On the phenomenological rate equation in a model exchange reaction

The limits of validity of the phenomenological rate equation, and the phenomenological rate constant k_f and k_b are found for the exchange reaction $\text{A + BC}\textcolor{red}{}\text{AB + C}$ within the framework of simple multistate models. The possibility of macroscopically long transient times is discussed. The rate constants are shown to depend on the total densities n_A, n_B and n_C of the chemical species present in the mixture, when nonequilibrium effects are important. The distinction between a rate constant and a flux coefficient is illustrated. Differences among exchange, dissociation-recombination, and isomerization reactions in that regard are discussed.     

A phenomenological approach to the equation of state of a unitary Fermi gas

We propose a phenomenological approach for the equation of state of a unitary Fermi gas. The universal equation of state is parametrized in terms of Fermi–Dirac integrals. This reproduces the experimental data over the accessible range of fugacity and normalized temperature, but cannot describe the superfluid phase transition found in the MIT experiment [Ku et al, Science 335, 563 (2012)]. The most sensitive data for compressibility and specific heat at phase transition can, however, be fitted by introducing into the grand partition function a pair of complex conjugate zeros lying in the complex fugacity plane slightly off the real axis.     

Solution of the BGK model kinetic equation for very hard particle interaction

We study the Bhatnagar-Gross-Krook model kinetic equation with a velocity-dependent collision frequency. We derive the conditions that must be verified in order to keep the main physical properties of the Boltzmann equation, i.e.,H-theorem and conservation laws. The particular case of the so-called VHP interaction is considered, and the resulting kinetic equation is solved for a homogeneous and isotropic gas. Overpopulation phenomena are observed and analyzed for some kinds of initial conditions. The results are compared, where possible, with the exact solution of the Boltzmann equation.     

Relaxation of a test particle in systems with long-range interactions: diffusion coefficient and dynamical friction

We study the relaxation of a test particle immersed in a bath of field particles interacting via weak long-range forces. To order 1/N in the N→+∞ limit, the velocity distribution of the test particle satisfies a Fokker-Planck equation whose form is related to the Landau and Lenard-Balescu equations in plasma physics. We provide explicit expressions for the diffusion coefficient and friction force in the case where the velocity distribution of the field particles is isotropic. We consider (i) various dimensions of space d=3,2 and 1; (ii) a discret spectrum of masses among the particles; (iii) different distributions of the bath including the Maxwell distribution of statistical equilibrium (thermal bath) and the step function (water bag). Specific applications are given for self-gravitating systems in three dimensions, Coulombian systems in two dimensions and for the HMF model in one dimension.     

On quantum kinetic equation for hierarchic systems

The energy dissipation in a gas of structured objects, e.g. molecules, is considered. It is shown that the macroscopic irreversibility of the kinetic processes can be considered as a consequence of the microscopic operator ordering. Our approach is free of any special assumptions on the space-time geometry, except for the general region causality assumptions [J.C. Christensen, L. Crane, J. Math. Phys. 46 (2005) 122502], it can be applied to a wide variety of processes, from the cosmological processes at Big Bang stage till the energy dissipation in molecular gases.     

The kinetic equation and the equation of momentum transfer for a plasma

An attempt is made to derive a simple form of the collision integral of the kinetic equation for a plasma, by using Rostoker’s equation which expresses the pair correlation function in terms of the distribution functions of the particles, and the conditional probability of one particle shielding the other. The conditional probability function is assumed to be given by its equilibrium value. By taking first order velocity-moment of the resulting kinetic equation, the equation of momentum transfer has been obtained.     

Calculation of the electromagnetic absorption by a small metal particle with a two-parameter kinetic equation

The behavior of the magnetic dipole absorption cross section in a small metal particle is considered. A modified kinetic equation with the collision integral taking into account the deviation of the properties of metals at low temperatures from the Wiedemann-Franz law is solved. Diffuse reflection of electrons from the sample surface is considered. The dependence of the absorption cross section of a particle on the parameter describing the deviation from the Wiedemann-Franz law is analyzed.     

Dual kinetic balance approach to basis-set expansions for the dirac equation

A new approach to finite basis sets for the Dirac equation is developed. It does not involve spurious states and improves the convergence properties of basis-set calculations. Efficiency of the method is demonstrated for finite basis sets constructed from B splines by calculating the one-loop self-energy correction for a hydrogenlike ion.     

On the solution of the kinetic equation for a heat flux between concentric spheres

The problem of calculating a heat flux in a spherical layer is considered. The results are obtained in terms of the Bhatnagar-Gross-Krook model and the Boltzmann collision integral. A general (independent of the form of the kinetic equation and the solution method) expression for the heat flux as a function of the energy accommodation coefficient is derived. The results are compared with the experiment and the analytical results obtained previously.     

On the phenomenon of dynamical friction for particles interacting by the inverse square law

The dynamics of the homogeneous positive background of a one-component plasma (OCP) is taken into account — compensated system — in order to modify the first coefficient of the coupling constant expansion of the average velocity of a test electron derived in a previous letter for an uncompensated OCP.     

On the phenomenon of dynamical friction for particles interacting by the inverse square law

Using the expansion of the exact solution of the N-body problem in terms of the coupling constant, the phenomenon of dynamical friction for a system of particles interacting by the inverse square law is studied. The only assumption made is that correlations are absent at the initial line.     

Exact kinetic transport equation solutions in the particle propagation theory in the scattering medium

Charged particle kinetics in an inhomogeneous medium (stochastic magnetic field) is investigated. Exact analytic expressions for Green function of kinetic equation in relaxation-time approximation are derived in one and three dimensions with arbitrary particle absorption. We separately consider the case of isotropic particle injection as well as the case of unidirectional instantaneous particle injection. The new way of solution makes it possible to get off any Cauchy-Principal Value integrals in some solutions which arise in the inverse Fourier-Laplace transform. Weak scattering regime and diffusion approximation is considered, and particle density is derived in three dimensions and arbitrary particle source.     

Macroscopic phenomenological relations for nonlinear processes in kinetic theory

Nonlinear irreversible processes between states which are not local equilibrium states are investigated by methods of the kinetic theory. The phenomenological equations for the second-order fluxes in a multicomponent mixture are derived, and relations between some of the second-order phenomenological coefficients are established. It is shown that new independent forces appear in the second-order equation, namely the gradients of the chemical potentials. Expressions for the entropy, entropy flux, and entropy source are evaluated. These expressions are related to the phenomenological equations and coefficients, e.g., all the second-order contributions of the forces in the equations for the fluxes can be obtained by differentiation of the expression for the second-order entropy source with respect to the coupled forces.     

On the phenomenological approach to dual resonance models for deep inelastic electroproduction

We outline a systematic procedure for constructing a dual resonance model for deep inelastic electroproduction which incorporates locality, current algebra, Regge behaviour, Bjorken scaling and the Dashen — Fubini — Gell-Mann sum rule. We illustrate the procedure in terms of a simple Ansatz for theW ₂ electroproduction structure function.     

The kinetic boundary layer for the Klein-Kramers equation; A new numerical approach

We explore a numerical technique for determining the structure of the kinetic boundary layer of the Klein-Kramers equation for noninteracting Brownian particles in a fluid near a wall that absorbs the Brownian particles. The equation is of interest in the theory of diffusion-controlled reactions and of the coagulation of colloidal suspensions. By numerical simulation of the Langevin equation equivalent to the Klein-Kramers equation we amass statistics of the velocities at the first return to the wall and of the return times for particles injected into the fluid at the wall with given velocities. The data can be used to construct the solutions of the standard problems at an absorbing wall, the Milne and the albedo problem. We confirm and extend earlier results by Burschka and Titulaer, obtained by a variational method vexed by the slow convergence of the underlying eigenfunction expansion. We briefly discuss some further boundary layer problems that can be attacked by exploiting the results reported here.     

Deuteron form factors in a phenomenological approach

The electromagnetic form factors of the deuteron, particularly its quadrupole form factor, are studied with the help of a phenomenological Lagrangian approach where the vertex of the deuteron–proton–neutron with D-state contribution is explicitly taken into account. The results show the importance of this contribution to the deuteron quadrupole form factor in the approach.     

Master equation for a quantum particle in a gas

The equation for the quantum motion of a Brownian particle in a gaseous environment is derived by means of S-matrix theory. This quantum version of the linear Boltzmann equation accounts nonperturbatively for the quantum effects of the scattering dynamics and describes decoherence and dissipation in a unified framework. As a completely positive master equation it incorporates both the known equation for an infinitely massive Brownian particle and the classical linear Boltzmann equation as limiting cases.     

On the derivation of the incompressible Mavier-Stokes equation for Hamiltonian particle systems

We consider a Hamiltonian paticle system interacting by means of a pair potetial. We look at the behavior of the system on a space scale of order ^-1, times of order ^-2 and mean velocities of order , with a scale parameter. Assuming that the phase space density of the particles is give by a series in (the analog of the Chapman-Enskog expansion), the behavior of the system under this rescaling is described, to the lowest order in , by the incompressible Navier-Stokes equations. The viscosity is given in terms of microscopic correlations, and its expression agrees with the Green-Kubo formula.