At the mesoscale, reaction networks are described in terms of stochastic processes. In well-stirred solutions, the time evolution is ruled by the chemical master equation for the probability distribution of the random numbers of molecules. The entropy production is obtained for these reactive processes in the framework of stochastic thermodynamics. The entropy production can be decomposed using the Hill–Schnakenberg cycle decomposition in terms of the affinities and the reaction rates of the stoichiometric cycles of the reaction network. The multivariate fluctuation relation is established for the reactive currents. The results are applied to several examples of reaction networks, in particular, describing autocatalytic bistability, noisy chemical clocks, enzymatic kinetics, and copolymerization processes.

The multivariate fluctuation relation is established for the full counting statistics of the energy and particle fluxes across an open quantum system in contact with several reservoirs on the basis of microreversibility The quantum version of the nonequilibrium work fluctuation relation is recovered in the presence of a single reservoir. In the long-time limit, the time-reversal symmetry relation is expressed in terms of the cumulative generating function for the full counting statistics. In systems with independent particles, the symmetry relation can be obtained in the scattering approach for the transport of bosons and fermions. The temporal disorder and its time asymmetry can be characterized by the quantum version of the entropy and coentropy per unit time. Their difference gives the thermodynamic entropy production rate. Furthermore, the stochastic approach is also considered for electron transport in quantum dots, quantum point contacts, and single-electron transistors.

The experimental observation of driven Brownian motion and an analogous electric circuit confirms that the thermodynamic entropy production can be measured using the probabilities of the paths and their time reversal, i.e., from time asymmetry in temporal disorder. In this way, irreversibility is observed down to the nanometric scale in the position of the driven Brownian particle and a few thousand electron charges in the driven electric circuit. In addition, underdamped and overdamped driven Langevin processes are shown to obey the fluctuation relation and its consequences are discussed. The following examples are considered: a particle moving in a periodic potential and driven by an external force, a driven noisy pendulum, a driven noisy Josephson tunneling junction, the stochastic motion of a charged particle in electric and magnetic fields, and heat transport driven by thermal reservoirs.

Starting from the principles of fluctuating chemohydrodynamics, several nonequilibrium systems are investigated in order to deduce fluctuation relations for particle transport, reactive events, and electric currents with the methods presented in the previous chapters. Moreover, finite-time fluctuation theorems are obtained for stochastic processes with rates linearly depending on the random variables. In this way, fluctuation relations can be established for transport by diffusion, diffusion-influenced surface reactions, ion transport, diodes, transistors, and Brownian motion ruled by the generalized Langevin equation deduced from fluctuating hydrodynamics.

The stroboscopic observation of stochastic processes records the history of the system as paths, which can be characterized by their probability distribution. Temporal disorder results in the exponential decay of the path probabilities as the observational time increases. The mean decay rate defines the so-called entropy per unit time, which measures the amount of temporal disorder in the process. At equilibrium, the probabilities of a path and its time reversal are equal by the principle of detailed balance. In contrast, they differ under nonequilibrium conditions, which is the manifestation of irreversibility. Remarkably, the ratio of the probabilities of opposite paths has a logarithm obeying a fluctuation relation and having a mean value related to the thermodynamic entropy production rate. These results show that temporal ordering can be generated in nonequilibrium processes as a corollary of the second law. These considerations shed new light on Landauer’s principle.