Nonseparable wave evolution equations in quantum kinetics

The Wigner formalism is a convenient way of describing quantum mechanical effects through a framework of distribution functions in phase space. Currently, there are stochastic and deterministic approaches in use. In our deterministic method, the critical discretization of the diffusion term is done through the utilization of an integral formulation of the Wigner equation. This deterministic method is studied in the context of superposed quantum states as a precursor to simulations of entangled states.

A plasma consists of charged particles moving relatively freely, but interacting through the electric and magnetic fields produced by each particle. It is well known that all except a very small (but nevertheless important) fraction of the detectable universe is a plasma. Most plasma physics can be expressed in terms of five fairly straightforward equations, the four Maxwell equations for electromagnetism and the equation of motion for charged particles in electromagnetic fields. Why then is plasma physics such a mathematically difficult subject and why is numerical simulation so important? The most basic reason is that any experimental plasma contains a very large number of particles, 1018–1020 in a typical fusion experiment. Moreover, particles in plasmas have a tendancy to act independently of each other. In conventional hydrodynamics, large numbers of particles can be described by a few simple numbers such as their temperature, density and mean velocity. Sometimes we can do this with a plasma but even then we have a few more numbers to include such as current and charge density, along with the electromagnetic fields. More usually, plasmas are not so simply described. Plasmas are not usually Maxwellian and we have to think in terms of distribution functions in phase space. Whenever we can, we model the non—Maxwellian nature of the plasma by transport models, e.g. for heat flow, but this is a specialised skill in itself. When one considers that conventional hydrodynamics is very complex and involves advanced numerical simulation, it is hardly surprising that simulation plays such an important role in plasma physics.

A new scheme for the calculation of the collision integral of the nonlinear Boltzmann kinetic equation is presented, based on a piecewise constant discrete approximation of the distribution function in phase space and corresponding precalculated collision rate coefficients. The computing effort required for evaluation of the fivefold collision integral is significantly reduced by the use of precalculated rate coefficients. The method was combined with the conservative splitting method to solve the Boltzmann equation for a number of test cases to examine the performance of this method and demonstrate its efficiency. 10 refs., 6 figs.

Analytical solution for self-consistent particle equilibrium distribution in RF field with transverse focusing is found. Solution is attained in approximation of high brightness beam. Distribution function in phase space is determined as a stationary function of the energy integral. Equipartitioning for beam distribution between degrees of freedom follows directly from the choice of stationary distribution function. Analytical expressions for r-z equilibrium beam profile and maximum beam current in RF field are obtained.

The Rouse model for dilute polymer solutions undergoing homogeneous flows has been generalized to include the inertia of the beads in the equations of motion. To obtain the correct ‘‘diffusion equation’’ for the probability density distribution function in phase space, we generalize the diffusion equation derived by Murphy and Aguirre [J. Chem. Phys. 57, 2098 (1972)] from Hamilton’s equations of motion for an arbitrary number of interacting Brownian particles at equilibrium. Material functions are found, and the noninertial case is seen to be obtained as the zero mass limit in all steps of the development. In particular, the steady-state shear results are unaffected by the inclusion of inertia. It is also shown how two assumptions, ‘‘equilibration in momentum space,’’ and the neglect of acceleration, made independently by Curtiss, Bird, and Hassager in their phase-space kinetic theory, are actually the result of assuming zero mass.

Self-accelerating beam is a kind of light beam capable of self-bending in free space without any external potential, of which a typical one is the well-known Airy beam. Such a beam has gained great attention for its extraordinary properties, including nondiffracting, self-accelerating and self-healing, which may have versatile applications in the delivery and guiding of energy, information and objects using light, such as particle manipulation, micro-machining, optical routing, super-resolution imaging, etc. However, since Airy beam can only propagate along parabolic trajectory, which reduces the flexibility in practical applications, thus how to design accelerating beams propagating along arbitrary trajectory is still a crucial problem in this area. One scheme is to keep on finding other analytical solutions of the wave equation besides Airy beam, such as semi-Bessel accelerating beams, Mathius beams, and Weber beams, moving along circular, elliptical, or parabolic trajectories, but it becomes increasingly difficult to find out any more solutions. A more effective solution to this problem is based on the caustic method, which associates the predesigned trajectory with an optical caustics and then obtains the necessary initial field distribution by performing a light-ray analysis of the caustics. This method has been implemented in real space and Fourier space based on Fresnel diffraction integral and angular-spectrum integral, respectively. It has been found recently that they can be unified by constructing Wigner distribution function in phase space. Based on the caustic method, accelerating beams were constructed to propagate along arbitrary convex trajectories in two-dimensional space at first. With continuous development of this method, the types of accelerating beams available have been extending from convex trajectories to nonconvex trajectories, from two-dimensional trajectories to three-dimensional trajectories, and from one main lobe to multiple main lobes, which opens up more possibilities for emerging applications based on accelerating beams. In future, previous researches and applications based on Airy beams will certainly be generalized to all these new types of accelerating beams, and owing to the great flexibility in designing accelerating beams, more application scenarios may emerge in this process with huge development potential. Thus in this paper, we review the principle and progress of the caustic method in designing accelerating beams.

A semiclassical independent electron model for ion-atom collisions

Landau theory of Fermi liquids and nuclear fluid dynamics

The entropy of an electronic system is defined in terms of the Husimi function, a nonnegative distribution function in phase space. The Husimi function is calculated by maximizing the entropy subject to the constraints that the Husimi function give a Gaussian convolution of the desity when integrating over the momentum coordinates and that its second moment with respect to momentum give a sum of Gaussian convolutions of the density and the kinetic energy density. The result is compared with the Wigner function. Equations are given for calculating the density matrix from the Husimi function. The resulting equation for the exchange energy requires a difficult numerical integration. An alternate method is used to obtain the density matrix from an approximate partially collapsed Husimi matrix that gives the maximum entropy Husimi function as its diagonal. The results are exact for the harmonic oscillator ground state. Exchange energies calculated for H and the He isoelectronic series through C+4 show slight improvements over those calculated using a maximum entropy Wigner function.

We develop a framework for Large Scale Structure (LSS) perturbation theory, that solves the Vlasov-Poisson system of equations for the distribution function in full phase space. This approach relaxes the usual apriori assumption of negligible velocity dispersion underlying the Standard Perturbation Theory (SPT). We apply the new method to rederive the usual SPT kernels up to third order in the perturbative expansion. We also show that a counterterm, identical to the one introduced by standard Effective Field Theory (EFT) methods, naturally arises within our framework. We finish by making a precise connection to EFT techniques, which reveals the necessity of the EFTofLSS to self-consistently model the long-wavelength fluid, and illustrates the importance of having theoretical control over short distance fluctuations.

Some general properties of the equilibrium Wigner distribution function are discussed. An invariance law under Galilean transformation is obtained for systems with no external forces. The reduced distribution functions in configurational, momentum, and phase space are studied, with emphasis on the correlations of the velocities among themselves and between them and the configurational coordinates. It is shown that up to linear terms in h2, the reduced distribution can be derived from a Hamiltonian with momentum-dependent potential.

Theory of novel nonlinear quantum transport effects in resonant tunneling structures

A diagnostic technique for measuring beam emittance is developed for electron beams with diverging envelopes under strong space-charge forces. Radial profiles of current density, local temperature, and divergence angle are measured by the slit–pinhole method for axisymmetric beams. The particle distribution function in transverse phase space is then constructed and the rms emittance is obtained by numerical integrations. A 5-kV, 200-mA, 3-μs, and 60-pps electron beam is used in the comparison between theory and this experimental diagnostic method.

Dynamics of modulated beams in spectral domain

We introduce the lattice-Boltzmann code LB3D, version 7.1. Building on a parallel program and supporting tools which have enabled research utilising high performance computing resources for nearly two decades, LB3D version 7 provides a subset of the research code functionality as an open source project. Here, we describe the theoretical basis of the algorithm as well as computational aspects of the implementation. The software package is validated against simulations of meso-phases resulting from self-assembly in ternary fluid mixtures comprising immiscible and amphiphilic components such as water–oil–surfactant systems. The impact of the surfactant species on the dynamics of spinodal decomposition are tested and quantitative measurement of the permeability of a body centred cubic (BCC) model porous medium for a simple binary mixture is described. Single-core performance and scaling behaviour of the code are reported for simulations on current supercomputer architectures. Program summaryProgram Title: LB3DProgram Files doi:http://dx.doi.org/10.17632/9g9x2wr8z8.1Licensing provisions: BSD 3-clauseProgramming language: FORTRAN90, Python, CNature of problem: Solution of the hydrodynamics of single phase, binary immiscible and ternary amphiphilic fluids. Simulation of fluid mixtures comprising miscible and immiscible fluid components as well as amphiphilic species on the mesoscopic scale. Observable phenomena include self-organisation of mesoscopic complex fluid phases and fluid transport in porous media.Solution method: Lattice-Boltzmann (lattice-Bhatnagar–Gross–Krook, LBGK) [1, 2, 3] method describing fluid dynamics in terms of the single particle velocity distribution function in a 3-dimensional discrete phase space (D3Q19) [4, 5, 6]. Multiphase interactions are modelled using a phenomenological pseudo-potential approach [7, 8] with amphiphilic interactions utilising an additional dipole field [9, 10]. Solid boundaries are modelled using simple bounce-back boundary conditions and additional pseudo-potential wetting interactions [11].Additional comments including Restrictions and Unusual features: The purpose of the release is the provision of a refactored minimal version of LB3D suitable as a starting point for the integration of additional features building on the parallel computation and IO functionality. [1]S. Succi, The Lattice Boltzmann Equation: For Fluid Dynamics and Beyond, Oxford University Press, 2001.[2]B. Dünweg, A. Ladd, Lattice Boltzmann simulations of soft matter systems, Adv. Poly. Sci. 221 (2009) 89–166[3]C. K. Aidun, J. R. Clausen, Lattice-Boltzmann Method for Complex Flows, Annual Review of Fluid Mechanics 42 (2010) 439.[4]X. He, L.-S. Luo, A priori derivation of the lattice-Boltzmann equation, Phys. Rev. E 55 (1997) R6333.[5]X. He, L.-S. Luo, Theory of the lattice Boltzmann method: from the Boltzmann equation to the lattice Boltzmann equation, Phys. Rev. E 56.[6]Y. H. Qian, D. D’Humiéres, P. Lallemand, Lattice BGK Models for Navier–Stokes Equation, Europhysics Letters 17 (1992) 479.[7]X. Shan, H. Chen, Lattice-Boltzmann model for simulating flows with multiple phases and components, Physical Review E 47 (1993) 1815.[8]X. Shan, G. Doolen, Multicomponent lattice-Boltzmann model with interparticle interaction, Journal of Statistical Physics 81 (1995) 379.[9]H. Chen, B. Boghosian, P.V. Coveney, M. Nekovee, A ternary lattice-Boltzmann model for amphiphilic fluids, Proceedings of the Royal Society of London A 456 (2000) 2043.[10]M. Nekovee, P. V. Coveney, H. Chen, B. M. Boghosian, Lattice-Boltzmann model for interacting amphiphilic fluids, Phys. Rev. E 62 (2000) 8282.[11]N. S. Martys, H. Chen, Simulation of multicomponent fluids in complex three-dimensional geometries by the lattice-Boltzmann method, Phys. Rev. E 53 (1996) 743.