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Journal Article

Relating the Thermodynamic Arrow of Time to the Causal Arrow

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Allahverdyan, A., & Janzing, D. (2008). Relating the Thermodynamic Arrow of Time to the Causal Arrow. Journal of Statistical Mechanics: Theory and Experiment, 2008: P04001, pp. 1-21. doi:10.1088/1742-5468/2008/04/P04001.

Cite as: https://hdl.handle.net/11858/00-001M-0000-0013-C9CF-B
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 microcanonical distributions P(S) and P(S|F). Their structure supports a causal inference principle proposed recently in machine learning.