Ellis, Richard
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Job Title
Professor, Department of Mathematics and Statistics
Last Name
Ellis
First Name
Richard
Discipline
Other Statistics and Probability
Expertise
Probability and analysis, theory of large deviations
Introduction
Professor Ellis studies the asymptotic behavior of random systems using the theory of large deviations, which focuses on the exponential decay of probabilities in those systems. He has applied the theory to a wide range of problems in which detailed information on rare events is required. These include queueing systems as well as systems arising in statistical mechanics, including spin models and models of coherent structures in turbulence. In recent work he has used the theory of large deviations and the theory of convex functions to understand the equivalence and nonequivalence of the microcanonical, canonical, and generalized canonical ensembles both at the thermodynamic level and at the level of equilibrium macrostates. These ensembles are probability distributions used to describe the behavior of particles in statistical mechanical systems.
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Publication LARGE DEVIATIONS FOR MARKOV-PROCESSES WITH DISCONTINUOUS STATISTICS, .1. GENERAL UPPER-BOUNDS(1991) DUPUIS, P; Ellis, RS; WEISS, AIn this paper we prove an upper large deviation bound for a general class of Markov processes, which includes processes with discontinuous statistics. We also specialize the results to a class of jump Markov processes that model scaled queuing systems.Publication Metastability within the generalized canonical ensemble(2006-01) Touchette, H; Costeniuc, M; Ellis, RS; Turkington, BWe discuss a property of our recently introduced generalized canonical ensemble [M. Costeniuc, R.S. Ellis, H. Touchette, B. Turkington, The generalized canonical ensemble and its universal equivalence with the microcanonical ensemble, J. Stat. Phys. 119 (2005) 1283]. We show that this ensemble can be used to transform metastable or unstable (nonequilibrium) states of the standard canonical ensemble into stable (equilibrium) states within the generalized canonical ensemble. Equilibrium calculations within the generalized canonical ensemble can thus be used to obtain information about nonequilibrium states in the canonical ensemble.Publication Large deviations for a random walk model with state-dependent noise(2003-01) Boue, M; Hernandez-Hernandez, D; Ellis, RSIn this paper we prove the large deviation principle for a class of random walks with state-dependent noise. This type of model has important applications in queueing and communication theory and in the area of stochastic approximation.Publication A statistical approach to the asymptotic behavior of a class of generalized nonlinear Schrodinger equations(2004-01) Ellis, RS; Jordan, R; Otto, P; Turkington, BA statistical relaxation phenomenon is studied for a general class of dispersive wave equations of nonlinear Schrödinger-type which govern non-integrable, non-singular dynamics. In a bounded domain the solutions of these equations have been shown numerically to tend in the long-time limit toward a Gibbsian statistical equilibrium state consisting of a ground-state solitary wave on the large scales and Gaussian fluctuations on the small scales. The main result of the paper is a large deviation principle that expresses this concentration phenomenon precisely in the relevant continuum limit. The large deviation principle pertains to a process governed by a Gibbs ensemble that is canonical in energy and microcanonical in particle number. Some supporting Monte-Carlo simulations of these ensembles are also included to show the dependence of the concentration phenomenon on the properties of the dispersive wave equation, especially the high frequency growth of the dispersion relation. The large deviation principle for the process governed by the Gibbs ensemble is based on a large deviation principle for Gaussian processes, for which two independent proofs are given.Publication CONTINUOUS SYMMETRY-BREAKING IN A MEAN-FIELD MODEL(1983) EISELE, T; Ellis, RSA magnetic system on the sites (j/n;j=1,...,n) of the circle T approximately=R (mod 1) is studied in the limit n to infinity . The interaction is defined in terms of a continuous function J(x, y),x,y in T. For any ferromagnetic J(J>0) which satisfies a normalisation condition, the thermodynamic behaviour is identical to that of the Curie-Weiss model (J identical to 1). This simple case is in contrast to the behaviour for a class of translation invariant, non-ferromagnetic J, for which a continuum of equilibrium states exists for sufficiently low temperatures. In both cases a probabilistic interpretation of the equilibrium states is given.Publication Large deviations for small noise diffusions with discontinuous statistics(2000-01) Boue, M; Dupuis, P; Ellis, RSPublication POLYMERS AS SELF-AVOIDING WALKS - DISCUSSION(1981) Ellis, RSPublication Asymptotic behavior of the magnetization near critical and tricritical points via Ginzburg-Landau polynomials(2008-01) Ellis, RS; Machta, J; Otto, PTHThe purpose of this paper is to prove connections among the asymptotic behavior of the magnetization, the structure of the phase transitions, and a class of polynomials that we call the Ginzburg–Landau polynomials. The model under study is a mean-field version of a lattice spin model due to Blume and Capel. It is defined by a probability distribution that depends on the parameters β and K, which represent, respectively, the inverse temperature and the interaction strength. Our main focus is on the asymptotic behavior of the magnetization m(β n ,K n ) for appropriate sequences (β n ,K n ) that converge to a second-order point or to the tricritical point of the model and that lie inside various subsets of the phase-coexistence region. The main result states that as (β n ,K n ) converges to one of these points (β,K), m(bn,Kn) ~ [`(x)]|b-bn|g® 0m(nKn)x−n0 . In this formula γ is a positive constant, and [`(x)]x is the unique positive, global minimum point of a certain polynomial g. We call g the Ginzburg–Landau polynomial because of its close connection with the Ginzburg–Landau phenomenology of critical phenomena. For each sequence the structure of the set of global minimum points of the associated Ginzburg–Landau polynomial mirrors the structure of the set of global minimum points of the free-energy functional in the region through which (β n ,K n ) passes and thus reflects the phase-transition structure of the model in that region. This paper makes rigorous the predictions of the Ginzburg–Landau phenomenology of critical phenomena and the tricritical scaling theory for the mean-field Blume–Capel model.Publication Spatializing random measures: Doubly indexed processes and the large deviation principle(1999) Boucher, C; Ellis, RS; Turkington, BThe main theorem is the large deviation principle for the doubly indexed sequence of random measures Abstract Here $\theta$ is a probability measure on a Polish space $\mathscr{X},{D_{r,k}k=1,\ldots,2^r}$ is a dyadic partition of $\mathscr{X}$ (hence the use of $2^r$ summands) satisfying $\theta{D_{r,k}}= 1/2^r$ and $L_{q,1}L_{q,2},\ldotsL_{q,2_r}$ is an independent, identically distributed sequesnce of random probability measures on a Ploish space$ \mathscr{Y}$ such that ${L_{q,k}q\in \mathsbb{N}}$ satisfies the large deviation principle with a convex rate function. A number of related asymptotic results are also derived. The random measures $W_{ r,q}$ have important applications to the statistical mechanics of turbulence. In a companion paper, the large deviation principle presented here is used to give a rigorous derivation of maximum entropy principles arising in the well-known Miller–Robert theory of two-dimensional turbulence as well as in a modification of that theory recently proposed by Turkington.Publication Thermodynamic versus statistical nonequivalence of ensembles for the mean-field Blume-Emery-Griffiths model(2004-01) Ellis, RS; Touchette, H; Turkington, BWe illustrate a novel characterization of nonequivalent statistical mechanical ensembles using the mean-field Blume–Emery–Griffiths (BEG) model as a test model. The novel characterization takes effect at the level of the microcanonical and canonical equilibrium distributions of states. For this reason it may be viewed as a statistical characterization of nonequivalent ensembles which extends and complements the common thermodynamic characterization of nonequivalent ensembles based on nonconcave anomalies of the microcanonical entropy. By computing numerically both the microcanonical and canonical sets of equilibrium distributions of states of the BEG model, we show that for values of the mean energy where the microcanonical entropy is nonconcave, the microcanonical distributions of states are nowhere realized in the canonical ensemble. Moreover, we show that for values of the mean energy where the microcanonical entropy is strictly concave, the equilibrium microcanonical distributions of states can be put in one-to-one correspondence with equivalent canonical equilibrium distributions of states. Our numerical computations illustrate general results relating thermodynamic and statistical equivalence and nonequivalence of ensembles proved by Ellis et al.