Scaling Limits of Interacting Particle Systems

Kipnis, Claude; Landim, Cláudio

doi:10.1007/978-3-662-03752-2

Cited by 1,052 publications

(1,936 citation statements)

References 0 publications

Supporting

Mentioning

1,911

Contrasting

Unclassified

Order By: Relevance

“…Note that these two equations represent the same relation for different k. We write this relation as 8) and verify that it is compatible with assertion (3.1): Since…”

Section: Remark 34mentioning

confidence: 76%

See 1 more Smart Citation

Nonequilibrium Energy Profiles for a Class of 1-D Models

Eckmann

Young

2005

Commun. Math. Phys.

181

View full text Add to dashboard Cite

As a paradigm for heat conduction in 1 dimension, we propose a class of models represented by chains of identical cells, each one of which containing an energy storage device called a "tank". Energy exchange among tanks is mediated by tracer particles, which are injected at characteristic temperatures and rates from heat baths at the two ends of the chain. For stochastic and Hamiltonian models of this type, we develop a theory that allows one to derive rigorously -under physically natural assumptions -macroscopic equations for quantities related to heat transport, including mean energy profiles and tracer densities. Concrete examples are treated for illustration, and the validity of the Fourier Law in the present context is discussed.

show abstract

“…Note that these two equations represent the same relation for different k. We write this relation as 8) and verify that it is compatible with assertion (3.1): Since…”

Section: Remark 34mentioning

confidence: 76%

“…For our stochastic models, a proof of this property seems within reach though technically involved (see e.g. [4,19,8]); no known techniques are available for Hamiltonian systems. Extra assumptions pertaining to ergodicity and mixing issues are needed for our Hamiltonian models.…”

Section: Introductionmentioning

confidence: 99%

Nonequilibrium Energy Profiles for a Class of 1-D Models

Eckmann

Young

2005

Commun. Math. Phys.

181

View full text Add to dashboard Cite

show abstract

“…Let us emphasize that the limits in (15) and (19) are valid for any finite collection of points, including shock locations. For the global results, (16) and part (ii) of Theorem 2.2, we integrate over space so that the values of the processes at shocks become immaterial because the shocks are a Lebesgue null set.…”

Section: Remarkmentioning

confidence: 99%

Diffusive Fluctuations for One-Dimensional Totally Asymmetric Interacting Random Dynamics

Seppäläinen

2002

Commun. Math. Phys.

View full text Add to dashboard Cite

We study central limit theorems for a totally asymmetric, one-dimensional interacting random system. The models we work with are the Aldous-Diaconis-Hammersley process and the related stick model. The A-D-H process represents a particle configuration on the line, or a 1-dimensional interface on the plane which moves in one fixed direction through random local jumps. The stick model is the process of local slopes of the A-D-H process, and has a conserved quantity. The results describe the fluctuations of these systems around the deterministic evolution to which the random system converges under hydrodynamic scaling. We look at diffusive fluctuations, by which we mean fluctuations on the scale of the classical central limit theorem. In the scaling limit these fluctuations obey deterministic equations with random initial conditions given by the initial fluctuations. Of particular interest is the effect of macroscopic shocks, which play a dominant role because dynamical noise is suppressed on the scale we are working.

show abstract

“…Zero-range processes (ZRP) are minimal models [1], often used as simplified realisations of more complex processes (for reviews, see [2,3,4]). For instance they are instrumental for the understanding of condensation transitions in driven diffusive systems [5,6].…”

Section: Introductionmentioning

confidence: 99%

Nonequilibrium phase transition in a non-integrable zero-range process

Godrèche¹

2006

J. Phys. A: Math. Gen.

View full text Add to dashboard Cite

Abstract. The present work is an endeavour to determine analytically features of the stationary measure of a non-integrable zero-range process, and to investigate the possible existence of phase transitions for such a nonequilibrium model. The rates defining the model do not satisfy the constraints necessary for the stationary measure to be a product measure. Even in the absence of a drive, detailed balance with respect to this measure is violated. Analytical and numerical investigations on the complete graph demonstrate the existence of a first-order phase transition between a fluid phase and a condensed phase, where a single site has macroscopic occupation. The transition is sudden from an imbalanced fluid where both species have densities larger than the critical density, to a critical neutral fluid and an imbalanced condensate.

show abstract

Scaling Limits of Interacting Particle Systems

Cited by 1,052 publications

References 0 publications

Nonequilibrium Energy Profiles for a Class of 1-D Models

Nonequilibrium Energy Profiles for a Class of 1-D Models

Diffusive Fluctuations for One-Dimensional Totally Asymmetric Interacting Random Dynamics

Nonequilibrium phase transition in a non-integrable zero-range process

Contact Info

Product

Resources

About