scholarly journals Expansion Formulae of the Free Energy and Distribution Functions in Powers of the One Particle Distribution Function

1959 ◽  
Vol 21 (4) ◽  
pp. 501-510 ◽  
Author(s):  
Tohru Morita
Keyword(s):  
Free Energy ◽  
Distribution Function ◽  
Particle Distribution ◽  
Distribution Functions ◽  
Particle Distribution Function ◽  
Expansion Formulae ◽  
The One

A study of the mass ratio dependence of the mixed species collision integral for isotropic velocity distribution functions

1982 ◽  
Vol 27 (1) ◽  
pp. 135-148 ◽  
Author(s):  
A. J. M. Garrett
Keyword(s):  
Distribution Function ◽  
Velocity Distribution ◽  
Particle Distribution ◽  
Velocity Distribution Function ◽  
Collision Integral ◽  
Distribution Functions ◽  
Velocity Space ◽  
Particle Distribution Function ◽  
Collision Term ◽  
Test Species

This paper is concerned with the Boltzmann collision integral for the one-particle distribution function of a test species of particle undergoing elastic collisions with particles of a second species which is in thermal equilibrium. This expression is studied as a function of the ratio of the masses of the test and host particles for the case when the test particle distribution function is isotropic in velocity space. The analysis can also be considered as referring to the zeroth-order spherical harmonic in velocity space of a general velocity distribution function. The resulting collision term, due originally to Davydov, is of Fokker–Planck form and effectively describes a diffusion in energy. The method of derivation employed here is more systematic than hitherto, and is used to calculate the first correction to the Davydov term. Differences between classical and quantum cross-sections are considered; the correction to the Davydov term is checked by means of a comparison with the exact solution of the associated eigenvalue problem for the special case of Maxwell interactions treated classically.

scholarly journals Correlation magnetodynamics equations taking into account the uniaxial quadratic correction in the approximation of the one-particle distribution function

2021 ◽  
pp. 1-16
Author(s):  
Anton Valerievich Ivanov
Keyword(s):  
Distribution Function ◽  
Particle Distribution ◽  
Nearest Neighbors ◽  
Particle Distribution Function ◽  
Phase Transition Region ◽  
Simulation Results ◽  
The One ◽  
Bogolyubov Chain ◽  
New System ◽  
Good Agreement

The system of equations for correlation magnetodynamics (CMD) is based on the Bogolyubov chain and approximation of the two-particle distribution function taking into account the correlations between the nearest neighbors. CMD provides good agreement with atom-for-atom simulation results (which are considered ab initio), but there is some discrepancy in the phase transition region. To solve this problem, a new system of CMD equations is constructed, which takes into account the quadratic correction in the approximation of the one-particle distribution function. The system can be simplified in a uniaxial case.

Statistical mechanics of transport coefficients

1976 ◽  
Vol 16 (3) ◽  
pp. 289-297 ◽  
Author(s):  
G. Vasu
Keyword(s):  
Distribution Function ◽  
Kinetic Equation ◽  
Statistical Mechanics ◽  
Particle Distribution ◽  
Transport Coefficients ◽  
Particle Distribution Function ◽  
Hydrodynamic Modes ◽  
The One ◽  
General Method

The problem of transport coefficients in statistical mechanics is reconsidered. A general method is given by which the hydrodynamical equations can straightforwardly obtained starting from the kinetic equation for the one-particle distribution function. From the statistical counterparts of the hydrodynamical equations so derived, the statistical expressions for the transport coefficients are immediately identified.Linearized hydrodynamic modes have recently been the object of very thorough reserach from the viewpoint of irreversible statistical mechanics; in particular, the Brussels school formalism has been used by Résibois to derive the eigenfrequencies of the hydrodynamical modes, whereby operatorial equations for transport coefficients have been obtained (Résibois 1970; see also the instructive book by Balescu (1975) on this subject).

A study of the mass ratio dependence of the mixed species collision integral

1982 ◽  
Vol 28 (2) ◽  
pp. 233-254 ◽  
Author(s):  
A. J. M. Garrett
Keyword(s):  
Distribution Function ◽  
Spherical Harmonic ◽  
Particle Distribution ◽  
Collision Integral ◽  
Distribution Functions ◽  
Particle Distribution Function ◽  
Isotropic Case ◽  
Mass Ratio ◽  
Test Species ◽  
Special Cases

This paper is concerned with the Boltzmann collision integral for the one-particle distribution function of a test species of particle undergoing elastic collisions with particles of a second species which is in thermal equilibrium. A previous paper studied this expression as a function of the mass ratio for the two species of particle when the test particle distribution function was isotropic in velocity space; this work generalizes that analysis to anisotropic distribution functions by expanding the distribution function in tensorial spherical harmonics. First the limit of zero mass ratio is considered: this simplifies the calculation dramatically. There is no contribution to the collision integral from the zeroth-order spherical harmonic in this limit. Then the main calculation shows how to find the terms arising from the existence of a finite mass ratio as an ascending power series in this quantity, and evaluates for each spherical harmonic the next term, linear in mass ratio. This is checked for two special cases: that of an isotropic distribution function, when the expression reduces to Davydov's form, and that arising from a cross-section inversely proportional to the collision velocity, when a comparison with the exact solution of the associated eigen problem can be made. As in the isotropic case, an exact representation of the collision integral as an expansion in mass ratio must include some terms non-analytic in this quantity and vanishing more quickly than any positive power: it is shown how these arise in the present formalism. The formulae derived here have applications to the transport theory of electrons and light ions in a predominantly neutral gas as governed by the Boltzmann equation.

IRREVERSIBILITY IN RELATIVISTIC STATISTICAL MECHANICS

1994 ◽  
Vol 08 (29) ◽  
pp. 1847-1860 ◽  
Author(s):  
URI BEN-YA’ACOV
Keyword(s):  
Distribution Function ◽  
Statistical Mechanics ◽  
Particle Distribution ◽  
Equations Of Motion ◽  
Hamiltonian Formalism ◽  
Distribution Functions ◽  
Relativistic Dynamics ◽  
Direct Computation ◽  
Lagrangian Equations ◽  
The One

Relativistic statistical mechanics should be manifestly Lorentz covariant. In the absence of a Hamiltonian formalism in relativistic dynamics, a different approach which is based on the (Lagrangian) equations of motion is presented. Without any Liouville equation, this approach provides the direct computation of all the reduced n-particle distribution functions. The trajectories in the fully interacting system and ensemble averages are defined with respect to the parameters that fix the trajectories in the interaction-free limit. Irreversibility may emerge from microscopic dynamics due to the choice as to which part of the particles’ history — past or future — contributes to the interaction. Irreversibility is explicitly demonstrated in the evolution of the one-particle distribution function.

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