Mathematical Methods for Hydrodynamic Limits

Masi, Anna De; Presutti, Errico

doi:10.1007/bfb0086457

Cited by 227 publications

(242 citation statements)

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“…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

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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.

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Section: Introductionmentioning

confidence: 99%

Nonequilibrium Energy Profiles for a Class of 1-D Models

Eckmann

Young

2005

Commun. Math. Phys.

181

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“…Overviews of some of these links appear in papers [2], [10], and [17]. General treatments of large scale behavior of interacting random systems can be found in [9], [21], [22], [23], [32], and [33].…”

Section: Introductionmentioning

confidence: 99%

The Random Average Process and Random Walk in a Space-Time Random Environment in One Dimension

Balázs

Rassoul-Agha

Seppäläinen

2006

Commun. Math. Phys.

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Abstract. We study space-time fluctuations around a characteristic line for a onedimensional interacting system known as the random average process. The state of this system is a real-valued function on the integers. New values of the function are created by averaging previous values with random weights. The fluctuations analyzed occur on the scale n 1/4 where n is the ratio of macroscopic and microscopic scales in the system. The limits of the fluctuations are described by a family of Gaussian processes. In cases of known product-form invariant distributions, this limit is a two-parameter process whose time marginals are fractional Brownian motions with Hurst parameter 1 4 . Along the way we study the limits of quenched mean processes for a random walk in a space-time random environment. These limits also happen at scale n 1/4 and are described by certain Gaussian processes that we identify. In particular, when we look at a backward quenched mean process, the limit process is the solution of a stochastic heat equation.

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“…No well-defined deterministic system results when we let for example the number of cells of a stochastic cellular automaton go to infinity while keeping the contact structure fixed! Only when we at the same time let individuals move faster or interact over larger distances will a good deterministic limit obtain (though this certainly has not been proved yet in some generality: a nice discussion of this sort of limits can be found in Spohn, 1991, andDe Masi &Presutti, 1991). …”

Section: A Larger Biological Perspectivementioning

confidence: 99%