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.     

Statistical Inference with Fractional Brownian Motion

We give a test between two complex hypothesis; namely we test whether a fractional Brownian motion (fBm) has a linear trend against a certain non-linear trend. We study some related questions, like goodness-of-fit test and volatility estimation in these models.     

Statistical Multiplexing of Homogeneous Fractional Brownian Streams

In this paper, the statistical multiplexing of independent fractional Brownian traffic streams with the same Hurst value 0.5<H<1 is studied. The buffer overflow probabilities based on steady-state and transient queue length tail distributions are used respectively as the common performance criterion. Under general conditions, the minimal buffer allocation to the merged traffic is identified in either case so that strictly positive bandwidth savings are realized. Impact of the common H value on multiplexing gains is investigated. The analytical results are applicable in data network engineering problems, where ATM is deployed as the transport network carrying long-range dependent data traffic.     

Variable electrical and thermal conductivity in the theory of generalized thermoelastic diffusion

This paper deals with the problem of a thermoelastic half-space with a permeating substance in contact with the bounding plane in the context of the theory of generalized thermoelastic diffusion with one relaxation time and with variable electrical and thermal conductivity. The bounding surface of the half-space is taken to be traction free and subjected to a time dependent thermal shock. The solution is obtained in the Laplace transform domain by a direct approach. A numerical technique is employed to obtain the solution in the physical domain. It is found that there exist two coupled waves, one of which is elastic and the other is thermal, and a third wave affects diffusion mainly. A comparison is made with the results obtained in a thermoelastic medium with and without diffusion in the following cases : (a) the electrical and thermal conductivities have constant values, (b) the presence of magnetic field and (c) the generalized theory in thermoelasticity. Received: June 1, 2005     

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     

Variable electrical and thermal conductivity in the theory of generalized thermoelastic diffusion

This paper deals with the problem of a thermoelastic half-space with a permeating substance in contact with the bounding plane in the context of the theory of generalized thermoelastic diffusion with one relaxation time and with variable electrical and thermal conductivity. The bounding surface of the half-space is taken to be traction free and subjected to a time dependent thermal shock. The solution is obtained in the Laplace transform domain by a direct approach. A numerical technique is employed to obtain the solution in the physical domain. It is found that there exist two coupled waves, one of which is elastic and the other is thermal, and a third wave affects diffusion mainly. A comparison is made with the results obtained in a thermoelastic medium with and without diffusion in the following cases : (a) the electrical and thermal conductivities have constant values, (b) the presence of magnetic field and (c) the generalized theory in thermoelasticity.     

Deterministic diffusion of particles

The diffusion of the particles is described in terms of a mean motion with a speed equal to the osmotic velocity associated with the diffusion process. Three numerical schemes are presented. The first two are based on the approximation of the gradient on an irregular mesh. The third is derived from a finite-element approach. Voronoi diagrams are used to handle the irregular grid of the particles. The convergence of the schemes is studied numerically, by comparing the results with the exact solution. Applications to the Fokker-Planck equation and to the problem of disposing particles according to a given probability distribution are presented.     

Genealogy of extremal particles of branching Brownian motion

Branching Brownian motion describes a system of particles that diffuse in space and split into offspring according to a certain random mechanism. By virtue of the groundbreaking work by M. Bramson on the convergence of solutions of the Fisher‐KPP equation to traveling waves, the law of the rightmost particle in the limit of large times is rather well understood. In this work, we address the full statistics of the extremal particles (first‐, second‐, third‐largest, etc.). In particular, we prove that in the large t‐limit, such particles descend with overwhelming probability from ancestors having split either within a distance of order 1 from time 0, or within a distance of order 1 from time t. The approach relies on characterizing, up to a certain level of precision, the paths of the extremal particles. As a byproduct, a heuristic picture of branching Brownian motion “at the edge” emerges, which sheds light on the still unknown limiting extremal process. © 2011 Wiley Periodicals, Inc.     

Sine-gordon transformations in nonequilibrium systems of Brownian particles

Finite volume grand canonical correlation functions of nonequilibrium systems of d-dimensional Brownian particles, interacting through a regular (long-range) pair potential with integrable first partial derivatives, are expressed in terms of the expectation values of a Gaussian random field. The initial correlation functions coincide with the Gibbs correlation functions corresponding to a more general pair long-range potential. Nonequilibrium Euclidean action is introduced, satisfying a thermodynamic stability property.     

Nevanlinna theory through the Brownian motion

In this paper, we introduce the Nevanlinna theory using stochastic calculus, following the works of Davis(1975), Carne(1986) and Atsuji(1995, 2005, 2008 and 2017), etc. In particular, we give(another) proofs of the classical result of Nevanlinna for meromorphic functions and the result of Cartan-Ahlfors for holomorphic curves by using the probabilistic method.     

Statistical tests of heterogeneity for anisotropic multifractional Brownian fields

In this paper, we deal with some anisotropic extensions of the multifractional Brownian fields that account for spatial phenomena whose properties of regularity and directionality may both vary in space. Our aim is to set statistical tests to decide whether an observed field of this kind is heterogeneous or not. The statistical methodology relies upon a field analysis by quadratic variations, which are averages of square field increments. Specific to our approach, these variations are computed locally in several directions. We establish an asymptotic result showing a linear Gaussian relationship between these variations and parameters related to regularity and directional properties of the model. Using this result, we then design a test procedure based on Fisher statistics of linear Gaussian models. Eventually we evaluate this procedure on simulated data.     

Rarefaction of moving diffusion particles

We investigate a flow of particles moving along a tube together with gas. The dynamics of particles is determined by a stochastic differential equation with different initial states. The walls of the tube absorb particles. We prove that if the incoming flow of particles is determined by a random Poisson measure, then the number of remained particles is characterized by the Poisson distribution. The parameter of this distribution is constructed by using a solution of the corresponding parabolic boundary-value problem.Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 5, pp. 691–694, May, 2004.     

On mathematical foundation of the Brownian motor theory

The paper contains mathematical justification of basic facts concerning the Brownian motor theory. The homogenization theorems are proved for the Brownian motion in periodic tubes with a constant drift. The study is based on an application of the Bloch decomposition. The effective drift and effective diffusivity are expressed in terms of the principal eigenvalue of the Bloch spectral problem on the cell of periodicity as well as in terms of the harmonic coordinate and the density of the invariant measure. We apply the formulas for the effective parameters to study the motion in periodic tubes with nearly separated dead zones.     

Two Brownian particles with rank-based characteristics and skew-elastic collisions

We construct a two-dimensional diffusion process with rank-dependent local drift and dispersion coëfficients, and with a full range of patterns of behavior upon collision that range from totally frictionless interaction, to elastic collision, to perfect reflection of one particle on the other. These interactions are governed by the left- and right-local times at the origin for the distance between the two particles. We realize this diffusion in terms of appropriate, apparently novel systems of stochastic differential equations involving local times, which we show are well posed. Questions of pathwise uniqueness and strength are also discussed for these systems.