Self-averaging Properties Research Articles

Analytic study of a model of diffusion on a random comblike structure

We present an analytic study of a model of diffusion on a random comblike structure in which no bias field does exist neither along the backbone nor along the branches. For any given disordered structure, our analytic treatment allows to compute in an exact manner the asymptotic behaviour at large time of the probability of presence of the particle at its initial site and on the backbone, and of the particle position and dispersion. The expressions of these quantities are shown to coincide asymptotically with those derived in simple “mean-field” treatments. The results for any given sample do not depend on the particular configuration (self-averaging property). The behaviours strongly depend on the distribution of the lengths of the branches. With an exponential distribution, transport is normal, while anomalous diffusion may take place for a power law distribution (when long branches are present with a sufficiently high weight).

Absence of self-averaging in shattering fragmentation processes.

We consider fluctuations in both the shattering and normal (nonshattering) fragmentation processes. The continuum shattering process is shown to lack the self-averaging property: the rms fluctuation in the total number of the finite-size fragements is larger than the mean value. The discrete shattering process exhibits strong fluctuations at the intermediate times when the total number of fragments is much greater than 1 but smaller than the number of the monomers contained in the primary particle. For the normal process, the fluctuations are small and self-averaging exists.

First-passage probability of a random walk on a disordered one-dimensional lattice

A method is presented which yields the exact solution of the generating function for the first-passage probability and related observables, for a one-dimensional random walk with random hopping rates on each site. A geometrical and statistical-mechanical interpretation of the problem is suggested and is used for obtaining the asymptotic properties of the occupation and first-passage probabilities. The class of values with the property of self-averaging is found.

Comparison of the Koster-Slater and the equation-of-motion method for calculation of the electronic structure of defects in compound semiconductors.

Traditional methods of calculating the electronic structure of defects in semiconductors rely on matrix-diagonalization methods which use the unperturbed crystalline wave functions as a basis. Equation-of-motion (EOM) methods, on the other hand, give excellent results with strong disorder and many defects and make no use of the basis of unperturbed wave functions, but require self-averaging properties of the wave functions which appear superficially to make them unsuitable for study of local properties. We show here that EOM methods are better than traditional methods for calculating the electronic structure of essentially any finite-range impurity potential. The reason is basically that the numerical cost of the traditional Green's-function methods grows approximately as ${\mathit{R}}^{7}$ o/Iper sitet/P, where R is the range of the potential, whereas the cost of the EOM methods per site is independent of the range of the potential. Our detailed calculations on a model of an oxygen vacancy in rutile ${\mathrm{TiO}}_{2}$ show that a crossover occurs very soon, so that equation-of-motion methods are better than the traditional ones in the case of potentials of realistic range.

Random infinite-volume Gibbs states for the Curie-Weiss random field Ising model

An approach to the definition of infinite-volume Gibbs states for the (quenched) random-field Ising model is considered in the case of a Curie-Weiss ferromagnet. It turns out that these states are random quasi-free measures. They are random convex linear combinations of the free product-measures “shifted” by the corresponding effective mean fields. The conditional self-averaging property of the magnetization related to this randomness is also discussed.

Absence of self-averaging of the order parameter in the Sherrington-Kirkpatrick model

We prove that ifĤ N is the Sherrington-Kirkpatrick (SK) Hamiltonian and the quantity $$\bar q_N = N^{ - 1} \sum \left\langle {S_l } \right\rangle _H^2 $$ converges in the variance to a nonrandom limit asN→∞, then the mean free energy of the model converges to the expression obtained by SK. Since this expression is known not to be correct in the low-temperature region, our result implies the “non-self-averaging” of the order parameter of the SK model. This fact is an important ingredient of the Parisi theory, which is widely believed to be exact. We also prove that the variance of the free energy of the SK model converges to zero asN→∞, i.e., the free energy has the self-averaging property.

Thermodynamics for the zero-level set of the Brownian bridge

The random set of instants where the Brownian bridge vanishes is constructed in terms of a random branching process. The Hausdorff measure supported by this set is shown to be equivalent to the partition function of a special class of disordered systems. This similarity is used to show rigorously the existence of a phase transition for this particular class of disordered systems. Moreover, it is shown that at high temperature the specific free energy has the strong self-averaging property and that at low temperature it has no self-averaging property. The unicity at high-temperature and the existence of many limits at low temperature are established almost surely in the disorder.

Exact results and self-averaging properties for random-random walks on a one-dimensional infinite lattice

We present new exact results for a one-dimensional asymmetric disordered hopping model. The lattice is taken infinite from the start and we do not resort to the periodization scheme used by Derrida. An explicit resummation allows for the calculation of the velocityV and the diffusion constantD (which are found to coincide with those given by Derrida) and for demonstrating thatV is indeed a self-averaging quantity; the same property is established forD in the limiting case of a directed walk.

Monte Carlo study of growth in the two-dimensional spin-exchange kinetic Ising model.

Results obtained from extensive Monte Carlo simulations of domain growth in the two-dimensional spin-exchange kinetic Ising model with equal numbers of up and down spins are presented. Using different measures of domain size---including the pair-correlation function, the energy, and circularly-averaged structure factor---the domain size is determined (at T=0.5${T}_{c}$) as a function of time for times up to ${10}^{6}$ Monte Carlo steps. The growth law R(t)=A+${\mathrm{Bt}}^{1/3}$ is found to provide an excellent fit (within 0.3%) to the data, thus indicating that at long times the classical value of (1/3 for the exponent is correct. It is pointed out that this growth law is equivalent to an effective exponent for all times (as given by Huse) ${n}_{\mathrm{eff}}$(t)=(1/3-1)/3 C/R(t). No evidence for logarithmic behavior is seen. The self-averaging properties of the various measures of domain size and the variation of the constants A and B with temperature are also discussed. In addition, the scaling of the structure factor and anisotropy effects due to the lattice are examined.

The mean spherical model in a random external field and the replica method

We provide a quick elementary solution of the mean spherical model in a random external field. This also allows an immediate proof of the self-averaging property of the free energy. We calculate the free energy by means of the replica method, i.e., for any (not necessarily integer) “replica number”n, and show that when a phase transition occurs the limits\((\lim _{H \to 0_ + } \lim _{N \to \infty } )\) andn → 0 are not interchangeable.