No abstract

No 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

We present a new approach to clustering, based on the physical properties of an inhomogeneous ferromagnet. No assumption is made regarding the underlying distribution of the data. We assign a Potts spin to each data point and introduce an interaction between neighboring points, whose strength is a decreasing function of the distance between the neighbors. This magnetic system exhibits three phases. At very low temperatures, it is completely ordered; all spins are aligned. At very high temperatures, the system does not exhibit any ordering, and in an intermediate regime, clusters of relatively strongly coupled spins become ordered, whereas different clusters remain uncorrelated. This intermediate phase is identified by a jump in the order parameters. The spin-spin correlation function is used to partition the spins and the corresponding data points into clusters. We demonstrate on three synthetic and three real data sets how the method works. Detailed comparison to the performance of other techniques clearly indicates the relative success of our method.

We evaluate by Monte Carlo simulations various singular thermodynamic quantities X, for an ensemble of quenched random Ising and Ashkin-Teller models. The measurements are taken at T c and we study how the distributions P (X) (and, in particular, their relative squared width, R X ) 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 ∼ l α ν → 0, where α < 0 and ν 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 R X tends to a universal constant independent of the amount of dilution (no self averaging). However this constant is different for canonical and grand canonical disorder. We identify the pseudo-critical temperatures T c (i, l) of each sample i, as the temperature at which the susceptibility reaches its maximal value. The distribution of these T c (i, l) over the ensemble was investigated; we found that its variance scales as (δT c (l)) 2 ∼ l − 2 ν . These results are in agreement with the recent predictions of Aharony and Harris. Our previously proposed finite size scaling ansatz for disordered systems was tested and found to hold. When we fit the data obtained for many samples of different sizes by a sample-independent form, the resulting scaling function was found to be universal and to behave similarly to pure systems. We did observe that to describe deviations from this universal function we do need sample-dependent scaling functions. These deviations are, however, relatively small and this led us to an interesting side result: sample-to-sample fluctuations of χ max , the susceptibility measured at T c (i, l), are smaller by a factor of 70 than those of χ(T c ). This indicates that to obtain a fixed statistical error it may be more computationally efficient to measure χ max . 05.50.+q, 75.10Nr, 75.40Mg, 75.50.Lk