Abstract
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 contains 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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Bernardin, C., Olla, S.: Fourier's law for a microscopic model of heat conduction. http://www.ceremade.dauphine.fr/olla/heatss-2.pdf, 2005, To appear in J. Stat. Phys
Bonetto, F., Lebowitz, J.L., Rey-Bellet, L.: Fourier's law: a challenge to theorists. In: Mathematical physics 2000, London: Imp. Coll. Press, 2000, pp. 128–150
Casati, G., Ford, J., Vivaldi, F., Visscher W.: One-dimensional classical many-body system having a normal thermal conduction. Phys. Rev. Lett. 52, 1861–1864 (1984)
De Groot, S., Mazur P.: Non-Equilibrium Thermodynamics, Amsterdam: North Holland, 1962
De Masi, A., Presutti, E.: Mathematical methods for hydrodynamic limits, Vol. 1501 Lecture Notes in Mathematics, Berlin: Springer-Verlag, 1991
Eckmann, J.-P., Young, L.-S.: Temperature profiles in Hamiltonian heat conduction. Europhysics Letters 68, 790–796 (2004)
Eyink, G., Lebowitz, J.L., Spohn, H.: Hydrodynamics of stationary nonequilibrium states for some stochastic lattice gas models. Commun. Math. Phys. 132, 253–283 (1990)
Garrido, P., Hurtado, P., Nadrowski, B.: Simple one-dimensional model of heat conduction which obeys fourier's law. Phys. Rev. Lett. 86, 5486–5489 (2001)
Gruber, C., Lesne, A.: Hamiltonian model of heat conductivity and Fourier law. Physica A, 351, 358–372 (2005)
Kipnis, C., Landim, C.: Scaling limits of interacting particle systems, Vol. 320 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Berlin: Springer-Verlag, 1999
Kipnis, C., Marchioro, C., Presutti, E.: Heat flow in an exactly solvable model. J. Stat. Phys. 27, 65–74 (1982)
Larralde, H., Leyvraz, F., Mejía-Monasterio, C.: Transport properties of a modified Lorentz gas. J. Stat. Phys. 113, 197–231 (2003)
Lepri, S., Livi, R., Politi, A.: Thermal conduction in classical low-dimensional lattices. Phys. Rep. 377, 1–80 (2003)
Li, B., Casati, G., Wang, J., Prosen, T.: Fourier law in the alternate mass hard-core potential chain. Phys. Rev. Lett. 92, 254–301 (2004)
Li, B., Casati, G., Wang, J., Prosen, T.: Fourier law in the alternate mass hard-core potential chain. Phys. Rev. Lett. 92, 254–301 (2004)
Liverani, C., Wojtkowski, M.P.: Ergodicity in Hamiltonian systems. In: Dynamics reported, Vol.4 of Dynam. Report. Expositions Dynam. Systems (N.S.), Berlin: Springer, 1995, pp. 130–202
Mejía-Monasterio, C., Larralde, H., Leyvraz, F.: Coupled normal heat and matter transport in a simple model system. Phys. Rev. Lett. 86, 5417–5420 (2001)
Posch, H.A.: Hoover, W.G.: Heat conduction in one-dimensional chains and nonequilibrium Lyapunov spctrum. Phys. Rev. E 58, 4344–4350 (1998)
Prosen, T., Robnik, M.: Energy transport and detailed verification of fourier heat law in a chain of colliding harmonic oscillators. J. Physics. A 25, 3449–3478 (1992)
Rateitschak, K., Klages, R., Nicolis, G.: Thermostating by deterministic scattering: the periodic Lorentz gas. J. Stat. Phys. 99, 1339–1364 (2000)
Sinai, J.G.: Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards. Uspehi Mat. Nauk 25, 141–192 (1970)
Spohn, H.: Large Scale Dynamics of Interacting Particles. Texts and Monographs in Physics, Heidelberg: Springer-Verlag, 1991
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Communicated by G. Gallavotti
JPE is partially supported by the Fonds National Suisse.
LSY is partially supported by NSF Grant #0100538.
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Eckmann, JP., Young, LS. Nonequilibrium Energy Profiles for a Class of 1-D Models. Commun. Math. Phys. 262, 237–267 (2006). https://doi.org/10.1007/s00220-005-1462-y
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DOI: https://doi.org/10.1007/s00220-005-1462-y