306 research outputs found
Large deviations in quantum lattice systems: one-phase region
We give large deviation upper bounds, and discuss lower bounds, for the
Gibbs-KMS state of a system of quantum spins or an interacting Fermi gas on the
lattice. We cover general interactions and general observables, both in the
high temperature regime and in dimension one.Comment: 30 pages, LaTeX 2
Ruelle-Lanford functions for quantum spin systems
We prove a large deviation principle for the expectation of macroscopic
observables in quantum (and classical) Gibbs states. Our proof is based on
Ruelle-Lanford functions and direct subadditivity arguments, as in the
classical case, instead of relying on G\"artner-Ellis theorem, and cluster
expansion or transfer operators as done in the quantum case. In this approach
we recover, expand, and unify quantum (and classical) large deviation results
for lattice Gibbs states. In the companion paper \cite{OR} we discuss the
characterization of rate functions in terms of relative entropies.Comment: 22 page
Decompositions of two player games: potential, zero-sum, and stable games
We introduce several methods of decomposition for two player normal form
games. Viewing the set of all games as a vector space, we exhibit explicit
orthonormal bases for the subspaces of potential games, zero-sum games, and
their orthogonal complements which we call anti-potential games and
anti-zero-sum games, respectively. Perhaps surprisingly, every anti-potential
game comes either from the Rock-Paper-Scissors type games (in the case of
symmetric games) or from the Matching Pennies type games (in the case of
asymmetric games). Using these decompositions, we prove old (and some new)
cycle criteria for potential and zero-sum games (as orthogonality relations
between subspaces). We illustrate the usefulness of our decomposition by (a)
analyzing the generalized Rock-Paper-Scissors game, (b) completely
characterizing the set of all null-stable games, (c) providing a large class of
strict stable games, (d) relating the game decomposition to the decomposition
of vector fields for the replicator equations, (e) constructing Lyapunov
functions for some replicator dynamics, and (f) constructing Zeeman games
-games with an interior asymptotically stable Nash equilibrium and a pure
strategy ESS
Asymptotic Behavior of Thermal Non-Equilibrium Steady States for a Driven Chain of Anharmonic Oscillators
We consider a model of heat conduction which consists of a finite nonlinear
chain coupled to two heat reservoirs at different temperatures. We study the
low temperature asymptotic behavior of the invariant measure. We show that, in
this limit, the invariant measure is characterized by a variational principle.
We relate the heat flow to the variational principle. The main technical
ingredient is an extension of Freidlin-Wentzell theory to a class of degenerate
diffusions.Comment: 40 page
Measuring the irreversibility of numerical schemes for reversible stochastic differential equations
Abstract. For a Markov process the detailed balance condition is equivalent to the time-reversibility of the process. For stochastic differential equations (SDEās) time discretization numerical schemes usually destroy the property of time-reversibility. Despite an extensive literature on the numerical analysis for SDEās, their stability properties, strong and/or weak error estimates, large deviations and infinite-time estimates, no quantitative results are known on the lack of reversibility of the discrete-time approximation process. In this paper we provide such quantitative estimates by using the concept of entropy production rate, inspired by ideas from non-equilibrium statistical mechanics. The entropy production rate for a stochastic process is defined as the relative entropy (per unit time) of the path measure of the process with respect to the path measure of the time-reversed process. By construction the entropy production rate is nonnegative and it vanishes if and only if the process is reversible. Crucially, from a numerical point of view, the entropy production rate is an a posteriori quantity, hence it can be computed in the course of a simulation as the ergodic average of a certain functional of the process (the so-called Gallavotti-Cohen (GC) action functional). We compute the entropy production for various numerical schemes such as explicit Euler-Maruyama and explicit Milsteinās for reversible SDEs with additive or multiplicative noise. Additionally, we analyze the entropy production for th
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