Statistical mechanical derivation of the second law of thermodynamics

In a macroscopic (quantum or classical) Hamiltonian system, we prove the second law of thermodynamics in the forms of the minimum work principle and the law of entropy increase, under the assumption that the initial state is described by a general equilibrium distribution. Therefore the second law is a logical necessity once we accept equilibrium statistical mechanics. Note added (December, 2000): We learned that the main observation in the present note (the inequality (8) in the case H = H ) is contained in A. Lenard, J. Stat. Phys. 19, 575 (1978). (See also W. Pusz and S. L. Woronowicz, Commun. Math. Phys. 58, 273 (1978), W. Thirring, Quantum Mechanics of Large Systems (Springer, 1983).) We note, however, that Lenard does not have the view point of micro-macro separation (which we believe necessary for thermodynamic discussions) and do not discuss results like our Theorem 2. We do not publish the present note as an original paper, but try to discuss these (and other related) points in future publications. To understand macroscopic irreversibility from microscopic mechanics is one of the unsolved fundamental problems in physics. One may roughly classify the problem of irreversibility into that of “equilibration” and of “operational irreversibility.” The former aims at justifying equilibrium statistical mechanics, and has a rich (but not yet satisfactory) history which goes back to Boltzmann [1]. The latter problem of “operational irreversibility” deals with a microscopic interpretation of the second law of thermodynamics. Briefly speaking, (a version of) the second law is a statement about the fundamental limitation on the possibility of adiabatic operation bringing one equilibrium state to another [2]. Recall that, although the initial and final states are assumed to be in perfect equilibriums, the process connecting the two can go far away from equilibrium. Although these two problems about irreversibility are intimately connected, we here concentrate only on the second problem of “operational irreversibility” and give a solution to it. More precisely, we assume that the initial sate of a macroscopic (quantum or classical) Hamiltonian system is described by a general equilibrium distribution, and rigorously derive the second law of thermodynamics in the forms of the minimum work principle and the law of entropy increase [3]. This establishes that the second law is a logical necessity once we accept equilibrium statistical mechanics and Hamiltonian mechanics . Setup and basic inequality: We examine an adiabatic process in thermodynamics, where an external (classical) agent performs an operation on a thermally isolated system. This may be modelled by a quantum mechanical system with an N -dimensional Hilbert space whose time evolution is determined by a time dependent Hamiltonian H(t). 1 Department of Physics, Gakushuin University, Mejiro, Toshima-ku, Tokyo 171, JAPAN electronic address: [email protected]