The second law of thermodynamics requires concave energy function of coupling constant in classical and quantum dynamics.

We study classical and quantum dynamics of particles confined in a bounded region by a time dependent confining potential. First, we prove that the time average over an oscillation period of work done by a classical single particle in a quench process cannot exceed the work in the corresponding quasi-static process, if the energy of the particle is a concave function of the coupling constant. We prove that the energy is indeed a concave function of the coupling constant for a classical single particle. Next, we prove the same theorem for quantum interacting particles.We prove that an energy eigenstate expectation value of work in a quench process cannot exceed the work in the corresponding quasi-static process, if its energy eigenvalue is a concave function of the coupling constant. We give a simple universal reason for the concavity, and prove that every energy eigenvalue is concave in some specific quantum systems. These results agree with the maximal work principle in the adiabatic environment as an expression of the second law of thermodynamics. Our result gives a simple example of an integrable system satisfying an analogue to the strong eigenstate thermalization hypothesis (ETH) with respect to the maximal work principle. All proofs are elementary and rigorous.