| dc.description.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. | en_US |
