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The kinetic limit of a system of coagulating Brownian particles. To appear in Arch. Rational Mech. Anal. Available at www.arxiv.org/math.PR/0408395. 29 Alan Hammond and Fraydoun Rezakhanlou. The kinetic limit of a system of coagulating planar Brownian par
"... Understanding the evolution in time of macroscopic quantities such as pressure or temperature is a central task in nonequilibrium statistical mechanics. We study this problem rigorously for a model of massbearing Brownian particles that are prone to coagulate when they are close, where the macrosc ..."
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Cited by 17 (7 self)
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Understanding the evolution in time of macroscopic quantities such as pressure or temperature is a central task in nonequilibrium statistical mechanics. We study this problem rigorously for a model of massbearing Brownian particles that are prone to coagulate when they are close, where the macroscopic quantity in this case is the density of particles of a given mass. Brownian motion
Kinetic limit for a system of coagulating planar Brownian particles
 J. Stat. Phys
"... We study a model of massbearing coagulating planar Brownian particles. Coagulation occurs when two particles are within a distance of order ε. We assume that the initial number of particles N is of order log ε. Under suitable assumptions of the initial distribution of particles and the microscopi ..."
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Cited by 9 (6 self)
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We study a model of massbearing coagulating planar Brownian particles. Coagulation occurs when two particles are within a distance of order ε. We assume that the initial number of particles N is of order log ε. Under suitable assumptions of the initial distribution of particles and the microscopic coagulation propensities, we show that the macroscopic particle densities satisfy a Smoluchowskitype equation. 1
Equilibrium Fluctuations for a Model of CoagulatingFragmenting planar Brownian Particles
, 2009
"... One of the main purposes of statistical mechanics is to explain the macroscopic behavior of various phenomena in terms of the statistics of their microscopic structures. Macroscopically we often have a PDE involving a small number of parameters. The microscopic description however involves a large n ..."
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One of the main purposes of statistical mechanics is to explain the macroscopic behavior of various phenomena in terms of the statistics of their microscopic structures. Macroscopically we often have a PDE involving a small number of parameters. The microscopic description however involves a large number of components that are evolving by either deterministic
RESEARCH OVERVIEW
"... Here I give an overview of the various strands in my research, which concerns rigorous statistical mechanics usually through the lens of probability theory. The order of topics is roughly made so that those of greatest present active interest to me are earlier. Citations of the form [α2] or [δ1] are ..."
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Here I give an overview of the various strands in my research, which concerns rigorous statistical mechanics usually through the lens of probability theory. The order of topics is roughly made so that those of greatest present active interest to me are earlier. Citations of the form [α2] or [δ1] are to the bibliography of others ’ work and of the form [A3] or [E1] to the appended publication list of mine. A: Phase boundary fluctuation and randomly growing interfaces This topic concerns the fluctuation exponents and scaling limits of static and dynamically defined random models of interfaces. In such systems, local Gaussian fluctuation competes with global curvature constraints to determine longitudinal and radial fluctuation exponents of 2/3 and 1/3. I have studied static models, partly in collaboration with Yuval Peres, and more recently dynamic ones, in large part with Ivan Corwin. Phase separation: To set the scene for explaining the static results first, consider the ferrogmagnetic Ising model, in which spins of two types, positive and negative, are assigned to the integerlattice sites in a domain such as a large box, with signs of opposite type in nearestneighbour pairs being penalized according to a fixed parameter. The two populations of spins each prefer their own type, and, at supercritical inverse temperature, a dominant phase forms, with one type in the majority. If we condition
A SIMPLE PARTICLE MODEL FOR A SYSTEM OF COUPLED EQUATIONS WITH ABSORBING COLLISION TERM
, 2013
"... Abstract. We study a particle model for a simple system of partial differential equations describing, in dimension d ≥ 2, a two component mixture where light particles move in a medium of absorbing, fixed obstacles; the system consists in a transport and a reaction equation coupled through pure abso ..."
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Abstract. We study a particle model for a simple system of partial differential equations describing, in dimension d ≥ 2, a two component mixture where light particles move in a medium of absorbing, fixed obstacles; the system consists in a transport and a reaction equation coupled through pure absorption collision terms. We consider a particle system where the obstacles, of radius ε, become inactive at a rate related to the number of light particles travelling in their range of influence at a given time and the light particles are instantaneously absorbed at the first time they meet the physical boundary of an obstacle; elements belonging to the same species do not interact among themselves. We prove the convergence (a.s. w.r.t. to the product measure associated to the initial datum for the light particle component) of the densities describing the particle system to the solution of the system of partial differential equationsinthe asymptoticsa d n n−κ → 0and a d n εζ → 0, for κ ∈ (0, 1 1 1
Coagulation and diffusion: a probabilistic perspective on the Smoluchowski PDE
, 2013
"... The Smoluchowski coagulationdiffusion PDE is a system of partial differential equations modelling the evolution in time of massbearing Brownian particles which are subject to shortrange pairwise coagulation. This survey presents a fairly detailed exposition of the kinetic limit derivation of the ..."
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The Smoluchowski coagulationdiffusion PDE is a system of partial differential equations modelling the evolution in time of massbearing Brownian particles which are subject to shortrange pairwise coagulation. This survey presents a fairly detailed exposition of the kinetic limit derivation of the Smoluchowski PDE from a microscopic model of many coagulating Brownian particles that was undertaken in [10]. It presents heuristic explanations of the form of the main theorem before discussing the proof, and presents key estimates in that proof using a novel probabilistic technique. The survey's principal aim is an exposition of this kinetic limit derivation, but it also contains an overview of several topics which either motivate or are motivated by this derivation.