Transformations among many of the diverse states of matter arise from microscopic interactions involving very many (approximately 10(23)) constituent particles and result in dramatic changes in macroscopic properties. The values of some physical parameters vanish, while others approach infinity. These changes and their evolution are strikingly similar in systems as apparently different as liquids, magnets, superconductors, ferroelectrics, and liquid crystals, which suggests that there is an underlying unity to phase transition phenomena. The basis and extent of this unity are reviewed for many-body systems in equilibrium, and analogies with instability phenomena in systems far from equilibrium (such as lasers, fluid flows, and avalanche electronic devices) are pointed out.
Transformations among many of the diverse states of matter arise from microscopic interactions involving very many (approximately 10(23)) constituent particles and result in dramatic changes in macroscopic properties. The values of some physical parameters vanish, while others approach infinity. These changes and their evolution are strikingly similar in systems as apparently different as liquids, magnets, superconductors, ferroelectrics, and liquid crystals, which suggests that there is an underlying unity to phase transition phenomena. The basis and extent of this unity are reviewed for many-body systems in equilibrium, and analogies with instability phenomena in systems far from equilibrium (such as lasers, fluid flows, and avalanche electronic devices) are pointed out.
Introduction In this chapter we provide a comprehensive and unified treatment of the physical concepts and quantities relevant to phase transitions and critical phenomena. Some of these concepts and quantities have already been discussed in the particular contexts of magnetism (Chapter 7), or liquid crystals (Chapter 9, We conclude the chapter with a brief reference to the application of renormalisation group ideas to critical phenomena. The resulting reconciliation of the predictions of classical theories with experiments and exact or quasi-exact solutions of model systems is a major triumph of twentieth-century physics. Phases and phase transitions Thermodynamics habitually deals with equilibrium systems that are uniform on the macroscopic scale – i.e., where the structure, density and composition are the same throughout the bulk of the system, which is many times greater than a single atom or molecule. Because the system is in equilibrium, intensive thermodynamic variables such as the temperature or chemical potential are also spatially uniform. Such a system is said to consist of a single phase – it is a one-phase system. Phases are what used to be called the states of matter. Solids, liquids, gases and plasmas are four commonly occurring phases – they correspond to the four elements of Aristotelian physics: earth, water, air and fire, respectively. There are, however, many other, less easily identified phases, some of which we have already encountered in this book, such as the ferromagnetic, antiferromagnetic, nematic or superconducting phases. On varying the temperature, pressure and chemical potential(s) of a system, properties such as its structure, density, composition, compressibility, specific heat, etc., in general change smoothly. Yet under certain conditions the changes may be abrupt – some of these properties may exhibit discontinuities, or even divergences, at a phase transition . Phase transitions are among the most spectacular phenomena of nature. Consider, for instance, the phase sequence of water on heating: from ice, an opaque, slippery, rock-like solid; to colourless, incompressible, liquid water, which nevertheless easily flows under gravity; to water vapour, an invisible and very low-density (greenhouse!) gas, hence of ill-defined volume.
Phase, as the term is used in thermodynamics, refers to a spatially uniform equilibrium system. Substances undergo phase transitions, changes in phase that occur upon variations of state variables. The chemical potential of substances in coexisting phases has the same value in each of the phases in which coexistence occurs. The specific heat is discontinuous at the van der Waals critical point—a second-order phase transition in the Ehrenfest classification scheme. This chapter shows that one-dimensional systems cannot support phase transitions. Just as with liquid-vapor phase transitions, exponents characterize the critical behavior of magnetic systems. Magnets and fluids are different physical systems, yet they have the same critical behavior. Through the concept of order parameter, Landau theory subsumes the phase-transition behavior of different systems into one theory. A more physical way of demonstrating the existence of a phase transition would be to calculate the order parameter.
Phase transitions are defined, and the concepts of order parameter and spontaneously broken symmetry are discussed. Simple models for magnetic phase transitions are introduced, together with some experimental examples. Critical exponents and the notion of universality are defined, and the consequences of the scaling assumptions are derived. Phase transitions and order parameters It is a fact of everyday experience that matter in thermodynamic equilibrium exists in different macroscopic phases. Indeed, it is difficult to imagine life on Earth without all three phases of water. A typical sample of matter, for example, has the temperature–pressure phase diagram presented in Fig. 1.1: by changing either of the two parameters the system may be brought into a solid, liquid, or gas phase. The change of phase may be gradual or abrupt. In the latter case, the phase transition takes place at well defined values of the parameters that determine the phase boundary. Phase transitions are defined as points in the parameter space where the thermodynamic potential becomes non-analytic. Such a non-analyticity can arise only in the thermodynamic limit, when the size of the system is assumed to be infinite. In a finite system the partition function of any system is a finite sum of analytic functions of its parameters, and is therefore always analytic. A sharp phase transition is thus a mathematical idealization, albeit one that describes the reality extremely well. Macroscopic systems typically contain ∼ 10 23 degrees of freedom, and as such are very close to being in the thermodynamic limit. The phase boundaries in Fig. 1.1, for example, for this reason represent reproducible physical quantities.
The basic concepts of statistical mechanics and its application to many-body systems, phase transitions and critical phenomena will be discussed briefly in this chapter. We will be using these ideas frequently in the subsequent lectures. Statistical mechanics describes the thermodynamic behaviour of macroscopic systems from the laws which govern the behaviour of the constituent elements at the microscopic level. The microscopic elements can be atoms, molecules, dipole moments or magnetic moments, etc. A macroscopic system is generally composed of a large number of these elements (of the order of Avogadro number N A ≈ 6.022 × 1023 per mole). Each element may have a large number of internal degrees of freedom associated with different types of motion such as translation, rotation, vibration etc. The constituent elements may interact with the external field applied to the system. There can also be very complex interaction among the constituent elements. The macroscopic properties of a system in the thermodynamic limit is thus determined by the properties of the constituent molecules, their interaction with external field as well as interaction among themselves. For a system of N elements, volume V and density ρ, the thermodynamic limit is defined as lim N → ∞, lim V → ∞ keeping ρ = N/V finite. In this limit, the extensive properties of the system become directly proportional to the size of the system (N or V), while the intensive properties become independent of the size of the system.
The theory of critical phenomena in systems at equilibrium is reviewed at an introductory level with special emphasis on the values of the critical point exponents α, β, γ,..., and their interrelations. The experimental observations are surveyed and the analogies between different physical systems - fluids, magnets, superfluids, binary alloys, etc. - are developed phenomenologically. An exact theoretical basis for the analogies follows from the equivalence between classical and quantal `lattice gases' and the Ising and Heisenberg-Ising magnetic models. General rigorous inequalities for critical exponents at and below T c are derived. The nature and validity of the `classical' (phenomenological and mean field) theories are discussed, their predictions being contrasted with the exact results for plane Ising models, which are summarized concisely. Padé approximant and ratio techniques applied to appropriate series expansions lead to precise critical-point estimates for the three-dimensional Heisenberg and Ising models (tables of data are presented). With this background a critique is presented of recent theoretical ideas: namely, the `droplet' picture of the critical point and the `homogeneity' and `scaling' hypotheses. These lead to a `law of corresponding states' near a critical point and to relations between the various exponents which suggest that perhaps only two or three exponents might be algebraically independent for any system.
Phase transitions represent fundamental transformations of matter, characterized by abrupt changes in macroscopic properties and often associated with singular behavior at critical points. While equilibrium aspects of phase transitions, particularly universal critical exponents and scaling laws, are well-understood, the non-equilibrium processes occurring near criticality, especially the production of entropy, remain a rich area of investigation. This paper explores the intricate relationship between critical entropy production and the emergence of scaling laws in systems undergoing phase transitions. We delve into how the rate of entropy production diverges or exhibits anomalous scaling near critical points, reflecting the system's attempts to dissipate energy and respond to external perturbations as it approaches a state of macroscopic change. By integrating concepts from non-equilibrium thermodynamics, statistical mechanics, and renormalization group theory, we aim to elucidate the underlying mechanisms that govern entropy generation in these critical regimes. We discuss theoretical frameworks that predict universal scaling behaviors for entropy production rates, analogous to those observed for equilibrium order parameters and susceptibilities. The implications of these scaling laws extend beyond theoretical physics, offering insights into complex systems, far-from-equilibrium dynamics, and the efficiency of energy conversion processes in critical environments. Our analysis highlights the critical entropy production as a unique signature of non-equilibrium dynamics during phase transitions, providing a deeper understanding of the interplay between fluctuations, dissipation, and universality near points of instability.