Dynamics and critical behavior of the q model

Correlations and critical behavior of the q model

The q model is a random walk model used to describe the flow of stress in a stationary granular medium. Here we derive the exact horizontal and vertical correlation functions for the q model in two dimensions. We show that close to a critical point identified in earlier work these correlation functions have a universal scaling form reminiscent of thermodynamic critical phenomena. We determine the form of the universal scaling function and the associated critical exponents ν and z.

On the distribution of elastic forces in disordered structures and materials. I. Computer simulation

Randomly structured materials and structures develop distributed internal forces when subjected to external loadings. Using granular packings, and random honeycombs and open–cell foams as prototypic examples, computer simulations were carried out to elucidate the statistical distribution of the internal forces in randomly structured materials. It was found that the sharpness of the force distribution depends critically on the degree of randomness of the structure. In general, the force distributions in these exemplary systems are found to range from the Maxwell–Boltzmann form to a sharply peaked Gaussian form. The results presented here form the basis for the development of a statistical mechanics theory to be presented elsewhere as part II.

Packing geometry and statistics of force networks in granular media

The relation between packing geometry and force network statistics is studied for granular media. Based on simulations of two-dimensional packings of Hertzian spheres, we develop a geometrical framework relating the distribution of interparticle forces P(f) to the weight distribution P(w), which is measured in experiments. We apply this framework to reinterpret recent experimental data on strongly deformed packings and suggest that the observed changes of P(w) are dominated by changes in contact network while P(f) remains relatively unaltered. We furthermore investigate the role of packing disorder in the context of the q model and address the question of how force fluctuations build up as a function of the distance beneath the top surface.

Replica model for an unusual directed polymer in 1+1 dimensions and prediction of the extremal parameter of gapped sequence alignment statistics

Sequence alignment is one of the most important bioinformatics tools for modern molecular biology. The statistical characterization of gapped alignment scores has been a long-standing problem in sequence alignment research. In this paper, we provide a self-contained exposition of sequence alignment, a short review about how this problem is related to the directed polymer problem in statistical physics, and some analytical results that can be used for predicting alignment score statistics. Basically, we present two classes of solutions for the gapped alignment statistics by explicitly calculating the evolution of the few-replica partition function in 1+1 dimensions. We have obtained the conditions under which the more important extremal parameter lambda, characterizing the alignment score statistics, becomes predictable.

Mechanical analog of temperature for the description of force distribution in static granular packings

It is shown that in a stressed granular packing, the effect of the applied pressure and structural randomness on the contact force distribution can be described accurately by a variational principle of minimizing energy subject to the constraint of keeping entropy at a fixed value. The constraint on entropy may be regarded as a measure of the degree of retained disorderness in the system. This procedure leads to the introduction of a parameter known as the "mechanical temperature." Similar to the role of the conventional thermal temperature in a thermal system, the mechanical temperature can be viewed as a parameter controlling the mixity between energy minimization and entropy maximization in the equilibrium condition.

Force correlations in the q model for general q distributions

We study force correlations in the q model for granular media at infinite depth for general q distributions. We show that there are no two-point force correlations as long as q values at different sites are uncorrelated. However, higher-order correlations can persist, and if they do, they only decay with a power of the distance. Furthermore, we find the entire set of q distributions for which the force distribution factorizes. It includes distributions ranging from infinitely sharp to almost critical. Finally, we show that two-point force correlations do appear whenever there are correlations between q values at different sites in a layer; various cases are evaluated explicitly.