An extension of the Kac ring model

We introduce a unitary dynamics for quantum spins which is an extension of a model introduced by Mark Kac to clarify the phenomenon of relaxation to equilibrium. When the number of spins gets very large, the magnetization satisfies an autonomous equation as function of time with exponentially fast relaxation to the equilibrium magnetization as determined by the microcanonical ensemble. This is proven as a law of large numbers with respect to a class of initial data. The corresponding Gibbs-von Neumann entropy is also computed and its monotonicity in time discussed. PACS numbers: 05.30.Ch, 05.40.-a 1 Relaxation to equilibrium The Kac ring model was introduced by Mark Kac to clarify how manifestly irreversible behaviour can be obtained from an underlying reversible dynamics, [2, 6, 9]. It explains via a simple model some of the conceptual subtleties in the problem of relaxation to equilibrium as for example are present in the derivation and the status of the Boltzmann equation for dilute gases. In particular, the Kac dynamics shares some basic features with a Hamiltonian time-evolution like being deterministic and dynamically reversible. In the present paper we extend that dynamics to a unitary evolution on a finite quantum spin system. Again, the dynamics remains far from realistic but it allows a precise formulation and discussion of some features of relaxation to equilibrium for a quantum dynamics. That is especially useful and relevant as, in the quantum domain, the problem of relaxation is beset with even greater conceptual difficulties. In our framework, relaxation to equilibrium becomes visible if one can select a small number of macroscopic variables that typically evolve via autonomous deterministic equations to take on values that correspond to equilibrium. Typical refers to a law of large numbers with respect to the initial data. Paradoxes are avoided by taking serious the fact that relaxation is a macroscopic phenomenon, involving a huge amount of degrees of freedom whose evolution is monitored over a realistic time-span. One should also keep in mind that relaxation to equilibrium goes beyond questions of return to equilibrium, see [7], which are mostly related to stability of equilibrium states. A more general introduction to that and various related problems can be found in the recent [8]. In section 2 we introduce the model and we state the basic result. Section 3 is devoted to a discussion of related issues. The proofs are postponed to the final section 4. Aspirant F.W.O. Vlaanderen, U. Antwerpen corresponding author, email: [email protected] Instituut voor Theoretische Natuurkunde, Rijksuniversiteit Groningen, The Netherlands 2 MODEL AND RESULTS 2 2 Model and results 2.1 The model Consider N sites on a ring (periodic boundary conditions). Between any two neighboring sites there is a fixed scattering mechanism to be specified below, and at each site, we find a spin 1/2 particle. Time is discrete and at each step the ring rotates in a fixed direction over one ring segment. Depending on the segment that each spin crosses, it is scattered to another state. For the Hilbert space HN we take the N -fold product of copies of C , the state space at each site: HN ≡ N ⊗