Figure 5 - available via license: Creative Commons Attribution 4.0 International
Content may be subject to copyright.
Comparison of the impact of Drude (solid lines) and algebraic (dashed lines) decay of the memory function. Panel (a): The probability distribution corresponding to the mean kinetic energy of the quantum harmonic oscillator is depicted for two values of α and for the frequency ω =. 0 1 0 . Panel (b): The probability density ∼ x ( ) p corresponding to the mean potential energy of the quantum harmonic oscillator is shown for different magnitudes of the parameter α and ω =. 0 5 0
Source publication
We reveal a new face of the old clichéd system: a dissipative quantum harmonic oscillator. We formulate and study a quantum counterpart of the energy equipartition theorem satisfied for classical systems. Both mean kinetic energy Ek and mean potential energy Ep of the oscillator are expressed as Ek = 〈εk〉 and Ep = 〈εp〉, where 〈εk〉 and 〈εp〉 are mean...
Context in source publication
Context 1
... in the contribution to both kinetic and potential energy. In this panel we also depict the impact of the eigenfrequency ω 0 on this characteristic. An increase of the latter parameter causes shifting of the curve towards larger values of x m , however, the overall shape of the functional dependence remains unchanged. Last but not least, in Fig. 5 we compare the probability ...
Similar publications
We investigate the frictional force arising from quantum fluctuations when two dissipative metallic plates are set in a shear motion. While early studies showed that the electromagnetic fields in the quantum friction setup reach nonequilibrium steady states, yielding a time-independent force, other works have demonstrated the failure to attain stea...
We extend previous work on the numerical diagonalization of quantum stress tensor operators in the Minkowski vacuum state, which considered operators averaged in a finite time interval, to operators averaged in a finite spacetime region. Since real experiments occur over finite volumes and durations, physically meaningful fluctuations may be obtain...
The exact statistics of an arbitrary measuring observable, describing specific features of an open quantum system, is analytically obtained. Due to the probabilistic nature of a sequence of intermediate measurements and stochastic fluctuations induced by the interaction with an external environment, the measurement outcomes at the end of the system...
RCS reconstruction is an important way to reduce the measurement time in anechoic chambers and expand the radar original data, which can solve the problems of data scarcity and a high measurement cost. The greedy pursuit, convex relaxation, and sparse Bayesian learning-based sparse recovery methods can be used for parameter estimation. However, the...
Citations
... A novel work [10] investigated the simplest quantum Brownian oscillator model to formulate the energy of the system in terms of the average energy of a quantum oscillator in a harmonic well. Based on this work, more researchers [11][12][13][14][15][16][17][18] tried to study quantum counterparts of the equipartition theorem in different versions of quadratic open quantum systems from various perspectives, including electrical circuits [11], dissipative diamagnetism [18], and focusing on kinetic energy for a more general setup [19]. ...
... It is evident that Eqs. (11) and (14) reduce to the results in Refs. [10,24] for the single-mode n S = 1 case. ...
... Eq. (9b)], Eqs. (13) and (14) reduce to the results presented in Ref. [19]. ...
The equipartition theorem is a fundamental law of classical statistical physics, which states that every degree of freedom contributes k B T / 2 to the energy, where T is the temperature and k B is the Boltzmann constant. Recent studies have revealed the existence of a quantum version of the equipartition theorem. In the present work, we focus on how to obtain the quantum counterpart of the generalized equipartition theorem for arbitrary quadratic systems in which the multimode Brownian oscillators interact with multiple reservoirs at the same temperature. An alternative method of deriving the energy of the system is also discussed and compared with the result of the quantum version of the equipartition theorem, after which we conclude that the latter is more reasonable. Our results can be viewed as an indispensable generalization of recent works on a quantum version of the equipartition theorem.
Published by the American Physical Society 2024
... In the context of the quantum counterpart of energy equipartition theorem [24][25][26] (see [27] for a threedimensional generalization), one can easily show that the following are probability distributions: ...
... (16)] falls off too, over the same timescale and reduces to a delta-function, characterizing Markovian, i.e., memoryless damping. Now, taking γ as defined in Eq. (18) to be finite, let us have a look at Eq. (25). Since the bath now has a fast dynamics, in the sense that the correlations must fall-off really quickly over the timescale Ω −1 which tends to zero, we may approximate the integral in Eq. (25) around the point ω = ω 0 as ...
The quantum Langevin equation as obtained from the independent-oscillator model describes a strong-coupling situation, devoid of the Born-Markov approximation that is employed in the context of the Gorini-Kossakowski-Sudarshan-Lindblad equation. The question we address is what happens when we implement such 'Born-Markov'-like approximations at the level of the quantum Langevin equation for a harmonic oscillator which carries a noise term satisfying a fluctuation-dissipation theorem. In this backdrop, we also comment on the rotating-wave approximation.
... These two energy functions are respectively dubbed as the mean energy and the internal energy. Remarkably, it is found that both the energy functions respect the quantum counterpart of the energy equipartition theorem, which has generated a considerable amount of interest in recent times [17][18][19][20][21][22][23][24][25][26][27]. We also compute the thermally-averaged energy from the partition function of the system and compare the result with the other two energy functions obtained from the fluctuation-dissipation theorem. ...
... Let us briefly discuss the quantum counterpart of energy equipartition theorem in the present context, mainly revisiting the material presented in earlier works [17][18][19][20][21][22][23][24][25][26][27]. for the dissipative oscillator, as a function γ/ω 0 , for α = βℏω 0 = 0.5 and ω cut = 10ω 0 . ...
... (1)]. One may in this case, split the mean energy into the kinetic and potential energy parts, and obtain the quantum counterpart of energy equipartition theorem individually for the kinetic and potential energies as has been studied in [19,24]. ...
In this paper, we discuss some aspects of the energetics of a quantum Brownian particle placed in a harmonic trap, also known as the dissipative quantum oscillator. Based on the fluctuation-dissipation theorem, we analyze two distinct notions of thermally-averaged energy that can be ascribed to the oscillator. These energy functions, respectively dubbed hereafter as the mean energy and the internal energy, are found to be unequal for arbitrary system-bath coupling strength, when the bath spectral function has a finite cutoff frequency, as in the case of a Drude bath. Remarkably, both the energy functions satisfy the quantum counterpart of the energy equipartition theorem, but with different probability distribution functions on the frequency domain of the heat bath. Moreover, the Gibbs approach to thermodynamics provides us with yet another thermally-averaged energy function. In the weak-coupling limit, all the above-mentioned energy expressions reduce to ϵ = ℏω 0 2 coth ℏω 0 2k B T , which is the familiar result. We generalize our analysis to the case of the three-dimensional dissipative magneto-oscillator, i.e., when a charged dissipative oscillator is placed in a spatially-uniform magnetic field.
... principle for the kinetic energy has recently been verified for a general quantum system coupled to a heat bath [33] (see also the previous works on the free Brownian particle and a dissipative harmonic oscillator in [6,37,7] and extensive literature therein). Motivated by E. Wigner's original vision to model any sufficiently complex quantum system by random matrices, a particularly strong microcanonical version of the equipartition principle for Wigner matrices was first formulated and proven in [5]. ...
The total energy of an eigenstate in a composite quantum system tends to be distributed equally among its constituents. We identify the quantum fluctuation around this equipartition principle in the simplest disordered quantum system consisting of linear combinations of Wigner matrices. As our main ingredient, we prove the Eigenstate Thermalisation Hypothesis and Gaussian fluctuation for general quadratic forms of the bulk eigenvectors of Wigner matrices with an arbitrary deformation.
... with Z S = Tr S (exp(−βĤ S )) being the canonical partition function. On the other hand, it is useful to note that there is a relationship between the propagator K(q f , t, q i , 0) = q f |exp(− it hĤ S )|q i formulated in terms of path integrals [11] and the matrix elements in the position basis of Equation (16). Such a relation is evident when we replace t → −ihβ in the propagator, i.e, K(q f , −ihβ, q i , 0) = q f |exp(−βĤ S )|q i , then one obtains ...
... This simple example shows that, for this particular case, if one expects to create a true correspondence between the classical equipartition theorem discussed in Section 1 and the theorem in quantum mechanics phase-space formulation, it must be done in the high-temperature regime because, in the low-temperature regime, one gets the modified density distribution. Furthermore, for a model whose coordinates are bilinearly coupled as in [16], explicit expressions are found for Equations (23) and (24) in both regimes for the weak coupling limit, where it is shown that the energy is not distributed equally for the damped harmonic oscillator. This can also be seen from [17] where the Wigner distribution function for Equations (40) and (39) does not correspond with the Gibbs distribution. ...
In classical physics, there is a well-known theorem in which it is established that the energy per degree of freedom is the same. However, in quantum mechanics, due to the non-commutativity of some pairs of observables and the possibility of having non-Markovian dynamics, the energy is not equally distributed. We propose a correspondence between what is known as the classical energy equipartition theorem and its counterpart in the phase-space formulation in quantum mechanics based on the Wigner representation. Further, we show that in the high-temperature regime, the classical result is recovered.
... Although it may come as a surprise that thermodynamics can be consistently formulated for a single Brownian particle, which is far from the thermodynamic limit that one invokes in traditional thermodynamical studies, it turns out that not only does a thermal description follow naturally from the framework of classical and quantum Brownian motion, it is also consistent with all the laws of thermodynamics, including the third law [22,26,27]. We will contrast the expressions of average energy between classical and quantum Brownian motion, which will lead us to the recently-proposed quantum counterpart of energy equipartition theorem [28,29,30,31,32,33,34,35,36,37,38]. As a condensed matter physics application of the framework of quantum Brownian motion, we shall discuss dissipative diamagnetism [38,39] and emphasize on the role of confining potentials. ...
... Subsequently, at a time t ≳ 0, the particle is released, so that further evolution of the system is governed by the Hamiltonian (19). Note that this scenario of introducing an ensemble of initial conditions is consistent with choosing the retarded solution (30). Corresponding to such an ensemble, one finds that the function f (t) is a random function of time, whose statistical properties may be deduced by using ⟨q j (0)⟩ = ⟨p j (0)⟩ = 0, and ...
... It may be shown that P k (ω) is both positive definite and is normalized, justifying its interpretation as a bona fide probability density (see Appendix A). In recent literature, Eq. (79) has been termed as the quantum counterpart of energy equipartition theorem [28,29,30,31,32,33,34,35,36,37,38]. Let us turn our attention to the potential energy. ...
We review in a pedagogic manner the topic of quantum Brownian motion, with an emphasis on its thermodynamic aspects. For the sake of completeness, we begin with the classical treatment of one-dimensional Brownian motion, discussing correlation functions and fluctuation-dissipation relations. The equation-of-motion approach, based on the Langevin equation, is mostly followed throughout the paper. A microscopic derivation of the generalized Langevin equation is outlined, based on the microscopic model of a heat bath as a collection of a large number of independent classical harmonic oscillators. We then consider a fully quantum-mechanical treatment of Brownian motion based on the quantum Langevin equation, where the bath is modelled as a collection of independent quantum harmonic oscillators. In the stationary state, we analyze the quantum counterpart of energy equipartition theorem, which has generated a considerable amount of interest in recent literature. The free energy, entropy and third law of thermodynamics are discussed for the one-dimensional quantum Brownian motion in a harmonic well. Following this, we explore some aspects of dissipative diamagnetism in the context of two-dimensional quantum Brownian motion. The role of an external magnetic field and confining potentials is discussed. We then briefly outline the path-integral approach to thermodynamics of a quantum Brownian particle. Finally, we devote a substantial part of the review to discussing stochastic thermodynamics and fluctuation theorems in the context of classical and quantum Langevin equation.
... The heat bath is taken to be composed of an infinitely many independent harmonic oscillators [7][8][9] and the reduced quantum dynamics of the system is described by the quantum Langevin equation [10]. We shall also discuss the result obtained here in the light of the quantum counterpart of energy equipartition theorem studied earlier [11][12][13][14][15][16]. ...
... The presence of a noise term also introduces a thermal interpretation to the averages. For instance, one can solve eqn (13) and compute x 2 and ẋ 2 explicitly. They are found to be equal and give [20,21] ...
... and therefore, the kinetic and potential energies are unequal in general [13,14,16]. It is the competition between I 1 and I 2 which will decide whether the mean kinetic energy exceeds the mean potential energy or vice versa. ...
In this short note, we study the celebrated virial theorem for dissipative systems, both classical and quantum. The classical formulation is discussed and an intriguing effect of the random force (noise) is made explicit in the context of the virial theorem. Subsequently, we derive a generalized virial theorem for a dissipative quantum oscillator, i.e. a quantum oscillator coupled with a quantum heat bath. Such a heat bath is modelled as an infinite collection of independent harmonic oscillators with a certain distribution of initial conditions. In this situation, the non-Markovian nature of the quantum noise leads to novel bath-induced terms in the virial theorem. We also consider the case of an electrical circuit with thermal noise and analyze the role of non-Markovian quantum noise in the context of the virial theorem.
... Thus, in a sense, eqn (17) generalizes the energy equipartition theorem for an electrical circuit, taking into account quantum mechanical considerations. Eqn (17) is the electrical analogue of the recently proposed quantum counterpart of energy equipartition theorem wherein, the mean energy of a quantum Brownian oscillator is expressible as a similar two-fold average [18][19][20][21][22][23][24][25][26]. ...
In this brief note, we demonstrate a generalized energy equipartition theorem for a generic electrical circuit with Johnson-Nyquist (thermal) noise. From quantum mechanical considerations, the thermal modes have an energy distribution dictated by Planck's law. For a resistive circuit with some inductance, it is shown that the real part of the admittance is proportional to a probability distribution function which modulates the contributions to the system's mean energy from various frequencies of the Fourier spectrum. Further, we analyze the case with a capacitor connected in series with an inductor and a resistor. The results resemble superstatistics, i.e. a superposition of two statistics and can be reformulated in the energy representation. The correct classical limit is obtained as $\hbar \rightarrow 0$.
... One particular aspect of dissipative quantum systems which has generated a considerable amount of interest in the recent times is the quantum counterpart of the energy equipartition theorem [15][16][17][18][19][20][21]. According to this result which has been proven under quite general considerations [19], the mean kinetic energy of a quantum particular interacting linearly with a heat bath can always be expressed as ...
... This exactly corresponds to the quantum counterpart of energy equipartition theorem explored in the recent years [15][16][17][18][19][20][21]. Since Re[γ(ω)] > 0, as a consequence of the second law (see, for example [10]), from eqn (21) Φ(ω) is positive definite by inspection. ...
... This corresponds to the quantum counterpart of energy equipartition theorem for the potential energy of the oscillator (see for example [16,17,21]). By the same arguments as above, the function P p (ω) is positive definite. ...
In this paper, we demonstrate a remarkable connection between the recently proposed quantum energy equipartition theorem and dissipative diamagnetism exhibited by a charged particle moving in a two dimensional harmonic potential in the presence of a uniform external magnetic field. The system is coupled to a quantum heat bath through coordinate variables with the latter being modelled as a collection of independent quantum oscillators. In the full frequency domain: $\omega \in (-\infty,\infty)$, the equilibrium magnetic moment $M_z$ can be expressed as an integral over the bath spectrum involving the relaxation function $\Phi(\omega)$, and subsequently, it is possible to propose a fruitful connection between the quantum counterpart of energy equipartition theorem and magnetic moment of the oscillator. We discuss an alternate picture, which emerges upon restricting the integration domain to $\omega \in [0,\infty)$. In these limits, the magnetic moment can be written as an average over a distribution function $P_M (\omega)$ which has two wings corresponding to positive and negative segments. At high temperatures, these two wings identically cancel each other. However, at low temperatures, a small asymmetry between the positive and negative wings results in non-zero diamagnetic moment. A comparative study of the present results with those obtained from the more traditional Gibbs approach is performed and a perfect agreement is obtained.
... Recently there has been a considerable amount of interest in studying the quantum counterpart of the energy equipartition theorem for open quantum systems [12][13][14][15][16][17]. In particular, it has been demonstrated that in the steady state, the average energy E of an open quantum system receives contributions from the bath degrees of freedom and can typically be cast in the form, E = sum of contributions from the degrees of freedom of the bath. ...
... Furthermore, our final results are fairly general and robust as long as one is in a nonequilibrium steady state. The mean energy of the system is expressed as a two fold average just as in the case of the dissipative oscillator (see [14]) where for the latter, E(ω, T ) in equation (1) is the mean energy of an individual thermostat oscillator of frequency ω and at temperature T obtained by averaging over the Gibbs canonical state of the bath. The second averaging occurs in equation (1) as an average over the bath frequencies ω. ...
... In the wide-band approximation, we take ξ j,k to be a real constant independent of other parameters. The hybridization strengths are therefore Γ j := γ j which are real constants and consequently equation (14) reads (also see [8]), ...
In this brief report, following the recent developments on formulating a quantum analogue of the classical energy equipartition theorem for open systems where the heat bath comprises of independent oscillators, i.e. bosonic degrees of freedom, we present an analogous result for fermionic systems. The most general case where the system is connected to multiple reservoirs is considered and the mean energy in the steady state is expressed as an integral over the reservoir frequencies. Physically this would correspond to summing over the contributions of the bath degrees of freedom to the mean energy of the system over a suitable distribution function ρ(ω) dependent on the system parameters. This result holds for nonequilibrium steady states, even in the nonlinear regime far from equilibrium. We also analyze the zero temperature behaviour and low temperature corrections to the mean energy of the system.
