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The Onset of Phase Transitions in Finite Systems
Author(s) -
Brézin E.
Publication year - 1986
Publication title -
physikalische blätter
Language(s) - English
Resource type - Journals
eISSN - 1521-3722
pISSN - 0031-9279
DOI - 10.1002/phbl.19860420707
Subject(s) - rounding , extrapolation , scaling , universality (dynamical systems) , gravitational singularity , phase transition , classification of discontinuities , renormalization group , mathematics , statistical physics , thermodynamic limit , widom scaling , critical exponent , critical phenomena , physics , mathematical physics , mathematical analysis , quantum mechanics , geometry , computer science , operating system
Abstract In finite systems singularities, such as discontinuities or divergences, associated with phase transitions are rounded. The extrapolation of finite size calculations to the singular thermodynamic limit makes use of scaling laws, introduced by Fisher, which involve the size of the system. The use and the theoretical basis of these laws are briefly reviewed here; some difficulties associated with the analytic calculation of the scaling functions are discussed. The correct treatment of zero‐momentum modes leads to a modified renormalization group picture; above the upper critical dimension (u.c.d.) finite size scaling does not hold but the properties of the finite system are easily calculable; below the u.c.d. the scaling functions are obtained in a systematic expansion in fractional powers of ϵ = 4–d. The rounding of first order transitions is also briefly considered; in spite of the lack of universality of the bulk transition itself, the rounding is universal because the correlation length of the finite system is large .