Exact Identity Linking Entropy Production and Mutual Information

Abstract

We establish an exact identity for overdamped Langevin dynamics: the total entropy production rate equals four times the mutual information rate between an infinitesimal displacement and its actual temporal midpoint, plus a mean-flow term. This provides a forward information-theoretic representation of irreversibility and makes the Shannon chain rule act directly on dissipation. For additive bipartitions, the chain rule reveals nonnegative self and interaction channels, with the self channel coinciding with apparent entropy production. This channel structure sharpens the learning-rate bound by showing that learning is constrained solely by the interaction channel. The same structure is also shown to resolve a recently reported coarse-graining reversal in hydrodynamically coupled colloids. Together, these results exhibit that, within this channel decomposition, learning and coarse-graining probe complementary parts of dissipation in overdamped Langevin systems.

45 pages, 16 figures (Main text: 7 pages, 5 figures; SM: 38 pages, 11 figures) Substantially revised; strengthened framing and consequences of the midpoint-information representation