Finitely coordinated models for low-temperature phases of amorphous systems

Abstract

We introduce models of heterogeneous systems with finite connectivity defined on random graphs to capture finite-coordination effects on the low-temperature behavior of finite dimensional systems. Our models use a description in terms of small deviations of particle coordinates from a set of reference positions, particularly appropriate for the description of low-temperature phenomena. A Born-von-Karman type expansion with random coefficients is used to model effects of frozen heterogeneities. The key quantity appearing in the theoretical description is a full distribution of effective single-site potentials which needs to be determined self-consistently. If microscopic interactions are harmonic, the effective single-site potentials turn out to be harmonic as well, and the distribution of these single-site potentials is equivalent to a distribution of localization lengths used earlier in the description of chemical gels. For structural glasses characterized by frustration and anharmonicities in the microscopic interactions, the distribution of single-site potentials involves anharmonicities of all orders, and both single-well and double well potentials are observed, the latter with a broad spectrum of barrier heights. The appearance of glassy phases at low temperatures is marked by the appearance of asymmetries in the distribution of single-site potentials, as previously observed for fully connected systems. Double-well potentials with a broad spectrum of barrier heights and asymmetries would give rise to the well known universal glassy low temperature anomalies when quantum effects are taken into account.

Spectra of sparse random matrices

“…exhibiting a Posisssonian degree distribution with average coordination c. We note at the outset that formal results carry over without modification to other cases [30]. There is no need at this point to specify the distribution of the K ij , but we shall typically look at Gaussian and bimodal distributions.…”

“…In [30] it is shown that they hold -unmodified -for non-Poissonian degree distributions as well, as long as the average connectivity in these systems remains finite.…”

“…Within the general class of finitely coordinated amorphous model systems considered in [30], the one represented by (2), (3) constitutes a particular sub-class, viz. that of harmonically coupled systems, for which the analysis was found to be much simpler than for systems involving anharmonic couplings.…”

“…3.3. As the formal structure of the self-consistency problem remains unaltered when the Poissonian random graphs are replaced by graphs with other degree distributions [30], we can exploit this fact to present results for regular and scale-free random graphs in Sec. 3.4.…”

The effect of load on agent-based algorithms for distributed task allocation

“…In other contexts where this problem occurs, such as continuous models on sparse random graphs (see e.g. Kühn et al [20] and references therein), population dynamics can be used for the theoretical analysis. For agent based models, however, this is tantamount to simulating the model.…”