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There is an intense debate in the recent literature about the correct generalization of Maxwell's velocity distribution in special relativity. The most frequently discussed candidate distributions include the Jüttner function as well as modifications thereof. Here we report results from fully relativistic one-dimensional molecular dynamics simulations that resolve the ambiguity. The numerical evidence unequivocally favors the Jüttner distribution. Moreover, our simulations illustrate that the concept of ''thermal equilibrium'' extends naturally to special relativity only if a many-particle system is spatially confined. They make evident that ''temperature'' can be statistically defined and measured in an observer frame independent way. DOI: 10.1103/PhysRevLett.99.170601 PACS numbers: 05.70.ÿa, 02.70.Ns, 03.30.+p At the beginning of the last century, it was commonly accepted that the one-particle velocity distribution of a dilute gas in equilibrium is described by the Maxwellian probability density function (PDF)[m: rest mass of a gas particle; v: velocity; T k B ÿ1 : temperature; k B : Boltzmann constant; d: space dimension; throughout, we adopt natural units such that the speed of light c 1]. When Einstein [1,2] had formulated the theory of special relativity (SR) in 1905, Planck and others noted immediately that f M is in conflict with the fundamental relativistic postulate that velocities cannot exceed the light speed c. A first solution to this problem was put forward by Jüttner [3]. Starting from a maximum entropy principle, he proposed the following relativistic generalization of Maxwell's PDF:[Z J Z J m; J ; d : normalization constant; E m v m 2 p 2 1=2 : relativistic particle energy; p mv v : momentum with Lorentz factor v 1 ÿ v 2 ÿ1=2 , jvj < 1]. Jüttner's distribution (2) became widely accepted among theorists during the first threequarters of the 20th century [4 -8]-although a rigorous microscopic derivation is lacking due to the difficulty of formulating a relativistically consistent Hamilton mechanics of interacting particles [9][10][11][12][13]. Doubts about the Jüttner function f J began to arise in the 1980s, when Horwitz et al. [14,15] proposed a ''manifestly covariant'' relativistic Boltzmann equation, whose stationary solution differs from Eq. (2) and, in particular, predicts a different mean energy-temperature relation in the ultrarelativistic limit T ! 1 [16]. Since then, partially conflicting results and proposals from other authors [17][18][19][20][21] have led to an increasing confusion as to which distribution actually represents the correct generalization of the Maxwellian (1). For example, a recently discussed alternative to Eq. (2) is the ''modified'' Jüttner function [18,19] The distribution (3) can be obtained, e.g., by combining a maximum relative entropy principle and Lorentz symmetry [20]. Compared with f J at the same parameter values J MJ & 1=m, the modified PDF f MJ exhibits a significantly lower particle population in the high energy tail because of the additional 1=E pre...

The homogeneous cooling state of a granular flow of smooth spherical particles described by the Boltzmann equation is investigated by means of the direct simulation Monte Carlo method. The velocity moments and also the velocity distribution function are obtained and compared with approximate analytical results derived recently. The accuracy of a Maxwell-Boltzmann approximation with a time-dependent temperature is discussed. Besides, the simulations show that the state of uniform density is unstable to long enough wavelength perturbations so that clusters and voids spontaneously form throughout the system. The instability has the characteristic features of the clustering instability which has been observed in molecular dynamics simulations of dense fluids and predicted by hydrodynamic models of granular flows.

A self-diffusionequation for a freely evolving gas of inelastic hard disks or spheres is derived starting from the Boltzmann–Lorentz equation, by means of a Chapman–Enskog expansion in the density gradient of the tagged particles. The self-diffusion coefficient depends on the restitution coefficient explicitly, and also implicitly through the temperature of the system. This latter introduces also a time dependence of the coefficient. As in the elastic case, the results are trivially extended to the Enskog equation. The theoretical predictions are compared with numerical solutions of the kinetic equation obtained by the direct simulation Monte Carlo method, and also with molecular dynamics simulations. An excellent agreement is found, providing mutual support to the different approaches.The Dirección General de Investigación Científica y Técnica (Spain) through Grant No. PB98-112

Abstract. Some transport properties of granular gases are investigated. Starting from a kinetic theory level of description, the hydrodynamic transport equations to Navier-Stokes order are presented. The equations are derived by means of the Chapman-Enskog procedure. To test the existence of a normal solution and the possibility of a hydrodynamic description, the theoretical predictions are compared with numerical simulations of the underlying kinetic equation for small deviations around the reference homogeneous state. An excellent agreement is found for all the range of dissipation in collisions considered. Similar analysis is presented for self-diffusion and Brownian motion. In the former case, also Molecular Dynamics results are shown to agree with the theoretical predictions. Quantitative and also qualitative differences with the elastic limit are discussed.

The physical mechanisms leading to the development of density inhomogeneities in a freely evolving low density granular gas are investigated. By means of the direct simulation Monte Carlo method, numerical solutions of the inelastic Boltzmann equation are constructed for both a perturbed system and also for an initially homogeneous state. Analysis of the Fourier components of the fields indicates that the nonlinear coupling contributions of the transversal velocity play a crucial role in the initial setup of clustering. A simple hydrodynamic model is proposed to describe what is observed in the simulations. Finally, the nature of the inhomogeneous state is briefly discussed. ͓S1063-651X͑99͒08909-6͔

We use a fast Fourier transform block Lanczos diagonalization algorithm to study the electronic states of excess electrons in fluid alkanes ͑methane, ethane, and propane͒ and in a molecular model of amorphous polyethylene ͑PE͒ relevant to studies of space charge in insulating polymers. We obtain a new pseudopotential for electron-PE interactions by fitting to the electronic properties of fluid alkanes and use this to obtain new results for electron transport in amorphous PE. From our simulations, while the electronic states in fluid methane are extended throughout the whole sample, in amorphous PE there is a transition between localized and delocalized states slightly above the vacuum level ͑ϳϩ0.06 eV͒. The localized states in our amorphous PE model extend to Ϫ0.33 eV below this level. Using the Kubo-Greenwood equation we compute the zero-field electron mobility in pure amorphous PE to be Ϸ2ϫ10Ϫ3 cm 2 /V s. Our results highlight the importance of electron transport through extended states in amorphous regions to an understanding of electron transport in PE.

We demonstrate theoretically and experimentally the phenomenon of vibrational resonance in a periodic potential, using cold atoms in an optical lattice as a model system. A high-frequency (HF) drive, with a frequency much larger than any characteristic frequency of the system, is applied by phase modulating one of the lattice beams. We show that the HF drive leads to the renormalization of the potential. We used transport measurements as a probe of the potential renormalization. The very same experiments also demonstrate that transport can be controlled by the HF drive via potential renormalization.

The Enskog-Boltzmann equation for a homogeneous freely evolving system of smooth hard disks colliding inelastically is solved by means of the direct simulation Monte Carlo method. The distribution function shows an exponential high velocity tail, while it is Gaussian for small velocities. The numerical results are compared with recent predictions of approximate analytical theories and quite good agreement is found.

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