Kinetic Theory of Soft Matter. The Penetrable-Sphere Model

The penetrable-sphere model has been introduced in the literature to describe the peculiar thermodynamic behavior of some colloidal systems. In this model the interaction potential is $\phi(r)=\epsilon>0$ if the two spheres are overlapped ($r\sigma$). In this paper the shear viscosity, thermal conductivity, and self-diffusion coefficients of a dilute gas of penetrable spheres are evaluated. It is found that the effective collision frequency $\nu(T^*)$ grows as $\sqrt{T^*}$ up to $T^*\equiv k_BT/\epsilon\simeq 0.25$, reaches a maximum at $T^*\simeq 0.415$ and then decays as ${T^*}^{-3/2}\log T^*$ for large temperatures. The results are applied to the hydrodynamic profiles in the steady Fourier and Couette flows.Comment: 6 pages, 4 figures; to appear in Rarefied Gas Dynamics: 24th International Symposium (AIP Conference Proceedings

The orbital counting problem for hyperconvex representations

We give a precise counting result on the symmetric space of a noncompact real algebraic semisimple group $G,$ for a class of discrete subgroups of $G$ that contains, for example, representations of a surface group on $\textrm{PSL}(2,\mathbb R)\times\textrm{PSL}(2,\mathbb R),$ induced by choosing two points on the Teichm\"uller space of the surface; and representations on the Hitchin component of $\textrm{PSL}(d,\mathbb R).$ We also prove a mixing property for the Weyl chamber flow in this setting

A simple model kinetic equation for inelastic Maxwell particles

The model of inelastic Maxwell particles (IMP) allows one to derive some exact results which show the strong influence of inelasticity on the nonequilibrium properties of a granular gas. The aim of this work is to propose a simple model kinetic equation that preserves the most relevant properties of the Boltzmann equation (BE) for IMP and reduces to the BGK kinetic model in the elastic limit. In the proposed kinetic model the collision operator is replaced by a relaxation-time term toward a reference Maxwellian distribution plus a term representing the action of a friction force. It contains three parameters (the relaxation rate, the effective temperature of the reference Maxwellian, and the friction coefficient) which are determined by imposing consistency with basic exact properties of the BE for IMP. As a consequence, the kinetic model reproduces the true shear viscosity and predicts accurate expressions for the transport coefficients associated with the heat flux. The model can be exactly solved for the homogeneous cooling state, the solution exhibiting an algebraic high-energy tail with an exponent in fair agreement with the correct one.Comment: 6 pages, 2 figures; presented in the 25th International Symposium on Rarefied Gas Dynamics (Saint-Petersburg, Russia, July 21-28, 2006

A note on the path integral representation for Majorana fermions

Majorana fermions are currently of huge interest in the context of nanoscience and condensed matter physics. Different to usual fermions, Majorana fermions have the property that the particle is its own anti-particle thus, they must be described by real fields. Mathematically, this property makes nontrivial the quantization of the problem due, for instance, to the absence of a Wick-like theorem. In view of the present interest on the subject, it is important to develop different theoretical approaches in order to study problems where Majorana fermions are involved. In this note we show that Majorana fermions can be studied in the context of field theories for constrained systems. Using the Faddeev-Jackiw formalism for quantum field theories with constraints, we derived the path integral representation for Majorana fermions. In order to show the validity of the path integral we apply it to an exactly solvable problem. This application also shows that it is rather simple to perform systematic calculations on the basis of the present framework.Comment: 7 pages, to be published in Journal of Physics A: Mathematical and Theoretica