Power-like decreasing solutions of the linearized Boltzmann equation and conservation of mass or energy

We study the power-like solutions of the spatially homogeneous linearized Boltzmann equation for a class of binary cross-sections proportional to $\text{|g|}{}^{\mathrm{<mtext>1?</mtext><mtext>4</mtext><mtext>s</mtext>}}$, s=4 or s < - 1, g being the relative speed. We show that these solutions violate the physical requirement of conservation of energy. A similar study for the associate thermalization problem leads to a violation of the conservation law of mass. We study the asymptotic behaviours of the eigenfunctions associated to non-discrete eigenvalues and corresponding to the regular spectrum. The main point, which was already present in our previous study of the hard sphere case is the link between a critical power like decreasing behaviour and conservation of energy. We proved that there exists a solution ($\text{R≈v}{}^{\mathrm{<mtext>-(6?</mtext><mtext>4</mtext><mtext>s</mtext><mtext>)</mtext>}}$) associated to this behaviour (as conjecture by Ernst, Hellesoe, Hauge) and it is the only one living outside the standard Hilbert space. A very interesting tool is provided by asymptotic kernels which carry the dominant part of the asymptotic behaviour of the solutions.     

Dirac spinor solitons or bags

Explicit solutions of a non-linear Dirac equation in the four-dimensional Minkowski space have been found. They are continuous eigenfunctions of spin $\text{1}\text{2}$, energy and parity, and are confined to the interior of a sphere of definite radius a, with no tails outside the sphere. They are called solitons in the paper though the name of spinor droplets or bags may be more appropriate. Because of the Lorentz covariance of the Dirac equations, arbitrarily moving solitons can be obtained from the rest system solutions by applying proper Lorentz transformations.     

A large-order analysis of one-dimensional lattice strong coupling expansions

A large order lattice strong coupling expansion for the ground state energy of the quantum mechanical hamiltonian $\text{1}\text{2}\text{p}{}^2\text{+ g|x|}{}^{\mathrm{\alpha }}\text{, α > 0}$, is provided by a perturbative solution of the transfer matrix eigenvalue equation. This algorithmic derivation allows one to perform either an analytic or a very large order (N ～ 50) numerical study of the expansion, which in particular is found to have a finite radius of convergence for α ? 2. These results, which can be useful for a better understanding of this model when treated as a quantum field theory, indicate that the continuum limit can be safely extracted but that the rate of convergence is generally very poor.     

The linear Boltzmann equation properties and solutions

This work presents results on the transport equation obtained recently. Properties and distribution functions obeying the Boltzmann equation in one and many-dimensional spaces are derived and discussed. New polynomials have been found that under certain conditions represent solutions of the linear transport equation. A number of structural and spectral theorems have been demonstrated permitting a better understanding of the equation. It is shown that the streaming operator, $\text{Ω·Δ + σ(v)Σ}{}_{\mathrm{tot}}\text{(v)}$, maps a class of functions {$\text{;ψ(}\text{x, v}\text{)}$} of two variables onto another class of functions, {$\text{ψ(}\text{x}\text{)}$}, depending only on one variable. The distribution functions obtained here satisfy rigorously homogeneous or inhomogeneous Dirichlet conditions on the boundary surface of the convex system for any order of the polynomial representation. This is obtained by using the new polynomials which are characterised by particular structural properties. In many-dimensional cases the polynomials become operators with tempered distributional character. Numerical evaluations are given in the one dimensional case for both isotropic and anisotropic scattering. An application of the theory is also given for the heterogeneous system of plane geometry. This paper is organized in three parts. Part A gives an introduction to the present method and indicates the way leading from the original to the linearised Boltzmann equation. Other solution methods are comparatively discussed. Part B is dealing with the one-dimensional equation and the eigenvalues and eigenfunctions are found for various physical conditions. Degenerate kernels have been used throughout. Both the critical and non-critical problems have been solved and results are presented in form of graphs and tables. Part C proceeds to the examination of some problems in spaces of dimension p > 1. The main advantages of the structural approach are that (i) the solution is found in an elementary way completely analytically. (ii) The boundary conditions are rigorously satisfied. (iii) The heterogeneous system is very easily solved. (iv) The eigenvalues are found algebraically.     

Multiplicative stochastic processes in statistical physics

For a large class of nonlinear stochastic processes with pure multiplicative fluctuations the corresponding time-dependent Fokker-Planck equation is solved exactly by means of analytic methods. We obtain a universal eigenvalue spectrum and the corresponding set of eigenfunctions.     

Renormalization techniques for quantum spin systems. Ground-state energies

Projection of the Hamiltonian of an antiferromagnetic lattice of spins $\text{1}\text{2}$, without external fields, onto a subspace of the total spinor space gives an approximation for the lowest eigenvalue of this Hamiltonian. Repeated projection results in a series expansion for this approximation. In each projection the form of the Hamiltonian is conserved. The formal structure of this projection technique shows a strong analogy with the Wilson theory or renormalization-group theory of phase transitions. Numerical results are given for linear chains and triangular lattice.Analogous techniques apply to Ising and isotropic XY models in transverse fields.     

Eigenvalues and eigenfunctions of the Fokker-Planck equation for the extremely underdamped Brownian motion in a double-well potential

The eigenvalues and eigenfunctions of the Fokker-Planck equation describing the extremely underdamped Brownian motion in a symmetric double-well potential are investigated. By transforming the Fokker-Planck equation to energy and position coordinates and by performing a suitable averaging over the position coordinate, a differential equation depending only on energy is derived. For finite temperatures this equation is solved by numerical integration, whereas in the weak-noise limit an analytic result for the lowest nonzero eigenvalue is obtained. Furthermore, by using a boundary-layer theory near the critical trajectory, the correction term to the zero-friction-limit result is found.     

Statistical flow-oscillator modeling of vortex-shedding

The very important engineering problem of modeling the fluid-structure interaction occurring during the shedding of vortices has defied, and will probably continue to defy, a closed form exact solution for the foreseeable future. Therefore, an attempt must be made to extract relevant information about the process in order to be able to have a basic understanding of it for the purpose of analysis. A useful method involves the flow-oscillator concepts of Hartlen and Currie [1] redefined here for stochastic processes. The fluid-structure system is assumed to be governed by the cross-coupled equations

$\text{x}\text{?}\text{(t)+2ξω}{}_{\mathrm{n}}\text{x}\text{?}\text{(t)+ω}{}^2{}_{\mathrm{n}}\text{=C}{}_{\mathrm{e}}\text{(t)pV}{}^2{}_0\text{(t)DL/2m (i)}$

$\text{C}\text{?}{}_{\mathrm{e}}\text{(t)+{α ? βC}{}^2{}_{\mathrm{e}}\text{(t)+γC}{}^4{}_{\mathrm{e}}\text{(t)}}\text{C}\text{?}{}_{\mathrm{e}}\text{(t)+ω}{}^2{}_0\text{C}{}_{\mathrm{e}}\text{(t)=b}\text{x}\text{?}\text{(t), (ii)}$

where these equations govern the structure and fluid oscillators, respectively. The fluid damping is non-linear. These equations are taken as stochastic differential equations because of the many unpredictable, random effects that determine the loading and response. The lift coefficient C_$\text{l}$(t) is assumed to be a zero mean, narrow band process and the velocity V₀, composed of a uniform, constant velocity current plus oscillating wave, a broad band process. The analysis is based on solving equation (i) for x(t) by using Duhamel's integral and substituting its derivative $\text{x}\text{?}\text{(t)}$ into equation (ii). This equation is then used to derive the Fokker-Planck equation for the process C_$\text{l}$(t). To obtain the Fokker-Planck equation, slowly varying variables are replaced by their long-time averages [2] and then the method of stochastic averaging is employed [3, 4]. The moment equation for the lift-oscillator process is derived from the Fokker-Planck equation and, as equation (ii) is non-linear, one finds the moment equation to be in terms of higher order moments. A truncation scheme [5] is used to derive the moment generating function. It is possible then to generate the first and second order statistics of the lift coefficient and the structure response in terms of the empirical parameters of fluid damping. This work was carried out in conjunction with an analysis of ocean wave-current forces with application to offshore fixed structures [6].     

Solutions of the Fokker-Planck equation in detailed balance

The solutions of the Fokker-Planck equation in detailed balance are investigated. Firstly the necessary and sufficient conditions obtained by Graham and Haken are derived by an alternative method. An equivalent form of these conditions in terms of an operator equation for the Fokker-Planck Liouville operator is given. Next, the transition probability is expanded in terms of an biorthogonal set of eigenfunctions of a certain operatorL. The necessary and sufficient conditions for detailed balance leads to a simple operator equation forL. This operator equation guarantees that on!y half of the biorthogonal set needs to be calculated. Finally the dependence of the eigenvalues on the reversible and irreversible drift coefficient is discussed.     

Dynamics of Gompertzian tumour growth under environmental fluctuations

We investigate the effect of correlated additive and multiplicative Gaussian white noise on the Gompertzian growth of tumours. Our results are obtained by solving numerically the time-dependent Fokker-Planck equation (FPE) associated with the stochastic dynamics. In our numerical approach we have adopted B-spline functions as a truncated basis to expand the approximated eigenfunctions. The eigenfunctions and eigenvalues obtained using this method are used to derive approximate solutions of the dynamics under study. We perform simulations to analyze various aspects, of the probability distribution, of the tumour cell populations in the transient- and steady-state regimes. More precisely, we are concerned mainly with the behaviour of the relaxation time (τ) to the steady-state distribution as a function of (i) of the correlation strength (λ) between the additive noise and the multiplicative noise and (ii) as a function of the multiplicative noise intensity (D) and additive noise intensity (α). It is observed that both the correlation strength and the intensities of additive and multiplicative noise, affect the relaxation time.     

An equation of state of the Ising systems

In the framework of the spin operators diagram technique the method of taking into account the correlation effects in magnetic system based on the idea of $\text{1}\text{z}$ expansion (z is the number of the nearest neighbours) is proposed. An equation of state of 3-dimensional Ising systems with the arbitrary spin values S in the first order in $\text{1}\text{z}$ is presented which considerably improves the mean field theory (being the zeroth order approximation). For various lattices the calculated values of Tc coinside within 1% with the results of the high-temperature expansions.     

Relativistic dynamics on a null plane

In view of possible applications to the quark model and to hadron spectroscopy, we investigate relativistic Hamiltonian quantum theories of finitely many degrees of freedom. We make use of the fact that if null planes are used as initial surfaces, the structure of the theory closely resembles nonrelativistic quantum mechanics: the inner variables that describe the structure of the system uncouple from the motion of the system as a whole. The dynamical content of such a theory resides in the operators M, $\text{j}$ of mass and spin that act in the space carrying the inner degrees of freedom. Relativistic invariance is equivalent to the requirement that M and $\text{j}$ generate a unitary representation of U(2). In contrast to this requirement, the condition that the wavefunctions of the system transform covariantly strongly restricts the dynamics. It is proven that for systems containing two constituents, covariance is equivalent to an algebraic relation that involves M and $\text{j}$ — the angular condition. A class of solutions of the angular condition is provided by a particular type of local manifestly covariant wave equations. One nontrivial solution of this class, a relativistic oscillator is given in detail. Confinement models of this type represent an interesting alternative to the solutions of the angular condition that result from the perturbation expansion of a local field theory through the three-dimensional quasipotential versions of the Bethe-Salpeter equation.     

Quantum effects in the Bertotti-Robinson metric

Scalar and spin $\text{1}\text{2}$ fields have been studied in the Bertotti-Robinson space-time and analytical solutions have been obtained for the Klein-Gordon equation and the Dirac equation. For large e particle creation takes place as in the pseudoeuclidean metric under the influence of a constant external field.     

A study of the relativistic coulomb problem in momentum space

The quantum-mechanical solutions of the operator equation corresponding to the classical equation $\text{[(p ? eA)}{}^2\text{+ m}{}^2\text{]}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{? (E ? eU) = 0}$ are studied in the particular case of the one-dimensional Coulomb potential and compared to the standard solutions of the Klein-Gordon equation.     

Electronic structure of the (111) and (111) surfaces of GaAs

Inward relaxation effects of the outermost Ga layer on the electronic structure of GaAs (111) Ga and outward expansion effects of the outermost As layer on that of GaAs ($\text{111}$) As are studied by extended Hückel theory. Three different surface geometries are examined for the respective surfaces. It is shown that upon relaxation on GaAs (111) or upon expansion on GaAs ($\text{111}$) new surface states associated with dangling- and back-bonds are revealed. The character and dispersion behaviour of strongly localized surface states are described.     

Extended Thomas-Fermi theory at finite temperature

We present a generalization of the extended Thomas-Fermi (ETF) theory to finite temperatures T. Starting from the Wigner-Kirkwood expansion of the Bloch density in powers of , we derive the gradient expansion of the free energy and entropy density functionals $\text{F}$[ρ] and σ[ρ] up to fourth order with their correct temperature-dependent coefficients. (Effective mass and spin-orbit contributions are taken into account up to second order.) For a harmonic-oscillator potential we show that both the h-expansion of the free energy and the entropy and the gradient expansion of the functionals [ρ] and σ[ρ] converge very fast and yield the exact quantum-mechanical results for kT ? 3 MeV, where the shell effects are washed out. Finally we discuss the Euler variational equation obtained with the new functionals and use its numerical solutions for semi-infinite symmetric nuclear matter to test the quality of parametrized trial densities. As an application, we present liquid-drop model parameters, calculated with a realistic Skyrme interaction, as functions of the temperature.     

On the functional integral for generalized Wiener processes and nonequilibrium phenomena

The attention will be focussed on a generalized Wiener diffusion process for which the macroscopic evolution $\text{y}\text{?}\text{= c}{}_1\text{(}\text{y}\text{)}$ equals zero, of course, and where the variance of the process obeys $\text{g}\text{?}\text{s}{}^2\text{= c}{}_2\text{(}\text{y}\text{)}$. The diffusion function c₂(y) may be state dependent in an arbitrary way. We invoke our treatment of the general time-local Gaussian process as presented in a previous paper. This process will be seen to define a generalized functional Wiener measure. This measure has already been used implicitly in earlier work being concerned with nonlinear, nonequilibrium Markov processes. The sum of the generalized measure over the entire function space will be shown to be exactly related to the general Fokker-Planck equation for the driftless diffusion process. The relation between the well-defined functional sum and its corresponding functional integral will be studied in detail. The analysis demonstrates in clear fashion the origin of the deviations from other approaches, and provides an extension of our previous results on nonequilibrium, nonlinear phenomena to include generalized diffusion processes.     

The Cauchy problem for the O(N), CP(N − 1), and GC(N,p) models

We study the Cauchy problem for the O(N), $\text{C}$P(N ? 1) and G$\text{C}$(N, p) models in n + 1 dimensional space-time. We prove the existence and uniqueness of solutions for small time intervals and for any n. In space-time dimension two, the previous solutions can be extended to all times by the method of a priori estimates. In space-time dimensions three and four, our estimates yield only partial results on the global existence problem. In all cases the solutions are required only to belong to local spaces, which means that they satisfy local regularity conditions but have no restrictions on their behaviour at infinity in space.     

Stochastic derivation of the Dirac equation in terms of a fluid of spinning tops endowed with random fluctuations at the velocity of light

If one analyzes the stochastic behaviour of classical “rigid” tops imbedded in Dirac's aether (relativistic thermostat) one obtains (for random jumps at the velocity of light) a probability distribution corresponding to the Feynman-Gell-Mann equation for relativistic spin $\text{1}\text{2}$ particles.