Relating the thermodynamic arrow of time to the causal arrow

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 itsthermodynamics 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 andthe back-action is negligible. This is because the back-action of F on S is described by anon-globally Hamiltonian Born–Oppenheimer term that violates the Liouvilletheorem, and makes the second law inapplicable to S. If S and F are mixing, underthe causal arrow condition they are described by microcanonical distributionsP(S) and P(S|F). Their structure supports a causal inference principle proposed recently in machinelearning.