A mathematical proof of the zeroth "law" of thermodynamics and the nonlinear Fourier "law" for heat flow.

What is now known as the zeroth "law" of thermodynamics was first stated by Maxwell in 1872: at equilibrium, "Bodies whose temperatures are equal to that of the same body have themselves equal temperatures." In the present paper, we give an explicit mathematical proof of the zeroth "law" for classical, deterministic, T-mixing systems. We show that if a body is initially not isothermal it will in the course of time (subject to some simple conditions) relax to isothermal equilibrium where all parts of the system will have the same temperature in accord with the zeroth "law." As part of the derivation we give for the first time, an exact expression for the far from equilibrium thermal conductivity. We also give a general proof that the infinite-time integral, of transient and equilibrium autocorrelation functions of fluxes of non-conserved quantities vanish. This constitutes a proof of what was called the "heat death of the Universe" as was widely discussed in the latter half of the 19th century.