Dynamical Derivation of Heat Transport

The theoretical derivation of the empirical Fourier law is still an open subject of investigation. We plan to contribute to the solution of this problem moving along the lines of the earlier work of [1]. A set of non-linearly interacting classical oscillators with energy E large enough generates chaos turning classical dynamics into thermodynamics. Following [1] we adopt the popular model proposed by Fermi, Pasta and Ulam (FPU), with a chain of N oscillators, in the condition where the Boltzmann principle S = k ln W applies, and we define temperature T according to the thermodynamic prescription 1/T = dS/dE. We add two additional oscillators, with vanishing kinetic energy, to the right and left end of the FPU chain and we study theoretically and numerically the equilibration process. The kinetic energy of the two additional oscillators is used as an indicator of the local temperature at the border of the FPU chain. We plan to apply iteratively this process to establish, with the help of the central limit theorem, the emergence of Fourier law.