Stable first-order particle-frame relativistic hydrodynamics for dissipative systems

We propose a stable first-order relativistic dissipative hydrodynamic equation in the particle frame (Eckart frame) for the first time. The equation to be proposed was in fact previously derived by the authors and a collaborator from the relativistic Boltzmann equation. We demonstrate that the equilibrium state is stable with respect to the time evolution described by our hydrodynamic equation in the particle frame. Our equation may be a proper starting point for constructing second-order causal relativistic hydrodynamics, to replace Eckart's particle-flow theory.     

Analytical solution of hyperbolic heat conduction equation in relation to laser short-pulse heating

In the present study, the hyperbolic heat conduction equation is derived from the Boltzmann transport equation and the analytical solution of the resulting equation appropriate to the laser short-pulse heating of a solid surface is presented. The time exponentially decaying pulse is incorporated as a volumetric heat source in the hyperbolic equation to account for the absorption of the incident laser energy. The Fourier transformation is used to simplify the hyperbolic equation and the analytical solution of the simplified equation is obtained using the Laplace transformation method. Temperature distribution in space and time are computed in steel for two laser pulse parameters. It is found that internal energy gain from the irradiated field, due to the presence of the volumetric heat source in the hyperbolic equation, results in rapid rise of temperature in the surface region during the early heating period. In addition, temperature decay is gradual in the surface region and as the depth below the surface increases beyond the absorption depth, temperature decay becomes sharp.     

Non-Markoffian effects in the laser

The atomic coordinates are eliminated exactly from the laser master equation with the help of a projector technique. The resulting integrodifferential equation for the field statistical operator with the kernel given correctly up to fourth order in the coupling constant is shown to be equivalent to a fourth order differential equation with respect to time. All theories footing on an “adiabatic” elimination procedure, especially the wellknown Fokker Planck equation treatment and a recently published theory containing a second order time derivative are shown to be successive approximations to our treatment. Non-Markoffian effects are discussed.     

Construction of a Langevin model from time series with a periodical correlation function: Application to wind speed data

A Langevin-type equation for stochastic processes with a periodical correlation function is introduced. A procedure of reconstruction of the equation from time series is proposed and verified on simulated data. The method is applied to geophysical time series–hourly time series of wind speed measured in northern Italy–constructing the macroscopic model of the phenomenon.     

Master Equation in Phase Space for a Uniaxial Spin System

A master equation, for the time evolution of the quasi-probability density function of spin orientations in the phase space representation of the polar and azimuthal angles is derived for a uniaxial spin system subject to a magnetic field parallel to the axis of symmetry. This equation is obtained from the reduced density matrix evolution equation (assuming that the spin-bath coupling is weak and that the correlation time of the bath is so short that the stochastic process resulting from it is Markovian) by expressing it in terms of the inverse Wigner-Stratonovich transformation and evaluating the various commutators via the properties of polarization operators and spherical harmonics. The properties of this phase space master equation, resembling the Fokker-Planck equation, are investigated, leading to a finite series (in terms of the spherical harmonics) for its stationary solution, which is the equilibrium quasi-probability density function of spin “orientations” corresponding to the canonical density matrix and which may be expressed in closed form for a given spin number. Moreover, in the large spin limit, the master equation transforms to the classical Fokker-Planck equation describing the magnetization dynamics of a uniaxial paramagnet.     

Mode coupling from linear and nonlinear kinetic equations

The calculation of mode coupling contributions to equilibrium time correlation functions from the nonlinear Boltzmann equation is reconsidered. It is suggested that the use of a nonlinear kinetic equation is not appropriate in this context, but instead such calculations should be reinterpreted in terms of the Klimontovich equation for the microscopic phase space density. For hard spheres the Klimontovich equation is formally similar to the nonlinear Boltzmann equation, and this similarity is exploited to explain the successful calculation of mode coupling effects from the latter. The relationship of the Klimontovich formulation to the linear ring approximation is also established.     

Radiative transfer theory with time delay for the effect of pulse entrapping in a resonant random medium: General transfer equation and point-like scatterer model

Abstract

A pulse propagation of a vector electromagnetic wave field in a discrete random medium under the condition of Mie resonant scattering is considered on the basis of the Bethe–Salpeter equation in the two-frequency domain in the form of an exact kinetic equation which takes into account the energy accumulation inside scatterers. The kinetic equation is simplified using the transverse field and far wave zone approximations which give a new general tensor radiative transfer equation with strong time delay by resonant scattering. This new general radiative transfer equation, being specified in terms of the low-density limit and the resonant point-like scatterer model, takes the form of a new tensor radiative transfer equation with three Lorentzian time-delay kernels by resonant scattering. In contrast to the known phenomenological scalar Sobolev equation with one Lorentzian time-delay kernel, the derived radiative transfer equation does take into account effects of (i) the radiation polarization, (ii) the energy accumulation inside scatterers, (iii) the time delay in three terms, namely in terms with the Rayleigh phase tensor, the extinction coefficient and a coefficient of the energy accumulation inside scatterers, respectively (i.e. not only in a term with the Rayleigh phase tensor). It is worth noting that the derived radiative transfer equation is coordinated with Poynting's theorem for non-stationary radiation, unlike the Sobolev equation. The derived radiative transfer equation is applied to study the Compton–Milne effect of a pulse entrapping by its diffuse reflection from the semi-infinite random medium when the pulse, while propagating in the medium, spends most of its time inside scatterers. This specific albedo problem for the derived radiative transfer equation is resolved in scalar approximation using a version of the time-dependent invariance principle. In fact, the scattering function of the diffusely reflected pulse is expressed in terms of a generalized time-dependent Chandrasekhar H-function which satisfies a governing nonlinear integral equation. Simple analytic asymptotics are obtained for the scattering function of the front and the back parts of the diffusely reflected Dirac delta function incident pulse, depending on time, the angle of reflection, the mean free time, the microscopic time delay and a parameter of the energy accumulation inside scatterers. These asymptotics show quantitatively how the rate of increase of the front part and the rate of decrease of the rear part of the diffusely reflected pulse become slower with transition from the regime of conventional radiative transfer to that of pulse entrapping in the resonant random medium.     

Interaction of nonlinearity and polarization mode dispersion

The derivation of the coupled nonlinear Schrödinger equation and the Manakov-PMD equation is reviewed. It is shown that the usual scalar nonlinear Schrödinger equation can be derived from the Manakov-PMD equation when polarization mode dispersion is negligible and the signal is initially in a single polarization state as a function of time. Applications of the Manakov-PMD equation to studies of the interaction of the Kerr nonlinearity with polarization mode dispersion are then discussed.     

Ion-acoustic waves in plasma of warm ions and isothermal electrons using time-fractional KdV equation

The ion-acoustic solitary wave in collisionless unmagnetized plasma consisting of warm ions-fluid and isothermal electrons is studied using the time fractional KdV equation. The reductive perturbation method has been employed to derive the Korteweg-de Vries equation for small but finite amplitude ion-acoustic wave in warm plasma. The Lagrangian of the time fractional KdV equation is used in a similar form to the Lagrangian of the regular KdV equation with fractional derivative for the time differentiation. The variation of the functional of this Lagrangian leads to the Euler-Lagrange equation that gives the time fractional KdV equation. The variational-iteration method is used to solve the derived time fractional KdV equation. The calculations of the solution are carried out for different values of the time fractional order. These calculations show that the time fractional can be used to modulate the electrostatic potential wave instead of adding a higher order dissipation term to the KdV equation. The results of the present investigation may be applicable to some plasma environments,such as the ionosphere plasma.     

Reversibility and irreversibility from an initial value formulation

From a time evolution equation for the single particle distribution function derived from the N-particle distribution function (A. Muriel, M. Dresden, Physica D 101 (1997) 297), an exact solution for the 3D Navier–Stokes equation – an old problem – has been found (A. Muriel, Results Phys. 1 (2011) 2). In this Letter, a second exact conclusion from the above-mentioned work is presented. We analyze the time symmetry properties of a formal, exact solution for the single-particle distribution function contracted from the many-body Liouville equation. This analysis must be done because group theoretic results on time reversal symmetry of the full Liouville equation (E.C.G. Sudarshan, N. Mukunda, Classical Mechanics: A Modern Perspective, Wiley, 1974). no longer applies automatically to the single particle distribution function contracted from the formal solution of the N-body Liouville equation. We find the following result: if the initial momentum distribution is even in the momentum, the single particle distribution is reversible. If there is any asymmetry in the initial momentum distribution, no matter how small, the system is irreversible.     

Empirical parameterization of CR-39 longitudinal track depth

In this work, new empirical equation describing the charged particles radiation track development against etching time and track longitudinal depth are presented. The equation involves four free fitting parameters. It is shown that this equation can reproduce tracks depth formed on the CR-39 by alpha particles at different energies and etching times. Parameters values obtained from experimental data can be used to predict etched track lengths at different energies and etching times. The empirical equation suggested is self consistent as far as reproducing all features of track depth development as a function of etching time and energy are concerned.     

Anomalous diffusion and charge relaxation on comb model: exact solutions

The random walks on the comb structure are considered. It is shown that due to fingers a diffusion has an anomalous character, that is an r.m.s. displacement depends on time by a power way with exponent . The generalized diffusion equation for an anomalous case is deduced. It essentially differs from a usual diffusion equation in the continuity equation form: instead of the first time derivative, the time derivative of fractal order appears. In the second part the charge relaxation on the comb structure is studied. A non-Maxwell character is established. The reason is that the electric field has three components, but a charge may relax only along some conducting lines.     

The Theory of Individual Based Discrete-Time Processes

A general theory is developed to study individual based models which are discrete in time. We begin by constructing a Markov chain model that converges to a one-dimensional map in the infinite population limit. Stochastic fluctuations are hence intrinsic to the system and can induce qualitative changes to the dynamics predicted from the deterministic map. From the Chapman–Kolmogorov equation for the discrete-time Markov process, we derive the analogues of the Fokker–Planck equation and the Langevin equation, which are routinely employed for continuous time processes. In particular, a stochastic difference equation is derived which accurately reproduces the results found from the Markov chain model. Stochastic corrections to the deterministic map can be quantified by linearizing the fluctuations around the attractor of the map. The proposed scheme is tested on stochastic models which have the logistic and Ricker maps as their deterministic limits.     

Some general problems of fast electrons transport

A generalization is given of the segments method in the form of a multistep method with generalized time for computing the transport of fast particles. The integral equation for a flow with generalized time in the phase space of variables is written under the assumption that the flow cuts the generalized time surface at right angles. The Green's function for the differential flow operator is the kernel of the integral equation. It is also shown that such an integral equation which can be obtained from a nonstationary kinetic equation provides a uniform consistent algorithm for solving either nonstationary or stationary problems. Examples of Green's functions are given for an operator of differential flow of fast electrons.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 110–114, August, 1974.The author would like to express his thanks to A. A. Vorob'ev and B. A. Kononov for their encouragement, to A. P. Yalovets for discussing the work with him, and to A. M. Kol'chuzhkin for going through the text.     

Stochastic theory of nonlinear rate processes with multiple stationary states. II. Relaxation time from a metastable state

We have developed a methodology for obtaining a Fokker-Planck equation for nonlinear systems with multiple stationary states that yields the correct system size dependence, i.e., exponential growth with system size of the relaxation time from a metastable state. We show that this relaxation time depends strongly on the barrier heightU(x) between the metastable and stable states of the system. For a Fokker-Planck (FP) equation to yield the correct result for the relaxation time from a metastable state, it is therefore essential that the free energy functionU(x) of the FP equation not only correctly locate the extrema of U(x), but also have the correct magnitudeU at these extrema. This is accomplished by so choosing the coefficients of the FP equation that its stationary solution is identical to that of the master equation that defines the nonlinear system.This work was supported in part by the National Science Foundation under Grant CHE 75-20624.     

A five-dimensional perspective on the Klein–Gordon equation

We discuss the Klein–Gordon (KG) equation using a path-integral approach in 5D space–time. We explicitly show that the KG equation in flat space–time admits a consistent probabilistic interpretation with positively defined probability density. However, the probabilistic interpretation is not covariant. In the non-relativistic limit, the formalism reduces naturally to that of the Schrödinger equation. We further discuss other interpretations of the KG equation (and their non-relativistic limits) resulting from the 5D space–time picture. Finally, we apply our results to the problem of hydrogenic spectra and calculate the canonical sum of the hydrogenic atom.     

Canonical transformations and exact invariants for time‐dependent Hamiltonian systems

An exact invariant is derived for n‐degree‐of‐freedom non‐relativistic Hamiltonian systems with general time‐dependent potentials. To work out the invariant, an infinitesimalcanonical transformation is performed in the framework of the extended phase‐space. We apply this approach to derive the invariant for a specific class of Hamiltonian systems. For the considered class of Hamiltonian systems, the invariant is obtained equivalently performing in the extended phase‐space a finitecanonical transformation of the initially time‐dependent Hamiltonian to a time‐independent one. It is furthermore shown that the invariant can be expressed as an integral of an energy balance equation. The invariant itself contains a time‐dependent auxiliary function ξ (t) that represents a solution of a linear third‐order differential equation, referred to as the auxiliary equation. The coefficients of the auxiliary equation depend in general on the explicitly known configuration space trajectory defined by the system's time evolution. This complexity of the auxiliary equation reflects the generally involved phase‐space symmetry associated with the conserved quantity of a time‐dependent non‐linear Hamiltonian system. Our results are applied to three examples of time‐dependent damped and undamped oscillators. The known invariants for time‐dependent and time‐independent harmonic oscillators are shown to follow directly from our generalized formulation.     

Fokker-Planck equation for a multiplicative stochastic process with a stationary gaussian noise

Starting from a Langevin equation with stationary gaussian noise of arbitrary correlation time, a corresponding Fokker-Planck equation is derived under the condition of small noise strength.     

Theory of the drift motion of charged particles

A differential equation is derived for the time dependence of the average random-motion energy of a particle drifting in an electric field. There is a discussion of the various versions of this equation, the equation of motion which follows from it, and particular solutions. There is a discussion of topics associated with the time dependence of the coordinate of a drifting particle, the average drift velocity, and the current-voltage characteristic of the drift region for the case of a space-charge limitation on the current.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii Fizika, No. 8, pp. 119–125, August, 1969.