Irina Mazilu

Applications of tridiagonal matrices in non-equilibrium statistical physics

Abstract. One dimensional stochastic problems on a finite lattice that model the time dependence of epidemics, particle deposition and voter influence can easily be cast into a simple form dV/dt = MV , where V is a vector with components representing the average occupation of the i-th cell and M is a matrix with coefficients drawn from the equations that give rates of evolution of a particular cell’s occupation due to its dependence upon other cells. These matrices are often in tridiagonal form (the only non-zero elements are along the main diagonal and the two diagonal rows to its right), or can be transformed via a unitary transformation into this form. In the tridiagonal form, eigenvalues and eigenvectors can be extracted via straightforward techniques, and the inverse of the matrix of eigenvectors can be inverted (in arbitrary finite dimension) so as to enforce the system’s initial conditions. Examples of such models are discussed and related to matrix theory.

Stochastic epidemic-type model with enhanced connectivity: exact solution

We present an exact analytical solution to a one-dimensional model of the susceptible–infected–recovered (SIR) epidemic type, with infection rates dependent on nearest-neighbor occupations. We use a quantum mechanical approach, transforming the master equation via a quantum spin operator formulation. We calculate exactly the time-dependent density of infected, recovered and susceptible populations for random initial conditions. Our results compare well with those of previous work, validating the model as a useful tool for additional and extended studies in this important area. Our model also provides exact solutions for the n-point correlation functions, and can be extended to more complex epidemic-type models.

Cooperative sequential adsorption models on a Cayley tree: analytical results and applications

We present a class of cooperative sequential adsorption models on a Cayley tree with constant and variable attachment rates and their possible applications for ionic self-assembly of thin films and drug encapsulation of nanoparticles. Using the empty interval method, and generalizing results known from reaction-diffusion processes on Cayley trees, we calculate a variety of quantities such as time-dependent surface coverage and time-dependent probabilities of certain particle configurations.

A multi-temperature kinetic Ising model and the eigenvalues of some perturbed Jacobi matrices

In this paper we analyze the eigenvalues of some perturbed Jacobi matrices. The results contain as particular cases the known spectra of several classes of tridiagonal matrices studied recently. As a motivation, we discuss a three and a four-temperature kinetic Ising model that can be analyzed using some perturbed Jacobi matrices. The analytical results can also be used for the associated reaction-diffusion systems to solve for the particle density.

From Complex to Simple: Interdisciplinary Stochastic Models.

We present two simple, one-dimensional, stochastic models that lead to a qualitative understanding of very complex systems from biology, nanoscience and social sciences. The first model explains the complicated dynamics of microtubules, stochastic cellular highways. Using the theory of random walks in one dimension, we find analytical expressions for certain physical quantities, such as the time dependence of the length of the microtubules, and diffusion coefficients. The second one is a stochastic adsorption model with applications in surface deposition, epidemics and voter systems. We introduce the ?empty interval method? and show sample calculations for the time-dependent particle density. These models can serve as an introduction to the field of non-equilibrium statistical physics, and can also be used as a pedagogical tool to exemplify standard statistical physics concepts, such as random walks or the kinetic approach of the master equation.

Exact energy spectrum of a two-temperature kinetic Ising model.

We develop the exact energy spectrum for a two-temperature kinetic Ising spin chain and its dual reaction-diffusion system with spatially alternating pair annihilation and creation rates. We also discuss the symmetries of the system pseudo-Hamiltonian and their role in developing a general solution. The surprisingly simple form for the eigenvalues leads to interesting physical consequences and to a possible numerical analysis of the dynamical properties of the system.

High Temperature Expansion for a Driven Bilayer System

Based directly on the microscopic lattice dynamics, a simple high temperature expansion can be devised for non-equilibrium steady states. We apply this technique to investigate the disordered phase and the phase diagram for a driven bilayer lattice gas at half filling. Our approximation captures the phases first observed in simulations, provides estimates for the transition lines, and allows us to compute signature observables of non-equilibrium dynamics, namely, particle and energy currents. Its focus on non-universal quantities offers a useful analytic complement to field-theoretic approaches.

Class of cooperative stochastic models: Exact and approximate solutions, simulations, and experiments using ionic self-assembly of nanoparticles.

We present exact and approximate results for a class of cooperative sequential adsorption models using matrix theory, mean-field theory, and computer simulations. We validate our models with two customized experiments using ionically self-assembled nanoparticles on glass slides. We also address the limitations of our models and their range of applicability. The exact results obtained using matrix theory can be applied to a variety of two-state systems with cooperative effects.

A two-state stochastic model for nanoparticle self-assembly: theory, computer simulations and applications

We introduce a stochastic cooperative model for particle deposition and evaporation relevant to ionic self-assembly of nanoparticles with applications in surface fabrication and nanomedicine, and present a method for mapping our model onto the Ising model. The mapping process allows us to use the established results for the Ising model to describe the steady-state properties of our system. After completing the mapping process, we investigate the time dependence of particle density using the mean field approximation. We complement this theoretical analysis with Monte Carlo simulations that support our model. These techniques, which can be used separately or in combination, are useful as pedagogical tools because they are tractable mathematically and they apply equally well to many other physical systems with nearest-neighbour interactions including voter and epidemic models.

Cooperative sequential-adsorption model in two dimensions with experimental applications for ionic self-assembly of nanoparticles.

Self-assembly of nanoparticles is an important tool in nanotechnology, with numerous applications, including thin films, electronics, and drug delivery. We study the deposition of ionic nanoparticles on a glass substrate both experimentally and theoretically. Our theoretical model consists of a stochastic cooperative adsorption and evaporation process on a two-dimensional lattice. By exploring the relationship between the initial concentration of nanoparticles in the colloidal solution and the density of particles deposited on the substrate, we relate the deposition rate of our theoretical model to the concentration.