Study of the rare B-s(0) and B-0 decays into the pi(+) pi(-) mu(+) mu(-) final state

A search for the rare decays $B_s^0 \to \pi^+\pi^-\mu^+\mu^-$ and $B^0 \to \pi^+\pi^-\mu^+\mu^-$ is performed in a data set corresponding to an integrated luminosity of 3.0 fb$^{-1}$ collected by the LHCb detector in proton-proton collisions at centre-of-mass energies of 7 and 8 TeV. Decay candidates with pion pairs that have invariant mass in the range 0.5-1.3 GeV/$c^2$ and with muon pairs that do not originate from a resonance are considered. The first observation of the decay $B_s^0 \to \pi^+\pi^-\mu^+\mu^-$ and the first evidence of the decay $B^0 \to \pi^+\pi^-\mu^+\mu^-$ are obtained and the branching fractions, restricted to the dipion-mass range considered, are measured to be $\mathcal{B}(B_s^0 \to \pi^+\pi^-\mu^+\mu^-)=(8.6\pm 1.5\,({\rm stat}) \pm 0.7\,({\rm syst})\pm 0.7\,({\rm norm}))\times 10^{-8}$ and $\mathcal{B}(B^0 \to \pi^+\pi^-\mu^+\mu^-)=(2.11\pm 0.51\,({\rm stat}) \pm 0.15\,({\rm syst})\pm 0.16\,({\rm norm}) )\times 10^{-8}$, where the third uncertainty is due to the branching fraction of the decay $B^0\to J/\psi(\to \mu^+\mu^-)K^*(890)^0(\to K^+\pi^-)$, used as a normalisation.Comment: 21 pages, 3 figures, 2 Table

The H-theorem for the physico-chemical kinetic equations with explicit time discretization

There is demonstrated in the present paper, that the H-theorem in the case of explicit time discretization of the physico-chemical kinetic equations, generally speaking, is not valid. We prove the H-theorem, when the system of the physico-chemical kinetic equations with explicit time discretization has the form of non-linear analogue of the Markov process with doubly stochastic matrix, and for more general cases. In these cases the proof is reduced to the proof of the H-theorem for Markov chains. The simplest discrete velocity models of the Boltzmann equation with explicit time discretization –the Carleman and Broadwell models are discussed and the H-theorem for them in the case of discrete time is proved. © 2017 Elsevier B.V

Kinetic Aggregation Models Leading to Morphological Memory of Formed Structures

Abstract: Kinetic equations describing the evolution of dispersed particles with different properties (such as the size, velocity, center-of-mass coordinates, etc.) are discussed. The goal is to develop an a priori mathematical model and to determine the coefficients of the resulting equations from experimentally obtained distribution functions. Accordingly, the task is to derive valid (physicochemically justified) aggregation equations. The system of equations describing the evolution of the discrete distribution function of dispersed particles is used to obtain continuum equations of the Fokker–Planck or Einstein–Kolmogorov type or a diffusion approximation to the distribution function of aggregating particles differing in the level of aggregation and in the number of constituent molecules. Distribution functions approximating experimental data are considered, and they are used to determine the coefficients of a Fokker–Planck-type equation. © 2022, Pleiades Publishing, Ltd

Кинетические уравнения Власова и Фоккера-Планка и модель агрегирования дисперсного твёрдого вещества

A multi-parameter molecular kinetic description of the formation of a nanosized solids from a supersaturated vapor or solution is considered. It is proposed to describe the long-range interaction of particles with a kinetic equation of the Vlasov [1]-[4] type, a particular case of which is the Fokker-Plank type equation that we consider to describe aggregation processes.Рассматривается многопараметрическое молекулярно-кинетическое описание образования нанодисперсного твёрдого вещества из пересыщенного пара или раствора. Предлагается описывать дальнодействующее взаимодействие частиц кинетическим уравнением типа Власова [1]-[4], частным случаем которого является рассматриваемое нами для описания процессов агрегации уравнение типа Фоккера-Планка

On the H-theorem for the Becker–Döring system of equations for the cases of continuum approximation and discrete time

In the present paper we make the transition from the Becker–Döring system of equations to the hybrid (discrete and continuum) description. This new type of system of equations consists of the equation of the Fokker–Planck–Einstein–Kolmogorov type added by the Becker–Döring equations. We consider the H-theorem for it. We also consider the H-theorem for the Becker–Döring system of equations with discrete time and showed that it is true for some partially implicit discretization in time. Due to generality of the kinetic approach the present work can be useful for specialists in different spheres engaged in modeling the evolution of structures differing by properties. © 2020 Elsevier B.V

The H-theorem for the physico-chemical kinetic equations with explicit time discretization

There is demonstrated in the present paper, that the H-theorem in the case of explicit time discretization of the physico-chemical kinetic equations, generally speaking, is not valid. We prove the H-theorem, when the system of the physico-chemical kinetic equations with explicit time discretization has the form of non-linear analogue of the Markov process with doubly stochastic matrix, and for more general cases. In these cases the proof is reduced to the proof of the H-theorem for Markov chains. The simplest discrete velocity models of the Boltzmann equation with explicit time discretization –the Carleman and Broadwell models are discussed and the H-theorem for them in the case of discrete time is proved. © 2017 Elsevier B.V

Approaches to determining the kinetics for the formation of a nano-dispersed substance from the experimental distribution functions of its nanoparticle properties

In the present paper, we discuss the kinetic equations for the evolution of particles of a nanodispersed substance, distinguishing by properties (sizes, velocities, positions, etc.). The aim of the present investigation is to determine the coefficients for the equations by the distribution functions, which are obtained experimentally. The experiment is characterized by the time interval, which is needed for the measurement of the distribution function. However, the nanodispersed substance is obtained from a highly supersaturated solution or vapor and this time interval is large, thus, one is able to measure distribution functions only when the processes of the integration and the fragmentation of the particles become rather slow. So it is advisable to reconstruct the kinetics for the formation of a nanodispersed substance by the experimental distribution functions measured when the processes are rather slow. The first problem that arises is the obtaining of correct equations, and, hence, the derivation of the equations from each other. From the discrete system of equations for the evolution of discrete distribution functions of particles of a nanodispersed substance, we obtain the continuum equation of the Fokker-Planck type, or of the Einstein-Kolmogorov type, or of the diffuse approximation on the distribution function of nanoparticles distinguishing by the numbers of molecules forming them. We consider the distribution functions, which approximate the experimental data. We determine the coefficients for the equation of the Fokker-Planck type by the stationary and non-stationary distribution functions of a nanodispersed substance. Due to unity of the kinetic approach, the present work may be useful for specialists of various areas, who study the evolution of structures (not only with nanosize) with differing properties

Generalized Boltzmann-Type Equations for Aggregation in Gases

The coalescence and fragmentation of particles in a dispersion system are investigated by applying kinetic theory methods, namely, by generalizing the Boltzmann kinetic equation to coalescence and fragmentation processes. Dynamic equations for the particle concentrations in the system are derived using the kinetic equations of motion. For particle coalescence and fragmentation, equations for the particle momentum, coordinate, and mass distribution functions are obtained and the coalescence and fragmentation coefficients are calculated. The equilibrium mass and velocity distribution functions of the particles in the dispersion system are found in the approximation of an active terminal group (Becker–Döring-type equation). The transition to a continuum description is performed. © 2017, Pleiades Publishing, Ltd

Generalized Boltzmann-Type Equations for Aggregation in Gases

The coalescence and fragmentation of particles in a dispersion system are investigated by applying kinetic theory methods, namely, by generalizing the Boltzmann kinetic equation to coalescence and fragmentation processes. Dynamic equations for the particle concentrations in the system are derived using the kinetic equations of motion. For particle coalescence and fragmentation, equations for the particle momentum, coordinate, and mass distribution functions are obtained and the coalescence and fragmentation coefficients are calculated. The equilibrium mass and velocity distribution functions of the particles in the dispersion system are found in the approximation of an active terminal group (Becker–Döring-type equation). The transition to a continuum description is performed. © 2017, Pleiades Publishing, Ltd