Equilibration of isolated many-body quantum systems with respect to general distinguishability measures

Abstract

We demonstrate equilibration of isolated many-body systems in the sense that, after initial transients have died out, the system behaves practically indistinguishable from a time-independent steady state, i.e., non-negligible deviations are unimaginably rare in time. Measuring the distinguishability in terms of quantum mechanical expectation values, results of this type have been previously established under increasingly weak assumptions about the initial disequilibrium, the many-body Hamiltonian, and the considered observables. Here, we further extend these results with respect to generalized distinguishability measures which fully take into account the fact that the actually observed, primary data are not expectation values but rather the probabilistic occurrence of different possible measurement outcomes.

Transportless equilibration in isolated many-body quantum systems

“…Equation (5) represents the completely general and formally exact solution of the dynamics, exhibiting the usual symmetry properties of quantum mechanics under time inversion. Moreover, the right hand side is a quasi-periodic function of t, giving rise to the well-known quantum revival effects [8]: A t á ñ r ( ) must return very close to A 0 á ñ r ( ) for certain, very rare times t. The problem of equilibration amounts to the question whether, in which sense, and under what conditions the expectation value (5) approaches some constant (time-independent) value for large t. Unless this expectation value is constant right from the beginning, which is not the case under generic (non-equilibrium) circumstances, the above mentioned revivals immediately exclude equilibration in the strict sense that (5) converges towards some well defined limit for t  ¥.On the other hand, 'practical equilibration' in the sense that (5) becomes virtually indistinguishable from a constant value for the overwhelming majority of all sufficiently large t has been demonstrated, for instance, in [9][10][11][12][13] under quite weak conditions on H, ρ(0), and A. In particular, equilibration in this sense still admits transient initial relaxation processes and is compatible with the above mentioned time inversion symmetry and quantum revival properties.…”

“…Strictly speaking, the relaxation of such an isolated system towards a steady long-time limit is immediately ruled out by the unitary time evolution and, in particular, by the well-know quantum revival effects [8]. Nevertheless, 'practical equilibration' (almost steady expectation values for the vast majority of all sufficiently large times) has been rigorously established in [9][10][11][12][13] under quite general conditions.…”

Dynamical Typicality for Initial States with a Preset Measurement Statistics of Several Commuting Observables

“…For instance, this case may be of interest for a system with a non-degenerate ground state, which is energetically separated by a gap from the first excited state and thus may exhibit an exceptionally large (macroscopic) population p n compared to all the other level populations p m /d m with m = n. The derivation of this generalization amounts to a straightforward combination of the approach in Refs. [23,24] and in the Appendix below, and is therefore omitted.…”

Dynamical Typicality Approach to Eigenstate Thermalization

“…Note that the function w(x) in (35) is normalized to unity and thus may be viewed as an eigenvalue probability distribution. Given the above assumptions (36), (37) are fulfilled, one can infer from (34), (36) the approximation νã = (a max −ã)w (38) and with (32), (37) it follows that…”