From Quantum Dynamics to the Second Law of Thermodynamics

In quantum systems which satisfy the hypothesis of equal weights for eigenstates [4], the maximum work principle (for extremely slow and relatively fast operation) is derived by using quantum dynamics alone. This may be a crucial step in establishing a firm connection between macroscopic thermodynamics and microscopic quantum dynamics. For special models introduced in [4, 5], the derivation of the maximum work principle can be executed without introducing any unproved assumptions. Although there is no doubt that the second law of thermodynamics is one of the most perfect and beautiful laws in physics, its connection to the rest of physics is still poorly understood. It should be stressed that equilibrium statistical mechanics does not lead to the second law. The second law deals with transformations between two equilibrium states caused by any macroscopically realizable processes which can be far from equilibrium. The second law sets sharp and highly nontrivial restrictions on the possibility of such transformations and on the energy exchange during the processes [1]. A traditional approach toward derivation of the second law, which goes back to Boltzmann [2], has been to start from certain stochastic description of microscopic dynamics. In the present note, we wish to concentrate on the possibility of deriving the second law from fully deterministic microscopic quantum dynamics. Such a link between quantum mechanics and thermodynamics (if established) should not only provide a further basis for thermodynamics but also give an indirect support to our belief (which can not been confirmed directly) that even macroscopic systems are governed by quantum mechanics. We shall here concentrate on the second law formulated as the maximum work principle (MWP) [3], and describe its derivation in quantum systems which satisfy the conditions stated in [4] for two limiting situations of infinitely slow and relatively (but not infinitely) fast operations. Here we describe only basic ideas of the derivation, and leave details (which are simply technical and not difficult) to [5]. Basic setup and previous results: Let us start by recalling the general ideas and results in [4], where we presented a scenario for deriving the canonical distribution from quantum dynamics, and an example in which such a derivation can be done without making any assumptions. (See [6] for related attempts of deriving statistical physics using quantum dynamics.) We consider an isolated quantum system which consists of a subsystem and a heat bath [7]. The subsystem alone is described by a Hamiltonian HS which is diagonalized as HSΨj = εjΨj for j = 1, 2, . . . , n, with ‖Ψj‖ = 1 and εj < εj+1. Similarly the bath has a 1 The first version May 8, 2000.