The Statistical Physics of Athermal Materials

At the core of equilibrium statistical mechanics lies the notion of statistical ensembles: a collection of microstates, each occurring with a given a priori probability that depends only on a few macroscopic parameters such as temperature, pressure, volume, and energy. In this review article, we discuss recent advances in establishing statistical ensembles for athermal materials. The broad class of granular and particulate materials is immune from the effects of thermal fluctuations because the constituents are macroscopic. In addition, interactions between grains are frictional and dissipative, which invalidates the fundamental postulates of equilibrium statistical mechanics. However, granular materials exhibit distributions of microscopic quantities that are reproducible and often depend on only a few macroscopic parameters. We explore the history of statistical ensemble ideas in the context of granular materials, clarify the nature of such ensembles and their foundational principles, highlight advances in testing key ideas, and discuss applications of ensembles to analyze the collective behavior of granular materials

Dimensional crossover of the fundamental-measure functional for parallel hard cubes

We present a regularization of the recently proposed fundamental-measure functional for a mixture of parallel hard cubes. The regularized functional is shown to have right dimensional crossovers to any smaller dimension, thus allowing to use it to study highly inhomogeneous phases (such as the solid phase). Furthermore, it is shown how the functional of the slightly more general model of parallel hard parallelepipeds can be obtained using the zero-dimensional functional as a generating functional. The multicomponent version of the latter system is also given, and it is suggested how to reformulate it as a restricted-orientation model for liquid crystals. Finally, the method is further extended to build a functional for a mixture of parallel hard cylinders.Comment: 4 pages, no figures, uses revtex style files and multicol.sty, for a PostScript version see http://dulcinea.uc3m.es/users/cuesta/cross.p

Understanding the Frequency Distribution of Mechanically Stable Disk Packings

Relative frequencies of mechanically stable (MS) packings of frictionless bidisperse disks are studied numerically in small systems. The packings are created by successively compressing or decompressing a system of soft purely repulsive disks, followed by energy minimization, until only infinitesimal particle overlaps remain. For systems of up to 14 particles most of the MS packings were generated. We find that the packings are not equally probable as has been assumed in recent thermodynamic descriptions of granular systems. Instead, the frequency distribution, averaged over each packing-fraction interval $\Delta \phi$, grows exponentially with increasing $\phi$. Moreover, within each packing-fraction interval MS packings occur with frequencies $f_k$ that differ by many orders of magnitude. Also, key features of the frequency distribution do not change when we significantly alter the packing-generation algorithm--for example frequent packings remain frequent and rare ones remain rare. These results indicate that the frequency distribution of MS packings is strongly influenced by geometrical properties of the multidimensional configuration space. By adding thermal fluctuations to a set of the MS packings, we were able to examine a number of local features of configuration space near each packing including the time required for a given packing to break to a distinct one, which enabled us to estimate the energy barriers that separate one packing from another. We found a positive correlation between the packing frequencies and the heights of the lowest energy barriers $\epsilon_0$. We also examined displacement fluctuations away from the MS packings to correlate the size and shape of the local basins near each packing to the packing frequencies.Comment: 21 pages, 20 figures, 1 tabl

Longitudinal dispersion in nonuniform isotropic porous media

A theoretical and experimental investigation has been made of the longitudinal dispersion of chemically and dynamically passive solutes during flow through nonuniform, isotropic porous media. Both theoretical and experimental results are limited to the high Peclet number, low Reynolds number flow regime. The goal of the theoretical investigation is to provide a quantitative method for calculating the coefficient of longitudinal dispersion using only measurable structural features of the porous medium and the characteristics of the carrying fluid and solute. A nonuniform porous medium contains variations in grain scale pore structure, but is homogeneous at the macroscopic level for quantities such as the permeability or porosity. A random capillary tube network model of nonuniform porous media is developed which uses a pore radius distribution and pore length distribution to characterize the grain scale structure of porous media. The analysis gives the asymptotic longitudinal dispersion coefficient in terms of integrals of kinematic properties of solute particles flowing through individual, random capillary tubes. However, shear dispersion within individual capillary tubes is found to have negligible impact on the overall longitudinal dispersion in porous media. The dispersion integrals are evaluated using a Monte Carlo integration technique. An analysis of the permeability in nonuniform porous media is used to establish a proper flow field for the analysis of longitudinal dispersion. The experimental investigation of longitudinal dispersion is carried out by measuring (with conductivity probes) the development of an initially sharp miscible displacement interface. The experimentally determined longitudinal dispersion coefficients are found to be greater in nonuniform media than in uniform media when compared using Peclet numbers based on the geometric mean grain diameter. The experimental breakthrough curves also display highly asymmetrical shapes, in which the "tail" of the breakthrough is longer than would be expected from advection-diffusion theory. Although the theoretical model does not predict the tailing behavior, it is found that the leading portion of the breakthrough curve is described by advection-diffusion theory. The theoretically determined longitudinal dispersion coefficients lie roughly within a factor of 1.35 of the measured values. The material presented in this report is essentially the same as the thesis submitted by the author in partial fulfillment of the requirements for the degree of Doctor of Philosophy