Equilibrium Fluctuations for Interacting Ornstein-Uhlenbeck Particles

 We study the hydrodynamic density fluctuations of an infinite system of interacting particles on ℝ ^d . The particles interact between them through a two body superstable potential, and with a surrounding fluid in equilibrium through a random viscous force of Ornstein-Uhlenbeck type. The stationary initial distribution is the Gibbs measure associated with the potential and with a given temperature and fugacity. We prove that the time-dependent density fluctuation field converges in law, under diffusive scaling of space and time, to the solution of a linear stochastic partial differential equation driven by white noise. Received: 10 July 2001 / Accepted: 9 September 2002 Published online: 8 January 2003 RID="*" ID="*" We thank J. Fritz for fruitful discussions, in particular about the existence of the infinite dynamics. A special thanks to L. Bertini for help in the proof of the spectral gap estimate (cf. Appendix B). Communicated by H. Spohn     

Kinetic Limit for a System of Coagulating Planar Brownian Particles

We study a model of mass-bearing coagulating planar Brownian particles. The coagulation occurs when two particles are within a distance of order ε. We assume that the initial number of particles N is of order |logε|. Under suitable assumptions of the initial distribution of particles and the microscopic coagulation propensities, we show that the macroscopic particle densities satisfy a Smoluchowski-type equation.     

Equilibrium Fluctuations for a Non Gradient Energy Conserving Stochastic Model

In this paper we study the equilibrium energy fluctuation field of a one-dimensional reversible non gradient model. We prove that the limit fluctuation process is governed by a generalized Ornstein-Uhlenbeck process, whose covariances are given in terms of the diffusion coefficient. The fact that the conserved, quantity (energy) is not a linear functional of the coordinates of the system: introduces new difficulties of a geometric nature when adapting the non gradient method introduced by Varadhan.     

Equilibrium Fluctuations for Zero-Range-Exclusion Processes

We introduce a Markovian particle system which is a kind of lattice gas on Z consisting of particles carrying energy and whose dynamics is a combination of those of an exclusion process (for particles) and a zero-range process (for energy). It has two conserved quantities, the number of particles and the total energy. The process is reversible relative to certain product probability measures, but of non-gradient type. It is proved that under hydrodynamic scaling the equilibrium fluctuation fields of two conserved quantities converge in law to an infinite dimensional Ornstein–Uhlenbeck process.     

Equilibrium fluctuations for interacting Brownian particles

We consider an infinitely extended system of Brownian particles interacting by a pair force-gradV. Their initial distribution is stationary and given by the Gibbs measure associated with the potentialV with fugacityz. We assume thatV is symmetric, finite range, three times continuously differentiable, superstable, and positive and that the fugacity is small in the sense that 0z0.28/edq(1-e ^–V(q)). In addition a certain essential self-adjointness property is assumed. We prove then that the time-dependent fluctuations in the density on a spatial scale of order ^–1 and on a time scale of order ^–2 converge as 0 to a Gaussian field with covariance dqg(q)(e ^(/2)|t| f)(q) withp the density and the compressibility.     

Loop-Erasure of Planar Brownian Motion

We use a coupling technique to prove that there exists a loop-erasure of the time-reversal of a planar Brownian motion stopped on exiting a simply connected domain, and that the loop-erased curve is a radial SLE₂ curve. This result extends to Brownian motions and Brownian excursions under certain conditioning in a finitely connected plane domain, and the loop-erased curve is a continuous LERW curve.     

Sedimentation of Brownian Particles in a Gravitational Potential

The settling of a Brownian particle in a semi-infinite fluid bounded by a bottom plane is studied on the basis of Smoluchowski's exact solution of the equation describing diffusion in the gravitational potential. Expressions are derived for the mean height and the variance of height at some time after starting at an initial height. These quantities show interesting behavior as a function of time. It is shown that for certain initial heights the Boltzmann entropy does not increase steadily. It increases at first but then decreases to its equilibrium value.     

Equilibrium Fluctuations for a System of Harmonic Oscillators with Conservative Noise

We investigate the harmonic chain forced by a multiplicative noise, the evolution is given by an infinite system of stochastic differential equations. Total energy and deformation are preserved, the conservation of momentum is destroyed by the noise. Gaussian product measures are the extremal stationary states of this model. Equilibrium fluctuations of the conserved fields at a diffusive scaling are described by a couple of generalized Ornstein-Uhlenbeck processes. 1991 Mathematics Subject Classification: Primary 60K31, secondary 82C22 Partially supported by Hungarian Science Foundation Grant T37685, the European Science Foundation Project RDSES, and by the ACI-NIM 168 ‘Transport hors équilibre.’     

A Mechanical Model of Brownian Motion for One Massive Particle Including Slow Light Particles

We provide a connection between Brownian motion and a classical mechanical system. Precisely, we consider a system of one massive particle interacting with an ideal gas, evolved according to non-random mechanical principles, via interaction potentials, without any assumption requiring that the initial velocities of the environmental particles should be restricted to be “fast enough”. We prove the convergence of the (position, velocity)-process of the massive particle under a certain scaling limit, such that the mass of the environmental particles converges to 0 while the density and the velocities of them go to infinity, and give the precise expression of the limiting process, a diffusion process.     

Free Energies and Fluctuations for the Unitary Brownian Motion

We show that the Laplace transforms of traces of words in independent unitary Brownian motions converge towards an analytic function on a non trivial disc. These results allow one to study the asymptotic behavior of Wilson loops under the unitary Yang–Mills measure on the plane with a potential. The limiting objects obtained are shown to be characterized by equations analogue to Schwinger–Dyson’s ones, named here after Makeenko and Migdal.     

Between Equilibrium Fluctuations and Eulerian Scaling: Perturbation of Equilibrium for a Class of Deposition Models

We investigate propagation of perturbations of equilibrium states for a wide class of 1D interacting particle systems. The class of systems considered incorporates zero range, K-exclusion, misanthropic, bricklayers models, and much more. We do not assume attractivity of the interactions. We apply Yau's relative entropy method rather than coupling arguments. The result is partial extension of T. Seppäläinen's recent paper. For 0<<1/5 fixed, we prove that, rescaling microscopic space and time by N, respectively N ¹⁺ , the macroscopic evolution of perturbations of microscopic order N ^– of the equilibrium states is governed by Burgers' equation. The same statement should hold for 0<<1/2 as in Seppäläinen's cited paper, but our method does not seem to work for 1/5.     

Directed Transport of Brownian Particles in a Periodic Channel

The transport of Brownian particles in the infinite channel within an external force along the axis of the channel has been studied. In this paper, we study the transport of Brownian particle in the infinite channel within an external force along the axis of the channel and an external force in the transversal direction. In this more sophisticated situation, some property is similar to the simple situation, but some interesting property also appears.     

Directed Transport of Brownian Particles in a Periodic Channel

The transport of Brownian particles in the infinite channel within an external force along the axis of the channel has been studied. In this paper, we study the transport of Brownian particle in the infinite channel within an external force along the axis of the channel and an external force in the transversal direction. In this more sophisticated situation, some property is similar to the simple situation, but some interesting property also appears.     

Equilibrium Fluctuations for the Totally Asymmetric Zero-Range Process

We consider the one-dimensional Totally Asymmetric Zero-Range process evolving on ? and starting from the Geometric product measure ν _ρ . On the hyperbolic time scale the temporal evolution of the limit density fluctuation field is deterministic, in the sense that the limit field at time t is a translation of the initial one. We consider the system in a reference frame moving at this velocity and we show that the limit density fluctuation field does not evolve in time until N ^4/3, which implies the current across a characteristic to vanish on this longer time scale.     

Equilibrium measure and Brownian escape process

The probability density of Brownian escape process has been obtained recently by R. K. Getoor. In this note we illustrate that Getoor's result can also be obtained by considering equilibrium charge distribution on a metal ball of radiusr>0. This approach reveals a connection between Brownian escape process and classical potential theory.     

Equilibrium Fluctuations in Lysozyme and Myoglobin

Hyperfine Interactions - The angular dependencies of inelastic intensities of Rayleigh scattering of Mössbauer radiation were measured for lysozyme and myoglobin (for different degrees of...     

Brownian Bridges for Late Time Asymptotics of KPZ Fluctuations in Finite Volume

Height fluctuations are studied in the one-dimensional totally asymmetric simple exclusion process with periodic boundaries, with a focus on how late time relaxation towards the non-equilibrium steady state depends on the initial condition. Using a reformulation of the matrix product representation for the dominant eigenstate, the statistics of the height at large scales is expressed, for arbitrary initial conditions, in terms of extremal values of independent standard Brownian bridges. Comparison with earlier exact Bethe ansatz asymptotics leads to explicit conjectures for some conditional probabilities of non-intersecting Brownian bridges with exponentially distributed distances between the endpoints.