Uniform TitleStatistical mechanics and combinatorics of some discrete lattice models
NameAyyer, Arvind (author), Lebowitz, Joel (chair), Gilman, Ronald (internal member), Horton, George (internal member), Zamolodchikov, Alexander (internal member), Zeilberger, Doron (outside member), Rutgers University, Graduate School - New Brunswick,
SubjectPhysics and Astronomy,
DescriptionMany problems in statistical physics involve enumeration of certain objects. In this thesis, we apply ideas from combinatorics and statistical physics to understand three different lattice models.
We investigate the structure of the nonequilibrium stationary state (NESS) of a system of first and second class particles on
L sites of a one-dimensional lattice in contact with first class particle reservoirs at the boundary sites and second class particles constrained to lie the system. The internal
dynamics are described by the usual totally asymmetric exclusion process (TASEP) with second class particles. We show in a conceptually simple way how pinned and unpinned (fat) shocks determine the general structure of the phase diagram. We also point out some unexpected features in the microscopic structure of the NESS both for finite L and in the limit of large L. In the latter case the local distribution of second class particles is given by an equilibrium pressure ensemble with a pair potential between neighboring particles which grows logarithmically with distance.
We model a long linear polymer constrained
between two plates as a walk on a two-dimensional lattice constrained to lie between two lines, x=y and x=y+w, which interacts with these lines via contact parameters s and t. The atomic steps of the walk can be taken to be from an arbitrary but fixed set S
with the only condition being that the first
coordinate of every element in S is strictly positive. For any such S and any w, we prescribe general algorithms (fully implemented in Maple) for the automated calculation of several mathematical and physical quantities of interest.
Ferrers (or Young) diagrams are very classical objects in representation theory, whose half-perimeter generating function of Ferrers diagrams is a straightforward rational function. We construct two new classes of Ferrers diagrams, which we call wicketed and gated Ferrers diagrams, which have internal voids in the shape of Ferrers diagrams, and calculate their half-perimeter generating functions, one of which is closely related to the generating function of the Catalan numbers, using a more abstract version of the usual transfer matrix method.
NoteIncludes bibliographical references (p. 81-85).
CollectionGraduate School - New Brunswick Electronic Theses and Dissertations
Organization NameRutgers, The State University of New Jersey
RightsThe author owns the copyright to this work.