Hydrodynamic limit for particle systems with nonconstant speed parameter

We establish the hydrodynamic limit for a class of particle systems on ℤ ^d with nonconstant speed parameter, assuming that the speed parameter is continuously differentiable in the spatial variable. If the particle system is on the one-dimensional latticeℤ and totally asymmetric, we derive the hydrodynamic equation for continuous speed parameters. We obtain nontrivial upper and lower bounds when either the speed parameter is discontinuous or there is a blockage at a fixed site.     

On the Two Species Asymmetric Exclusion Process with Semi-Permeable Boundaries

We investigate the structure of the nonequilibrium stationary state (NESS) of a system of first and second class particles, as well as vacancies (holes), on L sites of a one-dimensional lattice in contact with first class particle reservoirs at the boundary sites; these particles can enter at site 1, when it is vacant, with rate α, and exit from site L with rate β. Second class particles can neither enter nor leave the system, so the boundaries are semi-permeable. The internal dynamics are described by the usual totally asymmetric exclusion process (TASEP) with second class particles. An exact solution of the NESS was found by Arita. Here we describe two consequences of the fact that the flux of second class particles is zero. First, there exist (pinned and unpinned) fat shocks which determine the general structure of the phase diagram and of the local measures; the latter describe the microscopic structure of the system at different macroscopic points (in the limit L→∞) in terms of superpositions of extremal measures of the infinite system. Second, the distribution of second class particles is given by an equilibrium ensemble in fixed volume, or equivalently but more simply by a pressure ensemble, in which the pair potential between neighboring particles grows logarithmically with distance. We also point out an unexpected feature in the microscopic structure of the NESS for finite L: if there are n second class particles in the system then the distribution of first class particles (respectively holes) on the first (respectively last) n sites is exchangeable.     

Order of Current Variance and Diffusivity in the Rate One Totally Asymmetric Zero Range Process

We prove that the variance of the current across a characteristic is of order t ^2/3 in a stationary constant rate totally asymmetric zero range process, and that the diffusivity has order t ^1/3. This is a step towards proving universality of this scaling behavior in the class of one-dimensional interacting systems with one conserved quantity and concave hydrodynamic flux. The proof proceeds via couplings to show the corresponding moment bounds for a second class particle. We build on the methods developed in Balázs and Seppäläinen (Order of current variance and diffusivity in the asymmetric simple exclusion process, 2006) for simple exclusion. However, some modifications were needed to handle the larger state space. Our results translate into t ^2/3-order of variance of the tagged particle on the characteristics of totally asymmetric simple exclusion.     

One-dimensional hard rod caricature of hydrodynamics

We give here a rigorous deduction of the “hydrodynamic” equation which holds in the hydrodynamic limit, for a model system of one-dimensional identical hard rods interacting through elastic collisions. The equation should be considered as the analog of the Euler equation of real hydrodynamics. Owing to the degeneracy of the model, it is written in terms of a functiong(q, v, t) expressing the density of particles with velocityv at the pointq at timet. For this equation we prove an existence and uniqueness theorem in some natural class of functions. Our main result is the proof that if {^∈, ∈ >0} is a class of initial states which are homogeneous on a scale much less than ε^?1, and if the corresponding particle densities tend, asε→0, in the proper scale, to the initial hydrodynamic densityg _o (q,v), then, under some general assumptions on the states ∈^? and ong ₀, the particle densities of the evolved states at timeε ^?1 t, tend asε→0 to the unique solution of the hydrodynamic equation with initial conditiong ₀. The proof is completed by exhibiting a large class of initial families {^∈, ∈ >0} which possess the required properties.     

Metastability for the Exclusion Process with Mean-Field Interaction

We consider an exclusion particle system with long-range, mean-field-type interactions at temperature 1/β. The hydrodynamic limit of such a system is given by an integrodifferential equation with one conservation law on the circle $C$ : it is the gradient flux of the Kac free energy functional F _β. For β≤1, any constant function with value m ∈ [?1, +1] is the global minimizer of F _β in the space $\{ u:\int_C {u(x)} \,dx = m\} $ . For β>1, F _β restricted to $\{ u:\int_C {u(x)} \,dx = m\} $ may have several local minima: in particular, the constant solution may not be the absolute minimizer of F _β. We therefore study the long-time behavior of the particle system when the initial condition is close to a homogeneous stable state, giving results on the time of exit from (suitable) subsets of its domain of attraction. We follow the Freidlin–Wentzell approach: first, we study in detail F _β together with the time asymptotics of the solution of the hydrodynamic equation; then we study the probability of rare events for the particle system, i.e., large deviations from the hydrodynamic limit.     

Activated Random Walkers: Facts, Conjectures and Challenges

We study a particle system with hopping (random walk) dynamics on the integer lattice ? ^d . The particles can exist in two states, active or inactive (sleeping); only the former can hop. The dynamics conserves the number of particles; there is no limit on the number of particles at a given site. Isolated active particles fall asleep at rate λ>0, and then remain asleep until joined by another particle at the same site. The state in which all particles are inactive is absorbing. Whether activity continues at long times depends on the relation between the particle density ζ and the sleeping rate λ. We discuss the general case, and then, for the one-dimensional totally asymmetric case, study the phase transition between an active phase (for sufficiently large particle densities and/or small λ) and an absorbing one. We also present arguments regarding the asymptotic mean hopping velocity in the active phase, the rate of fixation in the absorbing phase, and survival of the infinite system at criticality. Using mean-field theory and Monte Carlo simulation, we locate the phase boundary. The phase transition appears to be continuous in both the symmetric and asymmetric versions of the process, but the critical behavior is very different. The former case is characterized by simple integer or rational values for critical exponents (β=1, for example), and the phase diagram is in accord with the prediction of mean-field theory. We present evidence that the symmetric version belongs to the universality class of conserved stochastic sandpiles, also known as conserved directed percolation. Simulations also reveal an interesting transient phenomenon of damped oscillations in the activity density.     

Hydrodynamic Limit of Multiple SLE

Recently del Monaco and Schleißinger addressed an interesting problem whether one can take the limit of multiple Schramm–Loewner evolution (SLE) as the number of slits N goes to infinity. When the N slits grow from points on the real line \(\mathbb {R}\) in a simultaneous way and go to infinity within the upper half plane \(\mathbb {H}\), an ordinary differential equation describing time evolution of the conformal map \(g_t(z)\) was derived in the \(N \rightarrow \infty \) limit, which is coupled with a complex Burgers equation in the inviscid limit. It is well known that the complex Burgers equation governs the hydrodynamic limit of the Dyson model defined on \(\mathbb {R}\) studied in random matrix theory, and when all particles start from the origin, the solution of this Burgers equation is given by the Stieltjes transformation of the measure which follows a time-dependent version of Wigner’s semicircle law. In the present paper, first we study the hydrodynamic limit of the multiple SLE in the case that all slits start from the origin. We show that the time-dependent version of Wigner’s semicircle law determines the time evolution of the SLE hull, \(K_t \subset \mathbb {H}\cup \mathbb {R}\), in this hydrodynamic limit. Next we consider the situation such that a half number of the slits start from \(a>0\) and another half of slits start from \(-a < 0\), and determine the multiple SLE in the hydrodynamic limit. After reporting these exact solutions, we will discuss the universal long-term behavior of the multiple SLE and its hull \(K_t\) in the hydrodynamic limit.     

Diffusion of tagged interacting spherically symmetric polymers

We consider the diffusion of two species of spherically symmetric macromolecules in solution under the influence of short range central pair potential interactions as well as two body hydrodynamic interactions. Starting from the N-particle Smoluchowski equation and using Felderhof's approach we derive, to linear order in densities, a pair of coupled diffusion equations for the single particle number densities. There are two independent diffusional modes each with an effective diffusion constant dependent in general upon both the interparticle potentials as well as the hydrodynamic model used for each type of macromolecule. However, in the limit that one species is present at very low density compared with the other species, one of the effective diffusion constants is dominated by hydrodynamic interactions. By tagging these tracer particles to observe their diffusion by light scattering, one can test both the mixed stick-slip boundary condition model and the permeable sphere model of the macromolecules.     

Hydrodynamic Limit of Condensing Two-Species Zero Range Processes with Sub-critical Initial Profiles

Two-species condensing zero range processes (ZRPs) are interacting particle systems with two species of particles and zero range interaction exhibiting phase separation outside a domain of sub-critical densities. We prove the hydrodynamic limit of nearest neighbour mean zero two-species condensing ZRP with bounded local jump rate for sub-critical initial profiles, i.e., for initial profiles whose image is contained in the region of sub-critical densities. The proof is based on H.T. Yau’s relative entropy method, which relies on the existence of sufficiently regular solutions to the hydrodynamic equation. In the particular case of the species-blind ZRP, we prove that the solutions of the hydrodynamic equation exist globally in time and thus the hydrodynamic limit is valid for all times.     

Single-particle Green function in the Calogero-Sutherland model for rational couplings β = p/q

We derive an exact expression for the single-particle Green function in the Calogero-Sutherland model for all the rational values of the coupling β. The calculation is based on Jack polynomial techniques and the results are given in the thermodynamical limit. Two types of intermediate states contribute. The first one consists of a particle propagating out of the Fermi sea and the second one consists of a particle propagating in one direction, q particles in the opposite direction and p holes.     

Semi-infinite TASEP with a Complex Boundary Mechanism

We consider a totally asymmetric exclusion process on the positive half-line. When particles enter the system according to a Poisson source, Liggett has computed all the limit distributions when the initial distribution has an asymptotic density. In this paper we consider systems for which particles enter according to a complex mechanism depending on the current configuration in a finite neighborhood of the origin. For this kind of models, we prove a strong law of large numbers for the number of particles which have entered the system at a given time. Our main tool is a new representation of the model as a multi-type particle system with infinitely many particle types.     

On the Small Mass Limit of Quantum Brownian Motion with Inhomogeneous Damping and Diffusion

We study the small mass limit (or: the Smoluchowski–Kramers limit) of a class of quantum Brownian motions with inhomogeneous damping and diffusion. For Ohmic bath spectral density with a Lorentz–Drude cutoff, we derive the Heisenberg–Langevin equations for the particle’s observables using a quantum stochastic calculus approach. We set the mass of the particle to equal \(m = m_{0} \epsilon \), the reduced Planck constant to equal \(\hbar = \epsilon \) and the cutoff frequency to equal \(\varLambda = E_{\varLambda }/\epsilon \), where \(m_0\) and \(E_{\varLambda }\) are positive constants, so that the particle’s de Broglie wavelength and the largest energy scale of the bath are fixed as \(\epsilon \rightarrow 0\). We study the limit as \(\epsilon \rightarrow 0\) of the rescaled model and derive a limiting equation for the (slow) particle’s position variable. We find that the limiting equation contains several drift correction terms, the quantum noise-induced drifts, including terms of purely quantum nature, with no classical counterparts.     

Non-equilibrium statistical mechanics in the general theory of relativity: II. Linear fields in a kinetic approximation

The first paper in this series introduced a new, manifestly covariant approach to non-equilibrium statistical mechanics in classical general relativity. The object of this second paper is to apply that formalism to the evolution of a collection of particles that interact via linear fields in a fixed curved background spacetime. Given the viewpoint adopted here, the fundamental objects of the theory are a many-particle distribution function, which lives in a many-particle phase space, and a many-particle conservation equation which this distribution satisfies. By viewing a composite N-particle system as interacting one- and (N ? 1)-particle subsystems, one can derive exact coupled equations for appropriately defined reduced one- and (N ? 1)-particle distribution functions. Alternatively, by treating all the particles on an identical footing, one can extract an exact closed equation involving only the one-particle distribution. The implementation of plausible assumptions, which constitute straightforward generalizations of standard non-relativistic “kinetic approximations”, then permits the formulation of an approximate kinetic equation for the one-particle distribution function. In the obvious non-relativistic limit, one recovers the well-known Vlasov-Landau equation. The explicit form for the relativistic expression is obtained for three concrete examples, namely, interactions via an electromagnetic field, a massive scalar field, and a symmetric second rank tensor field. For a large class of interactions, of which these three examples are representative, the kinetic equation will admit a relativistic Maxwellian distribution as an exact stationary solution; and, for these interactions, an H-theorem may be proved.     

One-Dimensional Traps,Two-Body Interactions,Few-Body Symmetries. II. N Particles

This is the second in a pair of articles that classify the configuration space and kinematic symmetry groups for N identical particles in one-dimensional traps experiencing Galilean-invariant two-body interactions. These symmetries explain degeneracies in the few-body spectrum and demonstrate how tuning the trap shape and the particle interactions can manipulate these degeneracies. The additional symmetries that emerge in the non-interacting limit and in the unitary limit of an infinitely strong contact interaction are sufficient to algebraically solve for the spectrum and degeneracy in terms of the one-particle observables. Symmetry also determines the degree to which the algebraic expressions for energy level shifts by weak interactions or nearly–unitary interactions are universal, i.e. independent of trap shape and details of the interaction. Identical fermions and bosons with and without spin are considered. This article analyzes the symmetries of N particles in asymmetric, symmetric, and harmonic traps; the prequel article treats the one, two and three particle cases.     

Exact solution of the master equation for the asymmetric exclusion process

Using the Bethe ansatz, we obtain the exact solution of the master equation for the totally asymmetric exclusion process on an infinite one-dimensional lattice. We derive explicit expressions for the conditional probabilitiesP(x₁,...,x_N;t/y ₁,...,y_N; 0) of findingN particles on lattices sitesx ₁,...,x_N at timet with initial occupationy ₁,...,y_N at timet=0.     

Effective Dynamics of a Tracer Particle Interacting with an Ideal Bose Gas

We study a system consisting of a heavy quantum particle, called the tracer particle, coupled to an ideal gas of light Bose particles, the ratio of masses of the tracer particle and a gas particle being proportional to the gas density. All particles have non-relativistic kinematics. The tracer particle is driven by an external potential and couples to the gas particles through a pair potential. We compare the quantum dynamics of this system to an effective dynamics given by a Newtonian equation of motion for the tracer particle coupled to a classical wave equation for the Bose gas. We quantify the closeness of these two dynamics as the mean-field limit is approached (gas density ${\to \infty}$ ). Our estimates allow us to interchange the thermodynamic with the mean-field limit.     

Rotational diffusion of a tracer colloid particle: I. Short time orientational correlations

We extend the generalized Smoluchowski equation to descrbe the diffusional relaxation of position and orientation in a suspension of interacting spherical colloid particles. Considering a tracer particle which interacts with other particles through spherically symmetric pair potentials and with an external field we obtain a cluster expansion representation of the orientational time correlation functions for the tracer. The one and two body cluster contributions are studied explicitly at short times. Working to first order in volume fraction φ we show that the initial slope of the time correlation functions is described by a modified diffusion coefficient D^r = D^r₀(1 −C^rφ) where C^r is a number determined by hydrodynamic and potential interactions. We evaluate C^r numerically for spheres with slip-stick hydrodynamic boundary conditions and also for permeable spheres.     

Interface motion in a planar spin-flip model derived from exclusion on the line

We consider a two-dimensional spin-flip model, which can be interpreted as the limit of the Ising model at low temperature and a small nonzero external field. In the hydrodynamic limit and for a special class of initial conditions, the motion of the interface is governed by a nonlinear partial differential equation with a lattice-distorted curvature term. In our proofs we use results about the hydrodynamic behavior of the weakly asymmetric exclusion process on the integers and also on the nonnegative integers with a trap at the boundary.