  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.
