Balance equation and the quasitemperature of channeled particles in equilibrium

Using the methods of nonequilibrium statistical thermodynamics, we obtain the equation for the transverse energy and momentum balance for fast atomic particles moving in the planar channeling regime. Based on the solution of this equation, we obtain an expression for the transverse quasitemperature in the quasiequilibrium in terms of the basic parameters of the theory. We show that the equilibrium quasitemperature of channeled particles is established because of particle diffusion in the space of transverse energies (subsystem “heating”), the dissipative process (“cooling”), and the anharmonic effects of particle oscillations between the channel walls (the redistribution of energies over the oscillatory degrees of freedom is the internal thermalization of the subsystem). According to the estimates for particles with an energy of the order of 1 MeV, the quasitemperature values are in the characteristic temperature range for a low-temperature plasma.     

Nonequilibrium statistical thermodynamics of channeled particles. The transverse quasitemperature

Fast charged particles moving in a crystal in the channeling regime are treated as an independent thermodynamic subsystem for which energy acnd momentum balance equations are derived in the comoving coordinate system. It is shown that the solution of these equations gives an expression for the transverse quasitemperature of the channeled particles in terms of the fundamental parameters of the nicroscopic theory. If the electron scattering is quasielastic, then at a penetration depth of the order of the coherence length the system as a whole remains in strong disequilibrium despite the attainment of internal equilibrium in the subsystem of the particles. There is a large difference between the thermodynamic parameters of the channeled particles and of the thermal reservoir.A. A. Baikov Institute of Metallurgy, Russian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 102, No. 1, pp. 106–118, January, 1995.     

Theorem on energy distribution over degrees of freedom for a quasistable symmetric anharmonic oscillator

We propose a method for describing quasiequilibrium processes in a quasistable anharmonic oscillator affected by heat noise. We introduce effective thermodynamic parameters analogous to those of an equilibrium system but differing from them in numerical value. We define the parameter characterizing the nonequilibrium of the system, prove the mathematical consistency of the proposed method, and obtain the quasiequilibrium analogue of the theorem on energy distribution over degrees of freedom.     

Two stages of motion of anharmonic oscillators modeling fast particles in crystals

We consider the evolution of the spatial distribution of fast atomic particles as they pass through a crystal. At small penetration depths where the particles move without significant loss of coherence, the joint probability density function becomes smoother at the expense of the gradient of the anharmonic potential of the planar channel. We show that as a result of this smoothing, there arises a quasiequilibrium (quasistationary) state of the subsystem of fast particles. The further evolution of the particle distribution is caused by nonelastic scattering and is described by the kinetic equation. At this stage, the anharmonic character of particle oscillations between the channel walls results in a renormalization of the total phonon scattering cross section of particles, and the renormalization is determined by the fourth-order anharmonic interaction.     

Statistical theory for fast particle channeling based on the local boltzmann equation. Correlation matrix of interactions and diffusion function of particles

The kinetic theory of motion for fast particles in a crystal is elaborated, based on the Bogoliubov chain of equations. A local kinetic equation is derived for the one-particle distribution function in conditions of particle interaction with thermal lattice oscillations and valence electrons. A characteristic of the particle subsystem in the de-channeling problem—the diffusion function B(ε_⊥) in the space of transverse energies—is determined, accounting for the explicit form of the collision term in the kinetic equation. It is found that the functional relationship described by B(ε_⊥) has different forms in the three variation intervals of ε_⊥ that are related to channeling, quasichanneling, and chaotic particle motion. Furthermore, it is shown that the diffusion function has a singularity for the value of the transverse energy equal to the potential barrier of the channel. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 111, No. 3, pp. 483–496, June, 1997.     

Nonequilibrium statistical thermodynamics of channeled particles: Resonance transitions and dechanneling

An elementary act of dechanneling includes diffusion of particles in the space of transverse energies and a resonance transition that occurs after a particle reaches the channeling-regime potential level. The dechanneling rate coefficient is defined using equations of nonequilibrium statistical thermodynamics. Physical quantities including the resonance transition matrix element, single-phonon and electron scattering relaxation rates, and the transverse quasi-temperature function related to the difference between the thermodynamic parameters of fast particles and the thermostat determining the dechanneling rate coefficient are expressed in terms of the basic parameters of the theory. Translated from Teoreticheskaya i Matematicheskaya Fizika. Vol. 116, No. 1, pp. 146–160, July, 1998.     

Nonequilibrium statistical thermodynamics of cascade processes in solids: the quasi temperature of cascade particles

Displacement cascades in solids are investigated in the framework of nonequilibrium statistical thermodynamics. The quasi temperature of a subsystem of cascade particles in metals and semiconductors is derived using the energy balance equation for a cascade process.     

Quantum-Classical Wigner-Liouville Equation

We consider a quantum system that is partitioned into a subsystem and a bath. Starting from the Wigner transform of the von Neumann equation for the quantum-mechanical density matrix of the entire system, the quantum-classical Wigner-Liouville equation is obtained in the limit where the masses M of the bath particles are large as compared with the masses m of the subsystem particles. The structure of this equation is discussed and it is shown how the abstract operator form of the quantum-classical Liouville equation is obtained by taking the inverse Wigner transform on the subsystem. Solutions in terms of classical trajectory segments and quantum transition or momentum jumps are described. __________ Published in Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 6, pp. 749–756, June, 2005.     

S-matrix description of nonequilibrium finite-temperature systems

We consider the “inclusive” (“partial”) method for describing nonequilibrium dissipative systems at the early (kinetic) evolution stage, when the temperature distribution is nonuniform. We formulate the perturbation theory in terms of space-time-local temperature Green’s functions and derive the Liouville equation for the one-particle partition function. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 149, No. 3, pp. 368–380, December, 2006.     

Optimization of Approximation Models for Solving Kinetic Equations

We consider the problem of cancer treatment using radiation therapy. A deterministic Boltzmann equation is considered as the particle transport model. The P_N system of moment equations is derived using angular discretization of the transport equation. This P_N system is solved using StaRMAP [3]. An optimization problem to get the desired particle distribution is formulated and solved using adjoint equations. The algorithm is validated with several simulations. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Electrical Conductivity of Charged Particle Systems and Zubarev’s Nonequilibrium Statistical Operator Method

One of the fundamental problems in physics that are not yet rigorously solved is the statistical mechanics of nonequilibrium processes. An important contribution to describing irreversible behavior starting from reversible Hamiltonian dynamics was given by D. N. Zubarev, who invented the method of the nonequilibrium statistical operator. We discuss this approach, in particular, the extended von Neumann equation, and as an example consider the electrical conductivity of a system of charged particles. We consider the selection of the set of relevant observables. We show the relation between kinetic theory and linear response theory. Using thermodynamic Green’s functions, we present a systematic treatment of correlation functions, but the convergence needs investigation. We compare different expressions for the conductivity and list open questions.     

Saltation in wind blown sand

The Boltzmann equation of the sand particle velocity distribution function in wind-blown sand two-phase flow is established based on the motion equation of single particle in air. And then, the generalized balance law of particle property in single phase granular flow is extended to gas-particle two-phase flow. The velocity distribution function of panicle phase is expanded into an infinite series by means of Grad’s method and the Gauss distribution is used to replace Maxwell distribution. In the case of truncation at the third-order terms, a closed third-order moment dynamical equation system is constructed. The theory is further simplified according to the measurement results obtained by stroboscopic photography in wind tunnel tests. Project supported by the National Natural Science Foundation of China (Grant Nos. 49301002, 19672022). This work was originally done in Xi’an Jiaotong University.     

Application of the particle method to simulate one-dimensional bounded plasma with a distributed sources

We consider the behavior of a plasma bounded in the longitudinal direction by absorbing walls. The model contains charged particles (electrons and ions) moving in the direction of an external magnetic field with two velocity components: longitudinal and transverse. The charged particles are created in pairs by a distributed source. The working model is based on the electrostatic “particles in a cell” method augmented by Emmert's model for a volume source and a model of binary Coulomb particle collisions using the Monte Carlo method. Calculation results are reported for a model with electron-ion collisions and for a collisionless plasma model. Translated from Chislennye Metody v Matematicheskoi Fizike, Published by Moscow University, Moscow, 1996, pp. 100–109.     

Nonequilibrium Statistical Operator Method and Generalized Kinetic Equations

We consider some principal problems of nonequilibrium statistical thermodynamics in the framework of the Zubarev nonequilibrium statistical operator approach. We present a brief comparative analysis of some approaches to describing irreversible processes based on the concept of nonequilibrium Gibbs ensembles and their applicability to describing nonequilibrium processes. We discuss the derivation of generalized kinetic equations for a system in a heat bath. We obtain and analyze a damped Schrödinger-type equation for a dynamical system in a heat bath. We study the dynamical behavior of a particle in a medium taking the dissipation effects into account. We consider the scattering problem for neutrons in a nonequilibrium medium and derive a generalized Van Hove formula. We show that the nonequilibrium statistical operator method is an effective, convenient tool for describing irreversible processes in condensed matter.     

The second law of thermodynamics for high-energy channeled particles

In microscopic theory, the number of kinetic equations underlying the proof of the second law of thermodynamics is quite restricted. We explicitly prove that the second law of thermodynamics is satisfied for high-energy particles moving in a crystal in the channeling regime. The proof involves a local Boltzmann equation for the distribution function of the particles written in the Bogoliubov form. In this, we take one statistical mechanism into account: the scattering of channeled particles on lattice atoms randomly displaced from the crystal sites.     

Approach to equilibrium of a rotating sphere in a Stokes flow

We study the unsteady rotary motion of a sphere immersed in a Stokes fluid. The equation of motion for the sphere leads to an integro-differential equation, and we are interested in the asymptotic behavior in time of the solution. Preparing initially the system (sphere + fluid) as a stationary state, we prove that the angular velocity of the sphere slows down with a law t ^−3/2 if no other forces than the one exerted by the fluid act on the sphere, while if the sphere is subject also to an elastic torque the asymptotic behavior of the angular position of the sphere is t ^−γ , with γ = 5/2 if the initial angular velocity is zero, γ = 3/2 otherwise. This behavior is due to the memory effect of the surrounding fluid. We discuss briefly other initial preparations of the system.     

Correlated Brownian Motions as an Approximation to Deterministic Mean-Field Dynamics

We analyze the transition from deterministic mean-field dynamics of several large particles and infinitely many small particles to a stochastic motion of the large particles. In this transition the small particles become the random medium for the large particles, and the motion of the large particles becomes stochastic. Under the assumption that the empirical velocity distribution of the small particles is governed by a probability density ψ, the mean-field force can be represented as the negative gradient of a scaled version of ψ. The stochastic motion is described by a system of stochastic ordinary differential equations driven by Gaussian space-time white noise and the mean-field force as a shift-invariant integral kernel. The scaling preserves a small parameter in the transition, the so-called correlation length. In this set-up, the separate motion of each particle is a classical Brownian motion (Wiener process), but the joint motion is correlated through the mean-field force and the noise. Therefore, it is not Gaussian. The motion of two particles is analyzed in detail and a diffusion equation is deduced for the difference in the positions of the two particles. The diffusion coefficient in the latter equation is spatially dependent, which allows us to determine regions of attraction and repulsion of the two particles by computing the probability fluxes. The result is consistent with observations in the applied sciences, namely that Brownian particles get attracted to one another if the distance between them is smaller than a critical small parameter. In our case, this parameter is shown to be proportional to the aforementioned correlation length. __________ Published in Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 6, pp. 757–769, June, 2005.     

Gibbs equation in the nonlinear nonequilibrium thermodynamics of dilute nonviscous gases

This paper deals with the derivation of the Gibbs equation for a nonviscous gas in the presence of heat flux. The analysis aims to shed some light on the physical interpretation of thermodynamic potentials far from equilibrium. Two different definitions for the chemical potential and thermodynamic pressure far from equilibrium are introduced: nonequilibrium chemical potential and nonequilibrium thermodynamic pressure at constant heat flux q and nonequilibrium chemical potential and nonequilibrium thermodynamic pressure at constant J = Vq, where V is the specific volume.