Nonequilibrium steady states arise if a system is driven in a time-independent way. This can be realized through contact with particle reservoirs at different (electro)chemical potential for enzymatic reactions and for transport through quantum dot structures. For molecular motors, an applied external force contributes to such an external driving. Formally, such systems are described by a master equation with time-independent transition rates that are constrained by the local detailed balance relation. Characteristic of such systems are persistent probability currents. This stationary state is unique and can be obtained either through a graph-theoretic method or as an eigenvector of the generator. These systems have a constant rate of entropy production. Moreover, this entropy production fulfills a detailed fluctuation theorem. The thermodynamic uncertainty relation provides a lower bound on entropy production in terms of the mean and dispersion of any current in the system. An important classification distinguishes unicyclic from multicyclic systems. In particular for the latter, the concept of cycles and their affinities are introduced and related to macroscopic or physical affinities driving an engine. In the linear response regime, Onsager coefficients are proven to obey a symmetry.

This chapter starts with a discussion of simple univariate chemical reactions networks emphasizing the need to impose thermodynamically consistent reaction rates. For a linear reaction scheme, the stationary distribution is given analytically as a Poisson distribution. Nonlinear schemes can lead to bistability. For large systems, the stationary solution can be expressed by an effective potential. Two types of Fokker–Planck descriptions are shown to fail in certain regimes. In the thermodynamic limit, the dynamics can be described by a simple rate equation. Entropy production is discussed on the various levels of description. A simple two-dimensional scheme, the Brusselator, can lead to persistent oscillations. Heat and entropy production are identified for an individual reaction event of a general multivariate reaction scheme.

Rare or extreme fluctuations beyond the Gaussian regime are treated through large deviation theory for the nonequilibrium steady state of discrete systems and of systems with Langevin dynamics. For both classes, we first develop the spectral approach that yields the scaled cumulant-generating function for state observables and currents in terms of the largest eigenvalue of the tilted generator. Second, we introduce the rate function of level 2.5 that can be determined exactly. Contractions then lead to bounds on the rate function for state observables or currents. Specialized to equilibrium, explicit results are obtained. As a general result, the rate function for any current is shown to be bounded by a quadratic function which implies the thermodynamic uncertainty relation.

The efficiency of classical heat engines is bounded by the Carnot efficiency leading to vanishing power. Efficiency at maximum power is often related to the Curzon–Ahlborn efficiency. As a paradigm for a periodic stochastic heat engine, a Brownian particle in a harmonic potential is sequentially coupled to two heat baths. For a simple steady-state heat engine, a two-state model coupled permanently to two heat baths leads to transport against an external force or against an imposed electrochemical potential. Affinities and Onsager coefficients in the linear response regime are determined. The identification of exchanged heat in the presence of particle transport is shown to be somewhat ambiguous.

Molecular motors that transform chemical energy into mechanical motion can be modeled in different ways. Thermodynamically consistent ratchet-type models lead to transport against an external force in a periodic potential that switches between different shapes. In a second class of models, the motor is described by a set of internal states that leads to discrete steps along a filamentous track. In the class of hybrid models, the motor cycles through internal states while pulling a cargo particle that follows a Langevin dynamics. For these motors, the first and second laws and the thermodynamic efficiency are discussed and illustrated with experimental data for a rotary motor, the F1-ATPase.

Active particles self-propel through some intrinsic mechanism. First, a simple one-dimensional model is introduced for which the density profile between confining walls and the pressure exerted on these walls can be calculated analytically. In three dimensions, run-and-tumble particles, active Brownian particles, and active Ornstein–Uhlenbeck particles constitute three classes of models that can be described by Langevin equations. The identification of entropy production in the steady state is shown to be ambiguous. The continuum limit of a thermodynamically consistent discrete model shows that Langevin descriptions contain some implicit coarse-graining which prevents the recovery of the full physical entropy production.

This chapter deals with correlation and response functions in equilibrium and in nonequilibrium steady states for a Langevin dynamics. First, the harmonic oscillator in equilibrium is discussed as a paradigmatic case. In the general nonlinear case, it is shown how time-derivatives in correlation functions can be replaced by state variables. The response function is derived within the path integral formalism. It can be expressed by various forms of a correlation function. One particularly transparent version restores the form of the equilibrium fluctuation-dissipation theorem for a nonequilibrium steady state. A second strategy to derive a response function starts with the perturbed Fokker–Plank operator. Causality imposes the Kramers–Kronig relations between the real and imaginary parts of the response function. Through the Harada–Sasa relation, the deviation from the equilibrium form of the fluctuation-dissipation relation can be related to the mean entropy production.

For the Brownian motion of a particle in a fluid, the Langevin equation for its momentum is introduced phenomenologically. The strength of the noise is shown to be related to friction, and, in a second step, to the diffusion coefficient. Excellent agreement with experiments on a levitated particle in gas is demonstrated. This phenomenological Langevin equation is then shown to follow from a general projection approach to the underlying Hamiltonian dynamics of the full system in the limit of an infinite mass ratio between Brownian particles and fluid molecules. For Brownian motion in liquids, additional time-scales enter that are discussed phenomenologically and illustrated with experiments.

The canonical description of an aqueous solution with an embedded enzyme is introduced. The mesostates of this enzyme comprise different conformations which are affected by binding and release of solute molecules. Thermodynamic potentials of these mesostates are identified. Heat and entropy production associated with transitions between these mesostates are determined both for a simple toy model and in the general case.

For a system in contact with a heat bath, it is shown how the distribution of any observable follows from a microcanonical description for the isolated system consisting of the system of interest and heat bath. The weak coupling approximation then leads to the standard expression for the canonical distribution. Free energy, canonical entropy, and pressure are introduced. For large systems, the equivalence of this canonical description with the microcanonical one is shown. For systems in contact with a particle reservoir, the grand-canonical distribution is derived. If the weak coupling approximation does not hold, the corrections due to strong coupling are determined. In particular, internal energy, free energy ,and entropy are identified such that the usual relations for these thermodynamic potentials hold true even in strong coupling.