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Journal of Mathematical Physics

On ergodic limits of normal states in quantum statistical mechanics

Eduard Prugovečki, A. Tip

Published Mar 1, 1974• 7 citations
Publisher

Abstract

The asymptotic behavior for t→±∞ of S(t)=exp(−i H t)Sexp(+i H t) and its time average S̄ (t)=t−1∫0tdu S(u) is discussed. Here is S an element of the Banach space B1, constituted by the trace class of operators on the (separable or nonseparable) Hilbert space H, and H is the Hamiltonian, i.e., a bounded or unbounded self-adjoint operator on H. Necessary and sufficient conditions are given for the existence of the limits S̄(± ∞) and S(± ∞) with respect to the weak topology on B1, for the latter under the assumption that the continuous spectrum of H is absolutely continuous. In addition it is shown that if, for a normal state (density operator) ρ, ρ̄(t) has a weak limit, then the limit is again a normal state. This provides further insight in the nature of Emch's ``first ergodic paradox'' [G. G. Emch, J. Math. Phys. 7, 1413 (1966)].

DOI
10.1063/1.1666637

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The asymptotic behavior for t→±∞ of S(t)=exp(−i H t)Sexp(+i H t) and its time average S̄ (t)=t−1∫0tdu S(u) is discussed. Here is S an element of the Banach space B1, constituted by the trace class of operators on the (separable or nonseparable) Hilbert space H, and H is the Hamiltonian, i.e., a bounded or unbounded self-adjoint operator on H. Necessary and sufficient conditions are given for the existence of the limits S̄(± ∞) and S(± ∞) with respect to the weak topology on B1, for the latter under the assumption that the continuous spectrum of H is absolutely continuous. In addition it is shown that if, for a normal state (density operator) ρ, ρ̄(t) has a weak limit, then the limit is again a normal state. This provides further insight in the nature of Emch's ``first ergodic paradox'' [G. G. Emch, J. Math. Phys. 7, 1413 (1966)].
The algebraic formulation of quantum statistical mechanics is extended so as to include local unbounded observables. We start, in a usual way, with a C*-algebra U of quasilocal bounded observables in Fock space HF, an arbitrary, locally normal state φ on U, and a corresponding Gel'fand-Naimark-Segal (GNS) representation Rφ of U in a Hilbert space Hφ. We then construct a set QL of local, closed operators whose domains are dense in HF, such that QL includes all the local observables of the system. The representation Rφ is then extended so as to provide a *-homomorphism of HL into the closed, densely defined operators in Hφ. Correspondingly, a number of results previously established for the local bounded observables are extended to the unbounded ones. For appropriate classes of locally normal states, these extended results include the Kubo-Martin-Schwinger boundary conditions, the spatially asymptotic and ergodic properties of space-correlation functions, and the temporally ergodic properties of time-correlation functions. It is also shown that, for locally normal Gibbs states, the time correlations between elements of a specified subset of QL are thermodynamical limits of the corresponding correlations for finite systems.
Abstract We rigorously examine, in generality, the ergodic properties of quantum lattice models with short range interactions, in the C∗C^*C∗ C ∗ algebra formulation of statistical mechanics. Ergodicity results, in the context of group actions on C∗C^*C∗ C ∗ algebras, assume that the algebra is asymptotically abelian, which is not generically the case for time evolution. The Lieb-Robinson bound tells us that, in a precise sense, the spatial extent of any time-evolved local operator grows linearly with time. This means that the algebra of observables is asymptotically abelian in a space-like region, and implies a form of ergodicity outside the light-cone. But what happens within it? We show that the long-time limit of the n-th moment of a ray-averaged observable, along space-time rays of almost every speed, converges to the n-th power of its expectation in the state (i.e. its ensemble average). Thus ray averages do not fluctuate in the long time limit. This is a statement of ergodicity, and holds in any state that is invariant under space-time translations and that satisfies weak clustering properties in space. The ray averages can be performed in a way that accounts for oscillations, showing that ray-averaged observables cannot sustainably oscillate in the long time limit. We also show that in the GNS representation of the algebra of observables, for any KMS state with the above properties, the long-time limit of the ray average of any observable converges (in the strong operator topology) to the ensemble average times the identity, again along space-time rays of almost every speed. This is a strong version of ergodicity, and indicates that, as operators, observables get “thinner” almost everywhere within the light-cone. A similar statement holds under oscillatory averaging.
An ergodic theorem is a statement of the equality of the time averages of the physical properties of a system and the averages of these quantities obtained from the consideration of a Gibbsian ensemble of identical systems. It is shown that, in quantum statistical mechanics, a necessary and sufficient condition for the existence of an ergodic theorem can be derived for a system in weak energetic interaction with its surroundings. This necessary and sufficient condition for the equality of time and (microcanonical) ensemble averages is that all operators invariant in time shall reduce to constant multiples of the unit operator when applied to the system itself. It is shown that the conditions for ergodicity are formally analogous to those derived by von Neumann for classical statistical mechanics. No assumption of "molecular chaos" is made.
Despite its long history, a canonical formulation of quantum ergodicity that applies to general classes of quantum dynamics, including driven systems, has not been fully established. Here we introduce and study a notion of quantum ergodicity for closed systems with time-dependent Hamiltonians, defined as statistical randomness exhibited in their longtime dynamics. Concretely, we consider the temporal ensemble of quantum states (time-evolution operators) generated by the evolution, and investigate the conditions necessary for them to be statistically indistinguishable from uniformly random states (operators) in the Hilbert space (space of unitaries). We find that the number of driving frequencies underlying the Hamiltonian needs to be sufficiently large for this to occur. Conversely, we show that statistical —indistinguishability up to some large but finite moment—can already be achieved by a quantum system driven with a single frequency, i.e., a Floquet system, as long as the driving period is sufficiently long. Our work relates the complexity of a time-dependent Hamiltonian and that of the resulting quantum dynamics, and offers a fresh perspective to the established topics of quantum ergodicity and chaos from the lens of quantum information. Published by the American Physical Society 2024
Abstract Random walks in a real separable Hilbert space correspond to a composition of random unitary operators in the space of functions square integrable with respect to a translation-invariant measure, as well as to a composition of random unitary channels in the space of quantum states. Conditions on the distribution of random walks that are necessary and sufficient for the continuity of the mathematical expectations of their representations in the Hilbert space and the space of quantum states are obtained. The dynamics of the set of quantum states obtained by averaging random shifts over the semigroup (with respect to the convolution operation) of centered Gaussian measures on the Hilbert space is studied. It is shown that, depending on the property of the covariance operator of the Gaussian measure to have a nuclear square root, such dynamics is either a strongly continuous quantum dynamical semigroup or instantaneously destroys the set of normal states. In the latter case, the dynamics of singular quantum states is described in terms of the evolution of mean values on an operator algebra that implements a representation of canonical commutation relations.