Relating the thermodynamic arrow of time to the causal arrow

Details Relating the thermodynamic arrow of time to the causal arrow Relating the thermodynamic arrow of time to the causal arrow

2007-08-08

arxiv.org/abs/0708.1175v1

Physics

Condensed Matter

Statistical Mechanics

10 pages

Scientific paper

10.1088/1742-5468/2008/04/P04001

Consider a Hamiltonian system that consists of a slow subsystem S and a fast subsystem F. The autonomous dynamics of S is driven by an effective Hamiltonian, but its thermodynamics is unexpected. We show that a well-defined thermodynamic arrow of time (second law) emerges for S whenever there is a well-defined causal arrow from S to F and the back-action is negligible. This is because the back-action of F on S is described by a non-globally Hamiltonian Born-Oppenheimer term that violates the Liouville theorem, and makes the second law inapplicable to S. If S and F are mixing, under the causal arrow condition they are described by microcanonic distributions P(S) and P(S|F). Their structure supports a causal inference principle proposed recently in machine learning.

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