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PrefaceIt is hard to find a field in theoretical physics which can compete with the theory of phase transitions and critical phenomena with respect to the number of written textbooks, reviews and monographs. In our opinion, writing a new book can be justified only by treating in a different manner new trends existing in the area. What do we have in mind in our case? There are just a few books on critical phenomena in systems with confined geometry: a collection of reprints [Cardy, ed. (1988)], a collection of reviews [Privman, ed. (1990)], and the monograph [Krech (1994)] on the Casimir effect. Indeed, in some modern texbooks on critical phenomena one can find special chapters devoted to this topic, see, e.g., [Cardy (1996)], [Domb (1996)], [Zinn-Justin (1996)], [Henkel (1999)]. Against the background of the numerous papers that appear annually, the gap in the monographic literature on the subject is obvious. The present book attempts to partially fill up this gap. We hope also to give our modest contribution in spreading the scaling ideas for fruitful interpretation and analysis of phase transitions in classical and quantum systems of finite volume. It is a well known fact that the volume is an irrelevant parameter for the local properties of a macroscopic system and, therefore, can be chosen arbi trary large. The conventional statistical mechanical theory studies abstract systems, consisting of infinitely many particles in an infinite volume, due to the essential simplifications that occur in their description. Moreover, it becomes possible to describe phase transitions mathematically in terms of discontinuous or singular behavior of some thermodynamic functions. In constructing the above, so called thermodynamic limit [Van Hove (1949)
VU1Preface tities. For example, in the canonical Gibbs ensemble the increase in the number of particles N has to be accompanied by a proportional increase in the volume V, so that the density p = N/V be constant. Any intensive quantity ay of a finite system can be written in the form ay = a,oo + Say, where a^ is the bulk value and Say is a finite-size correction which tends to zero as V -► oo. The finite-size correction Say contains a more detailed information about the shape of the system and the boundary conditions. Usually, the correction term becomes essential under rather special condi tions, e.g., in the vicinity of a second-order phase transition. When the relevant thermodynamic parameters approach a critical point, t...
