MATHPH
In the literature: open
Fourier's law of heat conduction for a deterministic anharmonic lattice
In plain words
Heat flow through a solid is proportional to the temperature gradient, but nobody has proven this from the reversible equations of motion of the atoms. In one dimension, numerics show the law fails.
Precise statement
Lattice of anharmonic oscillators in $d = 3$ (e.g. FPUT-beta) with Hamiltonian bulk dynamics and boundary heat baths at $T_{1}$ and $T_{2}$. Prove that the stationary heat current satisfies $J = \kappa (T_{1} - T_{2})/L + o(1/L)$ with finite $\kappa > 0$ as $L \to \infty$. Answer: a proof, or a proof of anomalous scaling.
What would settle it
A proof of finite thermal conductivity for one deterministic 3D lattice model.
Status in the literature
Posed as a challenge by Bonetto, Lebowitz and Rey-Bellet (2000); results exist mainly for models with stochastic noise added to the dynamics.