At the microscale, the motion of atoms and molecules composing matter is governed by Hamiltonian dynamics. For classical systems, this motion is described as trajectories in the phase space of the positions and momenta of the particles. Different equilibrium and nonequilibrium statistical ensembles can be introduced, each associated with some probability distribution, which is a solution of Liouville’s equation. The BBGKY hierarchy of equations is obtained for the multiparticle distribution functions. The presentation includes the properties of ergodicity and dynamical mixing, the Pollicott–Ruelle resonances, microreversibility, and the nonequilibrium breaking of time-reversal symmetry at the statistical level of description. The concept of entropy is introduced by coarse graining. Linear response theory is developed within the classical framework, leading to the Onsager–Casimir reciprocal relations and the fluctuation–dissipation theorem. The projection-operator methods are summarized.

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 overview of the principles of quantum statistical mechanics are given, emphasizing the fundamental differences with respect to classical statistical mechanics, as well as the analogies prevailing for the formulation of the properties. A functional time-reversal symmetry relation is presented, allowing the deduction of response theory. The Kubo formula is obtained for the linear response properties and the fluctuation–dissipation theorem is established. For weakly coupled systems, the quantum master equation and the corresponding stochastic Schrödinger equation are deduced. The slippage of initial conditions is discussed in relation to the positivity of the reduced statistical operator. The results are illustrated with the spin-boson model.

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.

At the mesoscale, the fluctuating phenomena are described using the theory of stochastic processes. Depending on the random variables, different stochastic processes can be defined. The properties of stationarity, reversibility, and Markovianity are defined and discussed. The classes of discrete- and continuous-state Markov processes are presented including their master equation, their spectral theory, and their reversibility condition. For discrete-state Markov processes, the entropy production is deduced and the network theory is developed, allowing us to obtain the affinities on the basis of the Hill–Schnakenberg cycle decomposition. Continuous-state Markov processes are described by their master equation, as well as stochastic differential equations. The spectral theory is also considered in the weak-noise limit. Furthermore, Langevin stochastic processes are presented in particular for Brownian motion and their deduction is carried out from the underlying microscopic dynamics.

Introduction, synopsis, and acknowledgments.

The fluctuations of energy and particle fluxes obey remarkable symmetries called fluctuation relations, which are valid arbitrarily far from equilibrium and find their origin in microreversibility. Yet they imply the nonnegativity of entropy production in accord with the second law of thermodynamics. They express the directionality of nonequilibrium processes, reducing at equilibrium to the conditions of detailed balance. The nonequilibrium work fluctuation relation and Jarzynski’s equality are presented in the absence and the presence of a magnetizing field and also for joint angular momentum transfer. Moreover, the multivariate fluctuation relation for all the fluxes across an open system in contact with several reservoirs is deduced from both the classical Hamiltonian microdynamics and the theory of stochastic processes. The multivariate fluctuation relation implies not only the fluctuation–dissipation theorem and the Onsager–Casimir reciprocal relations close to equilibrium but also their generalizations to the nonlinear response properties of relevance farther away from equilibrium.

Active nonequilibrium processes are characterized by the coupling of an ionic current or a mechanical motion to a chemical reaction. This coupling induces energy transduction, satisfying Onsager reciprocal relations in the linear regime close to equilibrium and the bivariate fluctuation relation in regimes farther away from equilibrium. These considerations concern, in particular, molecular motors as well as active colloidal particles that are self-propelled by chemical reactions catalyzed at their surface and diffusiophoresis. These active processes can be described by stochastic processes obeying bivariate fluctuation relations for the coupled currents. The mechanochemical coupling can be characterized in terms of the linear and nonlinear response coefficients, as well as the efficiencies defined in the different regimes of energy transduction, i.e., propulsion by the chemical reaction on the one hand, and the synthesis of fuel from products on the other hand.

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.

At the macroscale, thermodynamics rules the balances of energy and entropy. In nonisolated systems, the entropy changes due to the contributions from the internal entropy production, which is always nonnegative according to the second law, and the exchange of entropy with the environment. The entropy production is equal to zero at equilibrium and positive out of equilibrium. Thermodynamics can be formulated either locally for continuous media or globally for systems in contact with several reservoirs. Accordingly, the entropy production is expressed in terms of either the local or the global affinities and currents, the affinities being the thermodynamic forces driving the system away from equilibrium. Depending on the boundary and initial conditions, the system can undergo relaxation towards equilibrium or nonequilibrium stationary or time-dependent macrostates. As examples, thermodynamics is applied to diffusion, electric circuits, reaction networks, and engines.

Boltzmann’s equation ruling the time evolution of the one-particle distribution function is obtained by partitioning the phase space into the free-flight and collision domains in low-density gases. The expressions for the entropy production and the entropy exchange are related to the H-theorem. The transport properties and gas-surface interactions are discussed. Furthermore, the multivariate fluctuation relation for the energy and particle fluxes is deduced from the fluctuating Boltzmann equation. In addition, an integral fluctuation relation is established for the Boltzmann factorization of the multiparticle probability density into one-particle distribution functions.

The mathematical foundations of transport properties are analyzed in detail in several Hamiltonian dynamical models. Deterministic diffusion is studied in the multibaker map and the Lorentz gases where a point particle moves in a two-dimensional lattice of hard disks or Yukawa potentials. In these chaotic models, the diffusive modes are constructed as the eigenmodes of the Liouvillian dynamics associated with Pollicott–Ruelle resonances. These eigenmodes are distributions with a fractal cumulative function. As a consequence of this fractal character, the entropy production calculated by coarse graining has the expression expected for diffusion in nonequilibrium thermodynamics. Furthermore, Fourier’s law for heat conduction is shown to hold in many-particle billiard models, where heat conductivity can be evaluated with very high accuracy at a conductor-insulator transition. Finally, mechanothermal coupling is illustrated with models for motors propelled by a temperature difference.

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.