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# VI Scaling functions

[72.1.4.1] This section discusses how the general theory above may be used to obtain finite size scaling functions at the critical point.

[72.2.1.1] The finite size scaling function for the probability density of the observable is defined through an equation analogous to (1.3) by

 (6.1)

where is the anomalous dimension of . [72.2.1.2] The ensemble limit yields explicit analytical expressions for the scaling functions at the critical point. [72.2.1.3] This is seen from (4.11) as well as from (4.4) which become identical in the ensemble limit if . [72.2.1.4] If is identified as the macroscopic (thermodynamic) equivalent of the microscopic observable then it follows from (4.4) and (4.11) that the finite ensemble scaling functions are given as

 (6.2)

if . [72.2.1.5] The superscript is a reminder for the ensemble limit. [72.2.1.6] The point corresponding to first order transitions is singular and will not be discussed here. [72.2.1.7] For on the other hand the thermodynamic form (4.4) yields a simple Gaussian while the fieldtheoretic form (4.11) gives

 (6.3)

[72.2.1.8] This is the scaling function conjectured in [7] for the order parameter density on the basis of a Gaussian approximation. [72.2.1.9] Note that this scaling function, contrary to those for , does depend on the variable separately. [72.2.1.10] Note also that the order parameter generally has anomalous dimension and thus this scaling form for the order parameter distribution is expected to arise in the vicinity but not directly at the critical point.

[72.2.2.1] Another source for the dependence of the scaling function for the order parameter distribution on is the appearance of the nonuniversal cutoff function in the finite size scaling limit of equation (3.18). [72.2.2.2] With equation (3.18) and introducing the abbreviations , and the analogue of equation (6.2) reads

 (6.4)

for the finite size scaling limit. [72.2.2.3] Thus it is seen that the finite ensemble scaling function corresponds to the universal part of the finite size scaling function which is independent of while the cutoff function is responsible [page 73, §0]    for the dependence on and adds a nonuniversal part.

[73.1.1.1] The analytical expressions (3.5) and (3.12) for the universal part of critical finite size scaling functions can be employed to evaluate the scaling functions numerically. [73.1.1.2] In this effort the symmetry relation [28]

 (6.5)

reduces the computational effort. [73.1.1.3] Moreover equation (6.5) suggests a relation with the phenomenon of spontaneous symmetry breaking within the present approach. [73.1.1.4] In this view the two scaling functions represent the two pure phases, and thus on general thermodynamic grounds the full scaling function is expected to become a convex combination

 (6.6)

of two extremal phases. [73.1.1.5] The relation may be generalized to several phases or asymmetric situations.

[73.1.2.1] Consider now an ordinary critical point with a global symmetry such as in the Ising models. [73.1.2.2] Let be the order parameter which is assumed to be normalized such that . [73.1.2.3] Then becomes where is the equation of state exponent. [73.1.2.4] Abbreviating as the scaling function in equation (6.6) becomes

 (6.7)

[73.1.2.5] For the symmetric case the function is displayed in Figures 2, 3 and 4 for and several choices of . [73.1.2.6] The symmetrization in (6.7) corresponds to an “equal weight rule” which is known to apply for first order transitions [42]. [73.1.2.7] Figure 2 shows the case which is the value for the universality class of mean field models. [73.1.2.8] The six values for in Figure 2 through 4 are . [73.1.2.9] The case corresponds to the double peak structure with the widest peak separation while the value corresponds to the singly peaked function whose maximum has the smallest height. [73.1.2.10] Figure 3 shows the case which is close to the value of [16] for the threedimensional Ising model. [73.1.2.11] The value in Figure 4 is the value for the two dimensional Ising universality class.

[73.1.2.12] The scaling functions displayed in Figures 2 through 4 are consistent with published data on critical scaling functions [7, 43, 44]. [73.1.2.13] Moreover it is seen that the universal shape parameter is related to the type of boundary conditions. [73.1.2.14] Free boundary conditions apparently correspond to smaller values of the universal shape parameter than periodic boundary conditions. [73.1.2.15] This correspondence between the value of and the applied boundary conditions is not expected to be one to one. [73.1.2.16] The value of may be influenced by other universal factors such as the type or symmetry of the pure phases. [page 74, §0]    [74.1.0.1] On the other hand the boundary conditions may also influence other parameters such as the value of the symmetrization . [74.1.0.2] This is expected for boundary conditions which do not preserve the symmetry.

[74.1.1.1] Figure 5 shows that the scaling functions are not merely consistent but also in good quantitative agreement with Monte-Carlo simulations of the twodimensional Ising model [43, 44, 45] where the exact value of and the location of the critical point for the infinite system are known. [74.1.1.2] The open circles in Figure 5 represent the smooth interpolation through the data published in [43, 44, 45]. [74.1.1.3] The solid line is the analytical prediction shown in Figure 4 for . [74.1.1.4] For the comparison the nonuniversal scaling factors which were chosen to yield unit norm and variance in [43, 44, 45] were matched to those of the theoretical curve. [74.1.1.5] The excellent agreement between theory and simulation suggests to identify the value with periodic boundary conditions. [74.1.1.6] It is however not clear whether this identification will hold more generally.