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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Renormalization-group study of a hybrid driven diffusive system

Kevin E. Bassler, B. Schmittmann

Published May 1, 1994• 5 citations
Publisher

Abstract

We consider a d-dimensional stochastic lattice gas of interacting particles, diffusing under the influence of a short-ranged, attractive Ising Hamiltonian and a ``hybrid'' external field which is a superposition of a uniform and an annealed random drive, acting in orthogonal subspaces of dimensions one and m, respectively. Driven into a nonequilibrium steady state, the half-filled system phase segregates via a continuous transition at a field-dependent critical temperature. Using renormalization-group techniques, we show that its critical behavior falls into a new universality class with upper critical dimension ${\mathit{d}}_{\mathit{c}}$=5-m, characterized by two distinct anisotropy exponents, which, like all other indices, are computed exactly to all orders in perturbation theory.

DOI
10.1103/physreve.49.3614

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We consider a d-dimensional stochastic lattice gas of interacting particles, diffusing under the influence of a short-ranged, attractive Ising Hamiltonian and a ``hybrid'' external field which is a superposition of a uniform and an annealed random drive, acting in orthogonal subspaces of dimensions one and m, respectively. Driven into a nonequilibrium steady state, the half-filled system phase segregates via a continuous transition at a field-dependent critical temperature. Using renormalization-group techniques, we show that its critical behavior falls into a new universality class with upper critical dimension dc{\mathit{d}}_{\mathit{c}}dc​=5-m, characterized by two distinct anisotropy exponents, which, like all other indices, are computed exactly to all orders in perturbation theory.
We investigate the static and dynamic critical behaviour of a uniformly driven bilayer Ising lattice gas at half-filling. Depending on the strength of the inter-layer coupling J , phase separation occurs across or within the two layers. The former transitions are controlled by the universality class of model A (corresponding to an Ising model with Glauber dynamics), with upper critical dimension d c = 4. The latter transitions are dominated by the universality class of the standard (single-layer) driven Ising lattice gas, with d c = 5 and a nonclassical anisotropy exponent. These two distinct critical lines meet at a nonequilibrium bicritical point which also falls into the driven Ising class. At all transitions, novel couplings and dangerous irrelevant operators determine corrections to scaling.
We present a field theoretic renormalization group study for the critical behaviour of a uniformly driven diffusive system with quenched disorder, which is modelled by different kinds of potential barriers between sites. Due to their symmetry properties, these different realizations of the random potential barriers lead to three different models for the phase transition to transverse order and to one model for the phase transition to longitudinal order all belonging to distinct universality classes. In these four models that have different upper critical dimensions dc we find the critical scaling behaviour of the vertex functions in spatial dimensions d < dc. Its deviation from purely diffusive behaviour is characterized by the anomaly–exponent η that we calculate at first and second order, respectively in ǫ = dc − d. In each model η turns out to be positive which means superdiffusive spread of density fluctuations in the driving force direction. 1 1
Using field theoretic renormalization group methods we study the critical behavior of a driven diffusive system near a boundary perpendicular to the driving force. The boundary acts as a particle reservoir, which is necessary to maintain the critical particle density in the bulk. The scaling behavior of correlation and response functions is governed by a new exponent ${\ensuremath{\eta}}_{1},$ which is related to the anomalous scaling dimension of the chemical potential of the boundary. The new exponent and a universal amplitude ratio for the density profile are calculated at first order in $\ensuremath{\epsilon}=5\ensuremath{-}d.$ Some of our results are checked by computer simulations.
The ordering of charges on half-filled hypercubic lattices is investigated numerically, where electroneutrality is ensured by background charges. This system is equivalent to the s=1/2s=1/2s=1/2 Ising lattice model with antiferromagnetic 1/r1/r1/r interaction. The temperature dependences of specific heat, mean staggered occupation, and of a generalized susceptibility indicate continuous order-disorder phase transitions at finite temperatures in two- and three-dimensional systems. In contrast, the susceptibility of the one-dimensional system exhibits singular behavior at vanishing temperature. For the two- and three-dimensional cases, the critical exponents are obtained by means of a finite-size scaling analysis. Their values are consistent with those of the Ising model with short-range interaction, and they imply that the studied model cannot belong to any other known universality class. Samples of up to 1400, 1122{112}^{2}1122, and 223{22}^{3}223 sites are considered for dimensions 1 to 3, respectively.
It is explained how field-theoretic methods and the dynamic renormalisation group (RG) can be applied to study the universal scaling properties of systems that either undergo a continuous phase transition or display generic scale invariance, both near and far from thermal equilibrium. Part 1 introduces the response functional field theory representation of (nonlinear) Langevin equations. The RG is employed to compute the scaling exponents for several universality classes governing the critical dynamics near second-order phase transitions in equilibrium. The effects of reversible mode-coupling terms, quenching from random initial conditions to the critical point, and violating the detailed balance constraints are briefly discussed. It is shown how the same formalism can be applied to nonequilibrium systems such as driven diffusive lattice gases. Part 2 describes how the master equation for stochastic particle reaction processes can be mapped onto a field theory action. The RG is then used to analyse simple diffusion-limited annihilation reactions as well as generic continuous transitions from active to inactive, absorbing states, which
This paper introduces a position-space renormalization-group approach for nonequilibrium systems and applies the method to a driven stochastic one-dimensional gas with open boundaries. The dynamics are characterized by three parameters: the probability alpha that a particle will flow into the chain to the leftmost site, the probability beta that a particle will flow out from the rightmost site, and the probability p that a particle will jump to the right if the site to the right is empty. The renormalization-group procedure is conducted within the space of these transition probabilities, which are relevant to the system's dynamics. The method yields a critical point at alpha(c)=beta(c)=1/2, in agreement with the exact values, and the critical exponent nu=2.71, as compared with the exact value nu=2.00.