Scaling finite-size

According to finite-size scaling, the temperature, at which the maxima occur, T maxC- ) is shifted relative to the bulk critical temperature. The temperature 

The finite-size scaling expression for the anomalous part C L,Q of the finite-size heat capacity near the critical point reads  

If the system is restricted in only one dimension, by creating a thin film with the width L, for example, one should expect the finite-size scaling to compete with the crossover between three-dimensional and two-dimensional critical 

In a similar fashion one can describe a finite-size susceptibility, predicted as 

Meso-scale heterogeneities can be probed by the intensity of electromagnetic or neutron scattering at a selected wave number q, the instrumental scale. A good example of the scale-dependent meso-thermodynamic property is the isothermal compressibility of fluids or osmotic susceptibility of binary liquids near the critical point of phase separation. In the limit of zero wave number and/or when the correlation length is small (c g 1) the intensity becomes the thermodynamic susceptibility, which diverges at the critical point as 

By taking derivatives and/or setting the arbitrary scale factor b to appropriate values, Eq. [8] can be used to derive scaling forms of various observables. Eor instance, by setting b = L and h = 0, we obtain f r, L) = where 

Qf x) is a dimensionless scaling function. This can also be used to find how the critical point shifts as a function of L in geometries that allow a sharp transition at finite L (e.g., layers of finite thickness). The finite-L phase transition corresponds to a singularity in the scaling function at some nonzero argument a , . The transition thus occurs at = Xc, and the transition temperature 

Tc(L) of the finite-size system is shifted from the bulk value T by 

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 p that in general is different from 1 /v. 

Finite-size scaling has become one of the most powerful tools for analyzing computer simulation data of phase transitions. Instead of treating finite-size effects as errors to be avoided, one can simulate systems of varying size and test whether or not homogeneity relations such as Eq. [8] are fulfilled. Fits of the simulation data to the finite-size scaling forms of the observables then yield values for the critical exponents. We will discuss examples of this method later in the chapter. 

However, for large enough L where finite size scaling holds, one can show that certain reduced moment ratios should intersect exactly at Tc. This is seen by noting that the linear dimension L scales with the correlation length of concentration fluctuations  

From eq. (7.20) one now concludes that reduced moment ratios (where 

At first sight it is surprising that for larger chain length (e.g., N = 256, Fig. 7.12(b)) there is not such a well-defined intersection property, but rather, with increasing L the intersection point shifts somewhat to a lower 

This crossover is physically due to the fact that in a dense three-dimensional polymer melt the chain configurations are random-walk-like, i.e., the gyration radius behaves as i gyr ay/N, a being the size of an effective monomer, and hence the density of one chain within its own volume (kchain oc is pchain = N/Voam Since the melt density is of 

A more quantitative description of this problem shows that one can describe this crossover by comparing the reduced temperature distance t to the so-called Ginzburg number Gi, which for the Flory-Huggins theory of a symmetric polymer mixture reduces to  

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Binder K 1981 Finite size scaling analysis of Ising-model block distribution-functions Z. Phys. B. Oondens. Matter. 43 119-40... 

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 study a system with purely repulsive interactions which demonstrates the importance of entropy effects on the stability of phases when the effect of the corrugation potential due to the structured surface is completely neglected. The phase diagrams are determined by finite size scaling methods, in particular the methods of Sec. IV A. 

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.)... 

P. Nielaba, J. L. Lebowitz, H. Spohn, J. L. Valles. J Stat Phys 55 745, 1989. V. Privman, ed. Finite Size Scaling and Numerical Simulation. Singapore World Scientific, 1990. 

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... 

E. V. Albano. The critical behavior of dimer-dimer surface reaction models. Monte Carlo and finite-size scaling investigation. J Stat Phys 69 643-666,1992. 

With the availabihty of computers, the transfer matrix method [14] emerged as an alternative and powerful technique for the study of cooperative phenomena of adsorbates resulting from interactions [15-17]. Quantities are calculated exactly on a semi-infinite lattice. Coupled with finite-size scaling towards the infinite lattice, the technique has proved popular for the determination of phase diagrams and critical-point properties of adsorbates [18-23] and magnetic spin systems [24—26], and further references therein. Application to other aspects of adsorbates, e.g., the calculation of desorption rates and heats of adsorption, has been more recent [27-30]. Sufficient accuracy can usually be obtained for the latter without scaling and essentially exact results are possible. In the following, we summarize the elementary but important aspects of the method to emphasize the ease of application. Further details can be found in the above references. 

The inset illustrates the extrapolation of the simulation data to the thermodynamic limit according to mixed field finite size scaling for Na = 40 and Nb = 120. From Muller and Binder. ... 

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... 

Wilding, N. B., Critical-point and coexistence-curve properties of the Lennard-Jones fluid a finite-size scaling study, Phys. Rev. E1995, 52, 602-611... 

Errington, J. R., Evaluating surface tension using grand-canonical transition-matrix Monte Carlo simulation and finite-size scaling, Phys. Rev. E 2003, 67, 012102... 

Privman, V., (Ed.), Finite Size Scaling and Numerical Simulation of Statistical Mechanical Systems, World Scientific, Singapore, 1990... 

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 prefactor L in Eq. (36) is understood from the normalization condition, i d Pj ,( ) = 1, which must hold at all temperatures. From Eq. (36) one immediately obtains finite-size scaling relations for the order parameter < >T and ordering susceptibility by taking suitable moments of the distribution (note PJl ) is symmetric around ijf — O in the absence of symmetry-breaking fields and thus < > = 0) ... 

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. 6. Finite size scaling plot of the order parameter of the square...

The advantage of this analysis is twofold estimates for T v, and P/v are obtained from independent pieces of information and are hence not as correlated as in a fit such as shown in Fig. 6 in addition, corrections to finite size scaling can be taken into account systematically. In fact, Eqs. (34)-(42) are expected to hold only for L- oo for finite L correction terms described by correction exponents and amplitudes are present, e.g. 

This system was modelled in terms of the lattice gas with interactions shown in Fig. Ib. The phase diagram was first calculated by the transfer matrix finite size scaling technique for various choices of the interaction parameters [Pg.122]

Fig. 18. Phase diagram of the centered rectangular lattice gas model with ==0, 3/4 2 = V3> vJ

Fig. 21. Finite-size scaling plot of (a) order parameter m using Tc= T2 = 4.26 (b) the ordering susceptibility % using...

Fig. 22. Finite-size scaling plot of the order parameter correlation function at 0 == 1/2 and temperatures in the regime Tj < T < T for the model of Fig. 20. Resulting estimates i((T) are indicated. (From Landau. )...

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