Nonsymmetric first-order transitions: Finite-size scaling and tests for infinite-range models

Finite-size rounding of first-order transitions is studied for the general case of nonsymmetric phases and nonperiodic boundary conditions. The main features include the surface-induced shift of the rounded transition on the scale 1/L, while the order parameter discontinuity is rounded on the scale 1/L ^d. This rounding is described by the universal scaling forms with scaling functions identical to those for the periodic, symmetric case. The proposed formalism applies to scalar-order-parameter, single-domain systems. It is tested by exact calculations for a class of infinite-range models.     

Finite-size scaling for first-order transitions: Potts model

The finite-size scaling algorithm based on bulk and surface renormalization of de Oliveira is tesed onq-state Potts models in dimensionsD=2 and 3. Our Monte Carlo data clearly distinguish between first- and second-order phase transitions. Continuous-q analytic calculations performed for small lattices show a clear tendency of the magnetic exponentY=D-/v to reach a plateau for increasing values ofq, which is consistent with the first-order transition valueY=D. Monte Carlo data confirm this trend.     

Finite-size scaling for Potts models

Recently, Borgs and Kotecký developed a rigorous theory of finite-size effects near first-order phase transitions. Here we apply this theory to the ferromagneticq-state Potts model, which (forq large andd2) undergoes a first-order phase transition as the inverse temperature is varied. We prove a formula for the internal energy in a periodic cube of side lengthL which describes the rounding of the infinite-volume jumpE in terms of a hyperbolic tangent, and show that the position of the maximum of the specific heat is shifted by _m (L)=(Inq/E)L ^–d +O(L ^–2d ) with respect to the infinite-volume transition point _t . We also propose an alternative definition of the finite-volume transition temperature _t (L) which might be useful for numerical calculations because it differs only by exponentially small corrections from _t .     

Finite-size scaling analysis of isotropic-polar phase transitions in an amphiphilic fluid

We present Monte Carlo simulations of the isotropic-polar (IP) phase transition in an amphiphilic fluid carried out in the isothermal-isobaric ensemble. Our model consists of Lennard-Jones spheres where the attractive part of the potential is modified by an orientation-dependent function. This function gives rise to an angle dependence of the intermolecular attractions corresponding to that characteristic of point dipoles. Our data show a substantial system-size dependence of the dipolar order parameter. We analyze the system-size dependence in terms of the order-parameter distribution and a cumulant involving its first and second moments. The order parameter, its distribution, and susceptibility observe the scaling behavior characteristic of the 3D Ising universality class. Because of this scaling behavior and because all cumulants have a common intersection irrespective of system size we conclude that the IP phase transition is continuous. Considering pressures 1.3 ≤ P ≤ 3.0 we demonstrate that a line of continuous phase transitions exists which is analogous to the Curie line in systems exhibiting a ferroelectric transition. Our results are qualitatively consistent with Landau's theory of continuous phase transitions.     

First-order phase transitions in large entropy lattice models

We show the existence of a first-order phase transition in thev-dimensional Potts model forv≧2, when the number of states of a single spin is big enough. Low-temperature pure phases are proved to survive up to the critical temperature. Also the existence of a first-order transition in thev-dimensional Potts gauge model,v≧3, is obtained if the underlying gauge group is finite but large.     

Finite-size scaling for the mean spherical model with inverse power law interaction

A new analytical technique based on integral transformations with Mittag-Leffler-type kernels is used to derive the finite-size scaling function for the free energy per particle of the mean spherical model with inverse power law asymptotics of the interaction potential. The asymptotic formation of the singularities in the specific heat and magnetic susceptibility at the bulk critical point is studied.     

Finite-size scaling in extreme statistics

We study the deviations from the limit distributions in extreme value statistics arising due to the finite size (FS) of data sets. A renormalization method is introduced for the case of independent, identically distributed (iid) variables, showing that the iid universality classes are subdivided according to the exponent of the FS convergence, which determines the leading order FS shape correction function as well. It is found that, for the correlated systems of subcritical percolation and 1/f;(alpha) stationary (alpha<1) noise, the iid shape correction compares favorably to simulations. Furthermore, for the strongly correlated regime (alpha>1) of 1/f;(alpha) noise, the shape correction is obtained in terms of the limit distribution itself.     

Finite-size scaling of O(n) models in higher dimensions

We propose a finite-size scaling hypothesis for O(n) models, with n ⩾ 2, in geometry L^d−d′ × ∞^d′, with d > 4 and d′ ⩽ 2, subject to periodic boundary conditions. Several predictions, for T < T_c as well as T ≈ T_c, are made and are verified analytically for the special case of the spherical model (n = ∞).     

Finite-size scaling in complex networks

A finite-size-scaling (FSS) theory is proposed for various models in complex networks. In particular, we focus on the FSS exponent, which plays a crucial role in analyzing numerical data for finite-size systems. Based on the droplet-excitation (hyperscaling) argument, we conjecture the values of the FSS exponents for the Ising model, the susceptible-infected-susceptible model, and the contact process, all of which are confirmed reasonably well in numerical simulations.     

Finite-size scaling for mean-field percolation

By studying transfer matrix eigenvalues, correlation lengths for a mean field directed percolation model are obtained both near and far from the critical regime. Near criticality, finite-size scaling behavior is derived and an analytic technique is provided for obtaining the finite-size scaling function. Our methods involve the generating function, matched asymptotic expansions, and certain formulas developed for the study of eigenvalues of the transfer matrix for metastability.     

Finite-size scaling in a microcanonical ensemble

The finite-size scaling technique is extended to a microcanonical ensemble. As an application, equilibrium magnetic properties of anL×L square lattice Ising model are computed using the microcanonical ensemble simulation technique of Creutz, and the results are analyzed using the microcanonical ensemble finite-size scaling. The computations were done on the multitransputer system of the Condensed Matter Theory Group at the University of Mainz.     

Finite-size scaling in the multiparticle production

The finite-size scalling analysis of the scaled factorial moment data is proposed. This analysis allows to extract the scaling indices of the underlying higher-dimensional scale-invariant multiparticle distributions. Moreover, it exhibits the change of the effective scale involved in the dimensional projection with transverse momentum cuts applied to the data.     

Finite-size effects at first-order transitions

Finite-size rounding of a first-order phase transition is studied in “block”- and “cylinder”-shaped ferromagnetic scalar spin systems. Crossover in shape is investigated and the universal form of the rounded susceptibility peak is obtained. Scaling forms on the low-temperature side of the critical point are considered both above and below the borderline dimensionality,d _>=4. A method of phenomenological renormalization, applicable to both odd and even field derivatives, is suggested and used to estimate universal amplitudes for two-dimensional Ising models atT=T_c.     

Finite-size scaling in systems with long-range interaction

The finite-size critical properties of the (n) vector ϕ⁴ model, with long-range interaction decaying algebraically with the interparticle distance r like r ^{-d - σ}, are investigated. The system is confined to a finite geometry subject to periodic boundary condition. Special attention is paid to the finite-size correction to the bulk susceptibility above the critical temperature T _c. We show that this correction has a power-law nature in the case of pure long-range interaction i.e. 0 < σ < 2 and it turns out to be exponential in case of short-range interaction i.e.σ = 2. The results are valid for arbitrary dimension d, between the lower ( d _< = σ) and the upper ( d _> = 2σ) critical dimensions. Received 2 July 2001 and Received in final form 4 Septembre 2001