Nonequilibrium statistical thermodynamics of channeled particles: Thermal particles

The rate of atomic particles escaping a potential minimum as a result of interactions with the lattice thermal vibrations and the electron gas is investigated using nonequlibrium statistical thermodynamic methods. The physical properties of the theory are analyzed. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 116, No. 3, pp. 442–455, September, 1998.     

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.     

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.     

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.     

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.     

The second factor in the reciprocity law

The explicit formula for the symbol of the normed residue in a local field depends on the coefficients of two series. The second series is given by the choice of a prime element and a primitive root of unity. Recurrence relations are established in the work for the coefficients of this series which uniquely determine the series up to a factor from the ring of p-adic integers.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Maternaticheskogo Instituta im. V. A. Steklova. AN SSSR, Vol. 75, pp. 59–66, 1978.     

The second law of thermodynamics and multifractal distribution functions: Bin counting,pair correlations,and the Kaplan–Yorke conjecture

We explore and compare numerical methods for the determination of multifractal dimensions for a doubly-thermostatted harmonic oscillator. The equations of motion are continuous and time-reversible. At equilibrium the distribution is a four-dimensional Gaussian, so that all the dimension calculations can be carried out analytically. Away from equilibrium the distribution is a surprisingly isotropic multifractal strange attractor, with the various fractal dimensionalities in the range 1 < D < 4. The attractor is relatively homogeneous, with projected two-dimensional information and correlation dimensions which are nearly independent of direction. Our data indicate that the Kaplan–Yorke conjecture (for the information dimension) fails in the full four-dimensional phase space. We also find no plausible extension of this conjecture to the projected fractal dimensions of the oscillator. The projected growth rate associated with the largest Lyapunov exponent is negative in the one-dimensional coordinate space.     

Three-dimensional kinetic equation for high-energy particles in noncircular tokamak plasma

Translated from Chislennye Metody Resheniya Obratnykh Zadach Matematicheskoi Fiziki, pp. 209–213, 1988.     

Newton's second law in field theory

In this article we present a natural generalization of Newton's Second Law valid in field theory, i.e., when the parameterized curves are replaced by parameterized submanifolds of higher dimension. For it we introduce what we have called the geodesic k-vector field, analogous to the ordinary geodesic field and which describes the inertial motions (i.e., evolution in the absence of forces). From this generalized Newton's law, the corresponding Hamilton's canonical equations of field theory (Hamilton-De Donder-Weyl equations) are obtained by a simple procedure. It is shown that solutions of generalized Newton's equation also hold the canonical equations. However, unlike the ordinary case, Newton equations determined by different forces can define equal Hamilton's equations.     

The narrow fracture approximation by channeled flow

The singular problem of non-stationary Darcy flow in a region containing a narrow channel of width O(?) and high permeability is shown to be well approximated by a problem with flow concentrated on a weighted Sobolev space over a lower-dimensional interface. The convergence of the solution as ?→0 is proved for both the stationary case and the corresponding initial-boundary-value problem. The structure of the limiting problems is dependent on the rate of taper of the channel at its extremities.     

Continued fractions and the second Kepler law

In this paper we introduce a link between geometry of ordinary continued fractions and trajectories of points that moves according to the second Kepler law. We expand geometric interpretation of ordinary continued fractions to the case of continued fractions with arbitrary elements.     

A second order limit law for occupation times of the Cauchy process

The purpose of this note is to extend a second order limit law for one dimensional Cauchy process obtained in Kasahara (Y. Kasahara, Limit theorems for occupation times of Markov processes, Publ. RIMS, Kyoto Univ. 12 (1977), pp. 801–818), using the method of moments and some kind of chaining argument.     

Strong law of large numbers for supercritical superprocesses under second moment condition

Consider a supercritical superprocess X = {X_t, t≥0} on a locally compact separable metric space (E,m). Suppose that the spatial motion of X is a Hunt process satisfying certain conditions and that the branching mechanism is of the form ψ(x,λ)=-a(x)λ+b(x)λ2+∫(0,+∞)(e-λy-1+λy)n(x,dy),?x∈E,λ>0, where a∈Bb(E),b∈Bb+(E), and n is a kernel from E to (0,+∞) satisfying sup?x∈E∫0+∞y2n(x,dy)<+∞. Put Ttf(x)=Pδx?f,Xt?. Suppose that the semigroup {T_t; t≥0}is compact. Let λ₀ be the eigenvalue of the (possibly non-symmetric) generator L of {T_t}that has the largest real part among all the eigenvalues of L, which is known to be real-valued. Let ?0 and ?^0 be the eigenfunctions of L and L^(the dual of L) associated with λ₀, respectively. Assume λ₀>0. Under some conditions on the spatial motion and the ?0-transform of the semigroup {T_t}, we prove that for a large class of suitable functions f, lim?t→+∞e-λ0t?f,Xt?=W∞∫E?^0(y)f(y)m(dy),?Pμ-a.s., for any finite initial measure μ on E with compact support, where W_∞ is the martingale limit defined by W∞:=lim?t→+∞e-λ0t??0,Xt?. Moreover, the exceptional set in the above limit does not depend on the initial measure μ and the function f.     

Interacting Brownian particles and the Wigner law

Summary In this paper, we study interacting diffusing particles governed by the stochastic differential equationsdX _j (t)= _n dB _j (t) –D _jØ_n(X ₁,...,X _n)dt,j=1, 2,...,n. Here theB _jare independent Brownian motions in ^d , and Ø _n (X ₁,...,X _n)= _{n ij} V(X _iX _j) + _ni U(X ₁). The potentialV has a singularity at 0 strong enough to keep the particles apart, and the potentialU serves to keep the particles from escaping to infinity. Our interest is in the behaviour as the number of particles increases without limit, which we study through the empirical measure process. We prove tightness of these processes in the case ofd=1,V(x)=–log|x|,U(x)=x ²/2 where it is possible to prove uniqueness of the limiting evolution and deduce that a limiting measure-valued process exists. This process is deterministic, and converges to the Wigner law ast. Some information on the rates of convergence is derived, and the case of a Cauchy initial distribution is analysed completely.Supported by SERC grant number GR/H 00444