arXiv subjects

E. R. Speer

Publications and source records attributed to E. R. Speer.

At least 19 recordsLinked to original sources

arXiv2025

Approach to Hyperuniformity in the One-Dimensional Facilitated Exclusion Process

For the one-dimensional Facilitated Exclusion Process with initial state a product measure of density $ρ=1/2-δ$, $δ\ge0$, there exists an infinite-time limiting state $ν_ρ$ in which all particles are isolated and hence cannot move. We study the variance $V(L)$, under $ν_ρ$, of the number of particles in an interval of $L$ sites. Under $ν_{1/2}$ either all odd or all even sites are occupied, so that $V(L)=0$ for $L$ even and $V(L)=1/4$ for $L$ odd: the state is hyperuniform, since $V(L)$ grows more slowly than $L$. We prove that for densities approaching 1/2 from below there exist three regimes in $L$, in which the variance grows at different rates: for $L\ggδ^{-2}$, $V(L)\simeqρ(1-ρ)L$, just as in the initial state; for $A(δ)\ll L\llδ^{-2}$, with $A(δ)=δ^{-2/3}$ for $L$ odd and $A(δ)=1$ for $L$ even, $V(L)\simeq CL^{3/2}$ with $C=2\sqrt{2/π}/3$; and for $L\llδ^{-2/3}$ with $L$ odd, $V(L)\simeq1/4$. The analysis is based on a careful study of a renewal process with a long tail. Our study is motivated by simulation results showing similar behavior in higher dimensions; we discuss this background briefly.

math.PR↗

arXiv2024

Approach to Hyperuniformity of Steady States of Facilitated Exchange Processes

We consider the fluctuations in the number of particles in a box of size L^d in Z^d, d>=1, in the (infinite volume) translation invariant stationary states of the facilitated exclusion process, also called the conserved lattice gas model. When started in a Bernoulli (product) measure at density rho, these systems approach, as t goes to infinity, a "frozen" state for rho<=rho_c, with rho_c=1/2 for d=1 and rho_c<1/2 for d>=2. At rho=rho_c the limiting state is hyperuniform, that is, the variance of the number of particles in the box grows slower than L^d. We give a general description of how the variances at different scales of L behave as rho increases to rho_c. On the largest scale, L>>L_2, the fluctuations are normal (in fact the same as in the original product measure), while in a region L_1< =2.)

cond-mat.stat-mech↗

arXiv2022

Stationary States of the One-Dimensional Discrete-Time Facilitated Symmetric Exclusion Process

We describe the extremal translation invariant stationary (ETIS) states of the facilitated exclusion process on $\mathbb{Z}$. In this model all particles on sites with one occupied and one empty neighbor jump at each integer time to the empty neighbor site, and if two particles attempt to jump into the same empty site we choose one randomly to succeed. The ETIS states are qualitatively different for densities $ρ<1/2$, $ρ=1/2$, and $1/2<ρ<1$, but in each density region we find states which may be grouped into families, each of which is in natural correspondence with the set of all ergodic measures on $\{0,1\}^{\mathbb{Z}}$. For $ρ<1/2$ there is one such family, containing all the ergodic states in which the probability of two adjacent occupied sites is zero. For $ρ=1/2$ there are two families, in which configurations translate to the left and right, respectively, with constant speed 2. For the high density case there is a continuum of families. We show that all ETIS states at densities $ρ\le1/2$ belong to these families, and conjecture that also at high density there are no other ETIS states. We also study the possible ETIS states which might occur if the conjecture fails.

math.PR↗

arXiv2021

Stationary States of the One-dimensional Facilitated Asymmetric Exclusion Process

We describe the translation invariant stationary states (TIS) of the one-dimensional facilitated asymmetric exclusion process in continuous time, in which a particle at site $i\in\mathbb{Z}$ jumps to site $i+1$ (respectively $i-1$) with rate $p$ (resp. $1-p$), provided that site $i-1$ (resp. $i+1$) is occupied and site $i+1$ (resp. $i-1$) is empty. All TIS states with density $ρ\le1/2$ are supported on trapped configurations in which no two adjacent sites are occupied; we prove that if in this case the initial state is i.i.d.~Bernoulli then the final state is independent of $p$. This independence also holds for the system on a finite ring. For $ρ>1/2$ there is only one TIS. It is the infinite volume limit of the probability distribution that gives uniform weight to all configurations in which no two holes are adjacent, and is isomorphic to the Gibbs measure for hard core particles with nearest neighbor exclusion.

math.PR↗

arXiv2020

The Discrete-Time Facilitated Totally Asymmetric Simple Exclusion Process

We describe the translation invariant stationary states of the one dimensional discrete-time facilitated totally asymmetric simple exclusion process (F-TASEP). In this system a particle at site $j$ in $Z$ jumps, at integer times, to site $j+1$, provided site $j-1$ is occupied and site $j+1$ is empty. This defines a deterministic noninvertible dynamical evolution from any specified initial configuration on $\{0,1\}^{Z}$. When started with a Bernoulli product measure at density $ρ$ the system approaches a stationary state, with phase transitions at $ρ=1/2$ and $ρ=2/3$. We discuss various properties of these states in the different density regimes $0<ρ<1/2$, $1/2<ρ<2/3$, and $2/3<ρ<1$; for example, we show that the pair correlation $g(j)=\langleη(i)η(i+j)\rangle$ satisfies, for all $n\in Z$, $\sum_{j=kn+1}^{k(n+1)}g(j)=kρ^2$, with $k=2$ when $0 \le ρ\le 1/2$ and $k=3$ when $2/3 \le ρ\le 1$, and conjecture (on the basis of simulations) that the same identity holds with $k=6$ when $1/2 \le ρ\le 2/3$. The $ρ<1/2$ stationary state referred to above is also the stationary state for the deterministic discrete-time TASEP at density $ρ$ (with Bernoulli initial state) or, after exchange of particles and holes, at density $1-ρ$.

math-ph↗

arXiv2019

Exact Solution of the F-TASEP

We obtain the exact solution of the facilitated totally asymmetric simple exclusion process (F-TASEP) in 1D. The model is closely related to the conserved lattice gas (CLG) model and to some cellular automaton traffic models. In the F-TASEP a particle at site $j$ in $\mathbb{Z}$ jumps, at integer times, to site $j+1$, provided site $j-1$ is occupied and site $j+1$ is empty. When started with a Bernoulli product measure at density $ρ$ the system approaches a stationary state. This non-equilibrium steady state (NESS) has phase transitions at $ρ=1/2$ and $ρ=2/3$. The different density regimes $0<ρ<1/2$, $1/2<ρ<2/3$, and $2/3<ρ<1$ exhibit many surprising properties; for example, the pair correlation $g(j)=\langleη(i)η(i+j)\rangle$ satisfies, for all $n\in\mathbb{Z}$, $\sum_{j=kn+1}^{k(n+1)}g(j)=kρ^2$, with $k=2$ when $0\leρ\le1/2$, $k=6$ when $1/2\leρ\le2/3$, and $k=3$ when $2/3\leρ\le1$. The quantity $\lim_{L\to\infty}V_L/L$, where $V_L$ is the variance in the number of particles in an interval of length L, jumps discontinuosly from $ρ(1-ρ)$ to 0 when $ρ\to1/2$ and when $ρ\to2/3$.

cond-mat.stat-mech↗

arXiv2016

Translation invariant extensions of finite volume measures

We investigate the following questions: Given a measure $μ_Λ$ on configurations on a subset $Λ$ of a lattice $\mathbb{L}$, where a configuration is an element of $Ω^Λ$ for some fixed set $Ω$, does there exist a measure $μ$ on configurations on all of $\mathbb{L}$, invariant under some specified symmetry group of $\mathbb{L}$, such that $μ_Λ$ is its marginal on configurations on $Λ$? When the answer is yes, what are the properties, e.g., the entropies, of such measures? Our primary focus is the case in which $\mathbb{L}=\mathbb{Z}^d$ and the symmetries are the translations. For the case in which $Λ$ is an interval in $\mathbb{Z}$ we give a simple necessary and sufficient condition, local translation invariance (LTI), for extendibility. For LTI measures we construct extensions having maximal entropy, which we show are Gibbs measures; this construction extends to the case in which $\mathbb{L}$ is the Bethe lattice. On $\mathbb{Z}$ we also consider extensions supported on periodic configurations, which are analyzed using de~Bruijn graphs and which include the extensions with minimal entropy. When $Λ\subset\mathbb{Z}$ is not an interval, or when $Λ\subset\mathbb{Z}^d$ with $d>1$, the LTI condition is necessary but not sufficient for extendibility. For $\mathbb{Z}^d$ with $d>1$, extendibility is in some sense undecidable.

cond-mat.stat-mech↗

arXiv2015

The Truncated Moment Problem on $\mathbb{N}_0$

We find necessary and sufficient conditions for the existence of a probability measure on $\mathbb{N}_0$, the nonnegative integers, whose first $n$ moments are a given $n$-tuple of nonnegative real numbers. The results, based on finding an optimal polynomial of degree $n$ which is nonnegative on $\mathbb{N}_0$ (and which depends on the moments), and requiring that its expectation be nonnegative, generalize previous results known for $n=1$, $n=2$ (the Percus-Yamada condition), and partially for $n=3$. The conditions for realizability are given explicitly for $n\leq5$ and in a finitely computable form for $n\geq6$. We also find, for all $n$, explicit bounds, in terms of the moments, whose satisfaction is enough to guarantee realizability. Analogous results are given for the truncated moment problem on an infinite discrete semi-bounded subset of $\mathbb{R}$.

math.PR↗

arXiv2015

Central limit theorems, Lee-Yang zeros, and graph-counting polynomials

We consider the asymptotic normalcy of families of random variables $X$ which count the number of occupied sites in some large set. We write $Prob(X=m)=p_mz_0^m/P(z_0)$, where $P(z)$ is the generating function $P(z)=\sum_{j=0}^{N}p_jz^j$ and $z_0>0$. We give sufficient criteria, involving the location of the zeros of $P(z)$, for these families to satisfy a central limit theorem (CLT) and even a local CLT (LCLT); the theorems hold in the sense of estimates valid for large $N$ (we assume that $Var(X)$ is large when $N$ is). For example, if all the zeros lie in the closed left half plane then $X$ is asymptotically normal, and when the zeros satisfy some additional conditions then $X$ satisfies an LCLT. We apply these results to cases in which $X$ counts the number of edges in the (random) set of "occupied" edges in a graph, with constraints on the number of occupied edges attached to a given vertex. Our results also apply to systems of interacting particles, with $X$ counting the number of particles in a box $Λ$ whose size approaches infinity; $P(z)$ is then the grand canonical partition function and its zeros are the Lee-Yang zeros.

math.CO↗

arXiv2012

The blockage problem

We investigate the totally asymmetric exclusion process on Z, with the jump rate at site i given by r_i=1 for i nonzero, r_0=r. It is easy to see that the maximal stationary current j(r) is nondecreasing in r and that j(r)=1/4 for r>=1; it is a long outstanding problem to determine whether or not the critical value r_c of r such that j(r)=1/4 for r>r_c is strictly less than 1. Here we present a heuristic argument, based on the analysis of the first sixteen terms in a formal power series expansion of j(r) obtained from finite volume systems, that r_c=1 and that for r less than 1 and near 1, j(r) behaves as 1/4-γ\exp[-{a/(1-r)}] with a approximately equal to 2. We also give some new exact results about this system; in particular we prove that j(r)=J_max(r), with J_max(r) the hydrodynamic maximal current defined by Seppalainen, and thus establish continuity of j(r). Finally we describe a related exactly solvable model, a semi-infinite system in which the site i=0 is always occupied. For that system, the critical r is 1/2 and the analogue j_s(r) of j(r) satisfies j_s(r)=r(1-r) for r<=1/2; j_s(r) is the limit of finite volume currents inside the curve |r(1-r)|=1/4 in the complex r plane and we suggest that analogous behavior may hold for the original system.

math-ph↗

arXiv2007

Entropy of Open Lattice Systems

We investigate the behavior of the Gibbs-Shannon entropy of the stationary nonequilibrium measure describing a one-dimensional lattice gas, of L sites, with symmetric exclusion dynamics and in contact with particle reservoirs at different densities. In the hydrodynamic scaling limit, L to infinity, the leading order (O(L)) behavior of this entropy has been shown by Bahadoran to be that of a product measure corresponding to strict local equilibrium; we compute the first correction, which is O(1). The computation uses a formal expansion of the entropy in terms of truncated correlation functions; for this system the k-th such correlation is shown to be O(L^{-k+1}). This entropy correction depends only on the scaled truncated pair correlation, which describes the covariance of the density field. It coincides, in the large L limit, with the corresponding correction obtained from a Gaussian measure with the same covariance.

cond-mat.stat-mech↗

arXiv2007

Realizability of point processes

There are various situations in which it is natural to ask whether a given collection of $k$ functions, $ρ_j(\r_1,...,\r_j)$, $j=1,...,k$, defined on a set $X$, are the first $k$ correlation functions of a point process on $X$. Here we describe some necessary and sufficient conditions on the $ρ_j$'s for this to be true. Our primary examples are $X=\mathbb{R}^d$, $X=\matbb{Z}^d$, and $X$ an arbitrary finite set. In particular, we extend a result by Ambartzumian and Sukiasian showing realizability at sufficiently small densities $ρ_1(\mathbf{r})$. Typically if any realizing process exists there will be many (even an uncountable number); in this case we prove, when $X$ is a finite set, the existence of a realizing Gibbs measure with $k$ body potentials which maximizes the entropy among all realizing measures. We also investigate in detail a simple example in which a uniform density $ρ$ and translation invariant $ρ_2$ are specified on $\mathbb{Z}$; there is a gap between our best upper bound on possible values of $ρ$ and the largest $ρ$ for which realizability can be established.

math-ph↗

arXiv2002

Exact Large Deviation Functional of a Stationary Open Driven Diffusive System: The Asymmetric Exclusion Process

We consider the asymmetric exclusion process (ASEP) in one dimension on sites $i = 1,..., N$, in contact at sites $i=1$ and $i=N$ with infinite particle reservoirs at densities $ρ_a$ and $ρ_b$. As $ρ_a$ and $ρ_b$ are varied, the typical macroscopic steady state density profile $\bar ρ(x)$, $x\in[a,b]$, obtained in the limit $N=L(b-a)\to\infty$, exhibits shocks and phase transitions. Here we derive an exact asymptotic expression for the probability of observing an arbitrary macroscopic profile $ρ(x)$: $P_N(\{ρ(x)\})\sim\exp[-L{\cal F}_{[a,b]}(\{ρ(x)\});ρ_a,ρ_b]$, so that ${\cal F}$ is the large deviation functional, a quantity similar to the free energy of equilibrium systems. We find, as in the symmetric, purely diffusive case $q=1$ (treated in an earlier work), that $\cal F$ is in general a non-local functional of $ρ(x)$. Unlike the symmetric case, however, the asymmetric case exhibits ranges of the parameters for which ${\cal F}(\{ρ(x)\})$ is not convex and others for which ${\cal F}(\{ρ(x)\})$ has discontinuities in its second derivatives at $ρ(x) = \barρ(x)$; the fluctuations near $\barρ(x)$ are then non-Gaussian and cannot be calculated from the large deviation function.

cond-mat.stat-mech↗

arXiv2002

Exact Free Energy Functional for a Driven Diffusive Open Stationary Nonequilibrium System

We obtain the exact probability $\exp[-L {\cal F}(\{ρ(x)\})]$ of finding a macroscopic density profile $ρ(x)$ in the stationary nonequilibrium state of an open driven diffusive system, when the size of the system $L \to \infty$. $\cal F$, which plays the role of a nonequilibrium free energy, has a very different structure from that found in the purely diffusive case. As there, $\cal F$ is nonlocal, but the shocks and dynamic phase transitions of the driven system are reflected in non-convexity of $\cal F$, in discontinuities in its second derivatives, and in non-Gaussian fluctuations in the steady state.

cond-mat.stat-mech↗

arXiv2002

Large Deviation of the Density Profile in the Steady State of the Open Symmetric Simple Exclusion Process

We consider an open one dimensional lattice gas on sites $i=1,...,N$, with particles jumping independently with rate 1 to neighboring interior empty sites, the {\it simple symmetric exclusion process}. The particle fluxes at the left and right boundaries, corresponding to exchanges with reservoirs at different chemical potentials, create a stationary nonequilibrium state (SNS) with a steady flux of particles through the system. The mean density profile in this state, which is linear, describes the typical behavior of a macroscopic system, i.e., this profile occurs with probability 1 when $N \to \infty$. The probability of microscopic configurations corresponding to some other profile $ρ(x)$, $x = i/N$, has the asymptotic form $\exp[-N {\cal F}(\{ρ\})]$; $\cal F$ is the {\it large deviation functional}. In contrast to equilibrium systems, for which ${\cal F}_{eq}(\{ρ\})$ is just the integral of the appropriately normalized local free energy density, the $\cal F$ we find here for the nonequilibrium system is a nonlocal function of $ρ$. This gives rise to the long range correlations in the SNS predicted by fluctuating hydrodynamics and suggests similar non-local behavior of $\cal F$ in general SNS, where the long range correlations have been observed experimentally.

cond-mat↗

arXiv2001

Free Energy Functional for Nonequilibrium Systems: An Exactly Solvable Case

We consider the steady state of an open system in which there is a flux of matter between two reservoirs at different chemical potentials. For a large system of size $N$, the probability of any macroscopic density profile $ρ(x)$ is $\exp[-N{\cal F}(\{ρ\})]$; ${\cal F}$ thus generalizes to nonequilibrium systems the notion of free energy density for equilibrium systems. Our exact expression for $\cal F$ is a nonlocal functional of $ρ$, which yields the macroscopically long range correlations in the nonequilibrium steady state previously predicted by fluctuating hydrodynamics and observed experimentally.

cond-mat.stat-mech↗

arXiv1999

Spatial Particle Condensation for an Exclusion Process on a Ring

We study the stationary state of a simple exclusion process on a ring which was recently introduced by Arndt {\it et al} [J. Phys. A {\bf 31} (1998) L45;cond-mat/9809123]. This model exhibits spatial condensation of particles. It has been argued that the model has a phase transition from a ``mixed phase'' to a ``disordered phase''. However, in this paper exact calculations are presented which, we believe, show that in the framework of a grand canonical ensemble there is no such phase transition. An analysis of the fluctuations in the particle density strongly suggests that the same result also holds for the canonical ensemble.

cond-mat.stat-mech↗

arXiv1998

Exact solution of a cellular automaton for traffic

We present an exact solution of a probabilistic cellular automaton for traffic with open boundary conditions, e.g. cars can enter and leave a part of a highway with certain probabilities. The model studied is the asymmetric exclusion process (ASEP) with {\it simultaneous} updating of all sites. It is equivalent to a special case ($v_{\rm max}=1$) of the Nagel-Schreckenberg model for highway traffic, which has found many applications in real-time traffic simulations. The simultaneous updating induces additional strong short range correlations compared to other updating schemes. The stationary state is written in terms of a matrix product solution. The corresponding algebra, which expresses a system-size recursion relation for the weights of the configurations, is quartic, in contrast to previous cases, in which the algebra is quadratic. We derive the phase diagram and compute various properties such as density profiles, two point functions and the fluctuations in the number of particles (cars) in the system. The current and the density profiles can be mapped onto the ASEP with other time discrete updating procedures. Through use of this mapping, our results also give new results for these models.

cond-mat.stat-mech↗