This chapter deals with advanced topics for a multivariate Langevin and Fokker–Planck dynamics. For systems with multiplicative noise it is shown that neither the drift term in the Langevin equation nor the discretization parameter can be determined uniquely. If one of the two is fixed, the other one is determined. In contrast, the Fokker–Planck equation, which contains the physically observable distribution is unique. Experimental data for a particle near a wall illustrate the relevance of space-dependent friction. Martingales are introduced for a Langevin dynamics with a nonlinear expression of entropy production as a prominent example that with Doob’s optimal stopping theorem leads to universal results of its statistics. Finally, underdamped Langevin dynamics is described by the Klein–Kramers equation, for which entropy production is determined by the irreversible currents. A multi-time-scale analysis recovers the Smoluchowski equation in the overdamped limit even in the presence of an inhomogeneous temperature for which an anomalous contribution to entropy production is found.

Optimal protocols transform a given initial distribution into a given final one in finite time with a minimal amount of work or entropy production. We first analyze this optimization paradigmatically for a driven harmonic oscillator for which analytical results can be obtained. For a general Langevin dynamics, it is shown that the optimal protocol can be realized through a time-dependent potential with no need to use a nonconservative force. In contrast for discrete systems, nonconservative driving decreases the thermodynamic costs. For a broader perspective, we introduce concepts from information geometry which deals with the statistical manifold of distributions. The Fisher information provides a metric on this manifold from which the distance between two distributions as the minimal length connecting them can be derived. Speed limits yield relations between these quantities referring to an initial and a final distribution and the entropy production associated with the transformation of the former into the latter. For slow processes, cost along the optimal protocol or path is bounded by the distance between these distributions and the inverse of the allocated time.

This chapter deals with processes both from a macroscopic, thermodynamic point of view and from a dynamical perspective. For the latter, a class of processes is introduced that can be described through a Hamiltonian description with a time-dependent external control parameter. It is shown how the expressions of work and heat from classical thermodynamics can be obtained as an appropriate average over an initial distribution. The second law inequality relating work and free energy can then be proven as a consequence of a master inequality. With well-specified additional assumptions, second law inequalities for heat exchange and entropy production are derived.

Systems with a discrete set of mesostates and their canonical description in equilibrium are introduced. Observing trajectories in equilibrium yields the thermodynamic potentials of these mesostates. Time-scale separation allows one to describe the dynamics using a Markovian master equation. The ratio of transition rates is constrained by the free energy difference of the corresponding mesostates. First for relaxation and then for time-dependent driving, work, heat, and internal energy are identified along individual trajectories. Entropy production along such a trajectory is shown to contain three contributions given by the dissipated heat and the change in internal entropy and in stochastic entropy. The distributions of these thermodynamic quantities obey various exact fluctuation relations. For entropy production, the relation to the arrow of time and a putative identification within a Hamiltonian dynamics is discussed.

Superconductivity is a quantum state of matter that occurs through a phase transition driven by thermal fluctuations. In this state, materials show ideal electric conductivity and ideal diamagnetism to a very good approximation. Two main classes of superconductors, type I and type II, can be distinguished with regards to flux penetration under an applied magnetic field. The properties of these two types are first discussed in detail. Next, the Ginzburg–Landau theory is developed and it is shown that in the presence of a magnetic field, when the ratio of penetration and coherence lengths is smaller than 1⁄√2 the superconductor behaves as type I, while it behaves as type II when this ratio is larger than 1⁄√2. In this second case, the flux penetrates through vortices that form a hexagonal lattice. Finally, in the last part, the microscopic BCS theory is discussed in order to provide an understanding of the physical origin of superconductivity.

The chapter is an introduction to basic equilibrium aspects of phase transitions. It starts by reviewing thermodynamics and the thermodynamic description of phase transitions. Next, lattice models, such as the paradigmatic Ising model, are introduced as simple physical models that permit a mechano-statistical study of phase transitions from a more microscopic point of view. It is shown that the Ising model can quite faithfully describe many different systems after suitable interpretation of the lattice variables. Special emphasis is placed on the mean-field concept and the mean-field approximations. The deformable Ising model is then studied as an example that illustrates the interplay of different degrees of freedom. Subsequently, the Landau theory of phase transitions is introduced for continuous and first-order transitions, as well as critical and tricritical behaviour are analysed. Finally, scaling theories and the notion of universality within the framework of the renormalization group are briefly discussed.

The chapter starts by introducing the basic concepts of metastable and unstable states as well as time scales that control the occurrence of phase transitions. The limits for phase transitions taking place in equilibrium and out-of-equilibrium conditions are then established. In the latter case, thermally activated and athermal limits are distinguished associated with those situations where the transition is either driven or not driven by thermal fluctuations, respectively. Then the formal theory of the decay of metastable and unstable states in systems with conserved and non-conserved order parameters is developed. This general theory is in turn applied to the study of homogeneous and heterogeneous nucleation, spinodal decomposition and late stages of coarsening and domain growth.

The general concept of multiferroic materials as those with strong interplay between two or more ferroic properties is first introduced. Then, particular cases of materials with coupling magnetic and polar (magnetoelectric coupling), polar and structural (electrostructural coupling), and magnetic and structural (magnetostructural coupling) degrees of freedom are discussed in more detail. The physical origin of the interplay is analysed and symmetry-based considerations are used to determine the dominant coupling terms adequate to construct extended Ginzburg–Landau models that permit the determination of cross-response to multiple fields. The last part of the chapter is devoted to study morphotropic systems and morphotropic phase boundaries that separate crystallographic phases with different polar (magnetic) properties as examples of materials with electro(magneto)-structural interplay and that are expected to show giant cross-response to electric (magnetic) and mechanical fields.

Non-equilibrium phase transitions are non-thermal transitions that occur out-of-equilibrium. The chapter first discusses systems that are subjected and respond with hysteresis to an oscillating field due to a competition between driving and relaxation time scales. When the former is much shorter than the latter, a non-equilibrium transition occurs associated with the dynamical symmetry breaking due to hysteresis. A dynamical magnetic model is introduced and it is shown that the mean magnetization in a full cycle is the adequate order parameter for this transition. A mean-field solution predicting first-order, critical and tricritical behaviours is analysed in detail. The second example refers to externally driven disordered systems that respond intermittently through avalanches. The interesting aspect is that for a critical amount of disorder, avalanches occur with an absence of characteristic scales, which define avalanche criticality as reported in different ferroic materials. This behaviour can be accounted for by lattice models with disorder, driven by athermal dynamics.