Particle Number Conservation Research Articles

Nonlinear Boltzmann equation for the homogeneous isotropic case: Minimal deterministic Matlab program

The homogeneous isotropic Boltzmann equation (HIBE) is a fundamental dynamic model for many applications in thermodynamics, econophysics and sociodynamics. Despite recent hardware improvements, the solution of the Boltzmann equation remains extremely challenging from the computational point of view, in particular by deterministic methods (free of stochastic noise). This work aims to improve a deterministic direct method recently proposed [V.V. Aristov, Kluwer Academic Publishers, 2001] for solving the HIBE with a generic collisional kernel and, in particular, for taking care of the late dynamics of the relaxation towards the equilibrium. Essentially (a) the original problem is reformulated in terms of particle kinetic energy (exact particle number and energy conservation during microscopic collisions) and (b) the computation of the relaxation rates is improved by the DVM-like correction, where DVM stands for Discrete Velocity Model (ensuring that the macroscopic conservation laws are exactly satisfied). Both these corrections make possible to derive very accurate reference solutions for this test case. Moreover this work aims to distribute an open-source program (called HOMISBOLTZ), which can be redistributed and/or modified for dealing with different applications, under the terms of the GNU General Public License. The program has been purposely designed in order to be minimal, not only with regards to the reduced number of lines (less than 1000), but also with regards to the coding style (as simple as possible). Program summary Program title: HOMISBOLTZ Catalogue identifier: AEGN_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEGN_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: GNU General Public License No. of lines in distributed program, including test data, etc.: 23 340 No. of bytes in distributed program, including test data, etc.: 7 635 236 Distribution format: tar.gz Programming language: Tested with Matlab version ⩽6.5. However, in principle, any recent version of Matlab or Octave should work Computer: All supporting Matlab or Octave Operating system: All supporting Matlab or Octave RAM: 300 MBytes Classification: 23 Nature of problem: The problem consists in integrating the homogeneous Boltzmann equation for a generic collisional kernel in case of isotropic symmetry, by a deterministic direct method. Difficulties arise from the multi-dimensionality of the collisional operator and from satisfying the conservation of particle number and energy (momentum is trivial for this test case) as accurately as possible, in order to preserve the late dynamics. Solution method: The solution is based on the method proposed by Aristov (2001) [1], but with two substantial improvements: (a) the original problem is reformulated in terms of particle kinetic energy (this allows one to ensure exact particle number and energy conservation during microscopic collisions) and (b) a DVM-like correction (where DVM stands for Discrete Velocity Model) is adopted for improving the relaxation rates (this allows one to satisfy exactly the conservation laws at macroscopic level, which is particularly important for describing the late dynamics in the relaxation towards the equilibrium). Both these corrections make possible to derive very accurate reference solutions for this test case. Restrictions: The nonlinear Boltzmann equation is extremely challenging from the computational point of view, in particular for deterministic methods, despite the increased computational power of recent hardware. In this work, only the homogeneous isotropic case is considered, for making possible the development of a minimal program (by a simple scripting language) and allowing the user to check the advantages of the proposed improvements beyond Aristov's (2001) method [1]. The initial conditions are supposed parameterized according to a fixed analytical expression, but this can be easily modified. Running time: From minutes to hours (depending on the adopted discretization of the kinetic energy space). For example, on a 64 bit workstation with Intel CoreTM i7-820Q Quad Core CPU at 1.73 GHz and 8 MBytes of RAM, the provided test run (with the corresponding binary data file storing the pre-computed relaxation rates) requires 154 seconds.

Collisional damping of zonal flows due to finite Larmor radius effects

The collisional damping of seeded E×B zonal flows on the ion Larmor radius scale is studied using a gyrokinetic model. The focus is on flow damping due to finite Larmor radius effects, which cause a v∥/v anisotropy of the ion distribution function that is damped by ion-ion collisions. The gyrokinetic equations are solved in a slab geometry with no gradients or curvature, and a gyroaveraged Lorentz collision operator that conserves particle number, momentum, and energy is used. The solution of the gyrokinetic equations explores the dependence of the damping rate on the wavelength of the flows and the impact of the collisions on the ion distribution function. These numerical results can be used as a benchmark test during the implementation of finite Larmor radius effects in the collision operator of gyrokinetic codes.

Elimination of spurious 0+ states in the quasiparticle time blocking approximation

The quasiparticle time blocking approximation (QTBA) is considered as a model for the description of excitations in open-shell nuclei. Within the model the single-particle continuum, the pairing correlations and the quasiparticle-phonon coupling are included simultaneously. In the present version of the QTBA, the pairing correlations are included within the framework of the Bardeen-Cooper-Schrieffer approximation. In this approach, as is well known, the 0+ spurious states appear, which are caused by the breaking of the symmetry related to the particle-number conservation. In this work, the method is presented which solves the problem of the 0+ spurious states in the QTBA with the help of the projection technique. The method is illustrated by calculations of 0+ excitations in 120Sn nucleus.

Evolution of fluctuations near QCD critical point

We propose to describe the time evolution of quasi-stationary fluctuations near QCD critical point by a system of stochastic Boltzmann-Langevin-Vlasov-type equations. We derive the equations and study the system analytically in the linearized regime. Known results for equilibrium stationary fluctuations as well as the critical scaling of diffusion coefficient are reproduced. We apply the approach to the long-standing question of the fate of the critical point fluctuations during the hadronic rescattering stage of the heavy-ion collision after chemical freezeout. We find that if conserved particle number fluctuations survive the rescattering, so do, under a certain additional condition, the fluctuations of non-conserved quantities, such as mean transverse momentum. We derive a simple analytical formula for the magnitude of this "memory" effect.

Weibel instability and quasi-equilibria for collisionless plasmas

The Weibel instability for an electron–positron and an electron–ion plasma is investigated in a linear analysis of the Vlasov–Maxwell equations for temporal growth. Comparison with a computer simulation of Suzuki and Shigeyama confirms that a single exponentially temporally growing mode for the magnetic field dominates. A quasi-equilibrium one-particle distribution function for an electron–positron plasma is expressed as an expansion of even order Hermite polynomials multiplied by a Maxwellian. A quasi-equilibrium with current density linear in the vector potential is constrained by conservation of particle number, total energy and the wave equation for the vector potential. The sinh-Poisson equation, a nonlinear partial differential equation, is introduced for a more general quasi-equilibrium. An analytic expression is found for the vector potential in terms of Jacobian elliptic functions and the expansion coefficients for the expansion of the distribution function in Hermite polynomials are found.

Cluster properties and particle production in Poincaré invariant quantum mechanics

I outline the construction of exactly Poincar\'e invariant quantum models that satisfy cluster separability but do not conserve particle number.

Resonant States of Open Quantum Systems

We first show that quantum resonant states observe particle number conservation and hence are consistent with the probabilistic interpretation of quantum mechanics. We then present for a class of quantum open systems, a resonant-state expansion of the sum of the retarded and advanced Green's functions. The expansion is given purely in terms of all discrete eigenstates and does not contain any background integrals. Using the expansion, we argue that the Fano asymmetry of resonance peaks is interpreted as interference between discrete eigenstates. We microscopically derive the Fano parameters for several cases.

Quantum phase diagrams of fermionic dipolar gases in a planar array of one-dimensional tubes

We systematically study ground-state properties of fermionic dipolar gases in a planar array of one-dimensional potential tubes for arbitrary orientation of dipole moments. Using the Luttinger liquid theory with the generalized Bogoliubov transformation, we calculate the elementary excitations and the Luttinger scaling exponents for various relevant quantum orders. The complete quantum phase diagrams for arbitrary polar angle of the dipole moment are obtained including charge-density wave, $p$-wave superfluid, intertube gauge-phase density wave, and intertube $s$-wave superfluid, where the last two break the U(1) gauge symmetry of the system (conservation of particle number in each tube) and occur only when the intertube interaction is larger than the intratube interaction. We then discuss the physical properties and experimental features of these many-body phases.

Treatment of the intrinsic Hamiltonian in particle-number nonconserving theories

We discuss the implications of using an intrinsic Hamiltonian in theories without particle-number conservation, e.g., the Hartree–Fock–Bogoliubov approximation, where the Hamiltonian's particle-number dependence leads to discrepancies if one naively replaces the particle-number operator by its expectation value. We develop a systematic expansion that fixes this problem and leads to an a posteriori justification of the widely-used one- plus two-body form of the intrinsic kinetic energy in nuclear self-consistent field methods. The expansion's convergence properties as well as its practical applications are discussed for several sample nuclei.

Chemical nonequilibrium for interacting bosons: Applications to the pion gas

We consider an interacting pion gas in a stage of the system evolution where thermal but not chemical equilibrium has been reached, i.e., for temperatures between thermal and chemical freeze-out T(ther) < T < T(chem) reached in relativistic heavy-ion collisions. Approximate particle number conservation is implemented by a nonvanishing pion number chemical potential mu(pi) within a diagrammatic thermal field-theory approach, valid in principle for any bosonic field theory in this regime. The resulting Feynman rules are derived here and applied within the context of chiral perturbation theory to discuss thermodynamical quantities of interest for the pion gas such as the free energy, the quark condensate, and thermal self-energy. In particular, we derive the mu(pi) not equal 0 generalization of Luscher and Gell-Mann-Oakes-Renner-type relations. We pay special attention to the comparison with the conventional kinetic theory approach in the dilute regime, which allows for a check of consistency of our approach. Several phenomenological applications are discussed, concerning chiral symmetry restoration, freeze-out conditions, and Bose-Einstein pion condensation.

Levels of self-consistency in the GW approximation

We perform GW calculations on atoms and diatomic molecules at different levels of self-consistency and investigate the effects of self-consistency on total energies, ionization potentials, and particle number conservation. We further propose a partially self-consistent GW scheme in which we keep the correlation part of the self-energy fixed within the self-consistency cycle. This approximation is compared to the fully self-consistent GW results and to the GW(0) and the G(0)W(0) approximations. Total energies, ionization potentials, and two-electron removal energies obtained with our partially self-consistent GW approximation are in excellent agreement with fully self-consistent GW results while requiring only a fraction of the computational effort. We also find that self-consistent and partially self-consistent schemes provide ionization energies of similar quality as the G(0)W(0) values but yield better total energies and energy differences.

0νββnuclear matrix elements and the occupancy of individual orbits

The measured occupancies of valence orbits in 76Ge and 76Se are used as a guideline for modification of the effective mean field energies that results in better description of these quantities. With them, in combination with the self-consistent renormalized quasiparticle random phase approximation (SRQRPA) method that ensures conservation of the mean particle number in the correlated ground state, we show that the resulting 0νββ nuclear matrix element for the 76Ge-->76Se transition is reduced by ~25% compared to the previous QRPA value, and therefore the difference between the present approach and the interacting shell model predictions becomes correspondingly smaller. Analogous modification of the mean field energies for the A=82 system also results in a reduction of 0νββ matrix element for the 82Se-->82Kr transition, making it also closer to the shell model prediction.

High-order time-splitting Hermite and Fourier spectral methods

In this paper, we are concerned with the numerical solution of the time-dependent Gross–Pitaevskii Equation (GPE) involving a quasi-harmonic potential. Primarily, we consider discretisations that are based on spectral methods in space and higher-order exponential operator splitting methods in time. The resulting methods are favourable in view of accuracy and efficiency; moreover, geometric properties of the equation such as particle number and energy conservation are well captured. Regarding the spatial discretisation of the GPE, we consider two approaches. In the unbounded domain, we employ a spectral decomposition of the solution into Hermite basis functions; on the other hand, restricting the equation to a sufficiently large bounded domain, Fourier techniques are applicable. For the time integration of the GPE, we study various exponential operator splitting methods of convergence orders two, four, and six. Our main objective is to provide accuracy and efficiency comparisons of exponential operator splitting Fourier and Hermite pseudospectral methods for the time evolution of the GPE. Furthermore, we illustrate the effectiveness of higher-order time-splitting methods compared to standard integrators in a long-term integration.

Grand canonical Gutzwiller approximation for magnetic inhomogeneous systems

The Gutzwiller approximation (GA) for Gutzwiller-projected grand canonical wave functions with fugacity factors is investigated in detail. Our systems in general contain inhomogeneity and local magnetic moments. In deriving renormalization formulae, we also derive or estimate terms of higher powers of intersite contractions neglected in the conventional GA. We examine several different constraints, i.e., local/global spin-dependent/independent particle-number conservation. Out of the four, the local spin-dependent constraint seems the most promising at present. An improved GA derived from it agrees with the variational Monte Carlo method better than the conventional GA does. The corrections to the conventional GA can be interpreted as two-site correlation including the phase difference of configurations. Furthermore, projected quasi-particle excited states are orthogonal to each other within the GA. Using these states, spectral weights are calculated. We show that asymmetry between electron addition and removal spectra can appear by taking into account the higher powers of the intersite contractions in the case of the d-wave superconductors and the Fermi sea; the addition is smaller than the removal. However, the asymmetry is quite weak especially near the Fermi level. In contrast, projected s-wave superconductors can have the opposite asymmetry (addition larger than removal) especially near the Fermi level. In addition, formulae from the other three constraints are also derived, which may be useful depending on purposes.

Particle-number conservation in thermodynamic systems within the Richardson model

To describe thermal pairing systems, an approach that combines the modified BCS method and the Lipkin-Nogami particle-number projection method is proposed. The expressions of the thermal pairing gap and internal energy are derived. The latter are numerically studied as a function of the temperature within the Richardson model.

Coupled force-balance and particle-occupation rate equations for high-field electron transport

It is pointed out that in the framework of balance-equation approach, the coupled force-balance and particle-occupation rate equations can be used as a complete set of equations to determine the high-field transport of semiconductors in both strong and weak electron-electron interaction limits. We call to attention that the occupation rate equation conserves the total particle number and maintains the energy balance of the relative electron system, and there is no need to introduce any other term in it. The addition of an energy-drift term in the particle-occupation rate equation [Phys. Rev. B 71, 195205 (2005)] is physically inadequate for the violation of the total particle-number conservation and the energy balance. It may lead to a substantial unphysical increase of the total particle number by the application of a dc electric field.

Particle-number conservation in static-path approximation for thermal superfluid systems

By applying particle-number projection to the static-path approximation (SPA), the heat capacity and the breakdown of pairing correlations are investigated in the thermally excited, superfluid systems $^{172}\mathrm{Yb}$, $^{94}\mathrm{Mo}$, and $^{56}\mathrm{Fe}$. For the heavy nucleus $^{172}\mathrm{Yb}$, the heat capacities in both the SPA and the number-projected SPA (NPSPA) exhibit an $\mathcal{S}$ shape; the difference between the SPA and NPSPA heat-capacity curves is not very large and the particle-number projection thereby enhances the $\mathcal{S}$ shape already seen in the SPA. The temperature at which the $\mathcal{S}$-shape of heat capacity curve occurs parallels the temperature of the breakdown of pairing correlations as indicated by the effective pairing gap. However, for the comparatively lighter nuclei $^{94}\mathrm{Mo}$ and $^{56}\mathrm{Fe}$, the SPA does not produce an $\mathcal{S}$-shaped heat capacity on its own; only after particle-number projection the $\mathcal{S}$ shape appears in the heat-capacity curve. For $^{94}\mathrm{Mo}$, we compare the NPSPA result with thermal odd-even mass differences, which are regarded as a direct measure of the pairing gap.

Kinetic equation for finite systems of fermions with pairing

The solutions of the Wigner-transformed time-dependent Hartree–Fock–Bogoliubov equations are studied in the constant-Δ approximation. This approximation is known to violate particle-number conservation. As a consequence, the density fluctuation and the longitudinal response function given by this approximation contain spurious contributions. A simple prescription for restoring both local and global particle-number conservation is proposed. Explicit expressions for the eigenfrequencies of the correlated systems and for the density response function are derived and it is shown that the semiclassical analogous of the quantum single-particle spectrum has an excitation gap of 2Δ, in agreement with the quantum result. The collective response is studied for a simplified form of the residual interaction.

Collisional damping of plasma waves on a pure electron plasma column

The collisional damping of electron plasma waves (or Trivelpiece–Gould waves) on a pure electron plasma column is discussed. The damping in a pure electron plasma differs from that in a neutral plasma, since there are no ions to provide collisional drag. A dispersion relation for the complex wave frequency is derived from Poisson’s equation and the drift-kinetic equation with the Dougherty collision operator—a Fokker–Planck operator that conserves particle number, momentum, and energy. For large phase velocity, where Landau damping is negligible, the dispersion relation yields the complex frequency ω=(kzωp∕k)[1+(3∕2)(kλD)2(1+10iα∕9)(1+2iα)−1], where ωp is the plasma frequency, kz is the axial wavenumber, k is the total wavenumber, λD is the Debye length, ν is the collision frequency, and α≡νk∕ωpkz. This expression spans from the weakly collisional regime (α⪡1) to the moderately collisional regime (α∼1) and in the weakly collisional limit yields a damping rate which is smaller than that for a neutral plasma by the factor k2λD2⪡1. In the strongly collisional limit (α⪢1), the damping is enhanced by long-range interactions that are not present in the kinetic theory (which assumes pointlike interactions); the effect of these long-range collisions on the damping is discussed.

SOME ISSUES ON THE RELATIVISTIC BCS-BEC CROSSOVER

We discuss some issues about the relativistic BCS-BEC crossover in a boson-fermion model. These include particle number conservation, chemical equilibrium, the derivation of the thermodynamic potential, the regularization of the gap equation and the BCS ratio.