Fourier's Law: a Challenge for Theorists

By F. Bonetto, J. Lebowitz, Luc Rey-Bellet. 2000

326 citations recorded by the source corpus.

We present a selective overview of the current state of our knowledge (more precisely of ourignorance) regarding the derivation of Fourier's Law, ${\bf J}(\br) =-\kappa {\bf \nabla}T(\br)$; ${\bf J}$ the heat flux, $T$ the temperature and $\kappa$, the heat conductivity. This law is empirically well tested for both fluids and crystals, when the temperature varies slowly on the microscopic scale, with $\kappa$ 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 $\kappa$, involving integrals over equilibrium time correlations, for any system (or model) with a deterministic, e.g. Hamiltonian, microscopic evolution.

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Citations in the corpus

On differentiability of SRB states for partially hyperbolic systems (2004, background): “The question of differentiability plays important role in averaging, rigidity theory and statistical physics ([35, 31, 8 , 50]) but not much is known beyond the uniformly hyperbolic case (in [17, 6] very interesting results about the differentiability of SRB measures for uniformly hyperbolic system”
Fourier's Law for a Harmonic Crystal with Self-Consistent Stochastic Reservoirs (2003, background): “6), we can conclude that the temperature profile now converges to T(x)=TL−(TL−TR) L(x) L(1) ,”
Non-equilibrium steady states: fluctuations and large deviations of the density and of the current (2007, background): “When the two temperatures Ta and Tb are different (see figure 1), the system reaches in the long-time limit a non-equilibrium steady state [ 1 ]–[4], but there does not exist [5, 6] an expression which generalizes (1) for the steady state weights P (C) of the microscopic configurations”
On the (Boltzmann) entropy of non-equilibrium systems (2003, background): “c. 4) a ﬁlm of the particle’s motion run backwards will look the same as one run forward. The situation is diﬀerent when one considers open systems, e.g. systems in contact 29 with thermal reservoirs [19, 20]. The time evolution of the microstate X of such a system is then no longer given by a Hamil”
Heat Transport in Harmonic Lattices (2006, background): “One important question that has attracted a lot of attention is: what are the necessary and sufficient conditions for the validity of Fourier’s law ofheat conduction [ 2 , 3] ? We recall that Fourier’s law states that for a solid with a spatially varying temperature field T(x) inside it, the local h”
The Phonon Boltzmann Equation, Properties and Link to Weakly Anharmonic Lattice Dynamics (2005, background): “. For this reason we merely refer to a few review articles. Jackson [38] summarizes to work up to 1978, an authorative 2003 update being Lepri, Livi, and Politi [39]. Bonetto, Lebowitz and Rey-Bellet [40] emphasize more theoretical aspects, in particular large deviations and the ﬂuctuation theorem. ”