FOURIER'S LAW: A CHALLENGE TO THEORISTS

Mathematical Physics 2000, pp. 128-150 (2000) No AccessFOURIER'S LAW: A CHALLENGE TO THEORISTSF. BONETTO, J. L. LEBOWITZ, and L. REY-BELLETF. BONETTODepartment of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway NJ 08854, USAPresent address: IHES, 75 route de Chartres, 91440 Bures sur Yvette, France., J. L. LEBOWITZDepartment of Mathematics and Physics, Rutgers University, 110 Frelinghuysen Road, Piscataway NJ 08854, USAPresent address: IHES, 75 route de Chartres, 91440 Bures sur Yvette, France., and L. REY-BELLETDepartment of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway NJ 08854, USAPresent address: Department of Mathematics, University of Virginia, Kerchof Hall, Charlottesville VA 22903.https://doi.org/10.1142/9781848160224_0008Cited by:134 PreviousNext AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail Abstract: We present a selective overview of the current state of our knowledge (more precisely of our ignorance) regarding the derivation of Fourier's Law, J(r) = −κ∇T(r); J the heat flux, T the temperature and κ, the heat conductivity. This law is empirically well tested for both fluids and crystals, when the temperature varies slowly on the microscopic scale, with κ an intrinsic property which depends only on the system's equilibrium parameters, such as the local temperature and density. There is however at present no rigorous mathematical derivation of Fourier's law and ipso facto of Kubo's formula for κ, involving integrals over equilibrium time correlations, for any system (or model) with a deterministic, e.g. Hamiltonian, microscopic evolution. 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