Thermodynamic bounds on current fluctuations

Field: Physics and Astronomy (Physical Sciences)

Living systems, as well as useful artificial machines, operate under non-equilibrium conditions. This means that they are in contact with several reservoirs that are not in mutual thermodynamic equilibrium. These reservoirs provide resources such as food or fuel, or act as a thermal environment absorbing heat. Provided that the system under consideration is sufficiently small compared to the reservoirs, the state of the reservoirs will change negligibly on relevant time scales. If additionally the system is not manipulated externally, its dynamics becomes time-invariant, which is called a non-equilibrium steady state (NESS). Due to the thermal influence from its environment, the state of the system becomes erratic, which allows us to model its dynamics as a stochastic process, for which one can define several thermodynamic observables. Of particular interest throughout this Thesis are the input- and output-currents associated with a NESS. Examples for such currents include the consumption or production of a specific chemical species or the work associated with lifting a weight. A current of particular thermodynamic importance is the production of entropy in the total system, which quantifies its non-equilibrium character. In stark contrast to equilibrium systems, non-equilibrium systems are capable of maintaining non-zero average currents. In particular, the rate of entropy production is, due to the second law of thermodynamics, always greater than zero on average. However, again due to the thermal influence from the environment, the temporal evolution of these currents is superimposed by fluctuations. This means, that on short time scales, currents can deviate from the average intensity, and the entropy production can even become negative. The main objective of the work documented in this Thesis is to provide a comprehensive characterization of the statistics of current fluctuations. While an exact calculation of these statistical properties is possible, the results typically depend on all microscopic details of the system and on the driving forces associated with the reservoirs. Since such detailed information is practically neither available nor relevant, we focus on the derivation of bounds on the statistics of current fluctuations, which ideally depend on only a few thermodynamic properties of the system. Starting point for our work is a prominent inequality known as the “thermodynamic uncertainty relation” [A.C. Barato and U. Seifert, Phys. Rev. Lett. 114, 158101 (2015)]. It considers the uncertainty of a current, comparing the amplitude of its fluctuations to its mean, as a statistical measure and on the other hand the average rate of entropy production as a thermodynamic measure. The product of these two key quantities must always be greater than two, expressing a trade-off between precision and the thermodynamic cost for a non-equilibrium process. It holds for any current and for the huge class of systems that can be described in terms of Markov processes. We put this relation in a wider mathematical context, employing large deviation theory to derive it as a result of an equally general bound on the whole spectrum of current fluctuations. Our formalism allows for several refinements and generalizations of that bound and yields complementary, novel bounds on current fluctuations.