Thermodynamic temperature in linear and nonlinear Hamiltonian Systems

This paper introduces the temperature concept for Hamiltonian systems to describe the energy flow between two coupled sub-systems. As a result, a general and strict method to approach the energy analysis of linear and nonlinear systems, with potential applications both in theoretical mechanics as well as in engineering Statistical Energy Analysis is disclosed. The opportunity of a strict mathematical foundation to this important physical and engineering problem, is provided by the introduction of the Khinchin’s entropy. The analysis shows that, under (i) linearity, (ii) weak coupling and (iii) close-to-equilibrium conditions, a Fourier-like heat transmission law is obtained, where the thermodynamic temperature in proportional to the modal energy of the system, that is the ratio of its total energy and the number of its degrees of freedom. Generalized results for nonlinear systems are indeed derived in closed form for weak anharmonic potentials, showing in this case that the temperature depends on a series of integer and fractional powers of the system’s energy.