Maximum Entropy Principle for Lattice Kinetic Equations

Physical Review E

Kinetically constrained models are lattice-gas models that are used for describing glassy systems. By construction, their equilibrium state is trivial and there are no equal-time correlations between the occupancy of different sites. We drive such models out of equilibrium by connecting them to two reservoirs of different densities, and we measure the response of the system to this perturbation. We find that under the proper coarse-graining, the behavior of these models may be expressed by a nonlinear diffusion equation, with a modeland density-dependent diffusion coefficient. We find a simple approximation for the diffusion coefficient, and we show that the relatively mild discrepancy between the approximation and our numerical results arises due to non-negligible correlations that appear as the system is driven out of equilibrium, even when the density gradient is infinitesimally small. Similar correlations appear when such kinetically constrained models are driven out of equilibrium by applying a uniform external force. We suggest that these correlations are the reason for the same discrepancy between the approximate diffusion coefficient and the numerical results for a broader group of models-nongradient lattice-gas models-for which kinetically constrained models are arguably the simplest example thereof.

Physics Letters A, 2000

We study by numerical simulation the Kob-Andersen KA model of a lattice gas with kinetic constraints on a Ž . face-centered cubic FCC lattice, both in its canonical and in its grand-canonical version, and for different dynamical rules. The model exhibits a dynamical transition at a threshold value of the density, where the diffusion constant vanishes as a power law. We confirm that the corresponding exponent is independent of the details of the dynamics, while the threshold density is nonuniversal. On the other hand, the fluctuation-dissipation ratio X appears to be nonuniversal. q

Physical Review E, 2017

In kinetic theory, a system is usually described by its one-particle distribution function f (r, v, t), such that f (r, v, t)drdv is the fraction of particles with positions and velocities in the intervals (r, r + dr) and (v, v + dv), respectively. Therein, global stability and the possible existence of an associated Lyapunov function or H-theorem are open problems when non-conservative interactions are present, as in granular fluids. Here, we address this issue in the framework of a lattice model for granular-like velocity fields. For a quite general driving mechanism, including both boundary and bulk driving, we show that the steady state reached by the system in the long time limit is globally stable. This is done by proving analytically that a certain H-functional is non-increasing in the long time limit. Moreover, for a quite general energy injection mechanism, we are able to demonstrate that the proposed H-functional is non-increasing for all times. Also, we put forward a proof that clearly illustrates why the "classical" Boltzmann functional HB[f ] = dr dvf (r, v, t) ln f (r, v, t) is inadequate for systems with non-conservative interactions. Not only is this done for the simplified kinetic description that holds in the lattice models analysed here but also for a general kinetic equation, like Boltzmann's or Enskog's.

2000

In the last decade, minimal kinetic models, and primarily the Lattice Boltmann equation, have met with significant success in the simulation of complex hydrodynamic phenomena, ranging from slow flows in grossly irregular geometries to fully developed turbulence, to flows with dynamic phase transitions. Besides their practical value as efficient computational tools for the dynamics complex systems, these minimal models may also represent a new conceptual paradigm in modern computational statistical mechanics: instead of proceeding bottom-up from the underlying microdynamic systems, these minimal kinetic models are built top-down starting from the macroscopic target equations. This procedure can provide dramatic advantages, provided the essential physics is not lost along the way. For dissipative systems, one such essential requirement is the compliance with the Second Law of thermodynamics. In this Colloquium, we present a chronological survey of the main ideas behind the Lattice Boltzmann method, with special focus on the role played by the H theorem in enforcing compliance of the method with macroscopic evolutionary constraints (Second Law) as well as in serving as a numerically stable computational tool for fluid flows and other dissipative systems out of equilibrium. Contents I. Introduction 1 II. Statistical mechanics background 2 A. The BBGKY hierarchy 2 B. The atomistic level 2 C. Many-body kinetic level 3 D. The Boltzmann equation and the H-theorem 3 E. The macroscopic level 4 F. The top-down approach 4 III. Lattice Gas Cellular Automata 5 A. The Frisch-Hasslacher-Pomeau automaton 5 B. Boolean microdynamics 5 C. Merits and pitfalls of Lattice Gas Automata 6 IV. Lattice Boltzmann equations 7 A. The fully non-linear LBE 8 B. The LBE in scattering form 8 C. The self-consistent LBE 9 D. The lattice Bhatnagar-Gross-Krook equation 9 V. H theorem in discrete phase-space 9 A. Reflections on the continuum case 10 B. Galilean invariance 11 C. Realizability 11 D. Solvability 12 E. Discrete-time effects and the mirage of zero-viscosity 12 F. Over-constrained equilibria 13 VI. Directions for future research 13 A. Thermal LBE's

Reviews of Modern Physics, 2002

In the last decade, minimal kinetic models, and primarily the Lattice Boltmann equation, have met with significant success in the simulation of complex hydrodynamic phenomena, ranging from slow flows in grossly irregular geometries to fully developed turbulence, to flows with dynamic phase transitions. Besides their practical value as efficient computational tools for the dynamics complex systems, these minimal models may also represent a new conceptual paradigm in modern computational statistical mechanics: instead of proceeding bottom-up from the underlying microdynamic systems, these minimal kinetic models are built top-down starting from the macroscopic target equations. This procedure can provide dramatic advantages, provided the essential physics is not lost along the way. For dissipative systems, one such essential requirement is the compliance with the Second Law of thermodynamics. In this Colloquium, we present a chronological survey of the main ideas behind the Lattice Boltzmann method, with special focus on the role played by the H theorem in enforcing compliance of the method with macroscopic evolutionary constraints (Second Law) as well as in serving as a numerically stable computational tool for fluid flows and other dissipative systems out of equilibrium. Contents I. Introduction 1 II. Statistical mechanics background 2 A. The BBGKY hierarchy 2 B. The atomistic level 2 C. Many-body kinetic level 3 D. The Boltzmann equation and the H-theorem 3 E. The macroscopic level 4 F. The top-down approach 4 III. Lattice Gas Cellular Automata 5 A. The Frisch-Hasslacher-Pomeau automaton 5 B. Boolean microdynamics 5 C. Merits and pitfalls of Lattice Gas Automata 6 IV. Lattice Boltzmann equations 7 A. The fully non-linear LBE 8 B. The LBE in scattering form 8 C. The self-consistent LBE 9 D. The lattice Bhatnagar-Gross-Krook equation 9 V. H theorem in discrete phase-space 9 A. Reflections on the continuum case 10 B. Galilean invariance 11 C. Realizability 11 D. Solvability 12 E. Discrete-time effects and the mirage of zero-viscosity 12 F. Over-constrained equilibria 13 VI. Directions for future research 13 A. Thermal LBE's

2008

We apply the projection operator formalism to the problem of determining the asymptotic behavior of the lattice BGK equation in the hydrodynamic limit. As an alternative to the more usual Chapman-Enskog expansion, this approach offers many benefits. Most remarkably, it produces absolutely exact, though non-Markovian, hydrodynamic difference equations as an intermediate step. These are accurate to all orders in Knudsen number and hence contain all of the physics of the Burnett equations and beyond. If appropriate, these equations may then be Taylor expanded to second order in Knudsen number to obtain the usual hydrodynamic equations that result from the Chapman-Enskog analysis. The method offers the potential to derive hydrodynamic difference equations for complex fluids with sharp gradients, such as immiscible and amphiphilic flow, for which the assumptions underlying the Chapman-Enskog approach are generally invalid.

Communications in Mathematical Physics, 2010

Kinetically constrained lattice gases (KCLG) are interacting particle systems which show some of the key features of the liquid/glass transition and, more generally, of glassy dynamics. Their distintictive signature is the following: i) reversibility w.r.t. product i.i.d. Bernoulli measure at any particle density and ii) vanishing of the exchange rate across any edge unless the particle configuration around the edge satisfies a proper constraint besides hard core. Because of degeneracy of the exchange rates the models can show anomalous time decay in the relaxation process w.r.t. the usual high temperature lattice gas models particularily in the so called cooperative case, when the vacancies have to collectively cooperate in order for the particles to move through the systems. Here we focus on the Kob-Andersen (KA) model, a cooperative example widely analyzed in the physics literature, both in a finite box with particle reservoirs at the boundary and on the infinite lattice. In two dimensions (but our techniques extend to any dimension) we prove a diffusive scaling O(L 2) (apart from logarithmic corrections) of the relaxation time in a finite box of linear size L. We then use the above result to prove a diffusive decay 1/t (again apart from logarithmic corrections) of the spin-spin time autocorrelation function at any particle density, a result that has been sometimes questioned on the basis of numerical simulations. The techniques that we devise, based on a novel combination of renormalization and comparison with a long-range Glauber type constrained model, are robust enough to easily cover other choices of the kinetic constraints.

Physical Review E, 2006

We propose a discrete lattice version of the Fokker-Planck kinetic equation along lines similar to the Lattice-Boltzmann scheme. Our work extends an earlier one-dimensional formulation to arbitrary spatial dimension D. A generalized Hermite-Gauss procedure is used to construct a discretized kinetic equation and a Chapman-Enskog expansion is applied to adapt the scheme so as to correctly reproduce the macroscopic continuum equations. The stability of the algorithm with respect to the finite time-step ∆t is characterized by the eigenvalues of the collision matrix. A heuristic secondorder algorithm in ∆t is applied to investigate the time evolution of the distribution function of simple model systems, and compared to known analytical solutions. Preliminary investigations of sedimenting Brownian particles subjected to an orthogonal centrifugal force illustrate the numerical efficiency of the Lattice-Fokker-Planck algorithm to simulate non-trivial situations. Interactions between Brownian particles may be accounted for by adding a standard BGK collision operator to the discretized Fokker-Planck kernel.

Physical review. E, Statistical, nonlinear, and soft matter physics, 2012

We study the possibility of modifying collisions in the lattice Boltzmann method to keep the proper entropy balance. We demonstrate that in the space of distributions operated on by lattice Boltzmann methods which respect a Boltzmann type H theorem, there exists a vicinity of the equilibrium where collisions with entropy balance are possible and, at the same time, there exists a region of nonequilibrium distributions where such collisions are impossible. In particular, for a strictly concave and uniformly bounded entropy function with positive equilibria, we show that proper entropy balance is always possible sufficiently close to the local equilibrium and it is impossible sufficiently far from it, where additional dissipation has to appear. We also present some nonclassical entropies that do not share this concern. The cases where the distribution enters the region far from equilibrium typically occur in flows with low viscosity and/or high Mach number flows and in simulations on c...

Journal of Fluid Mechanics, 2009

Lattice kinetic equations incorporating the effects of external/internal force fields via a shift of the local fields in the local equilibria, are placed within the framework of continuum kinetic theory. The mathematical treatment reveals that, in order to be consistent with the correct thermo-hydrodynamical description, temperature must also be shifted, besides momentum. New perspectives for the formulation of thermo-hydrodynamic lattice kinetic models of non-ideal fluids are then envisaged. It is also shown that on the lattice, the definition of the macroscopic temperature requires the inclusion of new terms directly related to discrete effects. The theoretical treatment is tested against a controlled case with a non ideal equation of state.