The probability distribution (PD) of spin configurations in kinetic Ising models has been cast in the form of the canonical Boltzmann PD with a time-dependent effective Hamiltonian (EH). It has been argued that in systems with extensive energy EH depends linearly on the number of spins N leading to the exponential dependence of PD on the system size. In macroscopic systems the argument of the exponential function may reach values of the order of the Avogadro number which is impossible to deal with computationally, thus making unusable the linear master equation (ME) governing the PD evolution. To overcome the difficulty, it has been suggested to use instead the nonlinear ME (NLME) for the EH density per spin. It has been shown that in spatially homogeneous systems NLME contains only terms of order unity even in the thermodynamic limit. The approach has been illustrated with the kinetic Husimi-Temperley model (HTM) evolving under the Glauber dynamics. At finite N the known numerical results has been reproduced and extended to broader parameter ranges. In the thermodynamic limit an exact nonlinear partial differential equation of the Hamilton-Jacobi type for EH has been derived. It has been shown that the average magnetization in HTM evolves according to the conventional kinetic mean field equation.
Universality classes encompass the analogous thermodynamic behavior of unlike physical systems, at different spatial dimensions
d, in the vicinity of their critical point. Critical exponents define these classes, with the Ising model being the outstanding prototype that elucidates the differences from the mean-field category, believed to be valid above a critical dimension only. Here, in apparent striking contradiction to the Ising universality class, we demonstrate that the critical behavior of a finite Ising system of
N spins in
d=3 obeys mean-field Landau theory in the vicinity of its critical point, with classical critical exponents. Yet, when expressed in terms of the linear size
L of the system, the free energy unveils its proper finite-size scaling form, from which the thermodynamic limit critical temperature
Tc and the Ising critical exponents
ν,
γ and
β can be identified. We find that the larger the size
L, the smaller the mean-field region, shrinking to zero in the thermodynamic limit. These conclusions are achieved via the use of an alternative approach to collect data from a Monte Carlo simulation of a three-dimensional Ising model that allows for the evaluation of the free energy per spin
f=f(T,m;L) and of the coexistence curve, or spontaneous magnetization at zero magnetic field,
mcoex=m(T;L) as functions of temperature
T and magnetization per spin
m=M/N. Our results suggest a revision of the role of mean-field theory in the elucidation of critical phenomena.
The validity of finite-size scaling in the presence of an inhomogeneous external field vanishing in the thermodynamic-limit is studied using a fully finite three-dimensional mean spherical model. The external field is chosen to change sign stepwise in one space dimension and to be translationally invariant in the other two dimensions, in which the lattice is assumed periodic. The boundary conditions in the direction of broken translational invariance are (i) periodic, and (ii) free, and (iii) fixed. Exact expressions for the magnetization profile are derived and studied. An extended, coordinate-dependent finite-size scaling is found to hold near the shifted critical temperature. Different scaling forms hold near the bulk critical temperature: in case (ii) the distance from the boundary scales with the finite-size correlation length, and in case (iii) with the linear size of the system.
Universality classes encompass the analogous thermodynamic behavior of unlike physical systems, at different spatial dimensions d , in the vicinity of their critical point. Critical exponents define these classes, with the Ising model being the outstanding prototype that elucidates the differences from the mean-field category, believed to be valid above a critical dimension only. Here, in apparent striking contradiction to the Ising universality class, we demonstrate that the critical behavior of a finite Ising system of N spins in d = 3 obeys mean-field Landau theory in the vicinity of its critical point, with classical critical exponents. Yet, when expressed in terms of the linear size L of the system, the free energy unveils its proper finite-size scaling form, from which the thermodynamic limit critical temperature T c and the Ising critical exponents ν , γ and β can be identified. We find that the larger the size L , the smaller the mean-field region, shrinking to zero in the thermodynamic limit. These conclusions are achieved via the use of an alternative data analysis of Monte Carlo simulations of a three-dimensional Ising model that allows for the evaluation of the free energy per spin f = f ( T , m ; L ) and of the coexistence curve, or spontaneous magnetization at zero magnetic field, m coex = m ( T ; L ) as functions of temperature T and magnetization per spin m = M / N . Our results suggest a revision of the role of mean-field theory in the elucidation of critical phenomena.
Although the fully connected Ising model does not have a length scale, we show that the critical exponents for thermodynamic quantities such as the mean magnetization and the susceptibility can be obtained using finite size scaling with the scaling variable equal to N, the number of spins. Surprisingly, the mean value and the most probable value of the magnetization are found to scale differently with N at the critical temperature of the infinite system, and the magnetization probability distribution is not a Gaussian, even for large N. Similar results inconsistent with the usual understanding of mean-field theory are found at the spinodal. We relate these results to the breakdown of hyperscaling and show that hyperscaling can be restored by increasing N while holding the Ginzburg parameter rather than the temperature fixed, or by doing finite size scaling at the pseudocritical temperature where the susceptibility is a maximum for a given value of N. We conclude that finite size scaling for the fully connected Ising model yields different results depending on how the mean-field limit is approached.
Details are presented of an extension of the size-scaling hypothesis to systems in which each element interacts equally with all others (systems for which the mean-field approximation is valid in the thermodynamic limit). A simple argument, which relates the large-size critical behavior of physical quantities with the upper critical dimensionality of the corresponding short-range system, already presented in a Letter, is here made precise and checked either analytically or numerically on several examples. In particular, the scaling form for the magnetization is explicitly derived in the case of the infinitely coordinated Ising model, and a numerical study is presented of the infinitely coordinated
XY-Ising quantum model in a transverse field, with its extension in the presence of an imaginary longitudinal field (a model exhibiting a Yang-Lee edge singularity).