Kinetic equations for classical and quantum Brownian particles and eigenfunction expansions as generalized functions

A velocity-space orthogonal-function representation has been developed for the distribution function of a collisional plasma. The choice of functions is determined by several criteria: (1) the functions must provide an accurate representation of the Coulomb collision operator; (2) the representation must be efficient in that physical plasmas be represented by as few terms as possible in order to minimize both computer time and storage requirements; and (3) the collision and Vlasov operators must lead to simple expressions which can be rapidly and easily calculated. The collision operator in the representation used here is developed both for the nonlinear (general distribution functions) and approximately linear (approximately Maxwellian distribution functions) cases. Some numerical results are discussed. Expressions for macroscopic observables in terms of the expansion amplitudes are also given.

A unified implicit scheme for kinetic model equations. Part I. Memory reduction technique

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.

We use the spatially homogeneous linear Boltzmann equation to study the time evolution of an initial non-equilibrium distribution function of an ensemble of test particles dilutely dispersed in a background gas at thermal equilibrium. The systems considered are energetic N in He and Xe in He. We employ the quantum mechanical differential cross section to define the collision operator in the Boltzmann equation. The Boltzmann equation is solved with a moment method based on the expansion of the distribution function in the Sonine (Laguerre) polynomials as well as with a direct simulation Monte Carlo method. The moment method provides the approximate eigenvalues and eigenfunctions of the linear Boltzmann collision operator. The reciprocal of the eigenvalues is a measure of the relaxation times to equilibrium. For hard sphere cross sections, the relaxation of the average energy and the shape of the distribution function can be characterized by a single time scale determined by the momentum transfer cross section. We show that this is also the case for realistic quantum cross sections with dominant small angle scattering contributions.

We have developed and, to some extent, solved a number of kinetic equations for displaced correlation functions in a classical fluid. These functions, of which the Van Hove neutron scattering function is a special example, are one-particle distribution functions obtained from a Gibbs ensemble which is initially, at $t=0$, in equilibrium except for one labeled particle whose distribution $W(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}},\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}})$ at $t=0$ differs from its equilibrium value $\ensuremath{\rho}{h}_{0}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}})$, where $\ensuremath{\rho}$ is the average fluid density, and ${h}_{0}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}})$ is the Maxwellian velocity distribution function. We investigate the time evolution of the (self-) distribution function of this labeled particle, ${f}_{s}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}},\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}},t)$, as well as the deviation from equilibrium, $\ensuremath{\eta}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}},\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}},t)$, of the total one-particle distribution function. The latter represents the density of fluid particles, labeled and unlabeled, at position $\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}$ and velocity $\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}}$. Since both ${f}_{s}$ and $\ensuremath{\eta}$ are linear functionals of $W$, they will satisfy exactly a linear non-Markovian kinetic equation of the form $f=\mathrm{B}f+\ensuremath{\int}{0}^{t}d{t}^{\ensuremath{'}}\mathrm{M}({t}^{\ensuremath{'}})f(t\ensuremath{-}{t}^{\ensuremath{'}})$. B is a time-in-dependent and M a time-dependent (memory) operator (nonsingular in $t$). Our kinetic equations (first- and higher-order) are based on neglecting or approximating M in such a way that the short-time behavior of ${f}_{s}$ and $\ensuremath{\eta}$ is described exactly. The rationale behind this scheme is that our choice of initial ensemble is precisely of the type generally assumed in the "derivation" of kinetic equations. The calculation of B is straightforward and depends in a very important way on whether the interparticle potential in the fluid is smooth or contains a hard core. In the former case, the first-order kinetic equation is of the Vlasov type with an effective potential given by the equilibrium direct correlation function, while in the latter case, B contains, in addition, a linear Enskog-type collision term. We show that this Vlasov equation (also derived previously by many authors) gives a damping linear in the wave number $k$ for small $k$ instead of the hydrodynamic ${k}^{2}$ dependence. The kinetic equation for systems with hard cores does not give correct hydrodynamic behavior. (For a one-dimensional system of hard rods, the first-order kinetic equation is exact.) We also obtain and solve a second-order kinetic equation, which is a generalized Vlasov-Fokker-Planck-type equation, for systems with continuous potentials.

Semilinear operators on a complex Hilbert space are studied in a part of a program that aims to develop the theories of additive operators on complex and quaternionic Hilbert spaces for application to problems in mathematical physics. The more notable among the new results proved on the eigenvalue problem for semilinear operators are the following: (i) if α is an eigenvalue of a semilinear operator then so also is any complex number which has the same modulus as α; (ii) if a normal semilinear operator has two eigenvectors belonging to different eigenvalues, then either the two eigenvectors are orthogonal or two eigenvalues have the same moduli; and (iii) a normal semilinear operator has a complete set of eigenvectors if and only if it is self-adjoint. Further, it is shown that there exists a norm-preserving semilinear isomorphism between the spaces of bounded linear and semilinear operators on a complex Hilbert space. Finally it is demonstrated how the theory of semilinear operators can be exploited to solve the problems of finding three involutive mutually anticommuting self-adjoint two-by-two matrices and four four-by-four matrices with the same properties: the unusual and remarkably easy solution of this old familiar exercise establishes the relevance of the theory being developed here to physics.

Starting with the Uehling–Uhlenbeck quasiclassical generalized Boltzmann equation, a kinetic equation is derived for the distribution function f(ε,t) for charge carriers of energy ε and effective mass m in a semiconductor in the presence of an electric field E. Charge-carrier scattering with optical and acoustic phonons through strain and polar interactions are taken into account. Introducing a Legendre-polynomial expansion of the distribution function and assuming spherical constant energy surfaces, the following equation emerges in the limit of large charge carrier-to-phonon energy: (∂f0/∂t)+(eE/3) ((2ε/m))1/2 ((∂/∂ε) +(1/ε))(f1(ε,0)e−t/τ−T̂eE ((2ε/m))1/2 (∂/∂ε) f0)=(P−aQf0) (∂f0/∂ε) +(F−aGf0) (∂2f0/∂ε2)+Cf0 (1−af0). Collision integrals P,Q,F,G, and C, as well as the collision frequency τ, are evaluated explicitly. The quantum parameter a is 1 in the degenerate domain and 0 in the classical domain. The first two Legendre components of f(ε,t) are written f0 and f1, respectively, and T̂ is a time-dependent operator. Working in steady state and passing to the classical limit it is found that acoustic-strain interactions dominate over all others. Leading terms in this expansion give a new quasiclassical equation for the distribution of charge carriers of a semiconductor in an electric field. Further reducing this equation in the limits of equipartion and mu/p≪1, where p is carrier momentum and u is acoustic phonon velocity, gives a generalized Druyvesteyn equation which incorporates exclusion effects.

A new form of the collision operator for a Boltzmann gas of hard spheres and Coulomb plasma is proposed [1]. One‐component and many‐component systems are considered. The proposed collision operator properly takes into account the relaxation of the first 13 hydrodynamic moments. Besides this, it accounts for the non‐diagonal component contribution in the quadratic approximation in the expansion of the linearized collision operator with respect to the complete system of Hermite polynomials. It is shown that for a system of charged particles with the Coulomb interaction potential, these contributions are essential and lead to Spitzer corrections to the transport coefficients. An expression for the intensity of the Langevin source in the kinetic equation is obtained in the same approximation. A new form of the model collision operator for a Boltzmann gas of hard spheres is proposed. For a many‐component system we have reconstructed a non‐linear model collision integral by using the linearized collision integral found. Unlike previous ones, it does not contain complicated exponential dependence and avoids the coefficients ambiguity in the many‐component collision integral.

Views Icon Views Article contents Figures & tables Video Audio Supplementary Data Peer Review Share Icon Share Twitter Facebook Reddit LinkedIn Tools Icon Tools Reprints and Permissions Cite Icon Cite Search Site Citation V. V. Belyi; To model kinetic description. AIP Conf. Proc. 24 March 2011; 1332 (1): 269. https://doi.org/10.1063/1.3569530 Download citation file: Ris (Zotero) Reference Manager EasyBib Bookends Mendeley Papers EndNote RefWorks BibTex toolbar search Search Dropdown Menu toolbar search search input Search input auto suggest filter your search All ContentAIP Publishing PortfolioAIP Conference Proceedings Search Advanced Search |Citation Search

The kinetic equation proposed in [1,2] for describing the behavior of a system of particles in a gas flow differs from the usual Boltzmann equation with respect to the additional terms that take into account random variations of the particle velocity under the influence of the flow. As shown in [2], the collision operator and the Brownian-type operator in the starting kinetic equation describe essentially different simultaneous physical processes of change of state of the particle system: equalization of the mean kinetic energy of the particles and change of energy due to the action of the viscous forces associated with the suspending flow. Therefore the method of solving the kinetic equation used in [2], a direct generalization of the Chapman-Enskog method of solving the kinetic equation it is necessary to investigate method of solving the kinetic equation it is necessar y to investigate the relaxation processes in the system. Moreover, the relaxation of systems of the fluidized-bed type to the continuum state is also of independent interest in connection with the analysis of fast processes in the system, i.e., processes with a characteristic duration of the order of the mean free time.

The choice of functions is determined by several criteria: (1) the functions must provide an accurate representation of the Coulomb collision operator; (2) the representation must be efficient in that physical plasmas be represented by as few terms as possible in order to minimize both computer time and storage requirements; and (3) the collision and Vlasov operators must lead to simple expressions which can be rapidly and easily calculated. The collision operator is developed both for the non-linear and approximately linear cases.

A generalized virial theorem which expresses inverse compressibility in terms of integrals of virials and canonical distribution functions through the four-particle distribution is transformed to the grand canonical ensemble and becomes an expression for compressibility in terms of the same integrals formed with grand canonical distribution functions. The integrals are of a mixed (virial and fluctuation) type. While the thermodynamic functions expressed by the same integrals with canonical and grand canonical distribution functions are quite different, the two formulas agree in the thermodynamic limit because of the different asymptotic behavior of canonical and grand canonical distribution functions. We also derived an alternative form of the second virial theorem which expresses compressibility in terms of integrals over virials and grand canonical distribution functions through the three-particle distribution function only. It is shown that this form and its generalizations to higher derivatives of the density, as well as the hierarchy of fluctuation theorems and the fugacity expansions of distribution and correlation functions can all be very simply derived from a set of integro-differential equations satisfied by the grand canonical distribution functions. A generalization of the wall theorem (P/kT = ρwall) is derived and shown to be equivalent to the generalized virial theorem (canonical form).

In non-equilibrium classical thermostatistics, the state of a system may be described by not only dynamical/thermodynamical variables but also a kinetic distribution function. This 'double structure' bears some analogy with that in quantum thermodynamics, where both dynamical variables and the Hilbert space are involved. Recently, the concept of weak invariants has repeatedly been discussed in the context of quantum thermodynamics. A weak invariant is defined in such a way that its value changes in time but its expectation value is conserved under time evolution prescribed by a kinetic equation. Here, a new aspect of a weak invariant is revealed for the classical Fokker-Planck equation as an example of classical kinetic equations. The auxiliary field formalism is applied to the construction of the action for the kinetic equation. Then, it is shown that the auxiliary field is a weak invariant and is the Noether charge. The action is invariant under the transformation generated by the weak invariant. The result may shed light on possible roles of the symmetry principle in the kinetic descriptions of non-equilibrium systems. This article is part of the theme issue 'Fundamental aspects of nonequilibrium thermodynamics'.

The acceleration of energetic particles and their propagation in magnetic fields of the solar wind and the Galaxy is a topical astrophysical problem. Cosmic rays (CR) affect communications and the operation of on-board spacecraft electronics and disturb the magnetosphere and the ionosphere of the Earth. The scattering of particles by magnetic-field irregularities is the primary mechanism governing the CR propagation in the interplanetary medium. If the scattering of energetic particles in the interplanetary medium is relatively inefficient (i.e., the mean free path is comparable to the heliocentric distance), a kinetic equation should be used to characterize the CR propagation. The Fokker–Planck kinetic equation is used to analyze the propagation of charged energetic particles in a magnetic field represented as a superposition of a mean uniform field and magnetic inhomogeneities of various scales. This kinetic equation corresponds to multiple small-angle scattering, and its collision integral characterizes particle diffusion in the momentum space. A system of differential equations for the spherical harmonics of the CR distribution function is obtained based on the kinetic equation. The CR transport equations are derived and solved. The evolution of the CR distribution function under anisotropic scattering of particles by magnetic-field fluctuations is examined. It is demonstrated that the angular distribution function of particles depends to a considerable extent on the degree of their scattering anisotropy. The temporal dependence of the CR distribution function is analyzed, and the parameter characterizing the scattering anisotropy is estimated.

We establish a spectral representation for solutions to linear Hamilton equations with positive definite energy in a Hilbert space. Our approach is a special version of M. Krein's spectral theory of J-selfadjoint operators is the Hilbert spaces with an indefinite metric. Our main result is an application to the eigenfunction expansion for the linearized relativistic Ginzburg-Landau equation.