Monte Carlo Simulation in Statistical Physics by Unknown

Author:Unknown
Language: eng
Format: epub
Publisher: Springer-Verlag Wien 2012
Published: 2015-01-31T16:00:00+00:00

The field-driven transition in the Ising model is a particularly simple case, since the model possesses a symmetry χ(H,T,L) = χ(−H,T,L) even for finite L, and hence the transition is rounded but not shifted by the finite size of the system; it still occurs at H = 0. A more general case, which is also more interesting for practical applications, occurs for first-order transitions driven by temperature from an ordered phase at low temperature to a disordered phase at higher temperatures. Obviously, there is no symmetry between the high temperature and low temperature phases, and then we also expect a shift of the effective transition temperature Tc(L) (where the rounded peak of the specific heat representing the smeared delta function of the latent heat has its maximum) relative to the true transition temperature Tc(∞), (2.3.39)

where we have defined three exponents λ, αm, Θ for the shift, the height of the peak, and the temperature interval δT over which it is rounded. It turns out, however, that these finite-size effects can again be understood by a simple discussion in terms of thermodynamic fluctuation theory, similar to (2.3.36–38). We just have to extend (2.1.33) to the case where we have a superposition of two Gaussians [ΔT = T − Tc, Tc ≡ Tc(∞)] [2.104] (2.3.40)

where the specific heat in the infinite system near Tc behaves as

and the weights a+, a− are expressed in terms of the degeneracies of the two phases and their internal energies E+, E− as [2.104, 2.104a] (2.3.41)

From (2.3.40, 41) it is straightforward to obtain the energy 〈E〉L and specific heat C(T, L) as (2.3.42)

(2.3.43)

From (2.3.43) it is obvious that the maximum of the specific heat occurs at (2.3.44)

and has a height (2.3.45)

Since the temperature region δT over which rounding of the delta function peak occurs is given just by taking the argument of the exponential functions in (2.3.41) of order unity, , we conclude that the exponents λ, αm, θ defined in (2.3.39) are all equal to the dimensionality: (2.3.46)

Thus, since there is no diverging characteristic length to which the linear dimension L could be compared at a first order transition, it is simply the volume Ld that controls the size effects [2.98–104]. Figure 2.22 shows results obtained [2.104] for the q-state Potts model [2.106] with q = 10, whose Hamiltonian is (2.3.47)

For the square lattice E+,E− and C+, C− can be obtained from exact solutions [2.107] and hence a nontrivial test of (2.3.40–46) is possible. It is seen that the phenomenological theory outlined in (2.3.40–46) does in fact account for the behavior of the Monte Carlo results nicely. Again some “homework” by the reader (Exercise 3.50) is strongly recommended, but at the same time the warning is given that the accuracy of Fig. 2.22 was only reached with an effort of several hundred hours CPU on a supercomputer!

Fig. 2.22. (a) Temperature variation of the specific heat for various lattice sizes, for the 10-state Potts model on the square lattice. Data for some lattice sizes have been omitted in order to preserve the clarity of the figure.

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