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Dynamical typicality of quantum expectation values · Around

2009

Dynamical typicality of quantum expectation values

Bartsch, Christian, Gemmer, Jochen

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We show that the vast majority of all pure states featuring a common expectation value of some generic observable at a given time will yield very similar expectation values of the same observable at any later time.

This is meant to apply to Schroedinger type dynamics in high dimensional Hilbert spaces.
As a consequence individual dynamics of expectation values are then typically well described by the ensemble average.
Our approach is based on the Hilbert space average method.

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In a mathematical sense there is no relaxation since both the individual QEV’s and their average are quasiperiodic for finite dimensional systems, cf. J.V. Neumann, Z. Phys. 57, 30 (1929). However, in pertinent scenarios recurrences usually occur on timescales which are long compared to any practically relevant timescale
1929
Earlier work this paper cites.
E. Fick and G. Sauermann, The Quantum Statistics of Dynamic Processes (Springer, Berlin, 1990)
1990
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J. L. Lebowitz, Physics Today 46
1993
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P. Pechukas, Phys. Rev. Lett. 73
1994
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U. Weiss, Dissipative Quantum Systems (World Scientific, Singapore, 1999)
1999
Earlier work this paper cites.

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Then

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2006
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One may consider all below investigations as applying to a part of the Hilbert space that corresponds to a certain energy region of the system rather than to the full Hilbert space. That amounts to microcanonical scenarios of a given finite temperatur rather than infinite temperature

Cited in the paper.

S. Goldstein, J. L. Lebowitz, R. Tumulka, and N. Zanghi, J. Stat. Phys. 125

Cited in the paper.

P. Reimann, J. Stat. Phys. 132

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