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Non-Linear Problems in Interacting Particle Systems

Gabriel Nahum

Published Jan 29, 2024• 0 citations
OpenAlex

Abstract

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 $\Lambda_N = \{1, \dots, N\}$, evolving as the SSEP in $\{2, \dots, N - 1\}$ and coupled with two-site reaction-diffusion processes at the boundaries, acting on the pairs of sites $\{1,2\}$ and $\{N-1,N\}$. 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 $\partial_t \rho = \partial^2 u \rho^m$ for $m \in \mathbb{N}^+$, into a universal exclusion family parametrized by $m \in \mathbb{R}$, and, in particular, representing the transition from the slow diffusion regime ($m > 1$) to the fast diffusion regime ($m < 1$). We successfully treat the case $m \in (0,2)$, encompassing the phase transition $m = 1$, in this manner deriving the Porous Media Equation in the range $m \in (1,2)$, and the Fast Diffusion Equation for $m > 0$. Next, we generalize the PMM in another direction: constructing the diffusion coefficient $D(\rho) = \rho^n (1 - \rho)^k$ with $n,k \in \mathbb{N}^+$. 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.

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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.
We consider the general problem of determining the steady state of stochastic nonequilibrium systems such as those that have been used to model (among other things) biological transport and traffic flow. We begin with a broad overview of this class of driven diffusive systems - which includes exclusion processes - focusing on interesting physical properties, such as shocks and phase transitions. We then turn our attention specifically to those models for which the exact distribution of microstates in the steady state can be expressed in a matrix product form. In addition to a gentle introduction to this matrix product approach, how it works and how it relates to similar constructions that arise in other physical contexts, we present a unified, pedagogical account of the various means by which the statistical mechanical calculations of macroscopic physical quantities are actually performed. We also review a number of more advanced topics, including nonequilibrium free energy functionals, the classification of exclusion processes involving multiple particle species, existence proofs of a matrix product state for a given model and more complicated variants of the matrix product state that allow various types of parallel dynamics to be handled. We conclude with a brief discussion of open problems for future research.
In the preceding chapter, we have introduced several levels for the mathematical description of stochastic dynamics. We now use the kinetic Ising models introduced in Section 2.3.3 to formulate the dynamic scaling hypothesis which appropriately generalizes the homogeneity property of the static correlation function in the vicinity of a critical point, as established in Chapter 1. The dynamic critical exponent z is defined to characterize both the critical dispersion and the basic phenomenon of critical slowing-down. As a next step, and building on the results of Section 2.4, a continuum effective theory for the mesoscopic order parameter density, basically the dynamical analog to the Ginzburg–Landau approach, is constructed in terms of a non-linear Langevin equation. The distinction between dissipative and diffusive dynamics for the purely relaxational kinetics of either a non-conserved or conserved order parameter field, respectively, defines the universality classes A and B. Following the analysis of these models in the Gaussian approximation, they also serve to outline the construction of a dynamical perturbation theory for non-linear stochastic differential equations through direct iteration. In general, however, the order parameter alone does not suffice to fully capture the critical dynamics near a second-order phase transition. Additional hydrodynamic modes originating from conservation laws need to be accounted for as well. The simplest such situation is entailed in the relaxational models C and D, which encompass the static coupling of the order parameter to the energy density. Further scenarios emerge through reversible non-linear mode couplings in the Langevin equations of motion.
It is known that exact travelling wave solutions exist for families of (n + 1)-states stochastic one-dimensional non-equilibrium lattice models with open boundaries provided that some constraints on the reaction rates are fulfilled. These solutions describe the diffusive motion of a product shock or a domain wall with the dynamics of a simple biased random walker. The steady state of these systems can be written in terms of linear superposition of such shocks or domain walls. These steady states can also be expressed in a matrix product form. We show that in this case the associated quadratic algebra of the system has always a twodimensional representation with a generic structure. A couple of examples for n = 1 and n = 2 cases are presented. On the macroscopic level nonlinear hydrodynamic equations, such as the Burgers equation or the Fisher equation, might exhibit shocks in some cases [1, 2]. On the microscopic level some stochastic non-equilibrium one-dimensional lattice models with open boundaries might also develop shocks provided that some constrained on the reaction rates are fulfilled [3, 4, 5]. In [6] exact travelling wave solutions have been obtained for three families of two-states drivendiffusive models in one-dimension with nearest-neighbors interactions. These three models are the Partially Asymmetric Simple Exclusion Process (PASEP) [7, 8, 9], the Branching-Coalescing Random Walk (BCRW) and the Asymmetric Kawasaki-Glauber Process (AKGP). By taking a Bernoulli shock distribution as initial distribution, the shock distribution in these models evolves in time into a linear combination of similar distributions with different shock positions. The time evolution of the shock position in these models is similar to the dynamics of a random walker moving on a one-dimensional lattice with reflecting boundaries [6]. This property of the travelling wave solution is not limited to the two-states systems with nearest-neighbor interactions. Shocks in threestates
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.
Recent advances in the theory of non-equilibrium systems are reviewed. The emphasis is on collective behaviour, particularly non-equilibrium stochastic dynamics and steady state phase transitions. The models discussed are largely lattice-based 'exclusion' models (with hard core interactions) which can include a wide variety of collective stochastic processes and phenomena. The techniques employed range from intuitive reasoning and approximate mean field approaches to scaling and renormalization group methods, and to exact descriptions in terms of operator algebras, where many new results are included. Mappings to quantum systems, particularly lattice spin models, are also exploited, which provide new viewpoints and allow the use of standard techniques including Bethe ansatz and free fermion solutions.
A totally asymmetric exclusion process on a ring is investigated in which particles can move one or two sites. Special attention is spent to the high-speed case where particles are not allowed to move a single site if they could move two sites. The stationary state is calculated exactly in the framework of the matrix-product ansatz. Independently of the update this process evolves into subspace of the configuration space and can lead to the formation of an excess hole. One observes two phases where its velocity takes different values which are calculated exactly from the normalization-generating function. Numerically computed density profiles show an interesting algebraic form as a limit of a shock profile. For continuous time the process turns out to be related to the ASEP with a single defect particle and for synchronous update it leads to a natural defect dynamics. For the general definition of the process with maximum velocity two, from an exact analysis of the two-particle sector we find that the distribution of inter-particle distance can oscillate. This is underlined in the thermodynamic limit by an improved mean-field theory which shows a remarkably good agreement with simulations. The main focus of this work is the exact solution of lattice models by matrix-product ansatz. We finally obtain stationary states also for related processes. Especially considerable is a new formulation of steady states for parallel dynamics like the ASEP with open boundaries as a product of a pair-factorized and a matrix-product state.