  The Gibbs distribution universally characterizes states of thermal
equilibrium. In order to extend the Gibbs distribution to non-equilibrium
steady states, one must relate the self-information $\mathcal{I}(x) =
-\log(P_\text{ss}(x))$ of microstate $x$ to measurable physical quantities.
This is a central problem in non-equilibrium statistical physics. By
considering open systems described by stochastic dynamics which become
deterministic in the macroscopic limit, we show that changes $\Delta
\mathcal{I} = \mathcal{I}(x_t) - \mathcal{I}(x_0)$ in steady state
self-information along deterministic trajectories can be bounded by the
macroscopic entropy production $\Sigma$. This bound takes the form of an
emergent second law $\Sigma + k_b \Delta \mathcal{I}\geq 0$, which contains the
usual second law $\Sigma \geq 0$ as a corollary, and is saturated in the linear
regime close to equilibrium. We thus obtain a tighter version of the second law
of thermodynamics that provides a link between the deterministic relaxation of
a system and the non-equilibrium fluctuations at steady state. In addition to
its fundamental value, our result leads to novel methods for computing
non-equilibrium distributions, providing a deterministic alternative to
Gillespie simulations or spectral methods.
