Inevitable irreversibility in a quantum system consist- ing of many non-interacting “small” pieces

In statistical physics textbooks, one often encounters systems which consist of macroscopic numbers of identical small parts which do not interact with each other. This is not too unrealistic since there are many physical systems (such as certain spin systems, including nuclear spin systems) which are well approximated by such non-interacting models in some ranges of temperature and time. The question that we wish to examine here is the following: Does a system which consists of many non-interacting pieces behave as a “healthy” thermodynamic system? It is evident from exercises in statistical physics that the answer is “yes” when only equilibrium properties are concerned. If one focuses on certain non-equilibrium aspects, however, the situation may be different. In fact Sato, Sekimoto, Hondou, and Takagi recently proved that, in a system which consists of many non-interacting small parts, a simple quasi-static process involving contacts with two heat baths can never be reversible in general. It is remarkable that such a system fails to provide us with reversible processes, which are among the building blocks of conventional thermodynamics. Sato, Sekimoto, Hondou, and Takagi then raise an interesting question whether one can develop a new thermodynamic framework which is capable of describing these unavoidable irreversible processes. The basic idea of Sato, Sekimoto, Hondou, and Takagi is indeed quite simple. Suppose that a small piece (whose identical copies form the whole system) is a quantum system with n energy levels ε1, . . . , εn. We first assume that the whole system is in equilibrium with a heat bath at inverse temperature β. Then the probability of finding the small system in the i-th state is pi = e i/z(β) where z(β) is the partition function for the small system. We then gently decouple the system from the bath in such a way that the small system is still described by the same probability pi. Then we change a parameter in the model Hamiltonian very slowly, modifying the energy levels to ε′1, . . . , ε ′ n. The adiabatic theorem tells us that, if the parameter change is sufficiently slow, the probability of finding the small system in the i-th state is still given by the same pi. But this pi cannot be represented 1 Department of Physics, Gakushuin University, Mejiro, Toshima-ku, Tokyo 171, JAPAN electronic address: [email protected] 2 The comparison hypothesis as in [2] is also violated, if we include a contact with a heat bath (whose temperature is chosen carefully so that no net energy is exchanged) at the end of each “adiabatic process.” 3 We number the states so that εi ≤ εi+1 and εi ≤ εi+1.