Numerical Study of Nonequilibrium Diagram Theory

1972 ◽  
Vol 50 (4) ◽  
pp. 317-335 ◽  
Author(s):  
Gary R. Dowling ◽  
H. Ted Davis
Keyword(s):  
Distribution Function ◽  
Correlation Function ◽  
Kinetic Equation ◽  
Numerical Study ◽  
Pair Correlation ◽  
Transport Coefficients ◽  
Kinetic Equations ◽  
Collision Operator ◽  
Dense Fluid ◽  
Equilibrium Pair

In this paper we numerically analyze the first few diagrams in a Boltzmann-like collision operator that occurs in Severne's exact kinetic equation for the singlet distribution function. A similar analysis was used by Allen and Cole in deriving their singlet and doublet kinetic equations. Our analysis shows that the diagrams neglected by Allen and Cole in their kinetic equations are not negligible and these should be incorporated into dense fluid theories. The Allen–Cole kinetic transport coefficients and equilibrium pair correlation function are presented and calculated for dense argon. These results are not promising.

Collisional transport for a superthermal ion species in magnetized plasma

1986 ◽  
Vol 36 (3) ◽  
pp. 313-328 ◽  
Author(s):  
F. Cozzani ◽  
W. Horton
Keyword(s):  
Kinetic Equation ◽  
Transport Theory ◽  
A Priori ◽  
Transport Coefficients ◽  
Kinetic Equations ◽  
High Energy ◽  
Magnetized Plasma ◽  
Fractional Density ◽  
Background Ions ◽  
Ion Species

The transport theory of a high-energy ion species injected isotropically in a magnetized plasma is considered for arbitrary ratios of the high-energy ion cyclotron frequency to the collisional slowing down time. The assumptions of (i) low fractional density of the high-energy species and (ii) average ion speed faster than the thermal ions and slower than the electrons are used to decouple the kinetic equation for the high-energy species from the kinetic equations for background ions and electrons. The kinetic equation is solved by a Chapman–Enskog expansion in the strength of the gradients; an equation for the first correction to the lowest-order distribution function is obtained without scaling a priori the collision frequency with respect to the gyrofrequency. Various transport coefficients are explicitly calculated for the two cases of a weakly and a strongly magnetized plasma.

scholarly journals Projection-operator methods for classical transport in magnetized plasmas. Part 1. Linear response, the Braginskii equations and fluctuating hydrodynamics

2018 ◽  
Vol 84 (4) ◽  
Author(s):  
John A. Krommes
Keyword(s):  
Distribution Function ◽  
Projection Operator ◽  
Linear Response ◽  
Transport Coefficients ◽  
Collision Operator ◽  
Transport Equations ◽  
Transport Processes ◽  
Projection Operators ◽  
Covariant Representation ◽  
Classical Transport

An introduction to the use of projection-operator methods for the derivation of classical fluid transport equations for weakly coupled, magnetised, multispecies plasmas is given. In the present work, linear response (small perturbations from an absolute Maxwellian) is addressed. In the Schrödinger representation, projection onto the hydrodynamic subspace leads to the conventional linearized Braginskii fluid equations when one restricts attention to fluxes of first order in the gradients, while the orthogonal projection leads to an alternative derivation of the Braginskii correction equations for the non-hydrodynamic part of the one-particle distribution function. The projection-operator approach provides an appealingly intuitive way of discussing the derivation of transport equations and interpreting the significance of the various parts of the perturbed distribution function; it is also technically more concise. A special case of the Weinhold metric is used to provide a covariant representation of the formalism; this allows a succinct demonstration of the Onsager symmetries for classical transport. The Heisenberg representation is used to derive a generalized Langevin system whose mean recovers the linearized Braginskii equations but that also includes fluctuating forces. Transport coefficients are simply related to the two-time correlation functions of those forces, and physical pictures of the various transport processes are naturally couched in terms of them. A number of appendices review the traditional Chapman–Enskog procedure; record some properties of the linearized Landau collision operator; discuss the covariant representation of the hydrodynamic projection; provide an example of the calculation of some transport effects; describe the decomposition of the stress tensor for magnetised plasma; introduce the linear eigenmodes of the Braginskii equations; and, with the aid of several examples, mention some caveats for the use of projection operators.

Bootstrap-current computations for a stellarator-reactor plasma

1990 ◽  
Vol 44 (3) ◽  
pp. 431-453 ◽  
Author(s):  
W. D. D'Haeseleer ◽  
W. N. G. Hitchon ◽  
C. D. Beidler ◽  
J. L. Shohet
Keyword(s):  
Distribution Function ◽  
Kinetic Equation ◽  
Physical Interpretation ◽  
Collision Operator ◽  
Numerical Code ◽  
Parallel Flows ◽  
Bootstrap Current ◽  
The Difference ◽  
Rotational Transform ◽  
The Impact

Numerical results for the bootstrap current in a stellarator-reactor plasma are presented. The distribution function f is computed numerically from a kinetic equation that is averaged over the helical ripple. The parallel flows and the current are obtained as v‖ moments of f. The physics issues embedded in the code are discussed concisely, concentrating on the justification as to why the bootstrap current can be estimated from an averaged scheme. Results are presented for typical stellarator-reactor parameters. The numerical code FLOCS predicts that the momentum-restoring terms in the collision operator have no significant impact on the value of the bootstrap current (the difference being about 10%). The results obtained are related to the equilibrium flows, and a physical interpretation based on the kinetic picture is presented. Finally, an estimate for the impact of J‖ on the rotational transform is given.

scholarly journals A retarded time superposition principle and the relativistic collision operator

1983 ◽  
Vol 30 (2) ◽  
pp. 223-248 ◽  
Author(s):  
K. Hizanidis ◽  
K. Molvig ◽  
K. Swartz
Keyword(s):  
Correlation Function ◽  
Superposition Principle ◽  
Pair Correlation ◽  
Collision Operator ◽  
Retarded Time ◽  
Particle Correlation ◽  
Relativistic Collision ◽  
Point Correlation ◽  
Electro Magnetic ◽  
Klimontovich Equation

A retarded time superposition principle is formulated and proved for the two particle correlation function in a multi-species relativistic, and fully electro-magnetic, plasma. This principle is used to obtain the relativistic collision operator. Starting from the relativistic Klimontovich equation, the statistical (Liouville) average of the Klimontovich equation yields an expression for the collision operator in terms of the two-time two-point correlation function for two particles, G12(1, t12, t2). It is proved that G12(1, t12, t2) can be written in a retarded time superposition form in terms of the self-correlation W11(1, t12, t2) and the discreteness response function P(1, t12, t2). The equation for the pair correlation function G12(1, t12, t2), excluding triplet or higher-order correlations, is thus replaced by the simpler equation for P(1, t12, t2). This is the test particle problem which relates P(1, t12, t2) to the discreteness source term W11(1, t12, t2). The equations for P(1, t12, t2). and W11(1, t12, t2) are solved for stationary, homogeneous plasmas without external fields. With these solutions, the collision operator is expressed in terms of the relativistic dielectric properties of the plasma.

scholarly journals SPATIAL STATISTICS FOR SIMULATED PACKINGS OF SPHERES

2011 ◽  
Vol 20 (3) ◽  
pp. 203 ◽  
Author(s):  
Alexander Bezrukov ◽  
Dietrich Stoyan ◽  
Monika Bargieł
Keyword(s):  
Distribution Function ◽  
Correlation Function ◽  
Spatial Statistics ◽  
Volume Fraction ◽  
Pair Correlation ◽  
Simulation Methods ◽  
Coordination Numbers ◽  
Contact Distribution ◽  
Contact Distribution Function ◽  
Random Packings

This paper reports on spatial-statistical analyses for simulated random packings of spheres with random diameters. The simulation methods are the force-biased algorithm and the Jodrey-Tory sedimentation algorithm. The sphere diameters are taken as constant or following a bimodal or lognormal distribution. Standard characteristics of spatial statistics are used to describe these packings statistically, namely volume fraction, pair correlation function of the system of sphere centres and spherical contact distribution function of the set-theoretic union of all spheres. Furthermore, the coordination numbers are analysed.

scholarly journals Spatial patterns of primary seed dispersal and adult tree distributions: Genipa americana dispersed by Cebus capucinus – CORRIGENDUM

2015 ◽  
Vol 32 (1) ◽  
pp. 88-88
Author(s):  
Kim Valenta ◽  
Mariah E. Hopkins ◽  
Melanie Meeking ◽  
Colin A. Chapman ◽  
Linda M. Fedigan
Keyword(s):  
Distribution Function ◽  
Correlation Function ◽  
Pair Correlation Function ◽  
Pair Correlation ◽  
Nearest Neighbour ◽  
Study Region ◽  
Adult Tree ◽  
The Mean ◽  
K Function ◽  
Typical Point

Within the second paragraph of page 494 incorrect language was used to characterize the summary characteristics used. Sentences 3–11 of this paragraph should have read:Second, we calculated three univariate summary characteristics: the nearest neighbour distribution function D(r), the pair-correlation function g(r) and the K-function K(r). The use of multiple summary characteristics holds increased power to characterize variation in spatial patterns (Wiegand et al. 2013). The univariate nearest neighbour distribution function D(r) can be interpreted as the probability that the typical adult tree has its nearest neighbouring adult tree within radius r (or alternatively, the probability that the typical defecation has its nearest neighbouring defecation within radius r). The univariate pair-correlation function g(r) is a non-cumulative normalized neighbourhood density function that gives the expected number of points within rings of radius r and width w centred on a typical point, divided by the mean density of points λ in the study region (Wiegand et al. 2009). We applied g(r) to trees and defecation point patterns separately, using a ring width of 10 m. The K-function K(r) provides a cumulative counterpart to the non-cumulative pair-correlation function g(r) by analysing dispersion and aggregation up to distance r rather than at distance r (Weigand & Moloney 2004). The K-function can be defined as the number of expected points (i.e. either trees or defecations) within circles of radius r extending from a typical point, divided by the mean density of points λ within the study region. Here, we apply the square root transformation L(r) to the K-function to remove scale dependence and stabilize the variance: $L( r ) = \scriptstyle\sqrt {\frac{{K( r )}}{\pi }} - r$ (Besag 1977, Wiegand & Moloney 2014).

Examples of two point kinetic equations obeying the entropy principle

1981 ◽  
Vol 59 (9) ◽  
pp. 1241-1250 ◽  
Author(s):  
Miroslav Grmela ◽  
Jean Salmon
Keyword(s):  
Kinetic Equation ◽  
Large Time ◽  
Transport Coefficients ◽  
Kinetic Equations ◽  
Boltzmann Kinetic Equation ◽  
Qualitative Properties ◽  
Long Time Behaviour ◽  
Long Time ◽  
Entropy Principle ◽  
Qualitative Properties Of Solutions

Qualitative properties of solutions of a two point extension of the Boltzmann kinetic equation and of solutions of the Frey–Salmon two point kinetic equation are investigated. Our attention is focussed on: (i) compatibility of the long time behaviour of solutions with equilibrium thermodynamics, (ii) the thermodynamic equation of state implied by the two point kinetic equations, (iii) one point kinetic equations whose solutions approximate well the solutions to the two point kinetic equations for large time, and (iv) transport coefficients implied by the two point kinetic equations.

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).

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