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Scaling Theory for Finite-Size Effects in the Critical Region

Michael Ellis Fisher, Michael N. Barber

Published Jan 1, 1988• 2 citations
Publisher
DOI
10.1016/b978-0-444-87109-1.50007-8

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Critical phenomena in films of finite thickness are considered. A detailed scaling theory, with allowance for distinct exponents $\ensuremath{\lambda}$ and $\ensuremath{\theta}=\frac{1}{\ensuremath{\nu}}$ for the critical-point shift and rounding, respectively, is confirmed by exact calculations on ddd-dimensional ferromagnetic spherical models and ideal Bose fluids with various boundary conditions. Isingmodel results and existing data on real helium films are consistent with the theory.
We develop a scaling theory for the finite-size critical behavior of the microcanonical entropy (density of states) of a system with a critically divergent heat capacity. The link between the microcanonical entropy and the canonical energy distribution is exploited to establish the former, and corroborate its predicted scaling form, in the case of the 3d Ising universality class. We show that the scaling behavior emerges clearly when one accounts for the effects of the negative background constant contribution to the canonical critical specific heat. We show that this same constant plays a significant role in determining the observed differences between the canonical and microcanonical specific heats of systems of finite size, in the critical region.