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Hamilton-Jacobi Approach for Power-Law Potentials
The classical and relativistic Hamilton-Jacobi approach is applied to the
one-dimensional homogeneous potential, , where and
are continuously varying parameters. In the non-relativistic case, the
exact analytical solution is determined in terms of , and the total
energy . It is also shown that the non-linear equation of motion can be
linearized by constructing a hypergeometric differential equation for the
inverse problem . A variable transformation reducing the general problem
to that one of a particle subjected to a linear force is also established. For
any value of , it leads to a simple harmonic oscillator if , an
"anti-oscillator" if , or a free particle if E=0. However, such a
reduction is not possible in the relativistic case. For a bounded relativistic
motion, the first order correction to the period is determined for any value of
. For , it is found that the correction is just twice that one
deduced for the simple harmonic oscillator (), and does not depend on the
specific value of .Comment: 12 pages, Late
Demixing can occur in binary hard-sphere mixtures with negative non-additivity
A binary fluid mixture of non-additive hard spheres characterized by a size
ratio and a non-additivity parameter
is considered in infinitely many
dimensions. From the equation of state in the second virial approximation
(which is exact in the limit ) a demixing transition with a
critical consolute point at a packing fraction scaling as
is found, even for slightly negative non-additivity, if
. Arguments concerning the stability of the
demixing with respect to freezing are provided.Comment: 4 pages, 2 figures; title changed; final paragraph added; to be
published in PRE as a Rapid Communicatio
Computer simulation of uniformly heated granular fluids
Direct Monte Carlo simulations of the Enskog-Boltzmann equation for a
spatially uniform system of smooth inelastic spheres are performed. In order to
reach a steady state, the particles are assumed to be under the action of an
external driving force which does work to compensate for the collisional loss
of energy. Three different types of external driving are considered: (a) a
stochastic force, (b) a deterministic force proportional to the particle
velocity and (c) a deterministic force parallel to the particle velocity but
constant in magnitude. The Enskog-Boltzmann equation in case (b) is fully
equivalent to that of the homogeneous cooling state (where the thermal velocity
monotonically decreases with time) when expressed in terms of the particle
velocity relative to the thermal velocity. Comparison of the simulation results
for the fourth cumulant and the high energy tail with theoretical predictions
derived in cases (a) and (b) [T. P. C. van Noije and M. H. Ernst, Gran. Matt.
1, 57 (1998)] shows a good agreement. In contrast to these two cases, the
deviation from the Maxwell-Boltzmann distribution is not well represented by
Sonine polynomials in case (c), even for low dissipation. In addition, the high
energy tail exhibits an underpopulation effect in this case.Comment: 18 pages (LaTex), 10 figures (eps); to be published in Granular
Matte
A square-well model for the structural and thermodynamic properties of simple colloidal systems
A model for the radial distribution function of a square-well fluid of
variable width previously proposed [S. B. Yuste and A. Santos, J. Chem. Phys.
{\bf 101}, 2355 (1994)] is revisited and simplified. The model provides an
explicit expression for the Laplace transform of , the coefficients
being given as explicit functions of the density, the temperature, and the
interaction range. In the limits corresponding to hard spheres and sticky hard
spheres the model reduces to the analytical solutions of the Percus-Yevick
equation for those potentials. The results can be useful to describe in a fully
analytical way the structural and thermodynamic behavior of colloidal
suspensions modeled as hard-core particles with a short-range attraction.
Comparison with computer simulation data shows a general good agreement, even
for relatively wide wells.Comment: 23 pages, 10 figures; Figs. 4 and 5 changed, Fig. 6 new; to be
published in J. Chem. Phy
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