This paper is concerned with the ergodic subspaces of the state spaces of isolated quantum systems. We prove a new ergodic theorem for closed quantum systems which shows that the equilibrium state of the system takes the form of a grand canonical density matrix involving a complete commuting set of observables including the Hamiltonian. The result obtained, which is derived for a generic finite-dimensional quantum system, shows that the equilibrium state arising from unitary evolution is always expressible in the canonical form, without the consideration of a system-bath decomposition.
This paper is concerned with the ergodic subspaces of the state spaces of isolated quantum systems. We prove a new ergodic theorem for closed quantum systems which shows that the equilibrium state of the system takes the form of a grand canonical density matrix involving a complete commuting set of observables including the Hamiltonian. The result obtained, which is derived for a generic finite-dimensional quantum system, shows that the equilibrium state arising from unitary evolution is always expressible in the canonical form, without the consideration of a system-bath decomposition.
This paper is concerned with the ergodic subspaces of the state spaces of isolated quantum systems. We prove a new ergodic theorem for closed quantum systems which shows that the equilibrium state of the system takes the form of a grand canonical density matrix involving a complete commuting set of observables including the Hamiltonian. The result obtained, which is derived for a generic finite-dimensional quantum system, shows that the equilibrium state arising from unitary evolution is always expressible in the canonical form, without the consideration of a system-bath decomposition.
We prove the approach to equilibrium of quenched isolated quantum systems for which the change in the Hamiltonian brought about by the quench satisfies a certain closed commutator algebra with all the extensive integrals of motion of the system before the quench. The proof is carried out by following the exact unitary evolution of the entropy operator, defined as the negative of the logarithm of the nonequilibrium density matrix, and showing that, under the conditions implied by the assumed algebra, this entropy operator becomes, at infinite times, a linear combination of integrals of motion of the perturbed system. That is, we show how the nonequilibrium density matrix approaches a generalized Gibbs ensemble. The restricted class of systems for which the present results apply turn out to have degenerate spectra in general, as opposed to some generic systems for which a kind of ergodicity is expected, with a nontrivial dynamics believed to find instances in one-dimensional infinite super-integrable systems. Our findings constitute a direct demonstration of how a non-ergodic isolated quantum system may get to statistical equilibrium.
In joint work with J. L. Lebowitz, C. Mastrodonato, and N. Zanghì [2, 3, 4], we considered an isolated, macroscopic quantum system. Let H be a micro‐canonical “energy shell,” i.e., a subspace of the system’s Hilbert space spanned by the (finitely) many energy eigenstates with energies between E and E+δE. The thermal equilibrium macro‐state at energy E corresponds to a subspace Heq of H such that dimHeq/dimH is close to 1. We say that a system with state vector ψ ε H is in thermal equilibrium if ψ is “close” to Heq. We argue that for “typical” Hamiltonians, all initial state vectors ψ0 evolve in such a way that ψt is in thermal equilibrium for most times t. This is closely related to von Neumann’s quantum ergodic theorem of 1929.
In joint work with J. L. Lebowitz, C. Mastrodonato, and N. Zanghì [2, 3, 4], we considered an isolated, macroscopic quantum system. Let H be a micro‐canonical “energy shell,” i.e., a subspace of the system’s Hilbert space spanned by the (finitely) many energy eigenstates with energies between E and E+δE. The thermal equilibrium macro‐state at energy E corresponds to a subspace Heq of H such that dimHeq/dimH is close to 1. We say that a system with state vector ψ ε H is in thermal equilibrium if ψ is “close” to Heq. We argue that for “typical” Hamiltonians, all initial state vectors ψ0 evolve in such a way that ψt is in thermal equilibrium for most times t. This is closely related to von Neumann’s quantum ergodic theorem of 1929.
Descriptions of molecular systems usually refer to two distinct theoretical frameworks. On the one hand the quantum pure state, i.e., the wavefunction, of an isolated system is determined to calculate molecular properties and their time evolution according to the unitary Schrödinger equation. On the other hand a mixed state, i.e., a statistical density matrix, is the standard formalism to account for thermal equilibrium, as postulated in the microcanonical quantum statistics. In the present paper an alternative treatment relying on a statistical analysis of the possible wavefunctions of an isolated system is presented. In analogy with the classical ergodic theory, the time evolution of the wavefunction determines the probability distribution in the phase space pertaining to an isolated system. However, this alone cannot account for a well defined thermodynamical description of the system in the macroscopic limit, unless a suitable probability distribution for the quantum constants of motion is introduced. We present a workable formalism assuring the emergence of typical values of thermodynamic functions, such as the internal energy and the entropy, in the large size limit of the system. This allows the identification of macroscopic properties independently of the specific realization of the quantum state. A description of material systems in agreement with equilibrium thermodynamics is then derived without constraints on the physical constituents and interactions of the system. Furthermore, the canonical statistics is recovered in all generality for the reduced density matrix of a subsystem.
von Neumann has proved a quantum-mechanical ergodic theorem which deals with a closed macroscopic system and macroscopic observables. Certain unsatisfactory features of this theorem are probably due to the overly-strong idealization of a completely closed system.Since the claims of statistical mechanics go beyond macroscopic systems, there is a need for a more general theorem. The present paper deals with a system of any number of degrees of freedom and with any observable, during interaction with a temperature bath. In order to avoid an assumption of randomness, the bath must be described explicitly and precisely by a time-independent Hamiltonian. The model of the bath is obtained by taking the Gibbs ensemble seriously, i.e., as a set of N identical systems interacting through a potential $\ensuremath{\lambda}V$.The following theorem is proved. The time average of the quantum-mechanical expectation value of any observable with respect to any initial state is equal to its statistical average, in the double limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$, $\ensuremath{\lambda}\ensuremath{\rightarrow}0$, for the overwhelming majority of all interaction potentials V.