Geometric partition functions of cellular systems: Explicit calculation of the entropy in two and three dimensions

A method is proposed for the characterisation of the entropy of cellular structures, based on the compactivity concept for granular packings. Hamiltonian-like volume functions are constructed both in two and in three dimensions, enabling the identification of a phase space and making it possible to take account of geometrical correlations systematically. Case studies are presented for which explicit calculations of the mean vertex density and porosity fluctuations are given as functions of compactivity. The formalism applies equally well to two- and three-dimensional granular assemblies.Comment: 14 pages, 4 figures, to appear in The European Physical Journal E - Soft Matte

Inter-dependence of the volume and stress ensembles and equipartition in statistical mechanics of granular systems

We discuss the statistical mechanics of granular matter and derive several significant results. First, we show that, contrary to common belief, the volume and stress ensembles are inter-dependent, necessitating the use of both. We use the combined ensemble to calculate explicitly expectation values of structural and stress-related quantities for two-dimensional systems. We thence demonstrate that structural properties may depend on the angoricity tensor and that stress-based quantities may depend on the compactivity. This calls into question previous statistical mechanical analyses of static granular systems and related derivations of expectation values. Second, we establish the existence of an intriguing equipartition principle - the total volume is shared equally amongst both structural and stress-related degrees of freedom. Third, we derive an expression for the compactivity that makes it possible to quantify it from macroscopic measurements.Comment: 5 pages, including 2 figures, To appear in Phys. Rev. Let

Singular Laplacian Growth

The general equations of motion for two dimensional Laplacian growth are derived using the conformal mapping method. In the singular case, all singularities of the conformal map are on the unit circle, and the map is a degenerate Schwarz-Christoffel map. The equations of motion describe the motions of these singularities. Despite the typical fractal-like outcomes of Laplacian growth processes, the equations of motion are shown to be not particularly sensitive to initial conditions. It is argued that the sensitivity of this system derives from a novel cause, the non-uniqueness of solutions to the differential system. By a mechanism of singularity creation, every solution can become more complex, even in the absence of noise, without violating the growth law. These processes are permitted, but are not required, meaning the equation of motion does not determine the motion, even in the small.Comment: 8 pages, Latex, 4 figures, Submitted to Phys. Rev.

Successful Cessation Programs that Reduce Comorbidity May Explain Surprisingly Low Smoking Rates Among Hospitalized COVID-19 Patients

A recent, non-peer-reviewed meta-analysis suggests that smoking may reduce the risk of hospitalization with COVID-19 because the prevalence of smoking among hospitalized COVID-19 is less than that of the general population. However, there are alternative explanations for this phenomena based on (1) the failure to report, or accurately record, smoking history during emergency hospital admissions and (2) a pre-disposition to avoid smoking among COVID-19 patients with tobacco-related comorbidities (a type of “reverse” causation). For example, urine testing of hospitalized patients in Australia for cotinine showed that smokers were under-counted by 37% because incoming patients failed to inform staff about their smoking behavior. Face-to-face interviews can introduce bias into the responses to attitudinal and behavioral questions not present in the self-completion interviews typically used to measure smoking prevalence in the general population. Subjects in face-to-face interviews may be unwilling to admit socially undesirable behavior and attitudes under direct questioning. Reverse causation may also contribute to the difference between smoking prevalence in the COVID-19 and general population. Patients hospitalized with COVID-19 may be simply less prone to use tobacco than the general population. A potentially robust “reverse causation” hypothesis for reduced prevalence of smokers in the COVID-19 population is the enrichment of patients in that population with serious comorbidities that motivates them to quit smoking. We judge that this “smoking cessation” mechanism may account for a significant fraction of the reduced prevalence of smokers in the COVID-19 population. Testing this hypothesis will require a focused research program