Lack of self-averaging in critical disordered systems

Abstract

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 , V P , 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. Using 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 ∼ l −d (where d = 2 is the dimension). At criticality though, the results indicate that the magnetization M and the susceptibility χ are non self averaging, i.e.The energy E at criticality is clearly weakly self averaging, that is V E ∼ l −yv with 0 < y v < 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 α < 0 (α of the disordered model) then, for a quantity P which scales as l ρ at criticality, its variance V P will scale asymptotically as l 2ρ+ α ν . The theory is not applicable in the asymptotic limit (l → ∞) to the bond-disordered Ashkin-Teller model whereNonetheless in the accessible range of lattice sizes we found very good agreement between the theory and the data for V χ and V E . The theory may also be compatible with the data for the variance of the magnetization V M and the variance of the specific heat V C , but evidence for this is less convincing. 75.50.Lk 75.40Mg, 75.10Nr, 75.40Cx

Random Walks and Polymers in the Presence of Quenched Disorder

“…This behaviour was first believed to hold in general [102,110], but was later shown to be wrong in the case of random fixed points. In this case, it was argued [1,111] that eq.…”

“…Important progresses have been made recently in the understanding of finite size properties of random critical points [1,2,10,102,110,111] . To each disordered sample (i) of size L, one should first associate a pseudo-critical temperature T c (i, L) [10,102,110,111].…”

“…To each disordered sample (i) of size L, one should first associate a pseudo-critical temperature T c (i, L) [10,102,110,111]. The disorder averaged pseudo-critical critical temperature…”

“…At criticality however, this 'subdivision' argument breaks down because of the divergence of ξ at T c , and a lack of self-averaging has been found at criticality whenever disorder is relevant [1,110,111]. More precisely, for a given observable X, it is convenient to define its normalized width as…”

Three-dimensional Ising model confined in low-porosity aerogels: A Monte Carlo study

“…The independence of the width of distributions from L is clear, typical for systems lacking self-averaging, which is the expected behavior of thermodynamic quantities for any disordered system at criticality. 41 Self-averaging can be quantitatively checked by the normalized squared width 40 R A :…”