# Open Systems’ Density Matrix Properties in a Time Coarsened Formalism

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## Abstract

The concept of time-coarsened density matrix for open systems has frequently featured in equilibrium and non-equilibrium statistical mechanics, without being probed as to the detailed consequences of the time averaging procedure. In this work we introduce and prove the need for a selective and non-uniform time-sampling, whose form depends on the properties (whether thermalized or not) of the bath. It is also applicable when an open microscopic sub-system is coupled to another *finite* system. By use of a time-periodic minimal coupling model between these two systems, we present detailed quantitative consequences of time coarsening, which include initial state independence of equilibration, deviations from long term averages, their environment size dependence and the approach to classicality, as measured by a Leggett–Garg type inequality. An interacting multiple qubit model affords comparison between the time integrating procedure and the more conventional environment tracing method.

## Keywords

Foundations of quantum mechanics Density matrix Time coarsened formalism## Notes

### Acknowledgments

Thanks are due to Rafael Ruppin for substantial help in the calculations and to Ronnie Kosloff for a discussion. The article has benefited from the constructive comments of two referees, who also called our attention to several important references.

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