Dependence of equilibrium properties of channeled particles on the transverse quasitemperature

We use methods of nonequilibrium thermodynamics to investigate the quasiequilibrium and kinetic characteristics of channeled particles regarded as a separate thermodynamic subsystem. For the channeled particles, we derive the energy—momentum balance equation in the moving coordinate system and show that the solution of the balance equation provides an expression for the main thermodynamic parameter, the transverse quasitemperature of the channeled-particle subsystem. We study the quasiequilibrium angular distribution of particles after their passage through a thin single crystal, the quasiequilibrium distribution over the particle exit angles under backscattering conditions, and also the rate constant for the nonequilibrium (dechanneling) process at large deviations of the system as a whole from the thermodynamic equilibrium. We discuss a measurement method for the particle beam transverse temperature over the peak height of the angular particle distribution found in the framework of a “shoot-through” experiment. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 3, pp. 509–524, March, 2006.     

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.     

First order derivatives of thermodynamic functions under assumption of no chemical changes revisited

In the field of experimental chemistry, it is extremely useful and necessary to have fast access to equations that link the physical and chemical parameters to the state. In this regard, starting with Bridgmann's equations, an online application had been projected and implemented in order to generate all equations of first order derivatives of thermodynamic parameters for a closed system. MathJax open source and PHP language have been used for implementation. So a framework was conceived and implemented to automatically generate the first order derivatives of the thermodynamic parameters for closed system.     

Vibrations and heating of inelastic solids under harmonic loading

Two problem statements for the investigation of the thermomechanical behavior of inelastic materials under harmonic loading are developed. The “complete” problem statement consists of universal balance equations of thermodynamics and constitutive equations derived from the general thermodynamic theory of viscoplastic solids with internal state variables. An approximate problem statement is derived by the application of modified harmonic linearizing technique to the complete system of equations. Thereby the subsystem of mechanical equations is formulated in the complex-value form and material properties are described by means of amplitude-dependent complex-value moduli. The problems of vibration and dissipative heating of physically nonlinear solids are formulated. The main thermomechanical characteristics are analyzed for some classes of problems. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On complicated models of continuous media in the general theory of relativity

In the context of the general theory of relativity, the system of Euler's equations is obtained from the variational equation under the assumption that the Lagrangian of the material depends on supplementary (as compared with classical theories) thermodynamic parameters, and when possible irreversible processes are taken into account. It is shown that, for a thermodynamically closed system, the equations of momenta for a continuous medium are a consequence of the field equations. The form of the energymomentum tensor of the material is considered when the arguments include the Lagrangian of the derivatives of the supplementary thermodynamic parameters.     

Generating pareto-optimal alternatives by a nonfeasible hierarchical method

A hierarchical algorithm for generating Pareto-optimal alternatives for convex multicriteria problems is derived. At the upper level, values for Lagrange multipliers of the coupling constraints are first given. Then at the subsystems, Pareto-optimal values are determined for the subsystem objectives, whereby an additional term or an additional objective is included due to the Lagrange multipliers. In the subsystem optimizations, the coupling equations between the subsystems are not satisfied; therefore, the method is called nonfeasible. Finally, the upper level checks which of the subsystem solutions satisfy the coupling constraints; these solutions are Pareto-optimal solutions for the overall system.     

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.     

Kinetic Theory of Quantum Electrodynamic Plasma in a Strong Electromagnetic Field: II. The Covariant Mean-Field Approximation

A covariant kinetic equation for the matrix Wigner function is derived in the mean-field approximation from a general kinetic equation for the fermionic subsystem of a quantum electrodynamic plasma. We show that in the semiclassical limit, the equations for the components of the Wigner function can be transformed into closed kinetic equations for the Lorentz-invariant distribution functions of particles and antiparticles.     

An approach to constructing models of multiphase elastic porous media

A novel approach to describing the behaviour of multiphase elastic porous media is proposed. The average values of the physical quantities needed to describe the motions of porous media are formulated using an integral relation. The validity of this relation is taken as the fundamental hypothesis. The integral definition of the average values enables integral relations to be devised for the average values from the integral laws of conservation of mass, momentum and energy and the increase in entropy. Along with the average values, the integral relations contain new variables that can be identified with generalized thermodynamic forces, which can be used to take into account the phase interaction in a porous medium. The integral relations are used to derive differential equations for the rate of entropy change and Gibbs relations for a porous medium as a basis for obtaining the constitutive relations. Relationships between the thermomechanical parameters of the model are established from the Gibbs relations under additional assumptions. The equation for the rate of entropy change can be used to establish relations between the generalized thermodynamic forces and fluxes. A complete system of differential equations in the defining parameters, which describes the motion of multiphase elastic porous media, is finally obtained.     

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.     

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.     

Hybrid decentralized maximum entropy control for large-scale dynamical systems

In the analysis of complex, large-scale dynamical systems it is often essential to decompose the overall dynamical system into a collection of interacting subsystems. Because of implementation constraints, cost, and reliability considerations, a decentralized controller architecture is often required for controlling large-scale interconnected dynamical systems. In this paper, a novel class of fixed-order, energy-based hybrid decentralized controllers is proposed as a means for achieving enhanced energy dissipation in large-scale lossless and dissipative dynamical systems. These dynamic decentralized controllers combine a logical switching architecture with continuous dynamics to guarantee that the system plant energy is strictly decreasing across switchings. The general framework leads to hybrid closed-loop systems described by impulsive differential equations. In addition, we construct hybrid dynamic controllers that guarantee that each subsystem–subcontroller pair of the hybrid closed-loop system is consistent with basic thermodynamic principles. Special cases of energy-based hybrid controllers involving state-dependent switching are described, and an illustrative combustion control example is given to demonstrate the efficacy of the proposed approach.     

Numerical method for a dynamic optimization problem arising in the modeling of a population of aerosol particles

A model coupling differential equations and a sequence of constrained optimization problems is proposed for the simulation of the evolution of a population of particles at equilibrium interacting through a common medium.The first order optimality conditions of the optimization problems relaxed with barrier functions are coupled with the differential equations into a system of differential-algebraic equations that is discretized in time with an implicit first order scheme. The resulting system of nonlinear algebraic equations is solved at each time step with an interior-point/Newton method. The Newton system is block-structured and solved with Schur complement techniques, in order to take advantage of its sparsity. Application to the dynamics of a population of organic atmospheric aerosol particles is given to illustrate the evolution of particles of different sizes. To cite this article: A. Caboussat, A. Leonard, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Modeling and simulation of nitrogen regulation in Corynebacterium glutamicum

Modeling the dynamic behaviour of biochemical systems at a molecular level aims at understanding and predicting the interactions of macromolecules inside the cell. Models of small subsystems based on differential equations not only prepare the way for the long-term goal of understanding a whole cell, but are inherently valuable due to their ability to predict the behaviour of the subsystem for varying external conditions or parameters. Nitrogen supply is essential for prokaryotes, thus the nitrogen uptake is an interesting target for model building. The goal is to provide new information about the interactions of the relevant proteins by performing various simulations.A model based on piecewise linear differential equations is formulated for the nitrogen uptake in Corynebacterium glutamicum. We theoretically derive a model for biochemical networks and introduce a general method for the parameter estimation which is also applicable in the case of very short time series. This approach is applied to a special system concerning the nitrogen uptake using Western blot experiments. The equations are set up for the main components of this system, the optimization problem for parameter estimation is formulated and solved, and simulations for the evaluation of the model as well as for predictions are carried out.We show that model building based on differential equations can also, when only a few measurements are performed, lead to a satisfactory model which provides valuable insights into the way it’s network components function. For example, we are able to make predictions about the maximal value of the time course as well as the steady-state level of the signal transduction protein GlnK in case of restricted activity of the proteases when considering the transition of nitrogen starvation to nitrogen excess or vice versa.     

Point interactions in the problem of three quantum particles with internal structure

A system of three quantum particles with internal structure in which the two-body interactions are point interactions and are described in terms of two-channel Hamiltonians is considered. It is established that in the cases when the parameters of the model are such that the total Hamiltonian of the three-particle system is semibounded the Faddeev equations are Fredholm equations. Boundary conditions are formulated for the differential Faddeev equations whose solutions are the scattering wave functions.Joint Institute for Nuclear Research, Dubna. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 102, No. 2, pp. 258–282, February, 1995.     

Thermodynamic and statistical principles of the theory of elastoviscoplastic deformation and strengthening of materials

We propose a thermodynamic method and a statistical one for constructing the constitutive equations of elastoviscoplastic deformation and strengthening of materials. The thermodynamic method is based on the energy conservation law as well as the equations of entropy balance and entropy generation in the presence of self-equilibrated internal microstresses, which are characterized by coupled strengthening parameters. The general constitutive equations consist of the relations between thermodynamic flows and forces, which follow from nonnegativity of entropy generation and satisfy the generalized Onsager principle, as well as the thermoelasticity relations and the expression for entropy, which follow from the energy conservation law. The specific constitutive equations are obtained on the basis of representation of the energy dissipation rate as a sum of two constituents that describe translational and isotropic strengthening and are approximated by power and hyperbolic sine laws. Starting from the stochastic microstructural concepts, we construct the constitutive equations of elastoviscoplastic deformation and strengthening on the basis of the linear model of thermoelasticity and the nonlinear Maxwell model for spherical and deviatoric components of microstresses and microstrains, respectively. The solution of the problem of the effective properties and stress-strain state of a three-component material is constructed with the use of the combined Voigt–Reuss scheme and leads to constitutive equations coinciding, as to their form, with similar equations constructed by the thermodynamic method.     

A steady-state system in non-equilibrium thermodynamics including thermal and electrical effects

The steady-state equations for a charged gas or fluid consisting of several components, exposed to an electric field, are considered. These equations form a system of strongly coupled, quasilinear elliptic equations which in some situations can be derived from the Boltzmann equation. The model uses the duality between the thermodynamic fluxes and the thermodynamic forces. Physically motivated mixed Dirichlet–Neumann boundary conditions are prescribed. The existence of generalized solutions is proven. The key of the proof is a transformation of the problem by using the entropic variables, or electro-chemical potentials, which symmetrize the equations. The uniqueness of weak solutions is shown under the assumption that the boundary data are not far from the thermal equilibrium. A general uniqueness result cannot be expected for physical reasons. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

Stokes flow of an assemblage of porous particles: stress jump condition

The present article investigates the overall bed permeability of an assemblage of porous particles. For the bed of porous particles, the fluid-particle system is represented as an assemblage of uniform porous spheres fixed in space. Each sphere, with a surrounding envelope of fluid, is uncoupled from the system and considered separately. This model is popularly known as cell model. Stokes equations are employed inside the fluid envelope and Brinkman equations are used inside the porous region. The stress jump boundary condition is used at the porous-liquid interface together with the continuity of normal stress and continuity of velocity components. On the surface of the fluid envelope, three different possible boundary conditions are tested. The obtained expression for the drag force is used to estimate the overall bed permeability of the assemblage of porous particles and the behavior of overall bed permeability is analyzed with various parameters like modified Darcy number (Da*), stress jump coefficient (??), volume fraction (??), and effective viscosity.     

Dynamics Equations for an Anisotropic Plastic Subsystem of a Fractal Medium

Model equations for the dynamics of an anisotropic plastic subsystem of a fractal medium are obtained in explicit form using fractional calculus. The slow dynamics of a number of parameters are analyzed and the results are represented graphically.     

Simulation of wave processes in a vapor-liquid medium

Numerical methods for the simulation of nonlinear wave processes in a vapor-liquid medium with a model two-phase spherical symmetric cell, with a pressure jump at its external boundary are considered. The viscosity and compressibility of the liquid, as well as the space variation of pressure in the vapor, are neglected. The problem is described by the heat equations in the vapor and liquid, and by a system of ODEs for the velocity, pressure, and radius at the bubble boundary. The equations are discretized in space by an implicit finite-volume scheme on a dynamic adaptive grid with grid refinement near the bubble boundary. The total time derivative is approximated by a method of backward characteristics. “Nonlinear” iterations are implemented at each time step to provide a specified high accuracy. The results of numerical experiments are presented and discussed for the critical thermodynamic parameters of water, for some initial values of the bubble radius and pressure jump.