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.

An isolated system is described by a classical Hamiltonian dynamics. In the long-time limit, the trajectory of such a system yields a histogram, i.e., a distribution for any observable. With one plausible assumption, introduced here as a fundamental principle, this histogram is shown to lead to the microcanonical distribution. Pressure, temperature, and chemical potential can then be identified microscopically. This dynamical approach thus recovers the results that are often obtained for equilibrium by minimizing a postulated entropy function.

For time-dependent driving, the key concepts of time-reversed and backward protocols are introduced. The reversibility of Hamiltonian dynamics is shown to imply that work is antisymmetric with respect to time-reversal. Integral fluctuation relations are introduced as a general property of certain distributions. For the work distributions, this yields the Jarzynski relation, which expresses free-energy differences as a particular nonlinear average over nonequilibrium work. Various limiting cases such as slow driving and the apparent counterexample of free expansion of a gas are discussed. The Bochkov–Kuzovlev relation is shown to be another variant of such an integral fluctuation relation. The Crooks fluctuation relation yields a symmetry of the work distributions for a forward and a backward process. As an important application, free energy differences and a free energy landscape based on exploiting the Hummer–Szabo relation are recovered as illustrated with experimental data for the unfolding of biopolymers.

The asymmetric random walk is introduced as a simple model for a molecular motor. Thermodynamic consistency imposes a condition on the ratio between the forward and the backward rate. Fluctuations in finite time can be derived analytically and are used to illustrate the thermodynamic uncertainty relation. For the long-time limit, concepts from large deviation theory like a rate function and a contraction can be determined explicitly.

For many systems, the full information of an underlying Markovian decription is not accessible due to limited spatial or temporal resolution. We first show that such an often inevitable coarse-graining implies that, rather than the full entropy production, only a lower bound can be retrieved from coarse-grained data. As a technical tool, it is derived that the Kullback–Leibler divergence decreases under coarse-graining. For a discrete time-series obtained from an underlying time-continuous Markov dynamics, it is shown how the analysis of n-tuples leads to a better estimate with increasing length of the tuples. Finally, state-lumping as one strategy for coarse-graining an underlying Markov model is shown explicitly to yield a lower bound for the entropy production. However, in general, it does not yield a consistent interpretation of the first law along coarse-grained trajectories as exemplified with a simple model.

The Maxwell demon and the Szilard engine demonstrate that work can be extracted from a heat bath through measurement and feedback in apparent violation of the second law. A systematic analysis shows that, by including the measurement process and the subsequent erasure of a memory according to Landauer’s principle, the second law is indeed restored. For such feedback-driven processes, the Sagawa–Ueda relation provides a generalization of the Jarzynski relation. For the general class of bipartite systems, the concepts from stochastic thermodynamics are developed. This framework applies to systems where one component “learns” about the changing state of the other one, as in simple models for bacterial sensing. The chapter closes with a simple information machine that shows how the ordered sequence of bits in a tape can be used to transform heat into mechanical work. Likewise, mechanical work can be used to erase information, i.e., randomize such a tape. These processes are shown to obey a second law of information processing.

The overdamped Langevin equation for a particle in a potential and, possibly, subject to a nonconservative force is introduced. The corresponding Fokker–Planck equation, the Smoluchowski equation, is derived. In a time-independent potential, any initial distribution finally approaches the equilibrium one. For a constant external force and periodic boundary condition like the motion along a ring, a nonequilibrium steady state is established. As an application, the Kramers escape from a meta-stable well can be discussed. The mean local velocity and the path integral representation are introduced. Thermodynamic quantities like work, heat, and entropy production are identified along individual trajectories and their ensemble averages are determined. Their distributions are shown to obey detailed fluctuation relations. A master integral fluctuation relation can be specialized to yield inter alia the Jarzynski relation, the integral fluctuation relation for entropy production, and the Hatano–Sasa relation.