H-Theorems from Autonomous Equations

The H-theorem is an extension of the Second Law to a time-sequence of states that need not be equilibrium ones. In this paper we review and we rigorously establish the connection with macroscopic autonomy. If for a Hamiltonian dynamics for many particles, at all times the present macrostate determines the future macrostate, then its entropy is non-decreasing as a consequence of Liouville's theorem. That observation, made since long, is here rigorously analyzed with special care to reconcile the application of Liouville's theorem (for a finite number of particles) with the condition of autonomous macroscopic evolution (sharp only in the limit of infinite scale separation); and to evaluate the presumed necessity of a Markov property for the macroscopic evolution.Comment: 13 pages; v1 -> v2: Sec. 1-2 considerably rewritten, minor corrections in Sec. 3-

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.Comment: 15 pages, v2 -> v3: minor typographic correctio

Quantum Brownian Motion in a Simple Model System

We consider a quantum particle coupled (with strength λ) to a spatial array of independent non-interacting reservoirs in thermal states (heat baths). Under the assumption that the reservoir correlations decay exponentially in time, we prove that the motion of the particle is diffusive at large times for small, but finite λ. Our proof relies on an expansion around the kinetic scaling limit ( $${\lambda \searrow 0}$$ , while time and space scale as λ−2) in which the particle satisfies a Boltzmann equation. We also show an equipartition theorem: the distribution of the kinetic energy of the particle tends to a Maxwell-Boltzmann distribution, up to a correction of O(λ2

Extended Weak Coupling Limit for Friedrichs Hamiltonians

We study a class of self-adjoint operators defined on the direct sum of two Hilbert spaces: a finite dimensional one called sometimes a ``small subsystem'' and an infinite dimensional one -- a ``reservoir''. The operator, which we call a ``Friedrichs Hamiltonian'', has a small coupling constant in front of its off-diagonal term. It is well known that under some conditions in the weak coupling limit the appropriately rescaled evolution in the interaction picture converges to a contractive semigroup when restricted to the subsystem. We show that in this model, the properly renormalized and rescaled evolution converges on the whole space to a new unitary evolution, which is a dilation of the above mentioned semigroup. Similar results have been studied before \cite{AFL} in more complicated models and they are usually referred to as "stochastic Limit".Comment: changes in notation and title, minor correction

Derivation of some translation-invariant Lindblad equations for a quantum Brownian particle

We study the dynamics of a Brownian quantum particle hopping on an infinite lattice with a spin degree of freedom. This particle is coupled to free boson gases via a translation-invariant Hamiltonian which is linear in the creation and annihilation operators of the bosons. We derive the time evolution of the reduced density matrix of the particle in the van Hove limit in which we also rescale the hopping rate. This corresponds to a situation in which both the system-bath interactions and the hopping between neighboring sites are small and they are effective on the same time scale. The reduced evolution is given by a translation-invariant Lindblad master equation which is derived explicitly.Comment: 28 pages, 4 figures, minor revisio