We simulated site dilute Ising models in
d=3 dimensions for several lattice sizes
L. For each
L singular thermodynamic quantities
X were measured at criticality and their distributions
P(X) were determined, for ensembles of several thousand random samples. For
L→∞ the width of
P(X) tends to a universal constant, i.e. there is no self averaging. The width of the distribution of the sample dependent pseudocritical temperatures
Tc(i,L) scales as
δTc(L)∼L−1/ν and NOT as
∼L−d/2. Finite size scaling holds; the sample dependence of
Xi(Tc) enters predominantly through
Tc(i,L).
We evaluate by Monte Carlo simulations various singular thermodynamic quantities
X for ensembles of quenched random Ising and Ashkin-Teller models. The measurements are taken at
Tc and we study how the distributions
P(X) (and, in particular, their relative squared width,
RX) over the ensemble depend on the system size
l. The Ashkin-Teller model was studied in the regime where bond randomness is irrelevant and we found weak self-averaging; ${R}_{X}\ensuremath{\sim}{l}^{\ensuremath{\alpha}/\ensuremath{\nu}}\ensuremath{\rightarrow}0,$ where $\ensuremath{\alpha}<0$ and $\ensuremath{\nu}$ are the exponents (of the pure model fixed point) governing the transition. For the site-dilute Ising model on a cubic lattice, known to be governed by a random fixed point, we find that
RX tends to a constant, as predicted by Aharony and Harris. We tested whether this constant is universal. However, this constant is different for canonical and grand canonical disorder. We identify the pseudocritical temperature of each sample
i, Tc(i,l), as the temperature at which the susceptibility reaches its maximal value ${\ensuremath{\chi}}^{\mathrm{max}}.$ The distribution of these sample dependent
Tc(i,l) was investigated; we found that its variance scales as $[\ensuremath{\delta}{T}_{c}(l){]}^{2}\ensuremath{\sim}{l}^{\ensuremath{-}2/\ensuremath{\nu}}.$ Our previously proposed finite size scaling ansatz for disordered systems was tested and found to hold. We did observe deviations from a single function, which imply that sample dependent scaling functions are needed. These deviations are, however, relatively small and hence to obtain a fixed statistical error it may be more computationally efficient to measure ${\ensuremath{\chi}}^{\mathrm{max}}$ than the commonly used $\ensuremath{\chi}{(T}_{c}^{\ensuremath{\infty}}).$
We consider the sample to sample fluctuations that occur in the value of a thermodynamic quantity
P in an ensemble of finite systems with quenched disorder, at equilibrium. The variance of
P,
VP, which characterizes these fluctuations is calculated as a function of the systems' linear size
l, focusing on the behavior at the critical point. The specific model considered is the bond-disordered Ashkin-Teller model on a square lattice [Phys. Rev. 64, 178 (1943)]. Using extensive Monte Carlo simulations, several bond-disordered Ashkin-Teller models were examined, including the bond-disordered Ising model and the bond-disordered four-state Potts model. It was found that far from criticality all thermodynamic quantities which were examined (energy, magnetization, specific heat, susceptibility) are strongly self-averaging, that is ${V}_{P}\ensuremath{\sim}{l}^{\ensuremath{-}d}$ (where
d=2 is the dimension). At criticality though, the results indicate that the magnetization
M and the susceptibility $\ensuremath{\chi}$ are non-self-averaging, i.e., $\frac{{V}_{\ensuremath{\chi}}}{{\ensuremath{\chi}}^{2}}, \frac{{V}_{M}}{{M}^{2}}\ensuremath{\nrightarrow}0$. The energy
E at criticality is clearly weakly self-averaging, that is ${V}_{E}\ensuremath{\sim}{l}^{\ensuremath{-}{y}_{v}}$ with
0<yv<d. Less conclusively, and possibly only as a transient behavior, the specific heat too is found to be weakly self-averaging. A phenomenological theory of finite size scaling for disordered systems is developed, based on physical considerations similar to those leading to the Harris criterion. Its main prediction is that when the specific heat exponent $\ensuremath{\alpha}<0$ ($\ensuremath{\alpha}$ of the disordered model) then, for a quantity
P which scales as ${l}^{\ensuremath{\rho}}$ at criticality, its variance
VP will scale asymptotically as ${l}^{2\ensuremath{\rho}+\frac{\ensuremath{\alpha}}{\ensuremath{\nu}}}$. The theory is not applicable in the asymptotic limit ($l\ensuremath{\rightarrow}\ensuremath{\infty}$) to the bond-disordered Ashkin-Teller model where $\frac{\ensuremath{\alpha}}{\ensuremath{\nu}}={0}_{+}$. Nonetheless in the accessible range of lattice sizes we found very good agreement between the theory and the data for ${V}_{\ensuremath{\chi}}$ and
VE. The theory may also be compatible with the data for the variance of the magnetization
VM and the variance of the specific heat
VC, but evidence for this is less convincing.
Self-averaging of singular thermodynamic quantities at criticality for randomly and thermally diluted three-dimensional Ising systems has been studied by the Monte Carlo approach. Substantially improved self-averaging is obtained for critically clustered (critically thermally diluted) vacancy distributions in comparison with the observed self-averaging for purely random diluted distributions. Critically thermal dilution, leading to maximum relative self-averaging, corresponds to the case when the characteristic vacancy ordering temperature (theta) is made equal to the magnetic critical temperature for the pure three-dimensional (3D) Ising systems (T(3D)(c)). For the case of a high ordering temperature (theta>>T(3D)(c)), the self-averaging obtained is comparable to that in a randomly diluted system.
We enlighten some critical aspects of the three-dimensional (d = 3) random-field Ising model from simulations performed at zero temperature.We consider two different, in terms of the field distribution, versions of model, namely a Gaussian random-field Ising model and an equal-weight trimodal random-field Ising model.By implementing a computational approach that maps the ground-state of the system to the maximum-flow optimization problem of a network, we employ the most up-to-date version of the push-relabel algorithm and simulate large ensembles of disorder realizations of both models for a broad range of random-field values and systems sizes V = L × L × L, where L denotes linear lattice size and L max = 156.Using as finite-size measures the sampleto-sample fluctuations of various quantities of physical and technical origin, and the primitive operations of the push-relabel algorithm, we propose, for both types of distributions, estimates of the critical field h c and the critical exponent ν of the correlation length, the latter clearly suggesting that both models share the same universality class.Additional simulations of the Gaussian random-field Ising model at the best-known value of the critical field provide the magnetic exponent ratio β/ν with high accuracy and clear out the controversial issue of the critical exponent α of the specific heat.Finally, we discuss the infinite-limit size extrapolation of energyand order-parameter-based noise to signal ratios related to the self-averaging properties of the model, as well as the critical slowing down aspects of the algorithm.