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Graphical solution of Eq.(74) for different values of the parameter α. The eigenvalue λs is a concave function of s, plotted for different values of the restitution coefficient α for the 2-D inelastic Maxwell model. The line y = sγ0 is plotted for α = 0.6, 0.8 and α = 1(top to bottom). The intersections with λs determine the points s0 (filled circles) and s1 (open circles). Here s1 = a determines the exponent of the power law tail. For the elastic case (α = 1, γ0 = 0, energy conservation) there is only one intersection point.
Source publication
Analytic solutions $F(v,t)$ of the nonlinear Boltzmann equation in $d$-dimensions are studied for a new class of dissipative models, called inelastic repulsive scatterers, interacting through pseudo-power law repulsions, characterized by a strength parameter $\nu$, and embedding inelastic hard spheres ($\nu=1$) and inelastic Maxwell models ($\nu=0$...
Contexts in source publication
Context 1
... illustrate the solution method of (74) with the graphical construction in Figure 2, where we look for intersections of the line y = 1 2 λ 2 = γ 0 s with the curve y = λ s for different values of α. ...
Context 2
... relevant properties of λ s are: (i) lim s→0 λ s = −1; (ii) λ s is a concave function, monotonically increasing with s, and (iii) all eigenvalues for positive integers n are positive (see Figure 2). As can be seen from the graphical construction, the transcendental equation (74) has two solutions, the trivial one (s 0 = 2) and the solution s 1 = a with a > 2. The numerical solutions for d = 2, 3 are shown in Figure 3 as a function of α, and the α-dependence of the root a (α) can be understood from the graphical construction. ...
Citations
... It is noteworthy that the first-order stress tensor (1) turns out to be intrinsically coupled to the first-order spin-spin tensor (1) , as Eq. (40) shows. Thus, the derivation of the shear viscosity η requires the parallel derivation of the spin viscosity η introduced in Eq. (41), this new transport coefficient being given by Eq. (43a). ...
... (1) i j in the first row of Eq. (40). This is equivalent to formally setting ψ 20|02 → 0 in Eq. (43b), with the result ...
Granular gases demand models capable of capturing their distinct characteristics. The widely employed inelastic hard-sphere model (IHSM) introduces complexities that are compounded when incorporating realistic features like surface roughness and rotational degrees of freedom, resulting in the more intricate inelastic rough hard-sphere model (IRHSM). This paper focuses on the inelastic rough Maxwell model (IRMM), presenting a more tractable alternative to the IRHSM and enabling exact solutions. Building on the foundation of the inelastic Maxwell model (IMM) applied to granular gases, the IRMM extends the mathematical representation to encompass surface roughness and rotational degrees of freedom. The primary objective is to provide exact expressions for the Navier–Stokes–Fourier transport coefficients within the IRMM, including the shear and bulk viscosities, the thermal and diffusive heat conductivities, and the cooling-rate transport coefficient. In contrast to earlier approximations in the IRHSM, our study unveils inherent couplings, such as shear viscosity to spin viscosity and heat conductivities to counterparts associated with a torque-vorticity vector. These exact findings provide valuable insights into refining the Sonine approximation applied to the IRHSM, contributing to a deeper understanding of the transport properties in granular gases with realistic features.
... It is worth noting that the relation between a Boltzmann kinetic equation and a master equation is not new [51]. In fact, this relation is behind the so-called DSMC method, a Monte Carlo method to solve kinetic equations [52,53]. ...
... The variables c i on the argument of Φ s are now defined in terms of the temperature as given by Eqs. (50)- (51). So far, the results given in this section are valid for any network topology. ...
... The case β = 0 is related to the so-called Maxwell model for granular gases. The resulting equation for φ has an analytical solution [57,51,58,59]: ...
A model for continuous-opinion dynamics is proposed and studied by taking advantage of its similarities with a mono-dimensional granular gas. Agents interact as in the Deffuant model, with a parameter $\alpha$ controlling the persuasibility of the individuals. The interaction coincides with the collision rule of two grains moving on a line, provided opinions and velocities are identified, with $\alpha$ being the so-called coefficient of normal restitution. Starting from the master equation of the probability density of all opinions, general conditions are given for the system to reach consensus. The case when the interaction frequency is proportional to the $\beta$-power of the relative opinions is studied in more detail. It is shown that the mean-field approximation to the master equation leads to the Boltzmann kinetic equation for the opinion distribution. In this case, the system always approaches consensus, which can be seen as the approach to zero of the opinion temperature, a measure of the width of the opinion distribution. Moreover, the long-time behaviour of the system is characterized by a scaling solution to the Boltzmann equation in which all time dependence occurs through the temperature. The case $\beta=0$ is related to the Deffuant model and is analytically soluble. The scaling distribution is unimodal and independent of $\alpha$. For $\beta>0$ the distribution of opinions is unimodal below a critical value of $|\alpha|$, being multimodal with two maxima above it. This means that agents may approach consensus while being polarized. Near the critical points and for $|\alpha|\ge 0.4$, the distribution of opinions is well approximated by the sum of two Gaussian distributions. Monte Carlo simulations are in agreement with the theoretical results.
... Maxwell gases with scalar velocities, with constant rate of collision and diffusive driving (dv/dt = η where η is gaussian white noise), may be solved exactly to yield an exponential decay for the velocity distribution with β = 1 [62][63][64]. Both these cases can be analysed using single framework by considering a general model where a pair of particles with relative velocity v collide with rate |v| δ [65][66][67][68]. Even though one is especially interested in physical limit of δ = 1 (also called hard-spheres), the general model is useful for more general kernels which may phenomenologically have other values of δ [42]. ...
... Note also, that in elastic kinetic theory δ = − 3 corresponds to Coulomb interaction. For the model with collision rate |v| δ the Boltzmann equation with diffusive driving may be analyzed to yield β = (2 + δ)/2 [65] retrieving the special limits of hardsphere (δ = 1) and Maxwell (δ = 1) gases. For a review on results obtained from kinetic theory, see Ref. [69]. ...
... It is straightforward to see that, by balancing the second and third terms and ignoring inter-particle collisions, one obtains β = 2 [69,74,75]. On the other hand, when r w = 1, then Γ = 0, and on balancing the first and third terms, one obtains β = (2 + δ)/2 as in kinetic theory [65]. The analysis of the moment equation was performed under the assumption that the leading asymptotic behaviour of the velocity distribution is a stretched exponential. ...
We determine the asymptotic behavior of the tails of the steady state velocity distribution of a homogeneously driven granular gas comprising of particles having a scalar velocity. A pair of particles undergo binary inelastic collisions at a rate that is proportional to a power of their relative velocity. At constant rate, each particle is driven by multiplying its velocity by a factor \(-r_w\) and adding a stochastic noise. When \(r_w <1\), we show analytically that the tails of the velocity distribution are primarily determined by the noise statistics, and determine analytically all the parameters characterizing the velocity distribution in terms of the parameters characterizing the stochastic noise. Surprisingly, we find logarithmic corrections to the leading stretched exponential behavior. When \(r_w=1\), we show that for a range of distributions of the noise, inter-particle collisions lead to a universal tail for the velocity distribution.
... The theoretical explanation for low-frequency waves is based on collisions: discrete particles interact via repulsive forces due to impact. This concept is particularly relevant for granular gases [1][2][3], granular liquids [4][5][6] and for the macroscopic dynamic behavior of materials with a high packing density, the so-called granular solids. The evolution of the dynamic contact force F during a collision of two granules is modeled with the nonlinear Hertzian contact law for nonconforming surfaces, defined by F = Kδ n . ...
Low-frequency waves generated by an impact on an isostatic granular
medium with vanishing confinement pressure are theorized to depend on
rigid-particle collisions. According to this model, an impact generates
a solitary wave spanning several particle diameters. High-frequency
waves with wavelengths smaller than a typical granule size behave
differently, and their propagation can be modeled as diffusive. In this
paper, experiments and simulations based on a one-dimensional chain of
spheres are performed to measure surface and bulk waves in individual
granules. Results show that an impact also generates high-frequency
waves, due to the dynamic behavior of the individual granules, appearing
simultaneously with the solitary wave. Intra-particle Rayleigh waves
play a key role in generating and transmitting macroscopic wave modes.
The exchange energy between intra-particle wave modes also influences
the decay of the solitary wave.
... This quasi-static formulation can be applied to collisions provided contact times are long compared to intra-particle dynamic deformations [1]. This nonlinear-collision model is employed to deduce the coefficient of restitution for inelastic collisions and is particularly relevant for granular gases [2,3,4] and granular liquids [5,6,7] where the particle interaction is usually small compared to time of flight between collisions. The dynamics of granular solids feature two regimes that have been observed experimentally: a low-frequency ballistic regime and a high-frequency diffusive regime. ...
Wave propagation in granular solids is found to have at least two
regimes: a low-frequency ballistic signal and a high-frequency diffusive
tail. The latter is attributed to granules clapping or elastic waves
within the granules themselves. The low frequency front is traditionally
attributed to the rigid-body motion of particles and their interactions
via quasi-static contact forces. We present results obtained with
experimental and numerical models indicating that the low-frequency
ballistic wavefront is itself the result of surface waves in the
granules. We find surface waves to be the principal mechanism of energy
transmission.
... This argument predicts for the hard spheres model tails like e −a|v| 3/2 for the heat bath, like e −a|v| 2 for heat bath and friction, and like e −a|v| for homogeneous cooling states. Such behaviors were predicted, on the basis of slightly more precise arguments, by Ernst and Brito (see the review papers [29, 28, 27]; e.g. [28, Section 4.4]). ...
... [28, Section 4.4]). There is all reason to believe that this is the correct theoretical answer, let apart the special case β = γ = 0 which is more subtle (as discussed in [28]). The shear flow case is more intricate since there is no reason for solutions to be radially symmetric. ...
... Since then these equations have been studied thoroughly by several authors. Good recent review papers have been written by Ben Na¨ımNa¨ım and Krapivsky [5], Ernst and Brito [28, 27], who survey the existing physical literature and physical results. In [28] the emphasis is put on the Maxwell model as a model arising from stochastic dynamics. ...
This is a short and somewhat informal review on the most mathematical parts of the kinetic theory of granular media, intended for physicists and for mathematicians outside the field.
... This is similar to the rapid velocity relaxation to a Maxwell distribution in normal gases. A great deal is known about this special solution, the "homogeneous cooling solution" (HCS), from both analytic [8,9,10] and Monte Carlo simulation studies [11]. ...
The Boltzmann equation for a gas of smooth, inelastic hard spheres is introduced and its homogeneous solution for an isolated system is discussed. The possibility of hydrodynamic excitations is explored using the linearized Boltzmann equation for small spatial perturbations of the homogeneous state. It is shown that the spectrum of the generator for this linear dynamics contains five points at long wavelengths that are the origin of hydrodynamic excitations. With this knowledge, response functions are introduced that allow the formal derivation of linear hydrodynamic
equations and associated Green‐Kubo expressions for the transport coefficients at Navier‐Stokes order.
A model for continuous-opinion dynamics is proposed and studied by taking advantage of its similarities with a mono-dimensional granular gas. Agents interact as in the Deffuant model, with a parameter α controlling the persuasibility of the individuals. The interaction coincides with the collision rule of two grains moving on a line, provided opinions and velocities are identified, with α being the so-called coefficient of normal restitution. Starting from the master equation of the probability density of all opinions, general conditions are given for the system to reach consensus. The case when the interaction frequency is proportional to the β-power of the relative opinions is studied in more detail. It is shown that the mean-field approximation to the master equation leads to the Boltzmann kinetic equation for the opinion distribution. In this case, the system always approaches consensus, which can be seen as the approach to zero of the opinion temperature, a measure of the width of the opinion distribution. Moreover, the long-time behaviour of the system is characterized by a scaling solution to the Boltzmann equation in which all time dependence occurs through the temperature. The case β=0 is related to the Deffuant model and is analytically soluble. The scaling distribution is unimodal and independent of α. For β>0 the distribution of opinions is unimodal below a critical value of |α|, being multimodal with two maxima above it. This means that agents may approach consensus while being polarized. Near the critical points and for |α|≥0.4, the distribution of opinions is well approximated by the sum of two Gaussian distributions. Monte Carlo simulations are in agreement with the theoretical results.
In this paper, a two-dimensional granular gas of inelastic, rough spheres subject to driving is examined. Either the translational
degrees of freedom are agitated proportional to a power | v([(x)\vec]) |d\left| {v(\vec x)} \right|^\delta of the local particle velocity v([(x)\vec])v(\vec x), or the rotational degrees are agitated randomly with respect to the angular velocity w([(x)\vec])\omega (\vec x).
The steady state properties of the model, with respect to energy, partition of energy, and velocity distributions, are examined
for different values of δ, and compared with the homogeneous driving case δ = 0. A driving linearly proportional to v([(x)\vec])v(\vec x) seems to reproduce some experimental observations which could not be reproduced by a homogeneous driving. Furthermore, we
obtain that the system can be homogenized even for strong dissipation, if a driving inversely proportional to the velocity
is used (δ < 0). In the case of rotational driving, the system is well randomized and clusters are hindered. Even though rotational
driving may be difficult to realize experimentally, this is an opportunity to avoid or delay the often unwanted effect of
clustering.
Tese (doutorado)—Universidade de Brasília, Instituto de Física, 2006. Neste trabalho, nós estudamos a dinâmica dissipativa de sistemas de esferas rígidas e lisas, com coeficiente de restituição constante e com baixo número de partículas, no Regime de Esfriamento Homogêneo. Nós nos concentramos na obtenção da função de distribuição de velocidades por meio de simulações em Dinâmica Molecular. O principal objetivo é apresentar uma metodologia baseada na aplicação da técnica da Função Característica Empírica e da chamada função W, introduzida por Lévy para medir a distância de distribuições para uma Gaussiana. Nós usamos esta metodologia para (i) caracterizar estados assintóticos estacionários independentemente das condições iniciais; (ii) estudar a múltipla dependência desses estados estacionários no número de partículas, livre caminho médio, coeficiente de restituição e condições de contorno; (iii) discutir a existência de um estado limite no limite termodinâmico; (iv) estudar o problema da convergência da expansão polinomial de Sonine no regime de alta inelasticidade; (v) propor um novo método de truncagem para resolver a dinâmica dos coeficientes de Sonine e (vi) medir as caudas de alta energia superpopuladas. Além disso, nós investigamos em que medida os resultados teóricos relacionados à equação de Boltzmann inelástica podem ser reproduzidos. _________________________________________________________________________________ ABSTRACT In this work we study the dissipative dynamics of systems of smooth hard spheres with a constant restitution coefficient and small number of particles in a Homogeneous Cooling Regime. The main goal is to present a methodology based on the application of the Empiric Characteristic Function technique and the so-caled W function, introduced by Lévi to measure the distance of distributions from the Gaussian. We use this methodology to (i) characterize asymptotic stationary states independently of initial conditions; (ii) to study the multiple dependence of these stationary states on the number of particles, mean free path, restitutions coefficient and boundary conditions; (iii) to discuss the existence of a limit state in the thermodynamic limit; (iv) to study the problem of the convergence of the sonine plynomial expansion at the highly dissipative regime; (v) to propose a new truncation scheme to solve the dynamics of the Sonine coefficients and (vi) measure teh over populated high energy tails. Moreover, we investigate in what sense the theoretical results related to the inelastic Boltzmann equation can be reproduced.
