Steady state properties of discrete and continuous models of nonequilibrium phenomena

Abstract

The understanding of nonequilibrium phenomena, of fundamental importance in statistical physics, has great implications for many physical, chemical, and biological systems. Such phenomena are observed almost everywhere in the natural world. These phenomena are characterized by complicated spatiotemporal evolution. To explore nonequilibrium phenomena we often study simple model systems that embody their essential characteristics. In this thesis, we report the results of our investigations of the statistically steady state properties of three one-dimensional models: multispecies asymmetric simple exclusion processes, the Kuramoto- Sivashinsky equation, and the Burgers equation. The thesis is divided into two parts: Part I and Part II.

In Chapters 2–5 of Part I, we present our results for multispecies exclusion models, principally the phase diagrams and statistical properties of their nonequilibrium steady state (NESS). We list below abstracts of these chapters.

• In Chapter 2, we consider a multispecies ASEP (mASEP) on a one-dimensional lattice with semipermeable boundaries in contact with particle reservoirs. The mASEP involves ¹2𝑟 ¸1º species of particles: 𝑟 species of positive charges and their negative counterparts as well as vacancies. At the boundaries, a species can replace or be replaced by its negative counterpart. We derive the exact nonequilibrium phase diagram for the system in the long time limit. We find two new phenomena in certain regions of the phase diagram: dynamical expulsion when the density of a species becomes zero throughout the system, and dynamical localization when the density of a species is nonzero only within an interval far from the boundaries. We give a complete explanation of the macroscopic features of the phase diagram using what we call nested fat shocks.

• In Chapter 3, we study an asymmetric exclusion process with two species and vacancies on an open one-dimensional lattice called the left-permeable ASEP (LPASEP). The left boundary is permeable for the vacancies but the right boundary is not. We find a matrix product solution for the stationary state and the exact stationary phase diagram for the densities and currents. By calculating the density of each species at the boundaries, we find further structure in the stationary phases. In particular, we find that the slower species can reach and accumulate at the far boundary, even in phases where the bulk density of these particles approaches zero.

• In Chapter 4, we study a multispecies generalization of the model in Chapter 3. We determine all phases in the phase diagram using an exact projection to the LPASEP solved earlier. In most phases, we observe the phenomenon of dynamical expulsion of one or more species. We explain the density profiles in each phase using interacting shocks. This explanation is corroborated by simulations.

• In Chapter 5, we investigate a multispecies generalization of the single-species asymmetric simple exclusion process defined on an open one-dimensional, finite lattice connected to particle reservoirs. At the boundaries, a species can be replaced with any other species. We devise an exact projection scheme to find the phase diagram in terms of densities and currents of all species. In most of the phases, one or more species are absent in the system due to dynamical expulsion. We observe shocks as well in some regions of the phase diagram. We explain the density profiles using a generalized shock structure that is substantiated by numerical simulations.

In Chapters 7 and 8 of Part II, we study the statistical properties of turbulent, but statistically steady, states of the Kuramoto-Sivashinsky and the Burgers equations in one dimension. Our main results are summarized below.

• In Chapter 7, we investigate the long time and large system size properties of the onedimensional Kuramoto-Sivashinsky equation. Tracy-Widom and Baik-Rains distributions appear as universal limit distributions for height fluctuations in the one-dimensional Kardar-Parisi-Zhang (KPZ) stochastic partial differential equation (PDE). We obtain the same universal distributions in the spatiotemporally chaotic, nonequilibrium, but statistically steady state of KS deterministic PDE, by carrying out extensive pseudospectral direct numerical simulations to obtain the spatiotemporal evolution of the KS height profile h(x,t) for different initial conditions. We establish, therefore, that the statistical properties of the one-dimensional (1D) KS PDE in this state are in the 1D KPZ universality class.

• In Chapter 8, we study the statistical properties of decaying turbulence in the onedimensional Burgers equation, in the vanishing-viscosity limit; we start with random initial conditions, whose energy spectra have simple functional dependences on the wavenumber k:

E_0(k) = A \mathcal{E}(k) exp[ - 2 k^2 / k^2_c ] ,

where A is a positive real number, and k_c is a cutoff wavenumber. The simplest case is the single-power law \mathcal{E}(k) = k^{n}. We focus here on the case of the Gaussian laws which are characterized by E_0(k) = exp[ - 2 (k-k_c)^2 / k^2_c +2 k^2 / k^2_c]; in addition, we consider initial spectra which are combinations of either two or four single-power law spectral regions. For all these initial conditions, we systematize (a) the temporal decay of the total energy, (b) the rich temporal evolution of the energy spectrum, and (c) the spatiotemporal evolution of the velocity field. We present our results in the context of earlier studies of this problem.