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.     

Interaction of Two Charges in a Uniform Magnetic Field: II. Spatial Problem

The interaction of two charges moving in ℝ³ in a magnetic field B can be formulated as a Hamiltonian system with six degrees of freedom. Assuming that the magnetic field is uniform and the interaction potential has rotation symmetry, we reduce this system to one with three degrees of freedom. For special values of the conserved quantities, choices of parameters or restriction to the coplanar case, we obtain systems with two degrees of freedom. Specialising to the case of Coulomb interaction, these reductions enable us to obtain many qualitative features of the dynamics. For charges of the same sign, the gyrohelices either “bounce-back”, “pass-through”, or exceptionally converge to coplanar solutions. For charges of opposite signs, we decompose the state space into “free” and “trapped” parts with transitions only when the particles are coplanar. A scattering map is defined for those trajectories that come from and go to infinite separation along the field direction. It determines the asymptotic parallel velocities, guiding centre field lines, magnetic moments and gyrophases for large positive time from those for large negative time. In regimes where gyrophase averaging is appropriate, the scattering map has a simple form, conserving the magnetic moments and parallel kinetic energies (in a frame moving along the field with the centre of mass) and rotating or translating the guiding centre field lines. When the gyrofrequencies are in low-order resonance, however, gyrophase averaging is not justified and transfer of perpendicular kinetic energy is shown to occur. In the extreme case of equal gyrofrequencies, an additional integral helps us to analyse further and prove that there is typically also transfer between perpendicular and parallel kinetic energy.      

Equilibrium in quantum systems of particles with magnetic interaction. Fermi and bose statistics

Quantum systems of particles interacting via an effective electromagnetic potential with zero electrostatic component are considered (magnetic interaction). It is assumed that the j th component of the effective potential for n particles equals the partial derivative with respect to the coordinate of the jth particle of “magnetic potential energy” of n particles almost everywhere. The reduced density matrices for small values of the activity are computed in the thermodynamic limit for d-dimensional systems with short-range pair magnetic potentials and for one-dimensional systems with long-range pair magnetic interaction, which is an analog of the interaction of three-dimensional Chern-Simons electrodynamics (“magnetic potential energy” coincides with the one-dimensional Coulomb (electrostatic) potential energy). Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Published in Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 5, pp. 691–698, May 1997.     

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.     

Compactness for a class of hit-and-miss hyperspaces

In the study of some kind of generalized Vietoris-type topologies for the hyperspace of all nonempty closed subsets of a topological space (X, τ), namely the so called Δ-hit-and-miss-topologies with Δ⊇Cl(X) (or Δ-topologies), which was initiated by the second author in 1965, it is obvious, that the non-compactness of such a hyperspace often depends on the non-compactness even in the lower-semifinite topology (induced by the “hit-sets”), which is contained in all hypertopologies of this type. Otherwise, compactness for these topologies is easily obtained from the compactness of (X, τ) by well-known theorems, if the “miss-sets” are induced either by compact or closed subsets. To obtain a similar result for topologies with “miss-sets” generated by subsets with a property which generalizes both, closedness and compactness especially in the non-Hausdorff case, we use consequently a quite set-theoretical lemma, stated at the beginning.     

A problem with directional derivative in the theory of galvanomagnetic effects

We construct an asymptotics of the solution the Laplace equation in a “long” rectangle with the directional derivative given on its “long sides” and Dirichlet data on its “short sides.” By using the asymptotics, we calculate one of the integral characteristics, namely, the magnetoresistance. We obtain new formulas for the low-magnetic field magnetoresistance. Translated fromMatematicheskie Zametki, Vol. 65, No. 4, pp. 520–532, April, 1999.     

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.     

Compositions of n as alternating sequences of weakly increasing and strictly decreasing partitions

Compositions and partitions of positive integers are often studied in separate frameworks where partitions are given by q-series generating functions and compositions exhibiting specific patterns are designated by generating functions for these patterns. Here, we view compositions as alternating sequences of weakly increasing and strictly decreasing partitions (i.e. alternating blocks). We obtain generating functions for the number of such partitions in terms of the size of the composition, the number of parts and the total number of “valleys” and “peaks”. From this, we find the total number of “peaks” and “valleys” in the composition of n which have the mentioned pattern. We also obtain the generating function for compositions which split into just two partition blocks. Finally, we obtain the two generating functions for compositions of n that start either with a weakly increasing partition or a strictly decreasing partition.     

A Concentration Theorem for the Heat Equation

 We compare the solution of to the solution of the same equation where f is replaced by a “concentrated” source . As a result we derive some estimates on the solution in spatial norm, locally uniformly in t, with respect to the norm of for any integer . In the case we obtain a critical inequality relating the norm of to an exponential norm of u. (Received 1 September 2000; in revised form 17 January 2001)     

Optimal buffer size for a stochastic processing network in heavy traffic

We consider a one-dimensional stochastic control problem that arises from queueing network applications. The state process corresponding to the queue-length process is given by a stochastic differential equation which reflects at the origin. The controller can choose the drift coefficient which represents the service rate and the buffer size b>0. When the queue length reaches b, the new customers are rejected and this incurs a penalty. There are three types of costs involved: A “control cost” related to the dynamically controlled service rate, a “congestion cost” which depends on the queue length and a “rejection penalty” for the rejection of the customers. We consider the problem of minimizing long-term average cost, which is also known as the ergodic cost criterion. We obtain an optimal drift rate (i.e. an optimal service rate) as well as the optimal buffer size b ^*>0. When the buffer size b>0 is fixed and where there is no congestion cost, this problem is similar to the work in Ata, Harrison and Shepp (Ann. Appl. Probab. 15, 1145–1160, 2005). Our method is quite different from that of (Ata, Harrison and Shepp (Ann. Appl. Probab. 15, 1145–1160, 2005)). To obtain a solution to the corresponding Hamilton–Jacobi–Bellman (HJB) equation, we analyze a family of ordinary differential equations. We make use of some specific characteristics of this family of solutions to obtain the optimal buffer size b ^*>0. A.P. Weerasinghe’s research supported by US Army Research Office grant W911NF0510032.     

An anyon model

We construct an infinite-dimensional dynamical Hamiltonian system that can be interpreted as a localized structure (“quasiparticle”) on the plane E ₂. The model is based on the theory of an infinite string in the Minkowski space E _1,3 formulated in terms of the second fundamental forms of the worldsheet. The model phase space H is parameterized by the coordinates, which are interpreted as “internal” (E(2)-invariant) and “external” (elements of T*E ₂) degrees of freedom. The construction is nontrivial because H contains a finite number of constraints entangling these two groups of coordinates. We obtain the expressions for the energy and for the effective mass of the constructed system and the formula relating the proper angular momentum and the energy. We consider a possible interpretation of the proposed construction as an anyon model.     

The Geometry of Standard Deontic Logic

Whereas geometrical oppositions (logical squares and hexagons) have been so far investigated in many fields of modal logic (both abstract and applied), the oppositional geometrical side of “deontic logic” (the logic of “obligatory”, “forbidden”, “permitted”, . . .) has rather been neglected. Besides the classical “deontic square” (the deontic counterpart of Aristotle’s “logical square”), some interesting attempts have nevertheless been made to deepen the geometrical investigation of the deontic oppositions: Kalinowski (La logique des normes, PUF, Paris, 1972) has proposed a “deontic hexagon” as being the geometrical representation of standard deontic logic, whereas Joerden (jointly with Hruschka, in Archiv für Rechtsund Sozialphilosophie 73:1, 1987), McNamara (Mind 105:419, 1996) and Wessels (Die gute Samariterin. Zur Struktur der Supererogation, Walter de Gruyter, Berlin, 2002) have proposed some new “deontic polygons” for dealing with conservative extensions of standard deontic logic internalising the concept of “supererogation”. Since 2004 a new formal science of the geometrical oppositions inside logic has appeared, that is “n-opposition theory”, or “NOT”, which relies on the notion of “logical bi-simplex of dimension m” (m = n − 1). This theory has received a complete mathematical foundation in 2008, and since then several extensions. In this paper, by using it, we show that in standard deontic logic there are in fact many more oppositional deontic figures than Kalinowski’s unique “hexagon of norms” (more ones, and more complex ones, geometrically speaking: “deontic squares”, “deontic hexagons”, “deontic cubes”, . . ., “deontic tetraicosahedra”, . . .): the real geometry of the oppositions between deontic modalities is composed by the aforementioned structures (squares, hexagons, cubes, . . ., tetraicosahedra and hyper-tetraicosahedra), whose complete mathematical closure happens in fact to be a “deontic 5-dimensional hyper-tetraicosahedron” (an oppositional very regular solid).      

Asymptotic behavior and symmetry of condensate solutions in electroweak theory

We study condensate solutions of a nonlinear elliptic equation in ℝ², which models a W-boson with a cosmic string background. The existence of condensate solutions and an energy identity are discussed, based on which the refined asymptotic behavior of condensate solutions is established by studying the corresponding evolution dynamical system. Applying the “shrinking-sphere” method, we also prove the symmetry under inversions of condensate solutions for some special cases.     

Ill-posedness for semilinear wave equations with very low regularity

In this paper, we study the ill-posdness of the Cauchy problem for semilinear wave equation with very low regularity, where the nonlinear term depends on u and ∂ _t u. We prove a ill-posedness result for the “defocusing” case, and give an alternative proof for the supercritical “focusing” case, which improves the result in Fang and Wang (Chin. Ann. Math. Ser. B 26(3), 361–378, 2005). Supported by NSF of China 10571158.     

Bose-Einstein condensation in the excited band and the energy spectrum of the Bose-Hubbard model

Based on the mean field approximation, we investigate the transition into the Bose-Einstein condensate phase in the Bose-Hubbard model with two local states and boson hopping in only the excited band. In the hard-core boson limit, we study the instability associated with this transition, which appears at excitation energies δ < |t ₀ |, where |t ₀ | is the particle hopping parameter. We discuss the conditions under which the phase transition changes from second to first order and present the corresponding phase diagrams (Θ,μ) and (|t ₀ |, μ), where Θ is the temperature and μ is the chemical potential. Separation into the normal and Bose-Einstein condensate phases is possible at a fixed average concentration of bosons. We calculate the boson Green’s function and one-particle spectral density using the random phase approximation and analyze changes in the spectrum of excitations of the “particle” or “hole” type in the region of transition from the normal to the Bose-Einstein condensate phase.     

Realizability of greedy algorithms

We discuss new models of an “affine” theory of gravity in multidimensional space-times with symmetric connections. We use and develop ideas of Weyl, Eddington, and Einstein, in particular, Einstein’s proposal to specify the space-time geometry by the use of the Hamilton principle. More specifically, the connection coefficients are determined using a “geometric” Lagrangian that is an arbitrary function of the generalized (nonsymmetric) Ricci curvature tensor (and, possibly, of other fundamental tensors) expressed in terms of the connection coefficients regarded as independent variables. Such a theory supplements the standard Einstein gravity with dark energy (the cosmological constant, in the first approximation), a neutral massive (or tachyonic) vector field (vecton), and massive (or tachyonic) scalar fields. These fields couple only to gravity and can generate dark matter and/or inflation. The new field masses (real or imaginary) have a geometric origin and must appear in any concrete model. The concrete choice of the geometric Lagrangian determines further details of the theory, for example, the nature of the vector and scalar fields that can describe massive particles, tachyons, or even “phantoms.” In “natural” geometric theories, which are discussed here, dark energy must also arise. We mainly focus on intricate relations between geometry and dynamics while only very briefly considering approximate cosmological models inspired by the geometric approach.     

Current Fluctuations for Independent Random Walks in Multiple Dimensions

Consider a system of particles evolving as independent and identically distributed (i.i.d.) random walks. Initial fluctuations in the particle density get translated over time with velocity [(v)\vec]\vec{v}, the common mean velocity of the random walks. Consider a box centered around an observer who starts at the origin and moves with constant velocity [(v)\vec]\vec{v}. To observe interesting fluctuations beyond the translation of initial density fluctuations, we measure the net flux of particles over time into this moving box. We call this the “box-current” process.     

Quasi-parabolic analytic transformations of <Emphasis Type="Bold">C</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>. Parabolic manifolds

In [Rong, F., Quasi-parabolic analytic transformations of C ⁿ , J. Math. Anal. Appl. 343 (2008), 99–109], we showed the existence of “parabolic curves” for certain quasi-parabolic analytic transformations of C ⁿ . Under some extra assumptions, we show the existence of “parabolic manifolds” for such transformations.     

Reduction of hugoniot-maslov chains for trajectories of solitary vortices of the “shallow water” equations to the hill equation

According to Maslov’s idea, many two-dimensional, quasilinear hyperbolic systems of partial differential equations admit only three types of singularities that are in general position and have the property of “structure self-similarity and stability.” Those are: shock waves, “narrow” solitons, and “square-root” point singularities (solitary vortices). Their propagation is described by an infinite chain of ordinary differential equations (ODE) that generalize the well-known Hugoniot conditions for shock waves. After some reasonable closure of the chain for the case of solitary vortices in the “shallow water” equations, we obtain a nonlinear system of sixteen ODE, which is exactly equivalent to the (linear) Hill equation with a periodic potential. This means that, in some approximations, the trajectory of a solitary vortex can be described by the Hill equation. This result can be used to predict the trajectory of the vortex center if we know its observable part. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 112, No. 1, pp. 47–66.