The second law of thermodynamics from concave energy in classical mechanics

A recently proposed quantum mechanical criterion `concavity of energy' 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 time average of work done by particles in a quench process cannot exceed that in the corresponding quasi-static process, 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 with certain properties. This result implies that the system satisfies the principle of maximum work in the adiabatic environment as an expression of the second law of thermodynamics.