48 research outputs found
Generalized Fokker-Planck equation, Brownian motion, and ergodicity
Microscopic theory of Brownian motion of a particle of mass in a bath of
molecules of mass is considered beyond lowest order in the mass ratio
. 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
and are found to make the stationary distribution non-Maxwellian.
Some relevant numerical simulations are also presented
Non-equilibrium phase transition in a sheared granular mixture
The dynamics of an impurity (or tracer particle) immersed in a dilute
granular gas under uniform shear flow is investigated. A non-equilibrium phase
transition is identified from an exact solution of the inelastic Boltzmann
equation for a granular binary mixture in the tracer limit, where the impurity
carries either a vanishing (disordered phase) or a finite (ordered phase)
fraction of the total kinetic energy of the system. In the disordered phase,
the granular temperature ratio (impurity "temperature" over that of the host
fluid) is finite, while it diverges in the ordered phase. To correctly capture
this extreme violation of energy equipartition, we show that the picture of an
impurity enslaved to the host fluid is insufficient
Velocity Distributions of Granular Gases with Drag and with Long-Range Interactions
We study velocity statistics of electrostatically driven granular gases. For
two different experiments: (i) non-magnetic particles in a viscous fluid and
(ii) magnetic particles in air, the velocity distribution is non-Maxwellian,
and its high-energy tail is exponential, P(v) ~ exp(-|v|). This behavior is
consistent with kinetic theory of driven dissipative particles. For particles
immersed in a fluid, viscous damping is responsible for the exponential tail,
while for magnetic particles, long-range interactions cause the exponential
tail. We conclude that velocity statistics of dissipative gases are sensitive
to the fluid environment and to the form of the particle interaction.Comment: 4 pages, 3 figure
Transport coefficients for dense hard-disk systems
A study of the transport coefficients of a system of elastic hard disks,
based on the use of Helfand-Einstein expressions is reported. The
self-diffusion, the viscosity, and the heat conductivity are examined with
averaging techniques especially appropriate for the use in event-driven
molecular dynamics algorithms with periodic boundary conditions. The density
and size dependence of the results is analyzed, and comparison with the
predictions from Enskog's theory is carried out. In particular, the behavior of
the transport coefficients in the vicinity of the fluid-solid transition is
investigated and a striking power law divergence of the viscosity in this
region is obtained, while all other examined transport coefficients show a drop
in that density range.Comment: submitted to PR
Thermal Segregation Beyond Navier-Stokes
A dilute suspension of impurities in a low density gas is described by the
Boltzmann and Boltzman-Lorentz kinetic theory. Scaling forms for the species
distribution functions allow an exact determination of the hydrodynamic fields,
without restriction to small thermal gradients or Navier-Stokes hydrodynamics.
The thermal diffusion factor characterizing sedimentation is identified in
terms of collision integrals as functions of the mechanical properties of the
particles and the temperature gradient. An evaluation of the collision
integrals using Sonine polynomial approximations is discussed. Conditions for
segregation both along and opposite the temperature gradient are found, in
contrast to the Navier-Stokes description for which no segregation occurs.Comment: 9 figure
Ballistic Annihilation
Ballistic annihilation with continuous initial velocity distributions is
investigated in the framework of Boltzmann equation. The particle density and
the rms velocity decay as and , with the
exponents depending on the initial velocity distribution and the spatial
dimension. For instance, in one dimension for the uniform initial velocity
distribution we find . We also solve the Boltzmann equation
for Maxwell particles and very hard particles in arbitrary spatial dimension.
These solvable cases provide bounds for the decay exponents of the hard sphere
gas.Comment: 4 RevTeX pages and 1 Eps figure; submitted to Phys. Rev. Let
Fluctuating hydrodynamics for dilute granular gases
Starting from the kinetic equations for the fluctuations and correlations of
a dilute gas of inelastic hard spheres or disks, a Boltzmann-Langevin equation
for the one-particle distribution function of the homogeneous cooling state is
constructed. This equation is the linear Boltzmann equation with a fluctuating
white noise term. Balance equations for the fluctuating hydrodynamic fields are
derived. New fluctuating forces appear as compared with the elastic limit. The
particular case of the transverse velocity field is investigated in detail. Its
fluctuations can be described by means of a Langevin equation, but exhibiting
two main differences with the Landau-Lifshitz theory: the noise is not white,
and its second moment is not determined by the shear viscosity. This shows that
the fluctuation-dissipation relations for molecular fluids do not
straightforwardly carry over to inelastic gases. The theoretical predictions
are shown to be in good agreement with molecular dynamics simulation results
Kinetic Theory of a Dilute Gas System under Steady Heat Conduction
The velocity distribution function of the steady-state Boltzmann equation for
hard-core molecules in the presence of a temperature gradient has been obtained
explicitly to second order in density and the temperature gradient. Some
thermodynamical quantities are calculated from the velocity distribution
function for hard-core molecules and compared with those for Maxwell molecules
and the steady-state Bhatnagar-Gross-Krook(BGK) equation. We have found
qualitative differences between hard-core molecules and Maxwell molecules in
the thermodynamical quantities, and also confirmed that the steady-state BGK
equation belongs to the same universality class as Maxwell molecules.Comment: 36 pages, 4 figures, 5 table
Maxwell Model of Traffic Flows
We investigate traffic flows using the kinetic Boltzmann equations with a
Maxwell collision integral. This approach allows analytical determination of
the transient behavior and the size distributions. The relaxation of the car
and cluster velocity distributions towards steady state is characterized by a
wide range of velocity dependent relaxation scales, , with
the ratio of the passing and the collision rates. Furthermore, these
relaxation time scales decrease with the velocity, with the smallest scale
corresponding to the decay of the overall density. The steady state cluster
size distribution follows an unusual scaling form . This distribution is primarily algebraic, , for , and is exponential otherwise.Comment: revtex, 10 page
Non-commutative Geometry and Kinetic Theory of Open Systems
The basic mathematical assumptions for autonomous linear kinetic equations
for a classical system are formulated, leading to the conclusion that if they
are differential equations on its phase space , they are at most of the 2nd
order. For open systems interacting with a bath at canonical equilibrium they
have a particular form of an equation of a generalized Fokker-Planck type. We
show that it is possible to obtain them as Liouville equations of Hamiltonian
dynamics on with a particular non-commutative differential structure,
provided certain geometric in character, conditions are fulfilled. To this end,
symplectic geometry on is developped in this context, and an outline of the
required tensor analysis and differential geometry is given. Certain questions
for the possible mathematical interpretation of this structure are also
discussed.Comment: 22 pages, LaTe