Finite-size scaling theory

In the case of a single patch, the size dependence of the system follows directly from the finite size scaling theory [133]. In particular, the critical point temperature scales with the system size as predicted by the equation... 

The theoretical foundation for describing critical phenomena in confined systems is the finite-size scaling approach [64], by which the dependence of physical quantities on system size is investigated. On the basis of the Ising Hamiltonian and finite-size scaling theory, Fisher and Nakanishi computed the critical temperature of a fluid confined between parallel plates of distance D [66]. The critical temperature refers to, e.g., a liquid/vapor phase transition. Alternatively, the demixing phase transition of an initially miscible Kquid/Kquid mixture could be considered. Fisher and Nakashini foimd that compared with free space, the critical temperature is shifted by an amoimt... 

For a second-order transition, this problem is conveniently studied in terms of the order parameter distribution function, Pi,( ). Finite size scaling theory implies that near the critical point P/.( ) longer depends on the three variables L, 1 — T/T separately but rather is a scaled function of two variables (1 — only where v is the critical exponent of... 

The finite-size scaling theory combined with transfer matrix calculations had been, since the development of the phenomenological renormalization in 1976 by Nightingale [70], one of the most powerful tools to study critical phenomena in two-dimensional lattice models. For these models the partition function and all the physical quantities of the system (free energy, correlation length, response functions, etc) can be written as a function of the eigenvalues of the transfer matrix [71]. In particular, the free energy takes the form... 

Determination of the critical exponents by Monte Carlo simulation is usually based on the aforementioned finite size scaling theory [123 - 126,131 - 134]. A detailed presentation of this theory is well beyond the scope of this chapter. Moreover, several excellent reviews concerning this theory are available [133,134]. Therefore, I confine the discussion to some basic facts showing that it is possible to obtain reliable information about the properties of macroscopic systems fi om the results obtained from computer simulations performed for finite systems. In a macroscopic system near the critical point the correlation length diverges to infinity and the following relationship is satisfied ... 

In general, the temperatures at which the heat capacity and the compressibility reach their respective maxima in finite systems are different. From the finite size scaling theory it follows that in the case of a second-order phase transition (e.g. at the critical point) Qm = a/t and 7 = 7/j/, while for any first-order phase transition ocm — Im — d, where d is the dimensionality of the system. The above predictions are often used to determine the nature of phase transitions studied by computer simulation methods [77]. The finite size scaling theory implies also that near the critical point the system free energy is given by... 

At this stage finite-size scaling theory [155, 193-197, 199] is applicable for the description of the specific heat maximum and of the onset of the superfluid density (see the beginning of Section 11), which characterize the rounded-off X transition. The singular free energy density, /, of the finite system (in the absence of external fields) can be described in terms of a universal function (T()) in the form [194—197] f = where is a metric factor,... 

The variation of these exponents with 6 for large 6 was explained in [28] using the finite size scaling theory. For large 6, the growth constant of SAWs on the 6-fractal would be close to the critical value in two dimensions. It is convenient to change variables from pi ) to a variable that is proportional to the departure from criticality in these systems. 

The limit b — oo was analyzed by Kumar et al [53] using the finite size scaling theory (see sec. 5) and it was found that in this limit... 

A more rigorous approach is the mixed field finite size scaling theory as used by Liu et al. (2010) for the model confined fluids. This approach utilizes an ordering operator M, which is a linear combination of the number of particles A and total configurational energy U... 

Aeeording to the finite-size scaling theory of the first-order phase transitions, the finite size of the erystaUite eauses a rounding of singularities in the behavior of isothermal compressibility and heat eapacity. The boundary of the two-phase region formed on finite crystallites has length proportional to its one-dimensional size (L). This introduces an additional term to the chemical potential of the surface film [336] ... 

Privman, V. (ed.) (1990) Finite-Size Scaling Theory Finite Size Scaling and... 

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