Diffusive Fluctuations for One-Dimensional Totally Asymmetric Interacting Random Dynamics

We study central limit theorems for a totally asymmetric, one-dimensional interacting random system. The models we work with are the Aldous–Diaconis–Hammersley process and the related stick model. The A-D-H process represents a particle configuration on the line, or a 1-dimensional interface on the plane which moves in one fixed direction through random local jumps. The stick model is the process of local slopes of the A-D-H process, and has a conserved quantity. The results describe the fluctuations of these systems around the deterministic evolution to which the random system converges under hydrodynamic scaling. We look at diffusive fluctuations, by which we mean fluctuations on the scale of the classical central limit theorem. In the scaling limit these fluctuations obey deterministic equations with random initial conditions given by the initial fluctuations. Of particular interest is the effect of macroscopic shocks, which play a dominant role because dynamical noise is suppressed on the scale we are working. Received: 4 October 2001 / Accepted: 12 March 2002     

10.1007/s10955-006-9040-z

We derive hydrodynamic equations describing the evolution of a binary fluid segregated into two regions, each rich in one species,which are separated (on the macroscopic scale) by a sharp interface. Our starting point is a Vlasov-Boltzmann (VB) equation describing the evolution of the one particle position and velocity distributions, f_i (x, v, t), i = 1, 2. The solution of the VB equation is developed in a Hilbert expansion appropriate for this system. This yields incompressible Navier-Stokes equations for the velocity field u and a jump boundary condition for the pressure across the interface. The interface, in turn, moves with a velocity given by the normal component of u.     

Extended Systems with Deterministic Local Dynamics and Random Jumps

We consider particle systems on lattices with internal dynamics at each site and random jumps between sites. Models with simple chaotic local dynamics, namely expanding circle maps, are considered. Results on mean drift rates, central limit theorems and dependences on jump parameters are proved. A version of most of the results in this paper is contained in this author’s Ph.D. thesis [K]. This research is partially supported by a grant from the NSF.     

Two Speed TASEP

We consider the TASEP on ? with two blocks of particles having different jump rates. We study the large time behavior of particles’ positions. It depends both on the jump rates and the region we focus on, and we determine the complete process diagram. In particular, we discover a new transition process in the region where the influence of the random and deterministic parts of the initial condition interact. Slow particles may create a shock, where the particle density is discontinuous and the distribution of a particle’s position is asymptotically singular. We determine the diffusion coefficient of the shock without using second class particles. We also analyze the case where particles are effectively blocked by a wall moving with speed equal to their intrinsic jump rate.     

Quantum hydrodynamic modes in one-dimensional polaron system

Dynamical relaxation process of one-dimensional polaron system weakly coupled with a thermal phonon field is theoretically investigated. In addition to the diffusion relaxation, we have found that there appears a new macroscopic quantum sound mode which stabilizes the wave packet of the quantum particle even under the random collision with the thermal phonon. This coherent sound mode is a new hydrodynamic mode obeying a macroscopic linear wave equation for the density of the particle, instead of wave function.     

Extremal behavior of a coupled continuous time random walk

Coupled continuous time random walks (CTRWs) model normal and anomalous diffusion of random walkers by taking the sum of random jump lengths dependent on the random waiting times immediately preceding each jump. They are used to simulate diffusion-like processes in econophysics such as stock market fluctuations, where jumps represent financial market microstructure like log returns. In this and many other applications, the magnitude of the largest observations (e.g. a stock market crash) is of considerable importance in quantifying risk. We use a stochastic process called a coupled continuous time random maxima (CTRM) to determine the density governing the maximum jump length of a particle undergoing a CTRW. CTRM are similar to continuous time random walks but track maxima instead of sums. The many ways in which observations can depend on waiting times can produce an equally large number of CTRM governing density shapes. We compare densities governing coupled CTRM with their uncoupled counterparts for three simple observation/wait dependence structures.     

SELF-CONSISTENT TREATMENT OF PLASMA EXPANSION IN THE PRESENCE OF ELECTRON PLASMA WAVE

The hydrodynamic modification of an expanding plasma near the reflection point of an electron plasma wave is investigated by simultaneously solving the wave equation together with the steady-state hydrodynamic equations.The electric field and electron density fluctuation of the plasma wave are derived in a self-consistently steepened density profile. while the density jump and its dependence on the wave amplitude are also determined.     

Fluctuations in the Weakly Asymmetric Exclusion Process with Open Boundary Conditions

We investigate the fluctuations around the average density profile in the weakly asymmetric exclusion process with open boundaries in the steady state. We show that these fluctuations are given, in the macroscopic limit, by a centered Gaussian field and we compute explicitly its covariance function. We use two approaches. The first method is dynamical and based on fluctuations around the hydrodynamic limit. We prove that the density fluctuations evolve macroscopically according to an autonomous stochastic equation, and we search for the stationary distribution of this evolution. The second approach, which is based on a representation of the steady state as a sum over paths, allows one to write the density fluctuations in the steady state as a sum over two independent processes, one of which is the derivative of a Brownian motion, the other one being related to a random path in a potential.     

循环流化床内颗粒团流动的研究

循环流化床颗粒相流动具有多尺度效应:单颗粒运动的微尺度、颗粒团运动的介尺度和固相运动的宏尺度。颗粒相流动参数受单颗粒运动和颗粒聚团运动的制约,同时影响气相流动。基于气体分子运动论和颗粒动理学,建立相平均稠密气固两相流流动模型。介尺度模型考虑颗粒团与单颗粒之间、颗粒团与气相之间的动量和能量的传递和耗散。模拟计算颗粒容积份额、速度等参数与实测值相吻合。     

Fermion current algebras and Schwinger terms in (3+1)-dimensions

We derive an ODE for the macroscopic evolution of a tagged particle in models such as asymmetric simple exclusions and zero range processes. The right-hand side of the ODE is discontinuous and its solutions are understood in the Filippov sense. We establish the uniqueness of the ODE, and explore its relationship with the hydrodynamic equation of the particle density.Research partially supported by National Science Foundation grant DMS-9208490.     

Scaling,propagation, and kinetic roughening of flame fronts in random media

We introduce a model of two coupled reaction-diffusion equations to describe the dynamics and propagation of flame fronts in random media. The model incorporates heat diffusion, its dissipation, and its production through coupling to the background reactant density. We first show analytically and numerically that there is a finite critical value of the background density below which the front associated with the temperature field stops propagating. The critical exponents associated with this transition are shown to be consistent with meanfield theory of percolation. Second, we study the kinetic roughening associated with a moving planar flame front above the critical density. By numerically calculating the time-dependent width and equal-time height correlation function of the front, we demonstrate that the roughening process belongs to the universality class of the Kardar-Parisi-Zhang interface equation. Finally, we show how this interface equation can be analytically derived from our model in the limit of almost uniform background density.     

Diffusive behavior of asymmetric zero-range processes

We investigate a new interpretation for the Navier-Stokes corrections to the hydrodynamic equation of asymmetric interacting particle systems. We consider a system that starts from a measure associated with a profile that is constant along the drift direction. We show that under diffusive scaling the macroscopic behavior of the process is described by a nonlinear parabolic equation whose diffusion coefficient coincides with the diffusion coefficient of the hydrodynamic equation of the symmetric version of the process.     

Diffusion and electromigration on disordered surfaces

We study a one-dimensional disordered solid-on-solid model in which neighboring columns are shifted by quenched random phases. The static height-difference correlation function displays a minimum at a nonzero temperature. The model is equipped with volume-conserving surface diffusion dynamics, including a possible bias due to an electromigration force. In the case of Arrhenius jump rates a continuum equation for the evolution of macroscopic profiles is derived and confirmed by direct simulation. Dynamic surface fluctuations are investigated using simulations and phenomenological Langevin equations. In these equations the quenched disorder appears in the form of time-independent random forces. The disorder does not qualitatively change the roughening dynamics of near-equilibrium surfaces, but in the case of biased surface diffusion with Metropolis rates it induces a new roughening mechanism, which leads to an increase of the surface width as . Received 7 February 2000     

Stochastic Burgers and KPZ Equations from Particle Systems

We consider two strictly related models: a solid on solid interface growth model and the weakly asymmetric exclusion process, both on the one dimensional lattice. It has been proven that, in the diffusive scaling limit, the density field of the weakly asymmetric exclusion process evolves according to the Burgers equation and the fluctuation field converges to a generalized Ornstein-Uhlenbeck process. We analyze instead the density fluctuations beyond the hydrodynamical scale and prove that their limiting distribution solves the (non linear) Burgers equation with a random noise on the density current. For the solid on solid model, we prove that the fluctuation field of the interface profile, if suitably rescaled, converges to the Kardar–Parisi–Zhang equation. This provides a microscopic justification of the so called kinetic roughening, i.e. the non Gaussian fluctuations in some non-equilibrium processes. Our main tool is the Cole-Hopf transformation and its microscopic version. We also develop a mathematical theory for the macroscopic equations. Received: 24 October 1995/Accepted: 9 July 1996     

Condensation in the Zero Range Process: Stationary and Dynamical Properties

The zero range process is of particular importance as a generic model for domain wall dynamics of one-dimensional systems far from equilibrium. We study this process in one dimension with rates which induce an effective attraction between particles. We rigorously prove that for the stationary probability measure there is a background phase at some critical density and for large system size essentially all excess particles accumulate at a single, randomly located site. Using random walk arguments supported by Monte Carlo simulations, we also study the dynamics of the clustering process with particular attention to the difference between symmetric and asymmetric jump rates. For the late stage of the clustering we derive an effective master equation, governing the occupation number at clustering sites.     

Lattice Boltzmann modeling with discontinuous collision components: Hydrodynamic and Advection-Diffusion Equations

Irrespective of the nature of the modeled conservation laws, we establish first the microscopic interface continuity conditions for Lattice Boltzmann (LB) multiple-relaxation time, link-wise collision operators with discontinuous components (equilibrium functions and/or relaxation parameters). Effective macroscopic continuity conditions are derived for a planar implicit interface between two immiscible fluids, described by the simple two phase hydrodynamic model, and for an implicit interface boundary between two heterogeneous and anisotropic, variably saturated soils, described by Richard’s equation. Comparing the effective macroscopic conditions to the physical ones, we show that the range of the accessible parameters is restricted, e.g. a variation of fluid densities or a heterogeneity of the anisotropic soil properties. When the interface is explicitly tracked, the interface collision components are derived from the leading order continuity conditions. Among particular interface solutions, a harmonic mean value is found to be an exact LB solution, both for the interface kinematic viscosity and for the interface vertical hydraulic conductivity function. We construct simple problems with the explicit and implicit interfaces, matched exactly by the LB hydrodynamic and/or advection-diffusion schemes with the aid of special solutions for free collision parameters.     

The Hydrodynamic Limit of a Deterministic Particle System with Conservation of Mass and Momentum

We study the hydrodynamic limit of a deterministic one-dimensional particle system with nearest neighbour interaction and an additional regularizing force. Under its evolution mass and momentum are conserved. In the limit with Euler scaling their macroscopic distributions are shown to be governed by the compressible Navier–Stokes equations with a density dependent viscosity.     

A lattice Boltzmann method for immiscible multiphase flow simulations using the level set method

We consider the lattice Boltzmann method for immiscible multiphase flow simulations. Classical lattice Boltzmann methods for this problem, e.g. the colour gradient method or the free energy approach, can only be applied when density and viscosity ratios are small. Moreover, they use additional fields defined on the whole domain to describe the different phases and model phase separation by special interactions at each node. In contrast, our approach simulates the flow using a single field and separates the fluid phases by a free moving interface. The scheme is based on the lattice Boltzmann method and uses the level set method to compute the evolution of the interface. To couple the fluid phases, we develop new boundary conditions which realise the macroscopic jump conditions at the interface and incorporate surface tension in the lattice Boltzmann framework. Various simulations are presented to validate the numerical scheme, e.g. two-phase channel flows, the Young–Laplace law for a bubble and viscous fingering in a Hele-Shaw cell. The results show that the method is feasible over a wide range of density and viscosity differences.     

First-order phase transition and phase coexistence in a spin-glass model

We study the mean-field static solution of the Blume-Emery-Griffiths-Capel model with quenched disorder, an Ising-spin lattice gas with random magnetic interaction. The thermodynamics is worked out in the full replica symmetry breaking scheme. The model exhibits a high temperature/low density paramagnetic phase. As temperature decreases or density increases, a phase transition to a full replica symmetry breaking spin-glass phase occurs. The nature of the transition can be either of the second order or, at temperature below a given critical value, of the first order in the Ehrenfest sense, with a discontinuous jump of the order parameter, a latent heat, and coexistence of phases.