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.

Effusion is one of the most elementary kinetic processes, to which the concepts of nonequilibrium statistical mechanics can be applied. Remarkably, the stationary probability distribution can be exactly constructed as a so-called Poisson suspension, explicitly showing that time reversal is broken at the statistical level of description under nonequilibrium conditions. The multivariate fluctuation relation for the energy and particle currents can be directly deduced from the underlying microscopic dynamics. Moreover, temporal disorder and its nonequilibrium time asymmetry can be fully characterized and shown to be related to the thermodynamic entropy production. The multivariate fluctuation relation can also be applied to mass separation by effusion.

Hydrodynamics is deduced from the microscopic dynamics using local equilibrium probability distributions for multicomponent normal fluids and the phases of matter with broken continuous symmetries such as crystals and liquid crystals. The Nambu–Goldstone modes resulting from continuous symmetry breaking are identified at the microscopic level of description. The entropy and the entropy production are introduced within the local equilibrium approach in agreement with the second law of thermodynamics. The Green–Kubo formulas are obtained for all the transport coefficients associated with the linear response properties, including the cross-coupling effects satisfying the Onsager–Casimir reciprocal relations as a consequence of microreversibility. The boundary conditions due to the presence of interfaces are discussed, as well as the hydrodynamic long-time tails and their consequences, especially, in low-dimensional systems.