Relating the thermodynamic arrow of time to the causal arrow

TL;DR

This paper demonstrates that a thermodynamic arrow of time emerges in a slow subsystem of a Hamiltonian system when a causal arrow from the slow to fast subsystem exists and back-action is negligible, linking thermodynamics and causality.

Contribution

It reveals how the thermodynamic arrow of time arises from causal relations in Hamiltonian systems with slow and fast subsystems, introducing a connection to causal inference.

Findings

Thermodynamic arrow emerges under causal arrow conditions.

Back-action described by a non-Hamiltonian term violates Liouville theorem.

Supports a causal inference principle in the context of mixing systems.

Abstract

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.