Nonequilibrium measures which exhibit a temperature gradient: Study of a model
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We give some rules to define measures which could describe heat flow in homogeneous crystals. We then study a particular model which is explicitly solvable: the one dimensional nearest neighborhood Ising model. We analyze two cases. In the first one the spins at the two boundaries interact with reservoirs at different temperatures; in the thermodynamical limit the measure we introduce converges locally to Gibbs measures and a temperature profile is so derived. We obtain an explicit expression for the thermal conductivity coefficient which depends on the temperature. In the second case we study the asymptotic behavior starting from an initial state in which each half of the space is at a different temperature. We find again a temperature profile which asymptotically obeys the heat equation with the thermal conductivity coefficient previously derived. From a mathematical point of view, the analysis of the invariant measure is made possible by studying a “time-reversed” process related to a graphical representation of an associated process. This provides us with an explicit formula for then-fold correlation function and we study the limiting behavior using both this representation (for proving an exchangeability result) and a Donsker-type, spacetime renormalization procedure.
KeywordsThermal Conductivity Correlation Function Asymptotic Behavior Heat Flow Temperature Profile
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- 2.Goldstein, S., Lebowitz, J.L., Presutti, E.: Mechanical systems with stochastic boundaries. In: Lecture at the Colloqium on “Random fields: rigorous results in statistical mechanics and quantum field theory”. Esztergon, Hungary (1979)Google Scholar
- 3.Georgii, H.O.: “Canonical Gibbs measures. In: Lecture Notes in Mathematics, Vol. 760. Berlin, Heidelberg, New York: Springer 1979Google Scholar
- 4.Ruelle, D.: Statistical mechanics. New York: Benjamin 1968Google Scholar
- 5.Revuz, D.: Markov chains. North Holland: American Elsevier 1975Google Scholar
- 7.Liggett, T.: The stochastic evolution of infinite systems of interacting particles. In: Lecture Notes in Mathematics, Vol. 598, pp. 188–248. Berlin, Heidelberg, New York: Springer 1976Google Scholar
- 8a.Griffeath, D.: Ann. Prob.6, 379–387 (1978); Harris, T.: Ann. Prob.6, 355–378 (1978); Griffeath, D.: Graphical treatment of associate process. Séminaire de l'Ecole Polytechnique (1975); Neveu, J.: Evolution Markoviennes. Séminaire sur les processus markoviens à une infinité de particules, Ecole Polytechnique (1974); Spitzer, F.: Random fields and interacting particle systems MAA Seminar. Williamstown, Massachusetts (1971)Google Scholar
- 9.Feller, W.: An introduction to probability theory and its applications, I, II, 3rd edition. New York: Wiley and Sons 1967; Simon, B.: Functional integration and quantum physics. London, New York: Academic Press 1979Google Scholar
- 10.Billingsley, P.: Convergence of probability measures. New York: Wiley and Sons 1968Google Scholar
- 11.Metivier, M.: Sufficient conditions for tightness and weak convergence of a sequence of processes. Internal report, University of Minnesota, Minneapolis; Rebolledo, R.: La méthode des martingales appliquées à la convergence en loi des processus. Mém. Soc. Math. Fr. (to appear)Google Scholar
- 12.Spitzer, F.: Trans. Am. Math. Soc.198, 191–199 (1974)Google Scholar
- 14.Kingman, J.F.C.: J. Appl. Prob.6, 1–18 (1969)Google Scholar