Lattice Gas Dynamics Research Articles

Exploring crossing times and congestion patterns at scramble intersections in pedestrian dynamics models: A statistical analysis

In this study, we present a model designed to unravel the statistical complexities of pedestrian crossing times and related physical quantities in a unique pedestrian intersection, specifically a double intersecting diagonal crossing. Our model employs an agent-based stochastic process based on a lattice gas dynamics to describe a four-species interaction on an underlying square lattice representing the intersection. This model incorporates four static floor fields guiding each species within the scenario. Our findings emphasize the critical role that the density of highly driven pedestrians plays in the statistics of crossing times by triggering the transition from a normal distribution for low densities to a uniform distribution for a crowded scenario. In the steady state, we mapped the system mobility and showed that an anomalous mobile phase emerges when considering a scenario with high density of poorly driven pedestrians. To monitor these phenomena, we focus on two variables: directed mobility and average distance between agents. We conclude our analysis by overcoming a crucial limitation of the lattice gas model with the introduction of viscosity effects among pedestrians. We observe that mixed state regime emerges as the system presents a mobile-to-immobile phase transition when viscosity effects faints. Our research sheds light on the nuanced dynamics of scramble intersections, contributing to a deeper understanding of urban mobility challenges.

Lattice Boltzmann method for axisymmetric turbulent flows

A lattice Boltzmann model for axisymmetric turbulent flows is developed. It is a further development of the enhanced axisymmetric lattice Boltzmann method (AxLAB®). The turbulent flow is efficiently and naturally simulated through incorporation of the standard subgrid-scale (SGS) stress model into the axisymmetric lattice Boltzmann equation in a consistent manner with the lattice gas dynamics. The model is verified by applying it to three typical cases in engineering: (i) pipe flow through an abrupt axisymmetric constriction, (ii) axisymmetric separated and reattached flow and (iii) pulsatile flows in a stenotic vessel. The numerical results obtained using the present method are compared with experimental data and other available numerical solutions, indicating good agreements. The model is simple and is able to predict axisymmetric turbulent flows at good accuracy.

Filling Dynamics of Closed End Nanocapillaries

We have studied the filling dynamics of model capillaries using dynamic mean field theory for a confined lattice gas and Kawasaki dynamics simulations. We have found two different scenarios for filling of capped nanocapillaries from the vapor phase. As compared to channels with macroscopic width, in which the filling process occurs by the detachment of the meniscus from the cap, in mesoscopic channels there is an alternative mechanism associated with the spontaneous condensation of the liquid close to the pore opening and its subsequent growth toward the closed pore end. We show that these two scenarios have totally different filling dynamics, providing an additional mechanism for slow capillary condensation kinetics in nanoscopic objects.

Transport on a lattice with dynamical defects

Many transport processes in nature take place on substrates, often considered as unidimensional lanes. These unidimensional substrates are typically nonstatic: Affected by a fluctuating environment, they can undergo conformational changes. This is particularly true in biological cells, where the state of the substrate is often coupled to the active motion of macromolecular complexes, such as motor proteins on microtubules or ribosomes on mRNAs, causing new interesting phenomena. Inspired by biological processes such as protein synthesis by ribosomes and motor protein transport, we introduce the concept of localized dynamical sites coupled to a driven lattice gas dynamics. We investigate the phenomenology of transport in the presence of dynamical defects and find a regime characterized by an intermittent current and subject to severe finite-size effects. Our results demonstrate the impact of the regulatory role of the dynamical defects in transport not only in biology but also in more general contexts.

Large deviations of the empirical currents for a boundary-driven reaction diffusion model

We derive a large deviation principle for the empirical currents of lattice gas dynamics which combine a fast stirring mechanism (Symmetric Simple Exclusion Process) and creation/annihilation mechanisms (Glauber dynamics). Previous results on the density large deviations can be recovered from this general large deviation principle. The contribution of external driving forces due to reservoirs at the boundary of the system is also taken into account.

Archimedes’ law and its corrections for an active particle in a granular sea

We study the origin of buoyancy forces acting on a larger particle moving in a granular medium subject to horizontal shaking and its corrections before fluidization. In the fluid limit, Archimedes’ law is verified, before the limit memory effects counteract buoyancy, as also found experimentally. The origin of the friction is an excluded volume effect between active particles, which we study more exactly for a random walker in a random environment. The same excluded volume effect is also responsible for the mutual attraction between bodies moving in the granular medium. Our theoretical modeling proceeds via an asymmetric exclusion process, i.e. via a dissipative lattice gas dynamics simulating the position degrees of freedom of a low density granular sea.

Superdiffusivity of Two Dimensional Lattice Gas Models

It was proved \cite{EMYa, QY} that stochastic lattice gas dynamics converge to the Navier-Stokes equations in dimension $d=3$ in the incompressible limits. In particular, the viscosity is finite. We proved that, on the other hand, the viscosity for a two dimensional lattice gas model diverges faster than $\log \log t$. Our argument indicates that the correct divergence rate is $(\log t)^{1/2}$. This problem is closely related to the logarithmic correction of the time decay rate for the velocity auto-correlation function of a tagged particle.

Thermoelectric phenomena via an interacting particle system

We present a mesoscopic model for thermoelectric phenomena in terms of an interacting particle system, a lattice electron gas dynamics that is a suitable extension of the standard simple exclusion process. We concentrate on electronic heat and charge transport in different but connected metallic substances. The electrons hop between energy-cells located alongside the spatial extension of the metal wire. When changing energy level, the system exchanges energy with the environment. At equilibrium the distribution satisfies the Fermi-Dirac occupation-law. Installing different temperatures at two connections induces an electromotive force (Seebeck effect) and upon forcing an electric current, an additional heat flow is produced at the junctions (Peltier heat). We derive the linear response behavior relating the Seebeck and Peltier coefficients as an application of Onsager reciprocity. We also indicate the higher order corrections. The entropy production is characterized as the anti-symmetric part under time-reversal of the space-time Lagrangian.

Statistical mechanical description and Monte Carlo simulation of diffusion in two-component lattice systems

Two different approaches based on linear response theory are considered in this paper. One of them results in quite general and exact expressions for kinetic diffusion coefficients in terms of mean square particle displacements that do not explicitly use any information concerning lattice gas dynamics or interparticle interactions. Relations between kinetic and chemical diffusion coefficients are considered as well. The other approach invokes the lattice gas master equation. It leads to a representation of the kinetic diffusion coefficient as a sum of static and memory-dependent terms. For the former, a closed form expression is derived in the case of thermally activated particle jumps in the model of traps. On this basis, the thermodynamic and memory aspects of multicomponent diffusion are considered. Monte Carlo (MC) simulation and calculation results are discussed. It is shown that memory effects are more pronounced as compared to one-component systems.

Lattice Gas Dynamics: Application to Driven Vortices in Two Dimensional Superconductors

A continuous time Monte Carlo lattice gas dynamics is developed to model driven steady states of vortices in two dimensional superconducting networks. Dramatic differences are found when compared to a simpler Metropolis dynamics. Subtle finite size effects are found at low temperature, with a moving smectic that becomes unstable to an anisotropic liquid on sufficiently large length scales.

Hydrodynamic limit of a disordered lattice gas

We consider a model of lattice gas dynamics in ℤd in the presence of disorder. If the particle interaction is only mutual exclusion and if the disorder field is given by i.i.d. bounded random variables, we prove the almost sure existence of the hydrodynamical limit in dimension d≥3. The limit equation is a non linear diffusion equation with diffusion matrix characterized by a variational principle.

Logarithmic Sobolev constant for the dilute Ising lattice gas dynamics below the percolation threshold

We consider a conservative stochastic lattice gas dynamics reversible with respect to the canonical Gibbs measure of the bond dilute Ising model on Z d at inverse temperature β. When the bond dilution density p is below the percolation threshold, we prove that, for any ε>0, any particle density and any β, with probability one, the logarithmic Sobolev constant of the generator of the dynamics in a box of side L centered at the origin cannot grow faster that L 2+ ε .

A LATTICE BOLTZMANN MODEL FOR THE SHALLOW WATER EQUATIONS WITH TURBULENCE MODELING

A lattice Boltzmann model for the shallow water equations with turbulence modeling (LABSWETM) is developed. The flow turbulence is efficiently and naturally taken into account by incorporating the standard subgrid-scale stress model into the lattice Boltzmann equation in a consistent manner with the lattice gas dynamics. The model is applied to solve two flow problems and is verified by comparing numerical predictions with analytical solutions and available experimental data. The results show that the LABSWETM is able to provide basic features of flow turbulence and produce good predictions for turbulent shallow water flows.

Diffusive scaling of the spectral gap for the dilute Ising lattice-gas dynamics below the percolation threshold

We consider a conservative stochastic lattice-gas dynamics reversible with respect to the canonical Gibbs measure of the bond dilute Ising model on ℤd at inverse temperature β. When the bond dilution density p is below the percolation threshold we prove that for any particle density and any β, with probability one, the spectral gap of the generator of the dyamics in a box of side L centered at the origin scales like L−2. Such an estimate is then used to prove a decay to equilibrium for local functions of the form \(\) where e is positive and arbitrarily small and α = ½ for d = 1, α=1 for d≥2. In particular our result shows that, contrary to what happes for the Glauber dynamics, there is no dynamical phase transition when β crosses the critical value βc of the pure system.

Droplet motion for the conservative 2D Ising lattice gas dynamics below the critical temperature

We consider the 2D Ising lattice gas in a square of side L with free boundary conditions, temperature below the critical one and particle density slightly above the density of the vapor phase. Typical configurations consist of a quarter of a Wulff droplet of the liquid phase centered at one of the corners of the given square. We then introduced a reversible Markovian spin exchange dynamics, also known as Kawasaki dynamics, on the configuration space and we discuss the heuristics of the transition of a bubble of the liquid phase from one corner to another. We then present some numerical evidence suggesting that the preferred mechanism to make the transition is via evaporation of the original bubble and simultaneous reconstruction of a new bubble around a new corner.

A lattice-gas model of microemulsions

We develop a lattice gas model for the non-equilibrium dynamics of microemulsions. Our model is based on the immiscible lattice gas of Rothman & Keller, which we reformulate using a microscopic, particulate description so as to permit generalization to more complicated interactions, and on the prescription of Chan & Liang for introducing such interparticle interactions into lattice gas dynamics. We present the results of simulations to demonstrate that our model exhibits the correct phenomenology, and we contrast it with both equilibrium lattice models of microemulsions, and to other lattice gas models.

Logarithmic Sobolev inequality for lattice gases with mixing conditions

Let $$\mu _{\Lambda _L ,\lambda }^{gc} $$ denote the grand canonical Gibbs measure of a lattice gas in a cube of sizeL with the chemical potential γ and a fixed boundary condition. Let $$\mu _{\Lambda _L ,n}^c $$ be the corresponding canonical measure defined by conditioning $$\mu _{\Lambda _L ,\lambda }^{gc} $$ on $$\Sigma _{x \in \Lambda } \eta _x = n$$ . Consider the lattice gas dynamics for which each particle performs random walk with rates depending on near-by particles. The rates are chosen such that, for everyn andL fixed, $$\mu _{\Lambda _L ,n}^c $$ is a reversible measure. Suppose that the Dobrushin-Shlosman mixing conditions holds for $$\mu _{L,\lambda }^{gc} $$ forall chemical potentials λ ∈ γ ∈ ℝ. We prove that $$\smallint f\log fd\mu _{\Lambda _L ,n}^c \leqq const.L^2 D(\sqrt f )$$ for any probability densityf with respect to $$\mu _{\Lambda _L ,n}^c $$ ; here the constant is independent ofn orL andD denotes the Dirichlet form of the dynamics. The dependence onL is optimal.

Kinetic limit of a conservative lattice gas dynamics showing long-range correlations

An anisotropic lattice gas dynamics is investigated for which particles on ℤ d jump to empty nearest neighbor sites with (fast) rate e−2 in a specified direction and some particular configuration-dependent rates in the other directions. The model is translation and reflection invariant and is particle conserving. The space coordinate in the “fast-rate” direction is rescaled by e−1. It follows that the density field converges in probability, as e↓0, to the corresponding solution of a nonlinear diffusion-type equation. The microscopic fluctuations about the deterministic macroscopic evolution are determined explicitly and it is found that the stationary fluctuations decay via a power law (∼1/r d ) with the direction dependence of a quadrupole field.

Bifurcations of a lattice gas flow under external forcing

We study the behavior of a Frisch-Hasslacher-Pomeau lattice gas automaton under the effect of a spatially periodic forcing. It is shown that the lattice gas dynamics reproduces the steady-state features of the bifurcation pattern predicted by a properly truncated model of the Navier-Stokes equations. In addition, we show that the dynamical evolution of the instabilities driving the bifurcation can be modeled by supplementing the truncated Navier-Stokes equation with a random force chosen on the basis of the automaton noise.