description. First, we consider an electron fluid
in a strong magnetic field for which translations are broken by the presence of
Charge Density Waves. Therefore, the low-energy theory contains Goldstone
modes associated with the broken symmetries, which modify the spectrum
and transport properties. We focus on a new regime at non-zero lattice
pressure and compare with holographic models, finding perfect agreement
between the two descriptions.
Next we consider a simple system that mimics the weakly-coupled Drude
model from a hydrodynamic perspective. Specifically, a charged fluid in an
external electric field in the presence of impurities that relax momentum and
energy. We look for steady states, and we find that stationarity constraints
should be modified to include relaxations, which consequently give novel
predictions for the thermoelectric transport.
Finally, we study the effect of the axial anomaly on the transport properties
of Weyl semimetals in the hydrodynamic regime. We suggest a better
approach to deal with the derivative counting of the magnetic field, correcting
mistakes in the literature. Subsequently, we discuss the DC values of the
conductivities and look for models that obey fundamental and phenomenological
considerations. We find that generalized relaxations, which we study in depth using variational methods and kinetic-theory approaches, are a
necessary ingredient to have finite DC conductivity, conserve electric charge,
and have the correlators obey Onsager relations. Moreover, our model provides
qualitatively new predictions for the thermoelectric transport, which
could be used to probe the hydrodynamic regime in such materials.
... Since P½ξ a is a Gaussian distribution function, higher order correlation functions are obtained by implementing Wick's theorem. This averaging is a manifestation of stochasticity, establishing a direct relation between nonequilibrium dynamics of quantum open systems and stochastic field theory [52,53]. ...
We introduce an effective field theory to study indirect mixing of two fields induced by their couplings to a common decay channel in a medium. The extension of the method of Lee, Oehme, and Yang, the cornerstone of analysis of CP violation in flavored mesons, to include the mixing of particles with different masses provides a guide to and benchmark for the effective field theory. The analysis reveals subtle caveats in the description of mixing in terms of the widely used non-Hermitian effective Hamiltonian, more acute in the nondegenerate case. The effective field theory describes the dynamics of field mixing where the common intermediate states populate a bath in thermal equilibrium, as an open quantum system. We obtain the effective action up to second order in the couplings, where indirect mixing is a consequence of off-diagonal self-energy components. We find that if only one of the mixing fields features an initial expectation value, indirect mixing induces an expectation value of the other field. The equal time two point correlation functions exhibit an asymptotic approach to a stationary thermal state, and the emergence of long-lived bath-induced coherence which displays quantum beats as a consequence of interference of quasinormal modes in the medium. The amplitudes of the quantum beats are resonantly enhanced in the nearly degenerate case with potential observational consequences.
... With the Martin-Siggia-Rose formalism [74] one constructs the action by introducing a set of so-called response fieldsψ a (η, k). 21 Utilising the Janssen-de Dominicis formalism [75][76][77], one constructs the generating functional ...
Large-scale structure formation is studied in a kinetic theory approach, extending the standard perfect pressureless fluid description for dark matter by including the velocity dispersion tensor as a dynamical degree of freedom. The evolution of power spectra for density, velocity and velocity dispersion degrees of freedom is investigated in a non-perturbative approximation scheme based on the Dyson-Schwinger equations. In particular, the generation of vorticity and velocity dispersion is studied and predictions for the corresponding power spectra are made, which qualitatively agree well with results obtained from N-body simulations. It is found that velocity dispersion grows strongly due to non-linear effects and at late times its mean value seems to be largely independent of the initial conditions. By taking this into account, a rather realistic picture of non-linear large-scale structure formation can be obtained, albeit the numerical treatment remains challenging, especially for very cold dark matter models.
... To overcome this drawback, a response field is introduced and thus the correlation (spontaneous fluctuation) as well as the response function can be easily derived (see below). This new field-theoretical approach is called the Martin-Sigga-Rose-De Dominics-Janssen (MSRDJ) formalism [8][9][10]. The MSRDJ formalism has been used to analyze the recurrent neural networks, e.g., studying the onset of chaos [11][12][13], and to analyze deep neural networks in recent years [14][15][16]. ...
Dynamical mean-field theory is a powerful physics tool used to analyze the typical behavior of neural networks, where neurons can be recurrently connected, or multiple layers of neurons can be stacked. However, it is not easy for beginners to access the essence of this tool and the underlying physics. Here, we give a pedagogical introduction of this method in a particular example of random neural networks, where neurons are randomly and fully connected by correlated synapses and therefore the network exhibits rich emergent collective dynamics. We also review related past and recent important works applying this tool. In addition, a physically transparent and alternative method, namely the dynamical cavity method, is also introduced to derive exactly the same results. The numerical implementation of solving the integro-differential mean-field equations is also detailed, with an illustration of exploring the fluctuation dissipation theorem.
... As a first step in this direction, a field theory for a non-interacting, non-switching Brownian gas has recently been constructed for the global density [65]. The basic idea is to apply a MSRJD path integral construction [66][67][68] to the DK equation obtained by setting A n (x) = 0 in equation (3.11). (For a complementary approach based on a Doi-Peliti path integral formulation [69][70][71], see [72]. ...
There are many processes in cell biology that can be modelled in terms of an actively switching particle. The continuous degrees of freedom of the particle evolve according to a hybrid stochastic differential equation (hSDE) whose drift term depends on a discrete internal or environmental state that switches according to a continuous time Markov chain. Examples include Brownian motion in a randomly switching environment, membrane voltage fluctuations in neurons, protein synthesis in gene networks, bacterial run-and-tumble motion, and motor-driven intracellular transport. In this paper we derive generalized Dean-Kawasaki (DK) equations for a population of actively switching particles, either independently switching or subject to a common randomly switching environment. In the case of a random environment, we show that the global particle density evolves according to a hybrid DK equation. Averaging with respect to the Gaussian noise processes in the absence of particle interactions yields a hybrid partial differential equation for the one-particle density. We use this to show how a randomly switching environment induces statistical correlations between the particles. We also discuss methods for handling the moment closure problem for interacting particles, including dynamical density functional theory and mean field theory. We then develop the analogous constructions for independently switching particles. In order to derive a DK equation, we introduce a discrete set of global densities that are indexed by the single-particle internal states, and take expectations with respect to the switching process. However, the resulting DK equation is no longer closed when particle interactions are included. We conclude by deriving Martin-Siggia-Rose-Janssen-de Dominicis (MSRJD) path integrals for the global density equations in the absence of interactions, and relate this to recent field theoretic studies of Brownian gases and run-and-tumble particles (RTPs).
... Assuming anisotropic systems with a y > 0, we consider the critical phase transition between the homogeneous state and anisotropic phase separation that occurs as a x is changed. Applying the approach by Martin, Siggia, Rose, Janssen, and de Dominicis [93][94][95][96] to Eq. (G1), we can obtain the probability density for a dynamical path of configurations {φ(r, t )} t∈[0,T ] as ...
Motility-induced phase separation (MIPS) is a nonequilibrium phase separation that has a different origin from equilibrium phase separation induced by attractive interactions. Similarities and differences in collective behaviors between these two types of phase separation have been intensely discussed. Here, to study another kind of similarity between MIPS and attraction-induced phase separation under a nonequilibrium condition, we perform simulations of active Brownian particles with uniaxially anisotropic self-propulsion (uniaxial ABPs) in two dimensions. We find that (i) long-range density correlation appears in the homogeneous state; (ii) anisotropic particle configuration appears in MIPS, where the anisotropy removes the possibility of microphase separation suggested for isotropic ABPs [X.-Q. Shi et al., Phys. Rev. Lett. 125, 168001 (2020)]; and (iii) critical phenomena for the anisotropic MIPS presumably belong to the universality class for two-dimensional uniaxial ferromagnets with dipolar long-range interactions. Properties (i)–(iii) are common to the well-studied randomly driven lattice gas (RDLG), which is a particle model that undergoes phase separation by attractive interactions under external driving forces, suggesting that the origin of phase separation is not essential for macroscopic behaviors of uniaxial ABPs and RDLG. Based on the observations in uniaxial ABPs, we construct a coarse-grained Langevin model that shows properties (i)–(iii) and corroborates the generality of the findings.
... perceptron, one hidden layer, committee machine [14][15][16]) are well known and understood, when the algorithmic dynamics is governed by equilibrium processes [17]. However, the out-of-equilibrium dynamics of the learning processes [18][19][20] are much more difficult to study and most of the results about it are restricted to dense models where the Martin-Siggia-Rose formalism [21] can be applied to derive DMFT equations [22,23]. ...
The use of mini-batches of data in training artificial neural networks is nowadays very common. Despite its broad usage, theories explaining quantitatively how large or small the optimal mini-batch size should be are missing. This work presents a systematic attempt at understanding the role of the mini-batch size in training two-layer neural networks.
Working in the teacher-student scenario, with a sparse teacher, and focusing on tasks of different complexity, we quantify the effects of changing the mini-batch size $m$.
We find that often the generalization performances of the student strongly depend on $m$ and may undergo sharp phase transitions at a critical value $m_c$, such that for $m<m_c$ the training process fails, while for $m>m_c$ the student learns perfectly or generalizes very well the teacher. Phase transitions are induced by collective phenomena firstly discovered in statistical mechanics and later observed in many fields of science. Observing a phase transition by varying the mini-batch size across different architectures raises several questions about the role of this hyperparameter in the neural network learning process.
Исследуются флуктуации завихренности, возбуждаемые внешней случайной силой в двумерной жидкости в присутствии сильного внешнего сдвигового потока. Задача мотивирована анализом больших когерентных вихрей, возникающих в результате обратного энергетического каскада в конечной ячейке при больших числах Рейнольдса. Развивается теория возмущений для расчета нелинейных поправок к корреляционным функциям флуктуаций потока в предположении, что внешняя сила имеет малое время корреляции. Проанализированы поправки к парной корреляционной функции завихренности и некоторым моментам. Проведенный анализ позволяет установить достоверность теории возмущений для лабораторных экспериментов и численного моделирования.
We construct, for the first time, a Bemfica-Disconzi-Noronha-Kovtun (BDNK) theory for linear stochastic fluctuations, which is proved to be mathematically consistent, causal, and covariantly stable. The Martin-Siggia-Rose action is shown to be bilocal in most cases, and the noise is not white. The presence of nonhydrodynamic modes induces long-range correlations in the primary fluid variables (temperature, chemical potential, and flow velocity). However, correlators of conserved densities remain localized in space, and coincide with those calculated within fluctuating Isreal-Stewart theory. We show that, in some cases, there is a nonlocal change of variables that maps the Israel-Stewart action into the BDNK action.
Stirring a fluid through a Gaussian forcing at a vanishingly small Reynolds number produces a Gaussian random field, while flows at higher Reynolds numbers exhibit non-Gaussianity, cascades, anomalous scaling and preferential alignments. Recent works (Yakhot & Donzis, Phys. Rev. Lett. , vol. 119, 2017, 044501; Khurshid et al. , Phys. Rev. E , vol. 107, 2023, 045102) investigated the onset of these turbulent hallmarks in low-Reynolds-number flows by focusing on the scaling of the velocity gradients and velocity increments. They showed that the onset of power-law scalings in the velocity gradient statistics occurs at low Reynolds numbers, with the scaling exponents being surprisingly similar to those in the inertial range of fully developed turbulence. In this work we address the onset of turbulent signatures in low-Reynolds-number flows from the viewpoint of the velocity gradient dynamics, giving insights into its rich statistical geometry. We combine a perturbation theory of the full Navier–Stokes equations with velocity gradient modelling. This procedure results in a stochastic model for the velocity gradient in which the model coefficients follow directly from the Navier–Stokes equations and statistical homogeneity constraints. The Fokker–Planck equation associated with our stochastic model admits an analytic solution that shows the onset of turbulent hallmarks at low Reynolds numbers: skewness, intermittency and preferential alignments arise in the velocity gradient statistics through a smooth transition as the Reynolds number increases. The model predictions are in excellent agreement with direct numerical simulations of low-Reynolds-number flows.
By using the field theoretic renormalization group technique together with the operator product expansion, simultaneous influence of the spatial parity violation and finite-time correlations of an electrically conductive turbulent environment on the inertial-range scaling behavior of correlation functions of a passively advected weak magnetic field is investigated within the corresponding generalized Kazantsev-Kraichnan model in the second order of the perturbation theory (in the two-loop approximation). The explicit dependence of the anomalous dimensions of the leading composite operators on the fixed point value of the parameter that controls the presence of finite-time correlations of the turbulent field as well as on the parameter that drives the amount of the spatial parity violation (helicity) in the system is found even in the case with the presence of the large-scale anisotropy. In accordance with the Kolmogorov's local isotropy restoration hypothesis, it is shown that, regardless of the amount of the spatial parity violation, the scaling properties of the model are always driven by the anomalous dimensions of the composite operators near the isotropic shell. The asymptotic (inertial-range) scaling form of all single-time two-point correlation functions of arbitrary order of the passively advected magnetic field is found. The explicit dependence of the corresponding scaling exponents on the helicity parameter as well as on the parameter that controls the finite-time velocity correlations is determined. It is shown that, regardless of the amount of the finite-time correlations of the given Gaussian turbulent environment, the presence of the spatial parity violation always leads to more negative values of the scaling exponents, i.e., to the more pronounced anomalous scaling of the magnetic correlation functions. At the same time, it is shown that the stronger the violation of spatial parity, the larger the anomalous behavior of magnetic correlations.
Randomly coupled phase oscillators may synchronize into disordered patterns of collective motion. We analyze this transition in a large, fully connected Kuramoto model with symmetric but otherwise independent random interactions. Using the dynamical cavity method, we reduce the dynamics to a stochastic single-oscillator problem with self-consistent correlation and response functions that we study analytically and numerically. We clarify the nature of the volcano transition and elucidate its relation to the existence of an oscillator glass phase.
We investigate a two-dimensional system of interacting active Brownian particles. Using the Martin-Siggia-Rose-Janssen-de Dominicis formalism, we built up the generating functional for correlation functions. We study in detail the hydrodynamic regime with a constant density stationary state. Our findings reveal that, within a small density fluctuations regime, an emergent U(1) gauge symmetry arises, originated from the conservation of fluid vorticity. Consequently, the interaction between the orientational order parameter and density fluctuations can be cast into a gauge theory, where the concept of “electric charge density” aligns with the local vorticity of the original fluid. We study in detail the case of a microscopic local two-body interaction. We show that, upon integrating out the gauge fields, the stationary states of the rotational degrees of freedom satisfy a nonlocal Frank free energy for a nematic fluid. We give explicit expressions for the splay and bend elastic constants as a function of the Péclet number (Pe) and the diffusion interaction constant (kd).
We explore the phase transitions at the onset of time-crystalline order in O(N) models driven out of equilibrium. The spontaneous breaking of time translation symmetry and its Goldstone mode are captured by an effective description with O(N)×SO(2) symmetry, where the emergent external SO(2) results from a transmutation of the internal symmetry of time translations. Using the renormalization group and the ε=4−d expansion in a leading two-loop analysis, we identify a new nonequilibrium universality class. Strikingly, it controls the long-distance physics no matter how small the microscopic breaking of equilibrium conditions is. The O(N=2)×SO(2) symmetry group is realized for magnon condensation in pumped yttrium iron garnet films and in exciton-polariton systems with a polarization degree of freedom.
We investigate fluctuations of vorticity inside a coherent vortex generated by the inverse energy cascade in two-dimensional turbulence. Temporal and spatial correlations can be characterized by the pair correlation function. The interaction of fluctuations leads to a nonzero third moment of vorticity. We analyze the pair correlation function and the third moment using a model in which the pumping is short-correlated in time and derive explicit expressions for the Gaussian spatial correlation function for the pumping force.
A d-dimensional elastic manifold at depinning is described by a renormalized field theory, based on the functional renormalization group (FRG). Here, we analyze this theory to 3-loop order, equivalent to third order in ε=4−d, where d is the internal dimension. The critical exponent reads ζ=ε3+0.04777ε2−0.068354ε3+O(ε4). Using that ζ(d=0)=2−, we estimate ζ(d=1)=1.266(20), ζ(d=2)=0.752(1), and ζ(d=3)=0.357(1). For Gaussian disorder, the pinning force per site is estimated as fc=Bm2ρm+fc0, where m2 is the strength of the confining potential, B a universal amplitude, ρm the correlation length of the disorder, and fc0 a nonuniversal lattice-dependent term. For charge-density waves, we find a mapping to the standard ϕ4 theory with O(n) symmetry in the limit of n→−2. This gives fc=Ã(d)m2ln(m)+fc0, with Ã(d)=−∂n[ν(d,n)−1+η(d,n)]n=−2, reminiscent of logarithmic conformal field theories.
We show that it is possible to define a timelike future-directed information current within relativistic first-order hydrodynamics. This constitutes the first step toward a covariantly stable and causal formulation of first-order fluctuating hydrodynamics based on thermodynamic principles. We provide several explicit examples of first-order theories with an information current, covering many physical phenomena, ranging from electric conduction to viscosity and elasticity. We use these information currents to compute the corresponding equal-time correlation functions, and we find that the physically relevant (equal-time) correlators do not depend on the choice of the hydrodynamic frame as long as the frame leads to causal and stable dynamics. In the example of chiral hydrodynamics, we find that circularly polarized shear waves have different probabilities of being excited depending on their handedness, generating net helicity in chiral fluids.
We investigate analytically the distribution tails of the area A and perimeter L of a convex hull for different types of planar random walks. For N noninteracting Brownian motions of duration T we find that the large-L and -A tails behave as P(L)∼e−bNL2/DT and P(A)∼e−cNA/DT, while the small-L and -A tails behave as P(L)∼e−dNDT/L2 and P(A)∼e−eNDT/A, where D is the diffusion coefficient. We calculated all of the coefficients (bN,cN,dN,eN) exactly. Strikingly, we find that bN and cN are independent of N for N≥3 and N≥4, respectively. We find that the large-L (A) tails are dominated by a single, most probable realization that attains the desired L (A). The left tails are dominated by the survival probability of the particles inside a circle of appropriate size. For active particles and at long times, we find that large-L and -A tails are given by P(L)∼e−TΨNper(L/T) and P(A)∼e−TΨNarea(A/T), respectively. We calculate the rate functions ΨN exactly and find that they exhibit multiple singularities. We interpret these as DPTs of first order. We extended several of these results to dimensions d>2. Our analytic predictions display excellent agreement with existing results that were obtained from extensive numerical simulations.
Thermal activation of a particle from a deep potential trap follows the Arrhenius law. Recently, this result has been generalized for interacting diffusive particles in the trap, revealing two universality classes—the Arrhenius class and the excluded volume class. The result was demonstrated with the aid of numerical analysis. Here, we present a perturbative hydrodynamic approach to analytically validate the existence and range of validity for the two universality classes.
Lecture notes after the doctoral school (Post)Modern Thermodynamics held at the Uni-
versity of Luxembourg, December 2022, 5-7, covering and advancing continuous-time
Markov chains, network theory, stochastic thermodynamics, large deviations, determin-
istic and stochastic chemical reaction networks, metastability, martingales, quantum
thermodynamics, and foundational issues.
Complex dynamical systems may exhibit multiple steady states, including time-periodic limit cycles, where the final trajectory depends on initial conditions. With tuning of parameters, limit cycles can proliferate or merge at an exceptional point. Here we ask how dynamics in the vicinity of such a bifurcation are influenced by noise. A pitchfork bifurcation can be used to induce bifurcation behavior. We model a limit cycle with the normal form of the Hopf oscillator, couple it to the pitchfork, and investigate the resulting dynamical system in the presence of noise. We show that the generating functional for the averages of the dynamical variables factorizes between the pitchfork and the oscillator. The statistical properties of the pitchfork in the presence of noise in its various regimes are investigated and a scaling theory is developed for the correlation and response functions, including a possible symmetry-breaking field. The analysis is done by perturbative calculations as well as numerical means. Finally, observables illustrating the coupling of a system with a limit cycle to a pitchfork are discussed and the phase-phase correlations are shown to exhibit nondiffusive behavior with universal scaling.
A field theory approach for the nonequilibrium relaxation dynamics in open systems at late times is developed. In the absence of conservation laws, all excitations are subject to dissipation. Nevertheless, ordered stationary states satisfy Goldstone's theorem. It implies a vanishing damping rate at small momenta, which in turn allows for competition between environment-induced dissipation and thermalization due to collisions. We derive the dynamic theory in the symmetry-broken phase of an O(N)-symmetric field theory based on an expansion of the two-particle irreducible (2PI) effective action to next-to-leading order in 1/N and highlight the analogies and differences to the corresponding theory for closed systems. A central result of this approach is the systematic derivation of an open-system Boltzmann equation, which takes a very different form from its closed-system counterpart due to the absence of well-defined quasiparticles. As a consequence of the general structure of its derivation, it applies to open, gapless field theories that satisfy certain testable conditions, which we identify here. Specifically for the O(N) model, we use scaling analysis and numerical simulations to show that interactions are screened efficiently at small momenta and, therefore, the late-time evolution is effectively collisionless. This implies that fluctuations induced by a quench dissipate into the environment before they thermalize. Goldstone's theorem also constrains the dynamics far from equilibrium, which is used to show that the order parameter equilibrates more quickly for quenches preserving the O(N) symmetry than those breaking it explicitly.
Clocks are inherently out-of-equilibrium because, due to friction, they constantly consume free energy to keep track of time. The thermodynamic uncertainty relation (TUR) quantifies the trade-off between the precision of any time-antisymmetric observable and entropy production. In the context of clocks, the TUR implies that a minimum entropy production is needed in order to achieve a certain level of precision in timekeeping. But the TUR has only been proven for overdamped systems. Recently, a toy model of a classical underdamped pendulum clock was proposed that violated this relation [Phys. Rev. Lett. 128, 130606 (2022)], thus demonstrating that the TUR does not hold for underdamped dynamics. We propose an electronic implementation of such a clock, using a resistor-inductor-capacitor circuit and a biased CMOS inverter (not gate), which can work at different scales. We find that in the nanoscopic single-electron regime of the circuit, we essentially recover the toy model violating the TUR bound. However, in different macroscopic regimes of the circuit, we show that the TUR bound is restored and analyze the thermodynamic efficiency of timekeeping.
Statistical field theories provide powerful tools to study complex dynamical systems. In this work those tools are used to analyze the dynamics of a kinetic energy harvester, which is modeled by a system of coupled stochastic nonlinear differential equations and driven by colored noise. Using the Martin-Siggia-Rose response fields we analytically approach the problem through path integrals in the phase space and represent the moments that correspond to physical observables through Feynman diagrams. This analysis method is tested by comparing the solution to the linear case with previous analytical results. Through a perturbative expansion it is calculated how the nonlinearity affects, to the first order, the energy harvest supporting the results through numerical simulations.
Quantum statistical mechanics is renormalized, that is, the thermodynamical functions and the ``bare'' ν-body potentials vν occurring in the Hamiltonian are expressed as functionals of the ν-body distribution functions Gν (the expectation values of 2ν field operators). The field operators are considered as having two components (creation and annihilation), and Gν thus has (2)2ν components. The renormalization is carried out for superfluid Bose systems for which it is necessary to consider the expectation values of 2ν = 1, 2, 3, 4 operators. Superfluid Fermi systems and normal systems appear as particular cases, described by expectation values of 2ν = 2, 4 operators.
This problem, dealt with by algebraic methods in part I, is attacked here by diagrammatic methods. These methods are more dependent on convergence properties but the resulting functionals are obtained explicitly as power series of the Gν′s, the general term being represented by a class of diagrams characterized by their topological structure.
In the final result, the entropy is exhibited as an explicit functional of the distribution functions Gν, (2ν = 1, 2, 3, 4), or more precisely, of functions directly related to them. This functional no longer involves the potentials (nor the equilibrium parameters). It is stationary under independent variations of the Gν′s, subject to the constraints of constant energy and particle number. The four equations of stationarity exhibit each function vν as a functional of the Gν′s, self-consistently defining these distribution functions.
It is argued that Eulerian formulations are intrinsically unsuited for deriving the Kolmogorov theory because low-order Eulerian moments do not express sufficiently well a statistical dependence of nonsimultaneous amplitudes that accompanies the convection of small spatial scales by large spatial scales. Illustration is made by applying the direct-interaction approximation and a related, higher Eulerian approximation to an idealized convection problem and to a modified Navier-Stokes equation. Convection effects of low wavenumbers on high wavenumbers are removed in the modified equation, and as a consequence the direct-interaction approximation for it yields the Kolmogorov spectrum. Low-order Lagrangian moments provide a promisingly more complete description of the convection of small spatial scales by large, and a search for satisfactory Lagrangian closure approxi-mations seems highly desirable.
This is the first of a series of papers dealing with many-particle systems from a unified, nonperturbative point of view. It contains derivations and discussions of various field-theoretical techniques which will be applied in subsequent papers. In a short introduction the general method of approach is summarized, and its relationship to other field-theoretic problems indicated. In the second section the macroscopic properties of the spectra of many-particle systems are described. Asymptotic evaluations are performed which characterize these macroscopic features in terms of intensive parameters, and the relationship of these parameters to thermodynamics is discussed. The special characteristics of the ground state are shown to follow as a limiting case of the asymptotic evaluations. The third section is devoted to the time-dependent field correlation functions, or Green's functions, which describe the microscopic behavior of a multiparticle system. These functions are defined, and related to intensive macroscopic variables when the energy and number of particles are large. Spectral representations and other properties of various one-particle Green's functions are derived. In the fourth section the treatment of non-equilibrium processes is considered. As a particular example, the electromagnetic properties of a system are expressed in terms of the special two-particle Green's function which describes current correlation. The discussion yields specifically a fluctuation-dissipation theorem, a sum rule for conductivity, and certain dispersion relations. The fifth section deals with the differential equations which determine the Green's functions. The boundary conditions that characterize the Green's function equations are exhibited without reference to adiabatic decoupling. A method for solving the equations approximately, by treating the correlations among successively larger numbers of particles, is considered. The first approximation in this sequence is shown to yield a generalized Hartree-like equation. A related, but rigorous, identity for the single-particle Green's function is then derived. A second approximation, which takes certain two-particle correlations into account, is shown to produce various additional effects: The interaction between particles is altered in a manner characterized by the intensive macroscopic parameters, and the modification and spread of the energy-momentum relation come into play. In the final section compact formal expressions for the Green's functions and other physical quantities are derived. Alternative equations and systematic approximations for the Green's functions are obtained.
The covariant quantum electrodynamics of Tomonaga, Schwinger, and Feynman is used as the basis for a general treatment of scattering problems involving electrons, positrons, and photons. Scattering processes, including the creation and annihilation of particles, are completely described by the $S$ matrix of Heisenberg. It is shown that the elements of this matrix can be calculated, by a consistent use of perturbation theory, to any desired order in the fine-structure constant. Detailed rules are given for carrying out such calculations, and it is shown that divergences arising from higher order radiative corrections can be removed from the $S$ matrix by a consistent use of the ideas of mass and charge renormalization.
Recently the authors have derived kinetic equations describing the behavior of the spin autocorrelation function Gammaq(t) in a Heisenberg system at infinite temperature. In the present paper, this derivation is extended to finite temperatures (above the critical point). It is shown that, in the Weiss limit where the number of neighbors Z-->∞, the effects of the equilibrium (Ornstein-Zernicke) correlations present in the system at the initial time can be entirely incorporated within an effective temperature-dependent interaction which governs the temporal behavior of the autocorrelation function (af). A non-Markoffian kinetic equation is obtained in which the kernel is highly nonlinear in the complete af Gammaq(t) this contrasts with the infinite-temperature case, where the kernel was a functional of the direct af only. This new feature leads to simple approximations near the critical point, as will be discussed in detail in the next paper of this series.
A perturbation theory for the determination of transport coefficients near the critical point is presented. This perturbation theory is based upon processes in which one transport mode decays into several low-wave-number modes. Scaling-law concepts are used to calculate the order of magnitude of the matrix elements and frequency denominators which appear in this theory. This permits the estimation of the order of magnitude of the transport coefficients near the critical point. In particular, this approach indicates that the thermal conductivity should diverge roughly as (T-Tc)-23 on the critical isochore and coexistence curve, while the viscosity eta should be either weakly divergent or strongly cusped at the critical point. On the other hand, the bulk viscosity zeta should diverge roughly as (T-Tc)-2 for low frequencies, and as (T-Tc)-23 for higher frequencies on the critical isochore near the critical point. Specific predictions are made for these quantities in terms of critical indices, and the connection between these relations and the scaling of frequencies is discussed.
A general theory is presented for the transport coefficients exhibiting anomalous peaks near the critical point, employing correlation-function expressions for transport coefficients. Recognizing that the anomaly should arise from the anomalous increase in the fluctuations of certain macroscopic variables, we attempt an expansion of the flux entering the correlation-function expression in powers of the macroscopic variables, which then are supposed to obey the macroscopic equations of motion. A general formula for the anomalous part of the transport coefficient is given, restricting ourselves to the quadratic terms in this expansion. The general theory is illustrated for the shear viscosity of critical mixtures, choosing local concentration and local temperature as macroscopic variables. The anomaly is attributed to the cooperation of the two effects: (1) anomalous increase in certain large-scale fluctuations of macroscopic variables contained in the flux, and (2) the anomalous increase in the lifetimes associated with these fluctuations. If we ignore the local temperature fluctuations, Fixman's result of the anomalous viscosity is obtained. Generalizing these results, the large frequency and wave-vector dependence which is expected for the anomalous transport coefficients near the critical point is studied explicitly for the shear viscosity. The thermal conductivity of the critical mixture is also examined, and is found to have no anomaly in the same approximation, in agreement with the existing experiments. In an appendix, a more rigorous and systematic treatment of the general theory is given with the help of Mori's general theory of Brownian motion for macroscopic variables, and we indicate a possibility of obtaining a self-consistent set of equations for general nonlocal transport coefficients.
The collision integral for the kinetic equation describing a qunatum mechanical plasma is set up. It is shown that the classical collision integral derived by Rostoker, Rosenbluth, Balescu, and Lenard is easily obtained from the more perspicuous quantum mechanical result by letting $$\hbar${}$\rightarrow${}0$. The kinetic equations of Pines and Schrieffer, which describe the interactions of the electrons with plasma oscillations, are obtained by isolating the contributions from the plasma oscillations.
The approach to equilibrium of electrons, plasmons, and phonons in finite-temperature plasmas is studied in the random phase approximation. It is first shown that for an electron plasma in equilibrium, the long-wavelength, well-defined, plasmons contribute a term to the free energy which is appropriate to a collection of independent bosons. In order to study nonequilibrium processes, an explicit plasmon distribution function is introduced. The matrix element for plasmon-electron coupling is calculated in the random phase approximation, and second-order perturbation theory is used to write down the equations which couple the electron and plasmon distribution functions. Equilibrium is shown to result from the competition between the spontaneous emission of plasma waves by single fast electrons and the Landau damping of the plasma waves due to the same group of particles. The equation for the time rate of change of the electron distribution function reduces, in the classical limit, to a Fokker-Planck equation in which there appear diffusion and friction terms associated with plasma waves, of the type first considered by Klimontovitch. When an initial arbitrary nonequilibrium electron distribution is considered, it is seen that a plasma wave instability corresponds to coherent excitation of plasma waves by the electrons in contrast to the incoherent excitation associated with spontaneous emission. The method is generalized to two-component electron-ion plasma in which well defined acoustic plasma waves exist, by introducing an explicit distribution function for the phonons (the acoustic plasmons). The approach to equilibrium and the two-stream instability are derived and discussed for the coupled electron-phonon system; results similar to those obtained for the electron-plasmon system are found.
The many-boson system with repulsive interactions is treated by a variational method based on a variational trial state of an exponential pair-excitation type obtained by a generalization of that of Bogoliubov; the treatment is closely related to an intermediate-coupling approximation with respect to pairs. The nonlinear integral equation which determines the variational ground state is derived, and various properties of this ground state and the associated excited states are examined. The resultant low-lying spectrum lies below that of Bogoliubov by an amount proportional to the total number of particles. The variational principle is shown to produce rigorous energy eigenvalue differences for the pair part of the Hamiltonian. The variational states, however, still exhibit unphysical features characteristic of pair-excitation states: The pair correlation function does not go to zero at zero particle separation and the phonon spectrum exhibits a gap above the ground state. It is suggested that these features can be removed by using states which take into account excitation of momentum-conserving groups of more than two particles.
The theory of hydromagnetic isotropic turbulence in incompressible fluids is formulated using methods similar to those of quantum field theory. The present formulation is a generalization of Wyld's formulation of ordinary istropic turbulence in incompressible fluids. Solutions for the velocity and magnetic field in the form of perturbation series are set up. Terms in the series are then represented by a one-to-one correspondence by diagrams similar to Feynman diagrams. A study of these diagrams reveals that they may be rearranged to give integral equations governing the second order correlation functions for the velocity and the magnetic field. These integral equations contain an infinite number of terms. To obtain manageable results, the equations are truncated at finite orders, yielding approximate equations. An approximation at the lowest order gives Chandrasekhar's equations; while a “second approximation” gives a more complicated set of equations.
The extended mode coupling theory is used to investigate the non-hydrodynamical behavior of the two-dimensional fluids.
The perturbation series for the ground-state energy of a many-fermion ; system was investigated to arbitrary order for the "isotropic" case. This is the ; case of over-all spherical symmetry. both in the interaction and in the ; unperturbed single particle energies. It was shown that for spin one-half ; feninions the Brueckner-Goldstone perturbation series is valid to all orders in ; the perturbation. For spins greater than one-half it is in general incorrect ; even in the isotropic case. unless the interactions are spin independent. The ; discussion to arbitrary order in the interaction ts carried out by means of a ; Feynman-like propagator formalism, which is developed in detail. (auth);
A unified development of the subject of quantum electrodynamics is outlined, embodying the main features both of the Tomonaga-Schwinger and of the Feynman radiation theory. The theory is carried to a point further than that reached by these authors, in the discussion of higher order radiative reactions and vacuum polarization phenomena. However, the theory of these higher order processes is a program rather than a definitive theory, since no general proof of the convergence of these effects is attempted. The chief results obtained are (a) a demonstration of the equivalence of the Feynman and Schwinger theories, and (b) a considerable simplification of the procedure involved in applying the Schwinger theory to particular problems, the simplification being the greater the more complicated the problem.
Results are presented which express the thermodynamical functions of a normal inhomogeneous system as stationary functionals of a matrix Φmn. The eigenvalues of Φ are “quasi-particle” occupation numbers. The relationship with Landau theory of Fermi liquids is discussed. The rules necessary to construct the relevant functionals of Φ, including the one-body Green's function, in powers of the two-body interaction, are explicited.
This paper examines the dynamical behaviour of a field of homogeneous turbulence in which the joint-probability distribution of the fluctuating velocity components at three points is approximately normal. In principle, the analysis is formulated entirely in terms of the mean values {uiuj'} and {uiuj'uk' '}, where the number of primes denotes the point at which the velocity components are taken. First, the kinematical properties of the three-point correlation are obtained by techniques similar to those used in the well-known theory of the two-point correlation. In the particular case of isotropic turbulence, the necessary extensions to the existing invariant theory lead to the result that the three-point correlation is completely defined by two scalar functions. Two independent dynamical relations between these correlations are then derived from the Navier-Stokes equation, and the remainder of the paper is based on this (determinate) system of equations. These remarks refer only to the principle of the calculations; in fact, most of the results are obtained in terms of the Fourier transforms of the correlations defined above. The first set of deductions from the governing equations refer to the decay of isotropic turbulence at large Reynolds numbers. In particular, the exact solution of the inviscid equations for the vorticity is obtained, and it is shown to be consistent with the predictions of Kolmogoroff's theory of local similarity after a sufficiently long time of decay. The distribution of energy transfer between eddies of different sizes is also examined for a special form of the energy spectrum of turbulence, and the general features of this distribution appear to be satisfactory in the main energy-containing range of the spectrum. The remaining results are concerned with energy transfer in the large eddies. It is shown, beyond all reasonable doubt, that the magnitude of this energy transfer is such that the large eddies are not permanent during decay. Immediate consequences of this result are that Loitsiansky's integral is not an invariant of the motion, and that the usual triple correlation function k(r) is proportional to r-4 for large values of r. These conclusions are inconsistent with the theory initiated by Loitsiansky, and developed by Lin and Batchelor. The cause of this inconsistency is attributed to the dynamical unlikelihood of the basic assumptions made by these earlier authors.
One of the most serious difficulties in the theory of homogeneous turbulence is the indeterminacy of the equation for the velocity correlation function of any order, each involving the correlation of higher-by-one order. In the present paper this difficulty is resolved by treating the two dynamical equations for the second- and third-order velocity correlations, and by introducing the assumption of the zero fourth-order cumulant of the velocity field which yields a relationship between the fourth- and second-order velocity correlations. Actual calculation, however, is carried out in the wave-number space, and a pair of simultaneous equations for the energy spectrum function are derived in part I. Another difficulty of the subject arises from the present lack of knowledge about the initial state of turbulence. In part II, some probable initial conditions for the energy spectrum are examined, among which the initial spectrum of single-line type is chosen as the most suitable for the present problem and its dynamical consequences are fully discussed. The power-series solution for the initial spectrum as well as the energy decay law due to it are computed and compared with experimental data. It is found that the solution, in so far as the approximate expression calculated in the present paper is concerned, corresponds to the earlier initial period of decay. A solution which would be essentially in agreement with experiments is expected to be given by extending the present solution to the further developed stage of decay.
The differential cross section for coherent scattering of thermal neutrons by a liquid is given in general by the Fourier transform of a time-displaced radial density function. It is suggested here that, to an adequate degree of approximation, this time-displaced function can be expressed as a convolution of the ordinary radial density function with a self-diffusion function describing the wandering of an atom from an arbitrary initial position. The neutron scattering cross section then becomes the product of the Fourier transforms of these two functions. One of the transforms is the differential cross section for x-ray scattering and describes interference effects, the other governs the energy changes upon scattering. In this development the scatterer can be treated either quantum mechanically or classically. Recoil effects are not provided by the classical treatment, but this is a significant deficiency only in liquids of low atomic weight. Several models for calculating the self-diffusion function are considered, and from these it is suggested that a Gaussian function with a time-dependent width is a reasonable approximation for the case of a simple liquid. The principal features of the width are deduced. Quantization of the scatterer effects the width at small times. At large times the width depends only on the coefficient of self-diffusion of the liquid, and inelastic scattering is suggested as a means of determining this coefficient, as well as other features of atomic movement. The accuracy of the static approximation for determining liquid structures by neutron diffraction is assessed by considering the typical case of liquid lead near its melting point, and is found to be moderately good. The extension of the entire formalism to the case of polyatomic liquids is outlined.
It is shown how the method of thermodynamic Green's functions can be used to approximate the memory function associated with the equilibrium-fluctuation function Sc(r⃗-r⃗′,p⃗p⃗′,t-t′)≡〈[f(r⃗p⃗t)-〈f(r⃗p⃗t)〉][f(r⃗′p⃗′t′)-〈f(r⃗′p⃗′t′)〉]〉, where f(r⃗p⃗t) is the phase-space distribution operator. We obtain an approximation for the memory function for a gas, in the low-density limit, that is valid for all distances and times, satisfied various relevant symmetry conditions and sum rules, reduces for long times and distances to the Boltzmann collision operator, and gives results completely consistent with the conservation laws governing the system. We also indicate how these methods can be extended to treat other types of systems.
Virtually all measurable properties of a classical fluid may be determined from the expectation value of the phase-space density operator f(r⃗p⃗t)=Σαδ(r⃗-r⃗α(t)) δ(p⃗-p⃗α(t)), and the phase-space density correlation function 〈f(r⃗p⃗t)f(r⃗′p⃗′t′)〉-〈f(r⃗p⃗t)〉 〈f(r⃗′p⃗′t′)〉, a matrix with indices (r⃗p⃗t). Systematic procedures for approximating this matrix, unhindered by secular effects, are always based on approximations to its inverse. For a weakly coupled fluid, the inverse can be expanded in powers of λ, the ratio of potential to kinetic energy. The leading term in this expansion gives rise to a Vlasov equation for the phase-space correlation function. The next term is the first that includes collisions, and results in relaxation towards equilibrium. This paper is concerned with the detailed study of the resulting fundamental nontrivial approximation. It is not Markovian and is perfectly reversible. Although the approximation is complicated, it is tractable analytically in various limits, and numerically for all wavelengths and frequencies. In this paper, only the behavior in certain limits is evaluated. Particular attention is directed toward its contractions - the density correlation function, which is measured by inelastic neutron and light scattering, and the momentum correlation function. Calculation of the former at long wavelengths corroborates the Landau-Placzek expression for light scattering, and therefore demonstrates that the kinetic equation predicts hydrodynamic behavior at long times. Since the correlation function is correct to order λ2, it has, in contrast to a solution to the Boltzmann equation, the correct long-wavelength velocity of sound, c2=(dp/dmn)s≠5/3kBT/m. It also predicts different transport coefficients than those deduced from a Boltzman equation. These include a nonvanishing bulk viscosity. The transport coefficients reduce to those derived from the Boltzmann equation at low densities. Some aspects of the short-time behavior are also discussed.
The Kadanoff theory of scaling near the critical point for an Ising ferromagnet is cast in differential form. The resulting differential equations are an example of the differential equations of the renormalization group. It is shown that the Widom-Kadanoff scaling laws arise naturally from these differential equations if the coefficients in the equations are analytic at the critical point. A generalization of the Kadanoff scaling picture involving an "irrelevant" variable is considered; in this case the scaling laws result from the renormalization-group equations only if the solution of the equations goes asymptotically to a fixed point.
The asymptotic time behavior (∼ct-d/2, where d is the dimensionality of the system) of the velocity autocorrelation function and of the kinetic parts of the correlation functions for the shear viscosity and the heat conductivity is derived on the basis of a local equilibrium assumption and the linearized Navier-Stokes equations. The coefficients c are expressed in terms of the transport coefficients and thermodynamic quantities. The physical mechanism responsible for the long-time tail is indicated, and the connections between the present work and investigations based on molecular dynamics and on kinetic theory are discussed.
Molecular-dynamic studies of the behavior of the diffusion coefficient after a long time s have shown that the velocity autocorrelation function decays as s-1 for hard disks and as s-3/2 for hard spheres, at least at intermediate fluid densities. A hydrodynamic similarity solution of the decay in velocity of an initially moving volume element in an otherwise stationary compressible viscous fluid agrees with a decay of (ηs)-d/2, where η is the viscosity and d is the dimensionality of the system. The slow decay, which would lead to a divergent diffusion coefficient in two dimensions, is caused by a vortex flow pattern which has been quantitatively compared for the hydrodynamic and molecular-dynamic calculations.
A kinetic theory has been proposed by several authors with the goal of eliminating the divergences which appear in the density expansion in nonequilibrium systems. Here, it is shown that for a two-dimensional simple gas the theory presents a new divergence, resulting from the fact that correlations propagate over long distances as a result of hydrodynamic transport. This divergence is discussed explicitly for a gas model: the Maxwell model. It will be indicated why the kinetic theory for a perfect Lorentz gas does not exhibit this new divergence.
A generalization of the Ising model is solved, qualitatively, for its critical behavior. In the generalization the spin Sn⃗ at a lattice site n⃗ can take on any value from -∞ to ∞. The interaction contains a quartic term in order not to be pure Gaussian. The interaction is investigated by making a change of variable Sn⃗=Σmψm(n)Sm′, where the functions ψm(n⃗) are localized wavepacket functions. There are a set of orthogonal wave-packet functions for each order-of-magnitude range of the momentum k⃗. An effective interaction is defined by integrating out the wave-packet variables with momentum of order 1, leaving unintegrated the variables with momentum <0.5. Then the variables with momentum between 0.25 and 0.5 are integrated, etc. The integrals are computed qualitatively. The result is to give a recursion formula for a sequence of effective Landau-Ginsberg-type interactions. Solution of the recursion formula gives the following exponents: η=0, γ=1.22, ν=0.61 for three dimensions. In five dimensions or higher one gets η=0, γ=1, and ν=1/2, as in the Gaussian model (at least for a small quartic term). Small corrections neglected in the analysis may make changes (probably small) in the exponents for three dimensions.
The long-time behavior of velocity-correlation functions ρ(d)(t) characteristic for self-diffusion, viscosity, and heat conductivity is calculated for a gas of hard disks or hard spheres on the basis of the kinetic theory of dense gases. In d dimensions one finds that ρ(d)(t), after an initial exponential decay for a few mean free times t0, exhibits for times up to at least ∼40t0 a decay ∼α(d)(ρ)(t0/t)d/2, where α(d) is of the order of ρd-1, ρ=nad with n the number density, and a the hard-disk or hard-sphere diameter. The α(d)(ρ) are determined by the same dynamical events that are responsible for the divergences in the virial expansion of the transport coefficients. In this paper the α(d)(ρ) are calculated to lowest order in ρ. In this order, they are identical to the low-density limit of the α(d)(ρ) that have been obtained by other authors on the basis of hydrodynamical considerations.
The equations of Kirkwood and Salzburg for distribution functions are generalized to multicomponent systems. The generalized equations serve to derive a recurrence relation between the coefficients of powers of particle number densities in a Maclaurin expansion of the distribution functions. This recurrence relation is then used to derive the general term in the expansion of the distribution functions in terms of modified irreducible integrals in multicomponent systems, which includes the original one-component expansion of Mayer and Montroll as a special case.
The corresponding expansion of the potentials of average force is derived. The use of the new expansion is illustrated by a relatively simple derivation of the Fuchs expansion of the grand potential in multicomponent systems.
Possible applications to ionic solutions, impurities in solids and x-ray diffraction in solutions of several solutes are briefly discussed.
Some new formulas in the theory of cumulants (Thiele semi-invariants) are presented.
A method for treating nonlinear stochastic systems is described which it is hoped will be useful in both the quantum‐mechanical many‐body problem and the theory of turbulence. In this method the true problem is replaced by models that lead to closed equations for correlation functions and averaged Green's functions. The model solutions are exact descriptions of possible dynamical systems, and, as a result, they display certain consistency properties. For example, spectral components of Green's functions which must be positive‐definite in the true problem automatically are so for the models. The models involve a new stochastic element: Random couplings are introduced among an infinite collection of similar systems, the true problem corresponding to the limit where these couplings vanish. The method is first applied to a linear oscillator with random frequency parameter. The mean impulse‐response function of the oscillator is obtained explicitly for two successive models. The results suggest the existence of a sequence of model solutions which converges rapidly to the exact solution of the true problem. Applications then are made to the Schrödinger equation of a particle in a random potential and to Burgers' analog for turbulence dynamics. For both problems, closed model equations are obtained which determine the average Green's function, the amplitude of the mean field, and the covariance of the fluctuating field. The model solutions can be expressed as sums of infinite classes of terms from the formal perturbation expansions of the solutions to the true problems. It is suggested that correspondence to stochastic models may be a useful criterion to help judge the validity of partial summations of perturbation series.
This paper shows how the dynamical and thermodynamical properties of an interacting quantum mechanical system with many degrees of freedom may be expressed and calculated solely in terms of renormalized propagators and renormalized vertices or interactions. The formulation employed is sufficiently general to encompass systems which have several components, with Fermi or Bose statistics, whether or not they exhibit superfluidity or superconductivity. The process of renormalization is the functional generalization of the thermodynamic transformation from the chemical potential and temperature to the energy and matter densities. With each set of variables (here, functions) is associated a natural thermodynamic function (here a functional). The natural functional for the unrenormalized potentials which occur in the Hamiltonian is the logarithm of the grand partition function; the natural functional for the fully renormalized variables, the distribution functions, is the entropy. In particular, a stationarity principle for a functional F(2) of distribution functions subject to constraints is shown to provide a fully renormalized description of the system. The numerical value of this functional, at the stationarity point at which the distribution functions take their actual value, is the entropy of the system. The equations of stationarity are expressions for the unrenormalized ν‐body potentials vν in terms of the ν′‐body distribution functions Gν′. The functionals F(2) and vν (of the distribution functions Gν′) are expressed as the solutions of closed functional differential equations which may be used to generate their power‐series expansions. For a superfluid Bose system, as for the electromagnetic field interacting with matter, it is necessary to consider expectation values of odd, as well as even, numbers of field operators. In particular it is necessary to employ the expectation values Gν for 2ν = 1, 2, 3, 4 field operators. For a fermion system, even if it is superconducting, only the functions Gν for 2ν = 2, 4 are required. In contrast to other thermodynamical functionals, the entropy functional F(2) makes no reference to equilibrium parameters such as temperature and chemical potential.
This paper investigates the criteria for maintenance of the macroscopic conservation laws of number, momentum, and energy by approximate two-particle correlation functions in many-body systems. The methods of generating such approximations are the same as in a previous paper. However, the derivations of the conservation laws given here clarify both why the approximation method works and the connection between the macroscopic conservation laws and those at the vertices. Conserving nonequilibrium approximations are based on self-consistent approximations to the one-particle Green's function. The same condition that ensures that the nonequilibrium theory be conserving also ensures that the equilibrium approximation has the following properties. The several common methods for determining the partition function from the one-particle Green's function all lead to the same result. When applied to a zero-temperature normal fermion system, the approximation procedure maintains the Hugenholtz-Van Hove theorem. Consequently, the self-consistent version of Brueckner's nuclear matter theory obeys this theorem.
A recapitulation is first given of a recent theory of homogeneous turbulence based on the condition that the Fourier amplitudes of the velocity field be as randomly distributed as the dynamical equations permit. This theory involves the average infinitesimal-impulse-response functions of the Fourier amplitudes and employs a new kind of perturbation method which yields what are belived to be exact expansions of third- and higher-order statistical moments of the Fourier amplitudes in terms of second-order moments and these response functions.
The steady distribution function for homogeneous turbulence is studied starting from Liouville's equation, modified by the introduction of an instantaneously fluctuating external force, which acts as a random source of energy. A new technique for solving Liouville's equation is presented giving a systematic development of the concepts of turbulent diffusion and turbulent viscosity. It amounts to a consistent generalization of the random phase approximation. When the rate of input of energy into the kth Fourier component uk has a power form h|k|−α, the functional form of the mean value [left angle bracket] uku−k [right angle bracket] can be determined exactly in the limit of large Reynolds number; it is $Ah^{\frac{2}{3}}|\bf K|^ {-{\frac {1}{3}}(5+2 \alpha)}$. Liouville's equation proves an inadequate basis for the steady time-dependent mean $\langle u_k(t)u_{-k}(t^ \prime) \rangle $ and a more general equation is derived. The new equation can be solved in a similar way and shows that the time-dependent correlation starts like a Gaussian in time, then passes through an exponentially decaying state, then eventually has a power dependence $|t-t^\prime|- \gamma^ {|k|}$.
We consider a nonlinear oscillator driven by random, Gaussian noise. The oscillator, which is damped and has linear and cubic terms in the restoring force, is often called the Duffing Equation. The Fourier transform of the response is expanded in a series in the coefficient of the nonlinear term. This series is then squared and averaged, and each term in the resulting response spectrum series is expressed in terms of the response spectrum of the linearized harmonic oscillator (i.e., without the cubic term). Since the forcing function is Gaussian, the linear solution is Gaussian. The terms in the series for the response spectrum are then regrouped so that common quantities can be factored out. This process leads to consolidated equations for the response spectrum and the common factors. These consolidated equations are truncated in various ways, and the corresponding solutions are compared with an analog computer experiment. This technique was proposed for turbulent flow by Kraichnan and followed up by Wyld, and has yielded some good results. The numerical results indicate that the truncated consolidated equations can provide a substantial improvement over some other methods used to solve this type of problem. The methods compared with it are (1) the traditional truncated parametric expansion, (2) statistical linearization, and (3) use of the joint-normal hypothesis to express the fourth and sixth moments in terms of the second.
A set of independent variables, called stripped correlation functions, closely related to Green’s functions, are introduced to characterize a given field theory. The parameters of the theory, which describe the interaction between fields (interaction potentials), are expressed in terms of these variables. The equations obtained by putting all the interaction potentials equal to zero are shown to be derivable from a stationarity principle on vacuum fluctuations with respect to stripped correlation functions. The resulting theory of self-generating interactions is discussed and it is shown that in such a theory single-particle propagators are not renormalized. These results are examined in connection with «bootstrap» theories and with a «quark» model of elementary particles.
The response of a system to an external disturbance can always be expressed in terms of time dependent correlation functions of the undisturbed system. More particularly the linear response of a system disturbed slightly from equilibrium is characterized by the expectation value in the equilibrium ensemble, of a product of two space- and time-dependent operators. When a disturbance leads to a very slow variation in space and time of all physical quantities, the response may alternatively be described by the linearized hydrodynamic equations. The purpose of this paper is to exhibit the complicated structure the correlation functions must have in order that these descriptions coincide. From the hydrodynamic equations the slowly varying part of the expectation values of correlations of densities of conserved quantities is inferred. Two illustrative examples are considered: spin diffusion and transport in an ordinary one-component fluid.Since the descriptions are equivalent, all transport processes which occur in the nonequilibrium system must be exhibited in the equilibrium correlation functions. Thus, when the hydrodynamic equations predict the existence of a diffusion process, the correlation functions will include a part which satisfies a diffusion equation. Similarly when sound waves occur in the nonequilibrium system, they will also be contained in the correlation functions.The description in terms of correlation functions leads naturally to expressions for the transport coefficients like those discussed by Kubo. The analysis also leads to a number of sum rules relating the dissipative linear coefficients to thermodynamic derivatives. It elucidates the peculiarly singular limiting behavior these correlations must have.
Near the critical point the characteristic time of motion associated with certain degrees of freedom experiences an enormous slowing-down. This circumstance gives rise to a possibility of constructing a kinetic equation to describe such slowed-down motions of the system, which corresponds to the Boltzmann equation of dilute gases which describes slow variations of the single-particle distribution function. We accomplish the derivation of kinetic equations with the aid of a generalized Langevin equation due to Mori. The theory is illustrated by deriving kinetic equations obeyed by critical fluctuations in isotropic and planar Heisenberg ferromagnets, an isotropic Heisenberg antiferromagnet, the liquid helium near the λ point, and a binary critical mixture. The kinetic equations conform to the dynamical scaling whenever it holds, and are valid in the hydrodynamic as well as in the critical regimes. The kinetic equations are then used to obtain selfconsistent closed equations to determine time correlation functions of critical fluctuations. In particular, in the case of a binary critical mixture, the selfconsistent equation can be used to obtain the diffusion constant and the decay rate of concentration fluctuation in the critical regime, which are expressed in terms of shear viscosity, and the results are in good numerical agreement with those of the recent light scattering experiments. The theory also leads to the modification of the Fixman correction to when q ⪡ k where q and k are the wavenumber and inverse correlation range of the concentration fluctuation, respectively.
The theory of turbulence in an incompressible fluid is formulated using methods similar to those of quantum field theory. A systematic perturbation theory is set up, and the terms in the perturbation series are shown to be in one to one correspondence with certain diagrams analogous to Feynman diagrams. From a study of the diagrams it is shown that the perturbation series can be rearranged and partially summed in such a way as to reduce the problem to the solution of three simultaneous integral equations for three functions, one of which is the second order velocity correlation function. The equations have the form of infinite power series integral equations, and the first few terms in the power series are derived from an analysis of the diagrams to sixth order. Truncation of the integral equations at the lowest nontrivial order yields Chandrasekhar's equation, and truncation at a higher order yields the equations discussed by Kraichnan.
In this paper two things are done. (1) It is shown that a considerable simplification can be attained in writing down matrix elements for complex processes in electrodynamics. Further, a physical point of view is available which permits them to be written down directly for any specific problem. Being simply a restatement of conventional electrodynamics, however, the matrix elements diverge for complex processes. (2) Electrodynamics is modified by altering the interaction of electrons at short distances. All matrix elements are now finite, with the exception of those relating to problems of vacuum polarization. The latter are evaluated in a manner suggested by Pauli and Bethe, which gives finite results for these matrices also. The only effects sensitive to the modification are changes in mass and charge of the electrons. Such changes could not be directly observed. Phenomena directly observable, are insensitive to the details of the modification used (except at extreme energies). For such phenomena, a limit can be taken as the range of the modification goes to zero. The results then agree with those of Schwinger. A complete, unambiguous, and presumably consistent, method is therefore available for the calculation of all processes involving electrons and photons.
The problem of the behavior of positrons and electrons in given external potentials, neglecting their mutual interaction, is analyzed by replacing the theory of holes by a reinterpretation of the solutions of the Dirac equation. It is possible to write down a complete solution of the problem in terms of boundary conditions on the wave function, and this solution contains automatically all the possibilities of virtual (and real) pair formation and annihilation together with the ordinary scattering processes, including the correct relative signs of the various terms.
The temporal development of quantized fields, in its particle aspect, is described by propagation functions, or Green’s functions. The construction of these functions for coupled fields is usually considered from the viewpoint of perturbation theory. Although the latter may be resorted to for detailed calculations, it is desirable to avoid founding the formal theory of the Green’s functions on the restricted basis provided by the assumption of expandability in powers of coupling constants. These notes are a preliminary account of a general theory of Green’s functions, in which the defining property is taken to be the representation of the fields of prescribed sources.
- R. H. Kraichnan
- J. R. Dorfman
- H. H. Ernst