Irwin Oppenheim Award Talk: The Thermodynamic Uncertainty Relation: Theoretical Introduction

In thermodynamic equilibrium, molecular motions cannot generate a non-vanishing average current—of particles, mass, charge, etc. This fact is elegantly seen as a consequence of the time-reversible nature of equilibrium dynamics, which obeys detailed balance. It is well known that detailed balance can be broken by external driving, either with a time-dependent protocol (as in flashing ratchets) or by coupling a system to multiple incommensurate reservoirs (say a high-chemical-potential bath on one side of a membrane and a low-chemical-potential bath on the other side). In this latter case, the system coupled to the baths experiences a time-independent thermodynamic driving force causing the system to relax into a nonequilibrium steady state (NESS) that generates current. Since the current is made up of individual transport events, it will fluctuate around the steady state value, and the scale of these fluctuations is constrained by the typical rate of dissipation into the reservoirs, a constraint which has been labeled a Thermodynamic Uncertainty Relation (TUR). In this talk I will outline this TUR and illustrate the large-deviation-theoretic arguments underlying its derivation.