Phase transitions finite-size scaling

Note that the simple form of finite-size scaling summarized above is only valid below the upper critical dimension of the phase transition. Finite-size scaling can be generalized to dimensions above df, but this requires taking dangerously irrelevant variables into account. One important consequence is that the shift of the critical temperature, T< (L) — T oc L" is controlled by an exponent

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Binder K and Landau D P 1984 Finite size scaling at Ist-order phase transitions Phys. Rev. B 30 1477-85... 

In this section we review several studies of phase transitions in adsorbed layers. Phase transitions in adsorbed (2D) fluids and in adsorbed layers of molecules are studied with a combination of path integral Monte Carlo, Gibbs ensemble Monte Carlo (GEMC), and finite size scaling techniques. Phase diagrams of fluids with internal quantum states are analyzed. Adsorbed layers of H2 molecules at a full monolayer coverage in the /3 X /3 structure have a higher transition temperature to the disordered phase compared to the system with the heavier D2 molecules this effect is... 

FIG. 14 Phase diagram of the quantum APR model in the Q -T plane. The solid curve shows the line of continuous phase transitions from an ordered phase at low temperatures and small rotational constants to a disordered phase according to the mean-field approximation. The symbols show the transitions found by the finite-size scaling analysis of the path integral Monte Carlo data. The dashed line connecting these data is for visual help only. (Reprinted with permission from Ref. 328, Fig. 2. 1997, American Physical Society.)... 

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... 

Figures 6-9 illustrate the use of these finite size scaling relations for the square lattice gas with repulsion between both nearest and next nearest neighbors. In Fig. 6 the raw data of Fig. 5 are replotted in scaled form, as suggested by Eq. (37). Note that neither = TJcc) nor the critical exponents are known in beforehand - the phase transition of the (2x1) phase falls in the universality class of the XY model with uniaxial anisotropy which has nonuniversal exponents depending on R. Clearly, it is desirable to estimate without being biased by the choice of the critical exponents. This is possible...

Fig. 30. Phase diagram of a model for Si/W(110) in the temperature versus 9 plane. Experimentally determined interactions J Jj,are used. Full dots are from Monte Carlo calculations, while triangles are based on transfer matrix finite size scaling using strip widths of 8 and 12. The point labelled L indicates approximate location of Lifshitz point. The dotted line indicates the transition region between the (5 x l)and(6 x 1) phases. (From...

It is known that the coil-globule transition in flexible polymers is well explained by the theory of the type discussed [22]. Note that the chain length and the solvent quality come into the theory in the following combined form x = BN1/2/l3, which is the only dimensionless parameter governing the transition. The presence of the master curve (see Fig. 3.5 below) implies that the phase behavior of the thermodynamic limit with N —> oo is readily discussed from the measurement of shorter chains via finite-size scaling. 

Barber, M.N. (1983). Finite-size Scaling. Phase Transitions and Critical Phenomena, vol. 8, p. 146-268, (Domb, C. and Lebowitz, J. L.). London Academic Press. 

Phase transitions in statistical mechanical calculations arise only in the thermodynamic limit, in which the volume of the system and the number of particles go to infinity with fixed density. Only in this limit the free energy, or any thermodynamic quantity, is a singular function of the temperature or external fields. However, real experimental systems are finite and certainly exhibit phase transitions marked by apparently singular thermodynamic quantities. Finite-size scaling (FSS), which was formulated by Fisher [22] in 1971 and further developed by a number of authors (see Refs. 23-25 and references therein), has been used in order to extrapolate the information available from a finite system to the thermodynamic limit. Finite-size scaling in classical statistical mechanics has been reviewed in a number of excellent review chapters [22-24] and is not the subject of this review chapter. 

The above simple large-D picture helps to establish a connection to phase transitions. However, the questions which remain to be addressed are How to carry out such an analogy to the IV-electron atoms at D = 3 and what are the physical consequences of this analogy These questions will be examined in the following sections by developing the finite size scaling method for atomic and molecular systems. 

Using the finite-size scaling method, study of the analytical behavior of the energy near the critical point shows that the open-shell system, such as the lithium-like atoms, is completely different from that of a closed-shell system, such as the helium-like atoms. The transition in the closed-shell systems from a bound state to a continuum resemble a first-order phase transition, while for the open-shell system the transition of the valence electron to the continuum is a continuous phase transition [9]. 

Molecular systems are challenging from the critical phenomenon point of view. In this section we review the finite-size scaling calculations to obtain critical parameters for simple molecular systems. As an example, we give detailed calculations for the critical parameters for Hj-like molecules without making use of the Born-Oppenheimer approximation. The system exhibits a critical point and dissociates through a first-order phase transition [11],... 

In order to obtain the stability diagram for the three-body Coulomb systems in the X — K)-plane, one has to calculate the transition line, Xc(k), which separates the stable phase from the unstable one. To carry out the finite-size scaling calculations, the following complete basis set was used [66] ... 

M. N. Barber, Finite-Size Scaling, in Phase Transitions and Critical Phenomena, C. Domb and J. L. Lebowitz, eds., Academic Press, New York, 1983. 

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