Emergent second law for non-equilibrium steady states

Abstract

Abstract 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 $${{{{{{{\mathcal{I}}}}}}}}(x)=-\!\log ({P}_{{{{{{{{\rm{ss}}}}}}}}}(x))$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>I</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mo>−</mml:mo> <mml:mspace/> <mml:mi>log</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>ss</mml:mi> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> 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 $${{\Delta }}{{{{{{{\mathcal{I}}}}}}}}={{{{{{{\mathcal{I}}}}}}}}({x}_{t})-{{{{{{{\mathcal{I}}}}}}}}({x}_{0})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Δ</mml:mi> <mml:mi>I</mml:mi> <mml:mo>=</mml:mo> <mml:mi>I</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>t</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>−</mml:mo> <mml:mi>I</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> in steady state self-information along deterministic trajectories can be bounded by the macroscopic entropy production Σ. This bound takes the form of an emergent second law $${{\Sigma }}+{k}_{b}{{\Delta }}{{{{{{{\mathcal{I}}}}}}}}\,\ge \,0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Σ</mml:mi> <mml:mo>+</mml:mo> <mml:msub> <mml:mrow> <mml:mi>k</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>b</mml:mi> </mml:mrow> </mml:msub> <mml:mi>Δ</mml:mi> <mml:mi>I</mml:mi> <mml:mspace/> <mml:mo>≥</mml:mo> <mml:mspace/> <mml:mn>0</mml:mn> </mml:math> , which contains the usual second law Σ ≥ 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.

Topics & Concepts