Variational derivation of improved KP-type of equations

The Kadomtsev-Petviashvili equation describes nonlinear dispersive waves which travel mainly in one direction, generalizing the Korteweg-de Vries equation for purely uni-directional waves. In this Letter we derive an improved KP-equation that has exact dispersion in the main propagation direction and that is accurate in second order of the wave height. Moreover, different from the KP-equation, this new equation is also valid for waves on deep water. These properties are inherited from the AB-equation (E. van Groesen, Andonowati, 2007 [1]) which is the unidirectional improvement of the KdV equation. The derivation of the equation uses the variational formulation of surface water waves, and inherits the basic Hamiltonian structure.     

Variational principle for derivation of macroscopic equations

It is pointed out that the Schwinger variational principle of scattering theory applies to the case of linear and nonlinear relaxation problems in quantum statistics. By means of this principle it is possible to derive closed sets of equations for expectation values. To illustrate this variational method and to clarify the connection to other standard approaches some simple examples are treated for which the equations of motion are already known.     

On the derivation of linearized hydrodynamics from kinetic theory

The relation of the linearized hydrodynamic equations to kinetic theory is discussed. We show that the theory of subdynamics provides a general framework for their derivation. This theory and the related transformation theory permit the construction of a Lyapounov functional that for the hydrodynamical processes, considered here, becomes the macroscopic entropy as given by Gibbs relation.     

Variational derivation of KdV-type models for surface water waves

Using the Hamiltonian formulation of surface waves, we approximate the kinetic energy and restrict the governing generalized action principle to a submanifold of uni-directional waves. Different from the usual method of using a series expansion in parameters related to wave height and wavelength, the variational methods retains the Hamiltonian structure (with consequent energy and momentum conservation) and makes it possible to derive equations for any dispersive approximation. Consequentially, the procedure is valid for waves above finite and above infinite depth, and for any approximation of dispersion, while quadratic terms in the wave height are modeled correctly. For finite depth this leads to higher-order KdV type of equations with terms of different spatial order. For waves above infinite depth, the pseudo-differential operators cannot be approximated by finite differential operators and all quadratic terms are of the same spatial order.     

Variational calculations of asymmetric nuclear matter

We report on variational calculations of the energy E(ρ, β) of asymmetric nuclear matter having ? = ?_n + ?_p = 0.05 to 0.35 fm^?3, and β = (?_n ? ?_p/g9 = 0 to 1. The nuclear h used in this work consists of a realistic two-nucleon interaction, called v₁₄, that fits the available nucleon-nucleon scattering data up to 425 MeV, and a phenomenological three nucleon interaction adjusted to reproduce the empirical properties of symmetric nuclear matter. The variational many-body theory of symmetric nuclear matter is extended to treat matter with neutron excess. Numerical and analytic studies of the β-dependence of various contributions to the nuclear matter energy show that at ? < 0.35 fm^?3 the β⁴ terms are very small, and that the interaction energy EI(ρ, β) defined as E(ρ, β) ? T_F(ρ, β), where T_F is the Fermi-gas energy, is well approximated by EI₀(?) + β²EI₂(ρ). The calculated symmetry energy at equilibrium density is 30 MeV and it increases from 15 to 38 MeV as ? increases from 0.05 to 0.35 fm^?3.     

Variational approach to hydrodynamics: Application to the sonoluminescence gas bubble

We develop a variational approximation allowing to study under realistic physical conditions the interior dynamics of cavitating gas bubbles in liquids. We discuss in some detail the intricacy of the gas bubble dynamics related to the small size of the system. We also show how to formulate the problem in the framework of the energy equation for non-adiabatic processes.     

Variational methods in nuclear pairing theory

Particle number conserving groundstate wave functions are constructed by use of the Thouless-Peierls formalism. A comparison with PBCS- and FBCS-solutions shows an encouraging advantage of this method: 1. the groundstate is lower in energy than the FBCS groundstate, 2. the computer time necessary to obtain the groundstate wave function as well as its energy is smaller than with the FBCS method.     

Microscopic derivation of quantum fluctuations in nuclear reactions

The coupling between collective and intrinsic states of a nucleus is treated with first order time-dependent perturbation theory at zero temperature. In addition to the friction constant and conservative potential corrections, general expressions for the diffusion constants and differential equations in time for the second moments are derived for position and momentum dependent couplings, and are calculated in a particular solvable model. As a result, the diffusion constants do not necessarily vanish at zero temperature but there remain pure quantum effects. For oscillatory collective motion, the low temperature limit of the Einstein relation is given and an existence theorem for damped pure quantum states is derived.     

Variational derivation of the adiabatic time-dependent Hartree-Fock formalism and generalized Hamiltonian dynamics

The three-particle collision process 3 → 3 with relative high-energy motion of each pair of particles described by a model with eikonal Hamiltonian is investigated. No additional restrictions on the motion of the particles (such as fixed scattering centre approximation) are imposed. It is shown that the three-particle problem in this case can be solved analytically. An explicit expression for 3 → 3 amplitude off the energy shell is obtained as the result of the exact summation of the multiple scattering series in the considered model. On the energy shell this series breaks off (there are no terms higher than triple). The formula for the mutual cancellation of the higher-order terms of the series is derived.     

Variational calculations of realistic models of nuclear matter

We report variational calculations of nuclear matter with a semi-realistic Reid v₁₂ model, and a realistic v₁₄ model of the two-nucleon interaction operator. The v₁₄ model fits the available nucleon-nucleon scattering data up to 425 MeV lab energy, and has relatively weak L² and $\text{(}\text{L}\text{·}\text{S}\text{)}{}^2$ interactions in addition to the standard central, tensor and ($\text{L}\text{·}\text{S}$). The L² and $\text{(}\text{L}\text{·}\text{S}\text{)}{}^2$ interactions are treated semiperturbatively; their contribution reduces the overbinding of nuclear matter. However, the equilibrium k_F = 1.7 fm^?1 and E₀ = ?17.5 MeV obtained with the v₁₄ model are both higher than their empirical values k_F = 1.33 fm^? and E₀ = ?16 MeV. We assume that the difference between the calculated and empirical E(ρ) is entirely due to three-nucleon interactions (TNI). The TNI contributions are phenomenologically added to the nuclear matter energy, and their parameters are adjusted to obtain the correct equilibrium energy, density and compressibility. The required TNI contributions appear to be of reasonable magnitude.     

Variational calculations of ν8 models of nuclear matter

We report variational calculations of ν8 models of nuclear matter which contain central, spin, isospin, tensor and spin-orbit potentials. These semi-realistic models can explain the nucleon-nucleon scattering in 1S0, 3S1?3D1, 1P1 and 3P2?3F2 states up to ～ 300 MeV. The variational wave function has two-body central, spin, isospin, tensor and spin-orbit correlations. The terms in the cluster expansion of the energy expectation value, that do not contain the spin-orbit correlations are summed by chain summation techniques developed for the ν6 models. Of the terms containing spin-orbit correlations, the two-body and three-body-separable ones are calculated, and the magnitude of the rest is estimated. Results for three phase-equivalent ν8 models, which differ significantly in the strength of tensor and spin-orbit potentials, are reported. They suggest that simple ν8 models may not be able to simultaneously explain the binding energy and density of nuclear matter.     

On classical hydrodynamics and collective motion as approximation to the nuclear many-body problem

Using time-dependent unitary transformations, one can cast a one-body equation of the time-dependent Hartree-Fock type into a form which is closely related to equations for a classical irrotational fluid. The hydrodynamic equation of state finds its counterpart in a stationary constrained field equation. The hydrodynamic equations in turn can be translated into a classical Hamiltonian formalism with an infinite number of generalised coordinates, which are given as all possible spatial moments of the density. The reduction to a few ones, the “natural collective coordinates” is possible by the choice of appropriate initial conditions. The lowest of the hydrodynamical frequencies can be calculated in closed form by harmonic approximations. For the quadrupole frequency a value ofω=31 A^?1/3 MeV/h is obtained. As expected, the value does not agree with the experiment, but rather is in between the characteristic frequency for theβ-vibration and the isoscalar giant quadrupole vibration.     

Two-fluid hydrodynamics applied to the deconfinement transition in the fragmentation region in ultra-relativistic nuclear collisions

A two-fluid model is presented which describes the interpenetration of nuclei colliding at ultra-relativistic energies. The two fluids are coupled by friction resulting from hadron-hadron collisions. The model calculations predict the deconfinement transition in the baryon-rich fragmentation region. The degree of the conversion and the maximum temperature depend sensitively on the characteristic time for the rearrangement of nuclear matter into the quark-gluon plasma.     

Quantum hydrodynamics

Quantum hydrodynamics in superfluid helium and atomic Bose–Einstein condensates (BECs) has been recently one of the most important topics in low temperature physics. In these systems, a macroscopic wave function (order parameter) appears because of Bose–Einstein condensation, which creates quantized vortices. Turbulence consisting of quantized vortices is called quantum turbulence (QT). The study of quantized vortices and QT has increased in intensity for two reasons. The first is that recent studies of QT are considerably advanced over older studies, which were chiefly limited to thermal counterflow in ⁴${}^4$He, which has no analog with classical traditional turbulence, whereas new studies on QT are focused on a comparison between QT and classical turbulence. The second reason is the realization of atomic BECs in 1995, for which modern optical techniques enable the direct control and visualization of the condensate and can even change the interaction; such direct control is impossible in other quantum condensates like superfluid helium and superconductors. Our group has made many important theoretical and numerical contributions to the field of quantum hydrodynamics of both superfluid helium and atomic BECs. In this article, we review some of the important topics in detail. The topics of quantum hydrodynamics are diverse, so we have not attempted to cover all these topics in this article. We also ensure that the scope of this article does not overlap with our recent review article (arXiv:1004.5458), “Quantized vortices in superfluid helium and atomic Bose–Einstein condensates”, and other review articles.     

Bi-velocity hydrodynamics

Theoretical evidence derived from linear irreversible thermodynamics (LIT) jointly with Burnett’s solution of Boltzmann’s gas-kinetic equation is used to show that fluid mechanics and transport processes in both gaseous and liquid continua require the use of two independent velocities rather than one in order to correctly quantify the physics of fluid motion. This finding, reflecting the coalescence of macroscopic and molecular perspectives, undermines the current foundations of continuum fluid mechanics. Of the two required context-specific velocities, one is the mass velocity appearing in the continuity equation. The other is the volume velocity entering into the constitutive equation for the mechanical rate-of-working term appearing in the energy equation, where it serves as the multiplier of the pressure tensor . While the analysis involves only linear constitutive principles, the fundamental need for two independent velocities is noted to apply even in non-linear circumstances. A major consequence of these findings is that the Navier-Stokes-Fourier equations governing continuum fluid physics are incomplete for both single- and multi-component fluids. Our results are independently supported by the work of others based upon the use of conventional single-velocity arguments accompanied by ad hoc extensions of LIT. Our bi-velocity findings point to the existence of novel mechanodiffusive phenomena in fluid continua, entailing coupling between viscous flow and diffusion, whether referring to the diffusion of thermal energy in single-component non-isothermal fluids or of chemical species in inhomogeneous multicomponent mixtures.     

Exact canonically conjugate momentum to the quadrupole tensor and a microscopic derivation of the nuclear collective hamiltonian

Two momenta conjugate to the mass quadrupole tensor are given. The first is a canonical momentum only in a subspace of the shell model space. A microscopic collective kinetic energy in terms of this momentum and the quadrupole tensor is then obtained and compared with that of Bohr's hamiltonian. The second momentum is, on the other hand, canonically conjugate to the quadrupole tensor in the entire state space.     

Nonextensive perfect hydrodynamics — a model of dissipative relativistic hydrodynamics?

We demonstrate that nonextensive perfect relativistic hydrodynamics (q-hydrodynamics) can serve as a model of the usual relativistic dissipative hydrodynamics (d-hydrodynamics) therefore facilitating considerably its applications. As an illustration, we show how using q-hydrodynamics one gets the q-dependent expressions for the dissipative entropy current and the corresponding ratios of the bulk and shear viscosities to entropy density, ζ/s and η/srespectively.      

Variational Study of

We use the variationad method to study the meson states in the lattice Schwinger Model,and to calculate the mass ratio of vector and scalar mesons as well as the mass of vector meson.Our results are consistent with the exactly values in the continuum.