Exact solution of a one-dimensional Boltzmann equation for a granular tracer particle

We consider a one-dimensional system consisting of a granular tracer particle of mass $M$ in a bath of thermalized particles each of mass $m$. When the mass ratio, $M/m$, is equal to the coefficient of restitution, $\alpha$, the system maps to a a one-dimensional elastic gas. In this case, Boltzmann equation can be solved exactly. We also obtain expressions for the velocity autocorrelation function and the diffusion coefficient. Numerical simulations of the Boltzmann equation are performed for $M/m\neq \alpha$ where no analytical solution is available. It appears that the dynamical features remain qualitatively similar to those found in the exactly solvable case.Comment: 17 pages, 3 figures, Accepted in Physica

Shear and Bulk Viscosities of a Gluon Plasma in Perturbative QCD: Comparison of Different Treatments for the gg<->ggg Process

The leading order contribution to the shear and bulk viscosities, \eta and \zeta, of a gluon plasma in perturbative QCD includes the gg -> gg (22) process, gg ggg (23) process and multiple scattering processes known as the Landau-Pomeranchuk-Migdal (LPM) effect. Complete leading order computations for \eta and \zeta were obtained by Arnold, Moore and Yaffe (AMY) and Arnold, Dogan and Moore (ADM), respectively, with the inelastic processes computed by an effective g gg gluon splitting. We study how complementary calculations with 22 and 23 processes and a simple treatment to model the LPM effect compare with the results of AMY and ADM. We find that our results agree with theirs within errors. By studying the contribution of the 23 process to \eta, we find that the minimum angle \theta among the final state gluons in the fluid local rest frame has a distribution that is peaked at \theta \sim \sqrt{\alpha_{s}}, analogous to the near collinear splitting asserted by AMY and ADM. However, the average of \theta is much bigger than its peak value, as its distribution is skewed with a long tail. The same \theta behavior is also seen if the 23 matrix element is taken to the soft gluon bremsstrahlung limit in the center-of-mass (CM) frame. This suggests that the soft gluon bremsstrahlung in the CM frame still has some near collinear behavior in the fluid local rest frame. We also generalize our result to a general SU(N_c) pure gauge theory and summarize the current viscosity computations in QCD.Comment: ReVTex 4, 18 pages, 7 figures, accepted version in Phys. Rev.

Tagged particle in a sheared suspension: effective temperature determines density distribution in a slowly varying external potential beyond linear response

We consider a sheared colloidal suspension under the influence of an external potential that varies slowly in space in the plane perpendicular to the flow and acts on one selected (tagged) particle of the suspension. Using a Chapman-Enskog type expansion we derive a steady state equation for the tagged particle density distribution. We show that for potentials varying along one direction only, the tagged particle distribution is the same as the equilibrium distribution with the temperature equal to the effective temperature obtained from the violation of the Einstein relation between the self-diffusion and tagged particle mobility coefficients. We thus prove the usefulness of this effective temperature for the description of the tagged particle behavior beyond the realm of linear response. We illustrate our theoretical predictions with Brownian dynamics computer simulations.Comment: Accepted for publication in Europhys. Let

Shear viscosity of a superfluid Fermi gas in the unitarity limit

We compute the shear viscosity of a superfluid atomic Fermi gas in the unitarity limit. The unitarity limit is characterized by a divergent scattering length between the atoms, and it has been argued that this will result in a very small viscosity. We show that in the low temperature T limit the shear viscosity scales as xi^5/T^5, where the universal parameter 'xi' relates the chemical potential and the Fermi energy, mu=xi E_F. Combined with the high temperature expansions of the viscosity our results suggest that the viscosity has a minimum near the critical temperature T_c. A naive extrapolation indicates that the minimum value of the ratio of viscosity over entropy density is within a factor of ~ 5 of the proposed lower bound hbar/(4\pi k_B).Comment: 9 pages, 7 figures, LaTeX2

N-body decomposition of bipartite networks

In this paper, we present a method to project co-authorship networks, that accounts in detail for the geometrical structure of scientists collaborations. By restricting the scope to 3-body interactions, we focus on the number of triangles in the system, and show the importance of multi-scientists (more than 2) collaborations in the social network. This motivates the introduction of generalized networks, where basic connections are not binary, but involve arbitrary number of components. We focus on the 3-body case, and study numerically the percolation transition.Comment: 5 pages, submitted to PR

Generalized Fokker-Planck equation, Brownian motion, and ergodicity

Microscopic theory of Brownian motion of a particle of mass $M$ in a bath of molecules of mass $m\ll M$ is considered beyond lowest order in the mass ratio $m/M$. The corresponding Langevin equation contains nonlinear corrections to the dissipative force, and the generalized Fokker-Planck equation involves derivatives of order higher than two. These equations are derived from first principles with coefficients expressed in terms of correlation functions of microscopic force on the particle. The coefficients are evaluated explicitly for a generalized Rayleigh model with a finite time of molecule-particle collisions. In the limit of a low-density bath, we recover the results obtained previously for a model with instantaneous binary collisions. In general case, the equations contain additional corrections, quadratic in bath density, originating from a finite collision time. These corrections survive to order $(m/M)^2$ and are found to make the stationary distribution non-Maxwellian. Some relevant numerical simulations are also presented

Strong Shock Waves and Nonequilibrium Response in a One-dimensional Gas: a Boltzmann Equation Approach

We investigate the nonequilibrium behavior of a one-dimensional binary fluid on the basis of Boltzmann equation, using an infinitely strong shock wave as probe. Density, velocity and temperature profiles are obtained as a function of the mixture mass ratio \mu. We show that temperature overshoots near the shock layer, and that heavy particles are denser, slower and cooler than light particles in the strong nonequilibrium region around the shock. The shock width w(\mu), which characterizes the size of this region, decreases as w(\mu) ~ \mu^{1/3} for \mu-->0. In this limit, two very different length scales control the fluid structure, with heavy particles equilibrating much faster than light ones. Hydrodynamic fields relax exponentially toward equilibrium, \phi(x) ~ exp[-x/\lambda]. The scale separation is also apparent here, with two typical scales, \lambda_1 and \lambda_2, such that \lambda_1 ~ \mu^{1/2} as \mu-->0$, while \lambda_2, which is the slow scale controlling the fluid's asymptotic relaxation, increases to a constant value in this limit. These results are discussed at the light of recent numerical studies on the nonequilibrium behavior of similar 1d binary fluids.Comment: 9 pages, 8 figs, published versio

Alignment of Rods and Partition of Integers

We study dynamical ordering of rods. In this process, rod alignment via pairwise interactions competes with diffusive wiggling. Under strong diffusion, the system is disordered, but at weak diffusion, the system is ordered. We present an exact steady-state solution for the nonlinear and nonlocal kinetic theory of this process. We find the Fourier transform as a function of the order parameter, and show that Fourier modes decay exponentially with the wave number. We also obtain the order parameter in terms of the diffusion constant. This solution is obtained using iterated partitions of the integer numbers.Comment: 6 pages, 4 figure

One-particle and few-particle billiards

We study the dynamics of one-particle and few-particle billiard systems in containers of various shapes. In few-particle systems, the particles collide elastically both against the boundary and against each other. In the one-particle case, we investigate the formation and destruction of resonance islands in (generalized) mushroom billiards, which are a recently discovered class of Hamiltonian systems with mixed regular-chaotic dynamics. In the few-particle case, we compare the dynamics in container geometries whose counterpart one-particle billiards are integrable, chaotic, and mixed. One of our findings is that two-, three-, and four-particle billiards confined to containers with integrable one-particle counterparts inherit some integrals of motion and exhibit a regular partition of phase space into ergodic components of positive measure. Therefore, the shape of a container matters not only for noninteracting particles but also for interacting particles

Singular forces and point-like colloids in lattice Boltzmann hydrodynamics

We present a second-order accurate method to include arbitrary distributions of force densities in the lattice Boltzmann formulation of hydrodynamics. Our method may be used to represent singular force densities arising either from momentum-conserving internal forces or from external forces which do not conserve momentum. We validate our method with several examples involving point forces and find excellent agreement with analytical results. A minimal model for dilute sedimenting particles is presented using the method which promises a substantial gain in computational efficiency.Comment: 22 pages, 9 figures. Submitted to Phys. Rev.