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Multiple point interactions in lattice gas models far from equilibrium

Matthias Paeßens

Published Jan 1, 2004• 0 citations
OpenAlex

Abstract

In this work non-equilibrium phenomena in statistical models with more than nearest neighbor interactions are examined. The investigated models belong to three different types of non-equilibrium, i.e. systems undergoing aging, absorbing phase transitions and systems being intrinsically out of equilibrium. The behavior of the mean spherical model after a temperature quench from the disordered phase into the ordered phase in an external magnetic field is calculated. For a sufficiently small external field, the kinetics of the magnetization reversal transition from the metastable to the ground state is comparable to the aging behavior in the case without external field. For an oscillating external magnetic field the absence of a dynamic phase transition is demonstrated. The bosonic pair contact process with diffusion shows unusual behavior of the second moment of the probability distribution for which a phase transition from a constant asymptotic value to exponential divergence is found. The control parameter of this transition is the ratio of reaction and diffusion rates which gives insight into the role of diffusion in this process. The divergence of the second moment can lead to a negative correlation time while the correlation length is positive. Coupling conservative diffusion system to reaction dynamics leads to interesting behavior if the rates are tuned adequately. The general conditions for ergodicity breaking in those systems are derived and an example is constructed. For this example hysteresis is demonstrated. The behavior, including the exact steady state phase diagram, of this non-equilibrium many particle system is well described by an effective equilibrium one-particle problem. The motion of domain walls in reaction diffusion systems is investigated for three site interactions. A variety of models is found and they show phenomena like double shocks or additional symmetries not possible for two-site interaction models. While the two-site interaction models leading to exact shock measures are connected by the transformability to free fermion systems, this is not valid for the three site interaction models.

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Dynamic phase transitions and dynamic phase diagrams in the kinetic mixed spin-1 and spin-2 Ising system in an oscillating magnetic field

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Models of disorder

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Cambridge University Press eBooks•May 6, 1999•0 citations
In this work non-equilibrium phenomena in statistical models with more than nearest neighbor interactions are examined. The investigated models belong to three different types of non-equilibrium, i.e. systems undergoing aging, absorbing phase transitions and systems being intrinsically out of equilibrium. The behavior of the mean spherical model after a temperature quench from the disordered phase into the ordered phase in an external magnetic field is calculated. For a sufficiently small external field, the kinetics of the magnetization reversal transition from the metastable to the ground state is comparable to the aging behavior in the case without external field. For an oscillating external magnetic field the absence of a dynamic phase transition is demonstrated. The bosonic pair contact process with diffusion shows unusual behavior of the second moment of the probability distribution for which a phase transition from a constant asymptotic value to exponential divergence is found. The control parameter of this transition is the ratio of reaction and diffusion rates which gives insight into the role of diffusion in this process. The divergence of the second moment can lead to a negative correlation time while the correlation length is positive. Coupling conservative diffusion system to reaction dynamics leads to interesting behavior if the rates are tuned adequately. The general conditions for ergodicity breaking in those systems are derived and an example is constructed. For this example hysteresis is demonstrated. The behavior, including the exact steady state phase diagram, of this non-equilibrium many particle system is well described by an effective equilibrium one-particle problem. The motion of domain walls in reaction diffusion systems is investigated for three site interactions. A variety of models is found and they show phenomena like double shocks or additional symmetries not possible for two-site interaction models. While the two-site interaction models leading to exact shock measures are connected by the transformability to free fermion systems, this is not valid for the three site interaction models.
Among the many non-equilibrium processes encountered in nature we deal with two different but related aspects. One is the non-equilibrium relaxation process that is at the origin of \'aging phenomena••, and the other one is a non-equilibrium phase transition, called ••dynamic phase transition••. One of the main purposes of our research is to explore more realistic situations than studied previously. Indeed, in the study of aging phenomena certain kinds of disorder effects are considered, and we introduce the ••surface•• as a spatial boundary to the system undergoing the dynamic phase transition. In order to observe these processes as clearly as possible, we study in both cases simple spin systems. Using Monte Carlo simulations we first investigate aging in three-dimensional Ising spin glasses as well as in two-dimensional Ising models with disorder quenched to low temperatures. The time-dependent dynamical correlation length L(t) is determined numerically and the scaling behavior of various two-time quantities as a function of L(t)/L(s) is discussed where t and s are two different times. For disordered Ising models deviations of L(t) from algebraic growth law show up. The generalized scaling forms as a function of L(t)/L(s) reveal a generic simple aging scenario for Ising spin glasses as well as for disordered Ising ferromagnets. We also study the local critical phenomena at a dynamic phase transition by means of numerical simulations of kinetic Ising models with surfaces subjected to a periodic oscillating field. We examine layer-dependent quantities, such as the period-averaged magnetization per layer Q(z) and the layer susceptibility ¥ö(z), and determine local critical exponents through finite size scaling. Both for two and three dimensions, we find that the values of the surface exponents differ from those of the equilibrium critical surface. It is revealed that the surface phase diagram of the non-equilibrium system is not identical to that of the equilibrium system in three dimensions.
We present three works on Interacting Particle Systems where some form of non-linearity arises. In the context of the description of Non-Equilibrium Steady States (NESS), we formulate a Matrix Product Ansatz in order to describe the NESS of a process defined on the lattice ΛN={1,…,N}\Lambda_N = \{1, \dots, N\}ΛN​={1,…,N}, evolving as the SSEP in {2,…,N−1}\{2, \dots, N - 1\}{2,…,N−1} and coupled with two-site reaction-diffusion processes at the boundaries, acting on the pairs of sites {1,2}\{1,2\}{1,2} and {N−1,N}\{N-1,N\}{N−1,N}. These pairs of sites can take four different states each, leading to 12 possible transitions between them. We derive a set of constraints where the underlying quadratic algebra is consistent, which are related with the correlation of the reservoirs, present the representation of the objects in the formulation, and provide examples of rates satisfying these constraints. In the second work, our focus is in generalizing the Porous Media Model (PMM), associated with the hydrodynamic equation ∂tρ=∂2uρm\partial_t \rho = \partial^2 u \rho^m∂t​ρ=∂2uρm for m∈N+m \in \mathbb{N}^+m∈N+, into a universal exclusion family parametrized by m∈Rm \in \mathbb{R}m∈R, and, in particular, representing the transition from the slow diffusion regime (m>1m > 1m>1) to the fast diffusion regime (m<1m < 1m<1). We successfully treat the case m∈(0,2)m \in (0,2)m∈(0,2), encompassing the phase transition m=1m = 1m=1, in this manner deriving the Porous Media Equation in the range m∈(1,2)m \in (1,2)m∈(1,2), and the Fast Diffusion Equation for m>0m > 0m>0. Next, we generalize the PMM in another direction: constructing the diffusion coefficient D(ρ)=ρn(1−ρ)kD(\rho) = \rho^n (1 - \rho)^kD(ρ)=ρn(1−ρ)k with n,k∈N+n,k \in \mathbb{N}^+n,k∈N+. The generalized PMM that we introduce is a gradient model that inherits theoretical properties from PMM, as, for instance, the presence of mobile clusters and blocked configurations. The construction of the model is indirect and delicate, in order to maintain the gradient property. We then generalize this into a long-range dynamics inheriting the gradient property. The long-range generalization of the dynamics is simple and can be applied to any exclusion process.
In this thesis we examine universal scaling properties of strongly-correlated systems near and far from equilibrium. We discuss quantum phase transitions at vanishing temperature, multicritical and dynamic critical behavior near thermal equilibrium, and scaling properties of nonequilibrium steady states. We employ nonperturbative methods including the functional renormalization group as well as Monte Carlo simulations. A general outline of the functional renormalization group is given in the introductory chapters. \n \nIn the first part of this thesis, we investigate spinless fermions on the honeycomb lattice interacting via short-range repulsive interactions. Such a system can be seen as a simple model for suspended graphene. The short-range interactions control the ground state properties of the system that may lead to a chiral phase transition from the semimetal to the charge density wave (CDW)/Kekulé ordered state. We determine the universal scaling properties at the chiral transition, and establish the presence of large anomalous dimensions indicating the importance of strong fluctuations. \n \nThe competition of two nonvanishing order parameters and their corresponding multicritical behavior are investigated in the subsequent chapter. We characterize the bicritical and tetracritical behavior in the purely bosonic O(N_1) + O(N_2) symmetric model and comment on possible applications to condensed-matter and high-energy physics. \n \nIn the following chapter we discuss the long-time relaxational behavior at criticality of an order parameter with O(N) symmetry coupled to an additional conserved density. We find an anomalous diffusion phase with new dynamic scaling properties. Using the functional renormalization group we determine the complete dynamic critical behavior of the model in 2 < d < 4 dimensions and compare our results to experiments. \n \nFinally, we investigate the scaling properties of stationary states far from equilibrium. At the example of the one-dimensional Burgers’ equation we develop a novel approach to hydrodynamic turbulence using lattice Monte Carlo methods. We apply these techniques to determine the statistical properties of small-scale fluctuations in this model and identify the anomalous scaling behavior. \n
We study the nonequilibrium aspects of the kinetics of a mixed spin-1 and spin-2 Ising system including the biquadratic nearest-neighbor pair interaction ( K ) and crystal-field interaction ( D ) under the presence of a time-dependent oscillating external magnetic field within a mean-field approach. In particular, we calculate the dynamic phase transition (DPT) temperatures and present dynamic phase diagrams. The set of mean-field dynamic equations is obtained by employing Glauber transition rates. We investigate the time variations of average order parameters to find the phases in the system. We also study the thermal behavior of the dynamic order parameters to characterize the nature (continuous or discontinuous) of the phase transitions and obtain the DPT points. The dynamic phase diagrams are presented in three different planes. Phase diagrams contain disordered (d), ferrimagnetic (i), antiquadrupolar or staggered (a) phases, and four coexistence or mixed phase regions, namely i+d, i+a, i+a+d and a+d mixed phases, that strongly depend on interaction parameters.
We numerically investigate the long-time behavior of the density-density auto-correlation function in driven lattice gases, with particle exclusion and periodic boundary conditions in one, two, and three dimensions using precise Monte Carlo simulations of larger system sizes than previous studies. In the one-dimensional asymmetric exclusion process on a ring with half the lattice sites occupied, we find that correlations induce extremely slow relaxation to the asymptotic power law decay We compare the crossover functions obtained from our simulations with various analytic results in the literature, and analyze the characteristic oscillations that occur in finite systems away from half-filling. As expected, correlations are weak in three dimensions and consequently the mean-field description is adequate. We also investigate the relaxation towards the non-equilibrium steady state in the two-time density-density auto-correlations, starting from strongly correlated initial conditions. We obtain simple aging scaling behavior in one, two, and three dimensions, with the expected power laws. We numerically investigate the behavior of driven lattice gases with nearest neighbor interactions at half-filling with periodic boundary conditions below and at the critical temperature using Monte Carlo simulations of very large lattices in two dimensions. This work is one of few that explores the relaxation to a non-equilibrium steady state. We obtain data collapse for the finite-size scaling form of density-density auto-correlation function at the critical point. We achieve data collapse using finite-size scaling of the time-dependent order parameter during the transient regime starting from strongly correlated initial conditions. We present simple aging scaling of the density-density auto-correlation function at the critical point starting from strongly correlated initial conditions using Monte Carlo simulations of two different lattice anisotropies. We thus unambiguously confirm the critical exponents determined by renormalization group methods using measurement of dynamic quantities in the transient regime. Measuring these dynamic quantities in the transient regime provides more conclusive measurements of the critical exponents than previous studies measuring static quantities in the stationary state. We provide qualitative arguments that the lattice anisotropy determines the steady-state for sub-critical quenches.
The ordinary kinetic versions of the Ising model may be modified to exhibit steady nonequilibrium states. This is illustrated in chapter 4 where a conflict between two canonical mechanisms ( diffusion and reaction ) drives the configuration away from equilibrium. A more systematic investigation of this possibility, when the conflict is between different reaction processes only, is described here. We focus on spin systems evolving by a superposition of independent local processes of the kind variously known as spin flips, birth/death or creation/annihilation. The restriction to spin flip dynamics does not prevent the systems in this class from exhibiting a variety of nonequilibrium phase transitions and critical phenomena. Their consideration may therefore help in developing nonequilibrium theory. In addition, they have some practical interest, e.g., conflicting dynamics may occur in disordered materials such as dilute magnetic systems, and some of these situations can be implemented in the laboratory. The present chapter describes some exact, mean-field and MC results that together render an intriguing picture encouraging further study. It is argued in §7.1 that some of the peculiar, emergent, macroscopic behavior of microscopically disordered materials may be related to diffusion of disorder. This provides a physical motivation for the nonequilibrium random-field Ising model (NRFM).