Semiclassical limit of the Bogoliubov–de Gennes equation

In the paper [AIP Conf. Proc., Vol. 1333, Part I, p. 134-139 (2011)], the kinetic equation for two-particle distribution function was obtained by making use of exactly the same physical assumptions as Ludwig Boltzmann did. Instead of the collision integral, there are the linear scattering operator and the chaos projector in the right part of this equation. The Boltzmann equation follows from this two-particle equation without any additional assumptions after a simple integration. By now, only a few exact explicit solutions of the Boltzmann kinetic equation are discovered. They play an important role in the kinetic theory of rarefied gases. A well known example of such solution is the BKW solution for Maxwell molecules. In the report we presented a new exact solution for the two-particle kinetic equation that is intermediate asymptotic for space homogeneous relaxation problems. After reducing the two-particle distribution function to one-particle distribution function the solution is reduced to BKW solution.In the paper [AIP Conf. Proc., Vol. 1333, Part I, p. 134-139 (2011)], the kinetic equation for two-particle distribution function was obtained by making use of exactly the same physical assumptions as Ludwig Boltzmann did. Instead of the collision integral, there are the linear scattering operator and the chaos projector in the right part of this equation. The Boltzmann equation follows from this two-particle equation without any additional assumptions after a simple integration. By now, only a few exact explicit solutions of the Boltzmann kinetic equation are discovered. They play an important role in the kinetic theory of rarefied gases. A well known example of such solution is the BKW solution for Maxwell molecules. In the report we presented a new exact solution for the two-particle kinetic equation that is intermediate asymptotic for space homogeneous relaxation problems. After reducing the two-particle distribution function to one-particle distribution function the solution is reduced to BKW solution.

In the paper (Saveliev, in: AIP conference proceedings, vol 1333, p 134, 2011), the kinetic equation for two-particle distribution function was obtained by making use of exactly the same physical assumptions as Ludwig Boltzmann did. Instead of the collision integral, there are the linear scattering operator and the chaos projector in the right part of this equation. The Boltzmann equation follows from this two-particle equation without any additional assumptions after a simple integration. The article presents the method of generalized functions and considers the properties of the obtained exact solution for the two-particle kinetic equation for Maxwell's molecules, which is an intermediate asymptotics for problems of spatial homogeneous relaxation. After reducing the two-particle distribution function to a one-particle distribution function, the solution is reduced to the well-known Bobylev-Krook-Wu (BKW) mode.

This paper is devoted to the derivation of a kinetic equation for many-body one-component dissipative systems in an external random field. The system under consideration was discussed in the recent paper by Sliusarenko et al. [J. Math. Phys. 56, 043302 (2015)], where a generalization of the Vlasov kinetic equation was obtained. The potential interaction is assumed to be small (the corresponding small parameter λ), the dissipation interactions and the correlation functions of the external random field are considered as small quantities estimated by one small parameter μ for the simplicity. The kinetic equation is obtained up to the terms of the orders λ2μ0, λ1μ1, λ0μ2 inclusive. In weakly spatially nonuniform states of the system, this gives corrections to the Landau–Vlasov kinetic equation caused by the dissipation and external random field. In the case λ ≫ μ after the mean free time, the system reaches a state approximately described by the Maxwell distribution. The dissipation and random field will lead to the evolution of the system temperature. A time evolution equation for the temperature in spatially uniform states is derived on the basis of the obtained kinetic equation by a generalized Chapman–Enskog method which takes into account that the kinetic equation is an approximate one. This temperature time evolution equation is investigated up to the terms of the order μ3 inclusive. It is shown that under some conditions the final stage of the system evolution is a steady state. A fluctuation-dissipation theorem for this state is discussed. In this steady state, the system is described by a distribution function that contains corrections to the Maxwell distribution.

We consider the Wigner equation corresponding to a nonlinear Schrödinger evolution of the Hartree type in the semiclassical limit h \rightarrow 0 .Under appropriate assumptions on the initial data and the interaction potential, we show that the Wigner function is close in L^2 to its weak limit, the solution of the corresponding Vlasov equation. The strong approximation allows the construction of semiclassical operator-valued observables, approximating their quantum counterparts in Hilbert-Schmidt topology.The proof makes use of a pointwise-positivity manipulation, which seems necessary in working with the L^2 norm and the precise form of the nonlinearity. We employ the Husimi function as a pivot between the classical probability density and the Wigner function, which—as it is well known—is not pointwise positive in general.

FACt: FORTRAN toolbox for calculating fluctuations in atomic condensates

BBGKY-hierarchies and Vlasov's equations in postgalilean approximation

2 - Lagrangian Coordinates

Pair collisions are the main interaction process in the Boltzmann gas dynamics. By making use of exactly the same physical assumptions as was done by Ludwig Boltzmann we wrote the kinetic equation for two-particle distribution function of molecules in gas mixtures. Instead of the collision integral, there are the linear scattering operator and the chaos projector in the right part of this equation. We developed a new technique for factorization of the scattering operator on the bases of right inverses to the Casimir operator of the group of rotations. We exactly transformed the Boltzmann collision integral to the Landau-Fokker-Planck like form.

This paper is concerned with the investigation of a generalized kinetic equation describing the evolution of the density of a probability measure. In the general case this is a non-linear integro-differential equation. On the one hand, this equation includes as a special case the simpler linear Liouville equation (which underlies classical statistical mechanics) and the equation of a self-consistent field (the Vlasov kinetic equation). On the other hand, some other well-known equations also reduce to this equation, for instance, the vorticity equation for plane flows of an ideal incompressible fluid. The main aim of the paper is to study the problem of the weak limits, as the time tends to infinity, of solutions of the generalized kinetic equation. This problem plays a significant role in the transition from a micro- to a macrodescription, when the behaviour of the averages (most probable values) of dynamical quantities is considered. The theory of weak limits of solutions of the Liouville equation is closely connected with ideas and methods of ergodic theory. The case under consideration presents greater difficulties, which stem from the non-trivial problem of the existence of invariant countably-additive measures for dynamical systems in infinite-dimensional spaces. General results are applied to the analysis of continua of interacting particles and to the investigation of statistical properties of plane flows of an ideal fluid.

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.

Multiscale modeling of expanding plasmas is crucial for understanding the dynamics and evolution of various astrophysical plasma systems. In this context, the Expanding Box Model (EBM) was used to add the expansion into the kinetic equations, allowing us to describe plasma physics in a new system of reference non-expanding and co-moving with the plasma. This system allows us to maintain a constant volume through non-inertial forces, and its interpretation is fundamental to describing plasma physics.We have employed the EBM formalism to incorporate the expanding properties of the system into the plasma dynamics, which mainly affects transverse coordinates (i.e., y y/o z). Coordinate transformations were introduced within the co-moving frame system to obtain the modified Vlasov equation. Our main goal is to develop a plasma physics theory through a novel first principle description in the expanding frame, which is fundamentally based on the (collisionless) Vlasov equation for the evolution of the velocity distribution functions. Based on this, the expanding moments, such as the continuity, momentum, and energy equations, can then be derived, and an MHD model of the plasma expansion can be developed. Finally, coupling the obtained moments and Maxwell equations, a CGL-like plasma description is developed in the EB frame to study the evolution of macroscopic quantities (temperature, magnetic field, parallel beta, and anisotropy).Our results show the expansion affecting the kinetic and fluid equations through non-inertial and fictitious forces in the transverse directions, which contain all the information related to the expansion. These are thus reflected by the equations derived for the expanding moments of the distribution function, including density, bulk (drift) velocity, and pressure (or temperature). Furthermore, we developed an ideal expanding-MHD model based on these modified moments, providing a new interpretation and comparison with the existing results when expansion is considered. The EBM modifies the conservative form of the two adiabatic invariants in the CGL approximation. Equations are solved for radially decreasing magnetic fields and density profiles to study the relations between plasma parallel beta and anisotropy within the expansion. 

Solution of the Bogoliubov–de Gennes equations at zero temperature throughout the BCS–BEC crossover: Josephson and related effects

Can the Bogoliubov–de Gennes equation be interpreted as a ‘one-particle’ wave equation?

In fluid theory, the kinetic equation is converted into a system of moment equations. Any wanted moment is thereby coupled with other moments in an infinite set of equations. In plasma kinetic theory, the linearized Fourier-transformed kinetic equation is solved for the distribution function, and the wanted moments are obtained afterward as functions of the electric field strength. A complete knowledge of the frequency and wavenumber electric field spectrum is needed for these moments to be determined as functions of t and r. This standard kinetic approach is not suitable beyond the weak turbulence limit, in which the interaction of waves is treated in the random phase approximation. In situations where regular structures in (t, r) space arise due to phase correlations, recourse is commonly taken to fluid theory. In the present paper the plasma kinetic theory is treated differently by taking the electric field not as the cause of the process to be studied, but as a mediator between different moments. An arbitrary number of moment equations can be obtained in this way. Only those that, in the fluid limit, pass over to the fluid equations will be used. Outside the fluid limit, these so-called kinetic transport equations represent a continuation of the fluid equations into the kinetic regime. One purpose of kinetic transport theory is to extend the plasma kinetic theory beyond the weak turbulence limit. In contrast to standard plasma kinetic theory, the formulation of kinetic transport theory allows an application to neutral gases by setting the particle charge to zero. Therefore a second purpose of kinetic transport theory is to extend the transport equations for neutral gases beyond the fluid limit and to define the transport coefficients for highly rarefied gases.

We study the dispersion relation of the excitations of a dilute Bose-Einstein condensate confined in a periodic optical potential and its Bloch oscillations in an accelerated frame. The problem is reduced to one-dimensionality through a renormalization of the s-wave scattering length and the solution of the Bogolubov - de Gennes equations is formulated in terms of the appropriate Wannier functions. Some exact properties of a periodic one-dimensional condensate are easily demonstrated: (i) the lowest band at positive energy refers to phase modulations of the condensate and has a linear dispersion relation near the Brillouin zone centre; (ii) the higher bands arise from the superposition of localized excitations with definite phase relationships; and (iii) the wavenumber-dependent current under a constant force in the semiclassical transport regime vanishes at the zone boundaries. Early results by J. C. Slater [Phys. Rev. 87, 807 (1952)] on a soluble problem in electron energy bands are used to specify the conditions under which the Wannier functions may be approximated by on-site tight-binding orbitals of harmonic- oscillator form. In this approximation the connections between the low-lying excitations in a lattice and those in a harmonic well are easily visualized. Analytic results are obtained in the tight-binding scheme and are illustrated with simple numerical calculations for the dispersion relation and semiclassical transport in the lowest energy band, at values of the system parameters which are relevant to experiment.