The Second Law of Thermodynamics from Concave Energy in Classical Mechanics

A recently proposed quantum mechanical criterion "concavity of energy eigenvalues" for the second law of thermodynamics is studied also for classical particle systems confined in a bounded region by a potential with a time-dependent coupling constant. It is shown that the work done by particles in a quench process cannot exceed that in the corresponding quasi-static process, if and only if the energy is a concave function of the coupling constant. It is proven that the energy is indeed concave for a general confining potential satisfying certain conditions. This result implies that the system satisfies the principle of maximum work in an adiabatic environment as an expression of the second law of thermodynamics for two extreme cases of quasi-static and quench processes.