Survey of Equilibrium Critical Phenomena
Equilibrium and non-equilibrium critical phenomena are similar in many respects. Therefore, this book starts with a brief survey of some basic concepts of equilibrium critical phenomena, providing this background in a self-consistent way and establishing basic notations. This chapter may be used to recall the main features of modern theories of phase transitions for beginners who are not entirely familiar with the concepts of scaling and universality.
Although this introduction is kept as general as possible, for the sake of concreteness we shall use the language of ferromagnetic phase transitions. Following the historical perspective, we discuss the concepts of scaling and universality. In particular, critical exponents, generalised homogeneous functions, scaling forms and universal amplitude combinations are introduced. The basis for a deeper understanding of scaling and universality is provided by Wilson’s renormalisation group theory [629, 630], which is a topic on its own and not presented in this book. Instead, we focus on its implications on universal scaling and illustrate the main results, e.g. by sketching how renormalisation-group theory allows one to identify the relevant system parameters which determine the universality class. The renormalisation group also provides tools for computing critical exponents as well as universal scaling functions and explains the existence of an upper critical dimension. As a reference and preparation for the second volume of this book we also comment on the extension of scale-invariance in equilibrium systems to conformal invariance and recall the fluctuation-dissipation theorem in the context of relaxation phenomena. For a rigorous substantiation of scaling, universality and conformal invariance the interested reader is referred to established textbooks (e.g. [520, 525, 640, 122, 168, 270]) and review articles [632, 219, 621, 120, 112]. The fluctuation-dissipation theorem is discussed in various textbooks, see e.g. [144, 122].
KeywordsIsing Model Critical Exponent Conformal Transformation Conformal Invariance Universality Class
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