Canonical Gibbs Measure Research Articles

Finite-particle approximations for interacting Brownian particles with logarithmic potentials

We prove the convergence of $N$-particle systems of Brownian particles with logarithmic interaction potentials onto a system described by the infinite-dimensional stochastic differential equation (ISDE). For this proof we present two general theorems on the finite-particle approximations of interacting Brownian motions. In the first general theorem, we present a sufficient condition for a kind of tightness of solutions of stochastic differential equations (SDE) describing finite-particle systems, and prove that the limit points solve the corresponding ISDE. This implies, if in addition the limit ISDE enjoy a uniqueness of solutions, then the full sequence converges. We treat non-reversible case in the first main theorem. In the second general theorem, we restrict to the case of reversible particle systems and simplify the sufficient condition. We deduce the second theorem from the first. We apply the second general theorem to $\mathrm{Airy}_\beta$ interacting Brownian motion with $\beta=1, 2, 4$, and the Ginibre interacting Brownian motion. The former appears in the soft-edge limit of Gaussian (orthogonal/unitary/symplectic) ensembles in one spatial dimension, and the latter in the bulk limit of Ginibre ensemble in two spatial dimensions, corresponding to a quantum statistical system for which the eigen-value spectra belong to non-Hermitian Gaussian random matrices. The passage from the finite-particle stochastic differential equation (SDE) to the limit ISDE is a sensitive problem because the logarithmic potentials are long range and unbounded at infinity. Indeed, the limit ISDEs are not easily detectable from those of finite dimensions. Our general theorems can be applied straightforwardly to the grand canonical Gibbs measures with Ruelle-class potentials such as Lennard-Jones 6-12 potentials and and Riesz potentials.

Local microscopic behavior for 2D Coulomb gases

The study of two-dimensional Coulomb gases lies at the interface of statistical physics and non-Hermitian random matrix theory. In this paper we give a large deviation principle (LDP) for the empirical fields obtained, under the canonical Gibbs measure, by zooming around a point in the bulk of the equilibrium measure, up to the finest averaging scale \(N^{-1/2 + \varepsilon }\). The rate function is given by the sum of the “renormalized energy” of Serfaty et al. weighted by the inverse temperature, and of the specific relative entropy. We deduce a local law which quantifies the convergence of the empirical measures of the particles to the equilibrium measure, up to the finest scale.

Continuum percolation for Gibbsian point processes with attractive interactions

We study the problem of continuum percolation in infinite volume Gibbs measures for particles with an attractive pair potential, with a focus on low temperatures (large β). The main results are bounds on percolation thresholds ρ ± (β) in terms of the density rather than the chemical potential or activity. In addition, we prove a variational formula for a large deviations rate function for cluster size distributions. This formula establishes a link with the Gibbs variational principle and a form of equivalence of ensembles, and allows us to combine knowledge on finite volume, canonical Gibbs measures with infinite volume, grand-canonical Gibbs measures

Exponential Relaxation to Equilibrium for a One-Dimensional Focusing Non-Linear Schrödinger Equation with Noise

We construct generalized grand-canonical- and canonical Gibbs measures for a Hamiltonian system described in terms of a complex scalar field that is defined on a circle and satisfies a nonlinear Schrödinger equation with a focusing nonlinearity of order p < 6. Key properties of these Gibbs measures, in particular absence of “phase transitions” and regularity properties of field samples, are established. We then study a time evolution of this system given by the Hamiltonian evolution perturbed by a stochastic noise term that mimics effects of coupling the system to a heat bath at some fixed temperature. The noise is of Ornstein–Uhlenbeck type for the Fourier modes of the field, with the strength of the noise decaying to zero, as the frequency of the mode tends to ∞. We prove exponential approach of the state of the system to a grand-canonical Gibbs measure at a temperature and “chemical potential” determined by the stochastic noise term.

A NOTE ON AN INTEGRATION BY PARTS FORMULA FOR THE GENERATORS OF UNIFORM TRANSLATIONS ON CONFIGURATION SPACE

An integration by parts formula is derived for the first-order differential operator corresponding to the action of translations on the space of locally finite simple configurations of infinitely many points on ℝd. As reference measures, tempered grand canonical Gibbs measures are considered corresponding to a non-constant non-smooth intensity (one-body potential) and translation invariant potentials fulfilling the usual conditions. It is proven that such Gibbs measures fulfill the intuitive integration by parts formula if and only if the action of the translation is not broken for this particular measure. The latter is automatically fulfilled in the high temperature and low intensity regime.

Construction of a state evolution for Kawasaki dynamics in continuum

We consider conservative, non-equilibrium stochastic jump dynamics of interacting particles in continuum. These dynamics have a (grand canonical) Gibbs measure as invariant measure. The problem of existence of these dynamics is studied. The corresponding time evolution of correlation functions is constructed.

Mesoscopic Analysis of Droplets in Lattice Systems with Long-Range Kac Potentials

We investigate the geometry of typical equilibrium configurations for a lattice gas in a finite macroscopic domain with attractive, long range Kac potentials. We focus on the case when the system is below the critical temperature and has a fixed number of occupied sites. We connect the properties of typical configurations to the analysis of the constrained minimizers of a mesoscopic non-local free energy functional, which we prove to be the large deviation functional for a density profile in the canonical Gibbs measure with prescribed global density. In the case in which the global density of occupied sites lies between the two equilibrium densities that one would have without a constraint on the particle number, a "droplet" of the high (low) density phase may or may not form in a background of the low (high) density phase. We determine the critical density for droplet formation, and the nature of the droplet, as a function of the temperature and the size of the system, by combining the present large deviation principle with the analysis of the mesoscopic functional given in Nonlinearity 22, 2919---2952 (2009).

TAGGED PARTICLE PROCESS IN CONTINUUM WITH SINGULAR INTERACTIONS

We study the dynamics of a tagged particle in an environment of infinitely many Brownian particles in continuum. All the particles interact via the gradient of an interaction potential. Our strategy is to construct the infinite particle environment process at first and afterwards the process coupling the motion of the tagged particle and the motion of the environment. First we derive an integration by parts formula with respect to the standard gradient ∇Γon configuration spaces Γ for a general class of grand canonical Gibbs measures μ, corresponding to pair potentials ϕ and intensity measures σ = z exp (-ϕ)dx, 0 &lt; z &lt; ∞, having correlation functions fulfilling a Ruelle bound. Combining this with a second integration by parts formula with respect to the gradient ∇Γin direction γ ∈ Γ, by Dirichlet form techniques we can construct the environment process and the coupled process, respectively. Our results give the first mathematically rigorous and complete construction of the tagged particle process in continuum with interaction potential. In particular, we can treat interaction potentials which might have a singularity at the origin, nontrivial negative part and infinite range as e.g., the Lennard–Jones potential.

N/V-LIMIT FOR LANGEVIN DYNAMICS IN CONTINUUM

We construct an infinite particle/infinite volume Langevin dynamics on the space of simple configurations in ℝd having velocities as marks. The construction is done via a limiting procedure using N-particle dynamics in cubes (-λ, λ]d with periodic boundary condition. A main step to this result is to derive an (improved) Ruelle bound for the canonical correlation functions of N-particle systems in (-λ, λ]d with periodic boundary condition. After proving tightness of the laws of the finite particle dynamics, the identification of accumulation points as martingale solutions of the Langevin equation is based on a general study of properties of measures on configuration space fulfilling a uniform Ruelle bound (and their weak limits). Additionally, we prove that the initial/invariant distribution of the constructed dynamics is a tempered grand canonical Gibbs measure. All proofs work for a wide class of repulsive interaction potentials ϕ (including, e.g., the Lennard–Jones potential) and all temperatures, densities and dimensions d ≥ 1.

Phase Diagram of the ABC Model on an Interval

The three species asymmetric ABC model was initially defined on a ring by Evans, Kafri, Koduvely, and Mukamel, and the weakly asymmetric version was later studied by Clincy, Derrida, and Evans. Here the latter model is studied on a one-dimensional lattice of N sites with closed (zero flux) boundaries. In this geometry the local particle conserving dynamics satisfies detailed balance with respect to a canonical Gibbs measure with long range asymmetric pair interactions. This generalizes results for the ring case, where detailed balance holds, and in fact the steady state measure is known, only for the case of equal densities of the different species: in the latter case the stationary states of the system on a ring and on an interval are the same. We prove that in the limit N→∞ the scaled density profiles are given by (pieces of) the periodic trajectory of a particle moving in a quartic confining potential. We further prove uniqueness of the profiles, i.e., the existence of a single phase, in all regions of the parameter space (of average densities and temperature) except at low temperature with all densities equal; in this case a continuum of phases, differing by translation, coexist. The results for the equal density case apply also to the system on the ring, and there extend results of Clincy et al.

Non-ergodicity of Nosé–Hoover dynamics**This paper is published as part of a collection in honour of Todd Dupont's 65th birthday.

The Nosé–Hoover dynamics is a deterministic method that is commonly used to sample the canonical Gibbs measure. This dynamics extends the physical Hamiltonian dynamics by the addition of a ‘thermostat’ variable, which is coupled nonlinearly with the physical variables. The accuracy of the method depends on the dynamics being ergodic. Numerical experiments have been published earlier that are consistent with non-ergodicity of the dynamics for some model problems. The authors recently proved the non-ergodicity of the Nosé–Hoover dynamics for the one-dimensional harmonic oscillator.In this paper, this result is extended to non-harmonic one-dimensional systems. We also show that, for some multidimensional systems, the averaged dynamics for the limit of infinite thermostat ‘mass’ has many invariants, thus giving theoretical support for either non-ergodicity or slow ergodization. Numerical experiments for a two-dimensional central force problem and the one-dimensional pendulum problem give evidence for non-ergodicity.

Remarks on non-interacting conservative spin systems: The case of gamma distributions

We provide precise estimates of the spectral gap and logarithmic Sobolev constants of canonical Gibbs measures associated to gamma distributions with parameter α ≥ 1 . This is related to the Kannan–Lovász–Simonovits conjecture for simplices and unit balls of ℓ p n spaces.

Scaling limit of fluctuations for the equilibrium Glauber dynamics in continuum

The Glauber dynamics investigated in this paper are spatial birth and death processes in a continuous system having a grand canonical Gibbs measure of Ruelle type as an invariant measure. We prove that such processes, when appropriately scaled, have as scaling limit a generalized Ornstein–Uhlenbeck process. First we prove convergence of the corresponding Dirichlet forms. This convergence requires only very weak assumptions. The interaction potential ϕ only has to be stable (S), integrable (I), and we have to assume the low activity high temperature regime. Under a slightly stronger integrability condition ( I ∞ ) and a conjecture on the Percus–Yevick equation we even can prove strong convergence of the corresponding generators. Finally, we prove that the scaled processes converge in law. Here the hardest part is to show tightness of the scaled processes (note that the processes only have càdlàg sample path). For the proof we have to assume that the interaction potential is positive (P). The limiting process then is identified via the associated martingale problem. For this the above mentioned strong convergence of generators is essential.

N/V-limit for stochastic dynamics in continuous particle systems

We provide an N/V-limit for the infinite particle, infinite volume stochastic dynamics associated with Gibbs states in continuous particle systems on ℝ d ,d≥1. Starting point is an N-particle stochastic dynamic with singular interaction and reflecting boundary condition in a subset Λ⊂ℝ d with finite volume (Lebesgue measure) V=|Λ|<∞. The aim is to approximate the infinite particle, infinite volume stochastic dynamic by the above N-particle dynamic in Λ as N→∞ and V→∞ such that N/V→ρ, where ρ is the particle density. First we derive an improved Ruelle bound for the canonical correlation functions under an appropriate relation between N and V. Then tightness is shown by using the Lyons–Zheng decomposition. The equilibrium measures of the accumulation points are identified as infinite volume canonical Gibbs measures by an integration by parts formula and the accumulation points themselves are identified as infinite particle, infinite volume stochastic dynamics via the associated martingale problem. Assuming a property closely related to Markov uniqueness and weaker than essential self-adjointness, via Mosco convergence techniques we can identify the accumulation points as Markov processes and show uniqueness. I.e., all accumulation corresponding to one invariant canonical Gibbs measure coincide. The proofs work for general repulsive interaction potentials ϕ of Ruelle type and all temperatures, densities, and dimensions d≥1, respectively. ϕ may have a nontrivial negative part and infinite range as e.g. the Lennard–Jones potential. Additionally, our result provides as a by-product an approximation of grand canonical Gibbs measures by finite volume canonical Gibbs measures with empty boundary condition.

Around Multicolour Disordered Lattice Gas

We consider a system of multicolour disordered lattice gas, following closely the (monocolour) introduced by Faggionato and Martinelli(3,4). We study the projection on the monocolour system and we derive an estimate of the closeness between grand canonical and canonical Gibbs measures.

INFINITE SYSTEM OF BROWNIAN BALLS: EQUILIBRIUM MEASURES ARE CANONICAL GIBBS

We consider a system of infinitely many hard balls in ℝd undergoing Brownian motions and submitted to a smooth pair potential. It is modelized by an infinite-dimensional stochastic differential equation with a local time term. We prove that the set of all equilibrium measures, solution of a detailed balance equation, coincides with the set of canonical Gibbs measures associated to the hard core potential added to the smooth interaction potential.

Comparison of Finite Volume Canonical and Grand Canonical Gibbs Measures: The Continuous Case

We consider a continuous gas with finite range positive pair potential and we assume that the cluster expansion convergence condition holds. We prove a sharp bound on the difference between the finite volume grand canonical and canonical expectation of local observable. The bound is given in terms of the support of the observable, of its grand canonical variance and of the volume on which the system is confined.

Ergodicity of canonical Gibbs measures with respect to the diffeomorphism group

AbstractFor general potentials we prove that every canonical Gibbs measure on configurations over a manifold X is quasi‐invariant w.r.t. the group of diffeomorphisms on X. We show that this quasi‐invariance property also characterizes the class of canonical Gibbs measures. From this we conclude that the extremal canonical Gibbs measures are just the ergodic ones w.r.t. the diffeomorphism group. Thus we provide a whole class of different irreducible representations. (© 2004 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Logarithmic Sobolev constant for the dilute Ising lattice gas dynamics below the percolation threshold

We consider a conservative stochastic lattice gas dynamics reversible with respect to the canonical Gibbs measure of the bond dilute Ising model on Z d at inverse temperature β. When the bond dilution density p is below the percolation threshold, we prove that, for any ε>0, any particle density and any β, with probability one, the logarithmic Sobolev constant of the generator of the dynamics in a box of side L centered at the origin cannot grow faster that L 2+ ε .

The logarithmic Sobolev constant of Kawasaki dynamics under a mixing condition revisited

We consider a conservative stochastic spin exchange dynamics reversible with respect to the canonical Gibbs measure of a lattice gas model. We assume that the corresponding grand canonical measure satisfies a suitable strong mixing condition. Following previous work by two of us for the spectral gap, we provide an alternative and quite natural, from the physical point of view, proof of the well known result of Yau stating that the logarithmic Sobolev constant in a box of side L grows like L 2.