Eugene R. Speer

Statistical mechanics of probabilistic cellular automata

We investigate the behavior of discrete-time probabilistic cellular automata (PCA), which are Markov processes on spin configurations on ad-dimensional lattice, from a rigorous statistical mechanics point of view. In particular, we exploit, whenever possible, the correspondence between stationary measures on the space-time histories of PCAs on ℤd and translation-invariant Gibbs states for a related Hamiltonian on ℤ(d+1). This leads to a simple large-deviation formula for the space-time histories of the PCA and a proof that in a high-temperature regime the stationary states of the PCA are Gibbsian. We also obtain results about entropy, fluctuations, and correlation inequalities, and demonstrate uniqueness of the invariant state and exponential decay of correlations in a high-noise regime. We discuss phase transitions in the low-noise (or low-temperature) regime and review Tooms proof of nonergodicity of a certain class of PCAs.

Statistical mechanics of the nonlinear Schrödinger equation

AbstractWe investigate the statistical mechanics of a complex fieldø whose dynamics is governed by the nonlinear Schrödinger equation. Such fields describe, in suitable idealizations, Langmuir waves in a plasma, a propagating laser field in a nonlinear medium, and other phenomena. Their Hamiltonian

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

H(\phi ) = \int_\Omega {[\frac{1}{2}|\nabla \phi |^2 - (1/p) |\phi |^p ] dx}

Exact Solution of a Cellular Automaton for Traffic

is unbounded below and the system will, under certain conditions, develop (self-focusing) singularities in a finite time. We show that, whenΩ is the circle and theL2 norm of the field (which is conserved by the dynamics) is bounded byN, the Gibbs measureυ obtained is absolutely continuous with respect to Wiener measure and normalizable if and only ifp andN are such that classical solutions exist for all time—no collapse of the solitons. This measure is essentially the same as that of a one-dimensional version of the more realisitc Zakharov model of coupled Langmuir and ion acoustic waves in a plasma. We also obtain some properties of the Gibbs state, by both analytic and numerical methods, asN and the temperature are varied.

Renormalization and Ward identities using complex space‐time dimension

The microscopic structure of macroscopic shocks in the one-dimensional, totally asymmetric simple exclusion process is obtained exactly from the complete solution of the stationary state of a model system containing two types of particles-“first” and “second” class. This nonequilibrium steady state factorizes about any second-class particle, which implies factorization in the one-component system about the (random) shock position. It also exhibits several other interesting features, including long-range correlations in the limit of zero density of the second-class particles. The solution also shows that a finite number of second-class particles in a uniform background of first-class particles form a weakly bound state.

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

AbstractWe 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 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→∞. The probability of microscopic configurations corresponding to some other profile ρ(x), x=i/N, has the asymptotic form exp[−N

Asymmetric abeiian sandpile models

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