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Title: Fourier's law of heat conduction for a deterministic anharmonic lattice
Plain: 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: 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.
Settled by: A proof of finite thermal conductivity for one deterministic 3D lattice model.
Kind: well-posed
Answer type: proof
References: (empty)
Literature status: open
Status note: Posed as a challenge by Bonetto, Lebowitz and Rey-Bellet (2000); results exist mainly for models with stochastic noise added to the dynamics.
Posed since: (empty)