Field-theory methods in coagulation theory

Coagulating systems are systems of chaotically moving particles that collide and coalesce, producing daughter particles of mass equal to the sum of the masses involved in the respective collision event. The present article puts forth basic ideas underlying the application of methods of quantum-field theory to the theory of coagulating systems. Instead of the generally accepted treatment based on the use of a standard kinetic equation that describes the time evolution of concentrations of particles consisting of a preset number of identical objects (monomers in the following), one introduces the probability W(Q, t) to find the system in some state Q at an instant t for a specific rate of transitions between various states. Each state Q is characterized by a set of occupation numbers Q = {n ₁, n ₂, ..., n _g , ...}, where n _g is the total number of particles containing precisely g monomers. Thereupon, one introduces the generating functional Ψ for the probability W(Q, t). The time evolution of Ψ is described by an equation that is similar to the Schrödinger equation for a one-dimensional Bose field. This equation is solved exactly for transition rates proportional to the product of the masses of colliding particles. It is shown that, within a finite time interval, which is independent of the total mass of the entire system, a giant particle of mass about the mass of the entire system may appear in this system. The particle in question is unobservable in the thermodynamic limit, and this explains the well-known paradox of mass-concentration nonconservation in classical kinetic theory. The theory described in the present article is successfully applied in studying the time evolution of random graphs.     

Solvability of linear kinetic equations with multi-energetic inelastic scattering

In this paper we shall analyse the linear Boltzmann equation describing the motion of test particles through a background of heavy field particles that can appear in several energy states. The inelastic scattering process consists in the exchange of quanta of energy between the field and test particles. The well-posedness of the problem is investigated by means of the substochastic semigroup theory and the conditions on the scattering collision frequencies are given for the evolution to keep the collision rate finite and to preserve the total number of particles.     

固体炸药爆轰的一种考虑热学非平衡的反应流动模型

基于固体炸药爆轰过程中化学反应混合区内的固相反应物与气相生成物处于力学平衡状态及热学非平衡状态的事实,提出一种考虑热学非平衡效应的反应流动模型来描述固体炸药的爆轰流动现象.该爆轰流动模型的主要特点是,在反应混合物Euler方程和固相反应物质量守恒方程的基础上,通过附加一套关于固相反应物的组分物理量的流动控制方程来表达固相反应物与气相生成物之间的热学非平衡效应.根据反应混合区内固相反应物与气相生成物这两种化学组分保持各自内能守恒的混合规则,并借助它们具有压力相等的性质以及满足体积分数总和为1的条件,推导获得的附加方程有：固相反应物的内能演化方程、体积分数演化方程及反应混合物的压力演化方程.这样,建立的爆轰模型包括：反应混合物的质量守恒方程、动量守恒方程、总能量守恒方程、压力演化方程,以及固相反应物的质量守恒方程、内能演化方程、体积分数演化方程.对所获得的爆轰模型方程组采用一个时空二阶精度的有限体积法进行数值求解,典型爆轰问题算例结果表明本文提出的固体炸药爆轰模型是合理的.     

Arbitrary Lagrangian–Eulerian finite-element method for computation of two-phase flows with soluble surfactants

A finite-element scheme based on a coupled arbitrary Lagrangian–Eulerian and Lagrangian approach is developed for the computation of interface flows with soluble surfactants. The numerical scheme is designed to solve the time-dependent Navier–Stokes equations and an evolution equation for the surfactant concentration in the bulk phase, and simultaneously, an evolution equation for the surfactant concentration on the interface. Second-order isoparametric finite elements on moving meshes and second-order isoparametric surface finite elements are used to solve these equations. The interface-resolved moving meshes allow the accurate incorporation of surface forces, Marangoni forces and jumps in the material parameters. The lower-dimensional finite-element meshes for solving the surface evolution equation are part of the interface-resolved moving meshes. The numerical scheme is validated for problems with known analytical solutions. A number of computations to study the influence of the surfactants in 3D-axisymmetric rising bubbles have been performed. The proposed scheme shows excellent conservation of fluid mass and of the total mass of the surfactant.     

Deterministic Motion of the Controversial Piston in the Thermodynamic Limit

We consider the evolution of a system composed of N non-interacting point particles of mass m in a cylindrical container divided into two regions by a movable adiabatic wall (the adiabatic piston). We study the thermodynamic limit for the piston where the area A of the cross-section, the mass M of the piston, and the number N of particles go to infinity keeping A/M and N/M fixed. The length of the container is a fixed parameter which can be either finite or infinite. In this thermodynamic limit we show that the motion of the piston is deterministic and the evolution is adiabatic. Moreover if the length of the container is infinite, we show that the piston evolves toward a stationary state with velocity approximately proportional to the pressure difference. If the length of the container is finite, introducing a simplifying assumption we show that the system evolves with either weak or strong damping toward a well-defined state of mechanical equilibrium where the pressures are the same, but the temperatures different. Numerical simulations are presented to illustrate possible evolutions and to check the validity of the assumption.     

Cluster Coagulation

We introduce a general class of coagulation models, where clusters of given types may coagulate in more than one way and where the rate at which this happens may depend on the cluster types. In the continuum version of these models there is a generalization of Smoluchowski's coagulation equation. We introduce a notion of strong solution for this equation and prove the existence of a maximal strong solution, which while it persists is the only solution. When the total rate of coagulation for particles is bounded above and below by constant multiples of the product of their masses, we show that the maximal strong solution coincides with the maximal mass-conserving solution and does not persist for all time. Thus, for these models, loss of mass (to infinity) coincides with divergence of the second moment of the mass distribution and takes place in a finite time. When the total rate of coagulation of large particles is proportional to their masses, we establish the existence and uniqueness of solutions for all time. In a restricted class of "polymer" models, we allow coagulation of weighted shapes in a finite number of ways. For this class we establish a discrete approximation scheme for the continuum dynamics. For each continuum coagulation model, there is a corresponding finite-particle-number stochastic model. We show that, in the polymer case, which includes the case of simple mass coalescence, as the number of particles becomes large, the stochastic model converges weakly to the deterministic continuum model, at an exponential rate.     

Analytical Study of Giant Fluctuations and Temporal Intermittency

We study analytically giant fluctuations and temporal intermittency in a stochastic one-dimensional model with diffusion and aggregation of masses in the bulk, along with influx of single particles and outflux of aggregates at the boundaries. We calculate various static and dynamical properties of the total mass in the system for both biased and unbiased movement of particles and different boundary conditions. These calculations show that (i) in the unbiased case, the total mass has a non-Gaussian distribution and shows giant fluctuations which scale as system size (ii) in all the cases, the system shows strong intermittency in time, which is manifested in the anomalous scaling of the dynamical structure functions of the total mass. The results are derived by taking a continuum limit in space and agree well with numerical simulations performed on the discrete lattice. The analytic results obtained here are typical of the full phase of a more general model with fragmentation, which was studied earlier using numerical simulations.     

Diffusive Hydrodynamic Limits for Systems of Interacting Diffusions with Finite Range Random Interaction

We consider a system of interacting diffusive particles with finite range random interaction. The variables can be interpreted as charges at sites indexed by a periodic multidimensional lattice. The equilibrium states of the system are canonical Gibbs measures with finite range random interaction. Under the diffusive scaling of lattice spacing and time, we derive a deterministic nonlinear diffusion equation for the time evolution of the macroscopic charge density. This limit is almost sure with respect to the random environment. Received: 3 October 1996 / Accepted: 13 February 1997     

Dust trapping in vortex pairs

The motion of tiny heavy particles transported in a co-rotating point vortex pair, with or without particle inertia and sedimentation, is investigated. The dynamics of non-inertial sedimenting particles is shown to be chaotic, under the combined effects of gravity and of the circular displacement of the vortices. This phenomenon is very sensitive to the particles’ inertia, if any. By using a nearly hamiltonian dynamical system theory for the particles’ motion equation written in the rotating reference frame, one can show that small inertia terms of the particles’ motion equation strongly modify the Melnikov function of the homoclinic trajectories and heteroclinic cycles of the unperturbed system, as soon as the particles’ response time is of the order of the settling time (Froude number of order unity). The critical Froude number above which chaotic motion vanishes and a regular centrifugation takes place is obtained from this Melnikov analysis and compared to numerical simulations. Particles with a finite inertia, and in the absence of gravity, are not necessarily centrifuged away from the vortex system. Indeed, these particles can have various equilibrium positions in the rotating reference frame, like the Lagrange points of celestial mechanics, according to whether their Stokes number is smaller or larger than some critical value. An analytical stability analysis reveals that two of these points are stable attracting points, so that permanent trapping can occur for inertial particles injected in an isolated co-rotating vortex pair. Particle trapping is observed to persist when viscosity, and therefore vortex coalescence, is taken into account. Numerical experiments at large but finite Reynolds number show that particles can indeed be trapped temporarily during vortex roll-up, and are eventually centrifuged away once vortex coalescence occurs.     

Structural stability of disperse systems and finite nature of a coagulation front

A new discrete model of coagulation, which is a discrete analog of the Oort-van de Hulst-Safronov equation, is derived. It is shown that the familiar version, in contrast with Smoluchowski’s equation, can be used to calculate the propagation of a coagulation front. The relationship between compliance to the mass conservation law and the finite nature of the coagulation front is established, and then estimates of the time of violation of the mass conservation law are made for several classes of coagulation kernels. One of the conclusions is that the mass conservation law can be violated in cases where particles of roughly equal mass cannot coagulate, as occurs, for example, in gravitational coagulation. Estimates of the time for the appearance of structural instability of the system are made for multiplicative coagulation kernels. Zh. éksp. Teor. Fiz. 116, 717–730 (August 1999)     

Effect of negative masses in the statistical theory of multiple creation

The general statistical method of the microcanonical ensemble is used to calculate the statistical weights of a system which contains both positive and negative mass particles. It is shown that the probability of creation of an infinite number of particles in a system is equal to unity, whereas the probability of creation of a finite number of particles is zero. This reveals a further difficulty which appears when negative masses are introduced.In conclusion, I should like to take this opportunity to express my deep gratitude to Professor Ya. P. Terletskii for suggesting this work and for his interest in it.     

The Periodic Oscillation of an Adiabatic Piston in Two or Three Dimensions

We study a heavy piston of mass M that separates finitely many ideal, unit mass gas particles moving in two or three dimensions. Neishtadt and Sinai previously determined a method for finding this system’s averaged equation and showed that its solutions oscillate periodically. Using averaging techniques, we prove that the actual motions of the piston converge in probability to the predicted averaged behavior on the time scale M ^1/2 when M tends to infinity while the total energy of the system is bounded and the number of gas particles is fixed.     

Scaling limit for a mechanical system of interacting particles

A system of a large number of classical particles moving on a onedimensional segment with virtually reflecting boundaries is studied. The particles interact with one another through repulsive pair-potential forces and are subjected to resistance proportional to their velocities. Because of the latter it is only the number of particles that is conserved under the evolution of the system. It is proved that in the hydrodynamic limit of diffusion type scaling the normalized counting measure of particle locations converges and its limiting density is governed by a non-linear diffusion equation which in typical cases is of porous media equation type.     

Scaling solution for small cosmic string loops

The equation governing the time evolution of the number density of loops in a cosmic string network is a detailed balance determined by energy conservation. We solve this equation with the inclusion of the gravitational radiation effect, which causes the loops to shrink (and eventually decay) as time elapses. The solution approaches a scaling regime in which the total energy density in loops remains finite, converging both in the infrared and in the ultraviolet.     

Dissipative dynamics of superfluid vortices at nonzero temperatures

We consider the evolution and dissipation of vortex rings in a condensate at nonzero temperatures in the context of the classical field approximation, based on the defocusing nonlinear Schr?dinger equation. The temperature in such a system is fully determined by the total number density and the number density of the condensate. The collisions with noncondensed particles reduce the radius of a vortex ring until it completely disappears. We obtain a universal decay law for a vortex line length and relate it to mutual friction coefficients in the fundamental equation of vortex motion in superfluids.     

Electronic properties of a large quantum dot at a finite temperature

The physical properties of a two-dimensional parabolic quantum dot composed of large number of interacting electrons are numerically determined by the Thomas–Fermi (TF) method at a finite temperature. Analytical solutions are given for zero temperature for comparative purposes. The exact solution of the TF equation is obtained for the non-interacting system at finite temperatures. The effect of the number of particles and temperature on the properties are investigated both for interacting and non-interacting cases. The results indicate that the effect of e–e interaction on the density profile shows different temperature dependencies above and below a certain temperature T_c.     

Time-Dependent Transport in Nanoscale Devices

A method for simulating ballistic time-dependent device transport, which solves the time-dependent Sehrǒdinger equation using the finite difference time domain （FDTD） method together with Poisson＇s equation, is described in detail. The effective mass Schrǒdinger equation is solved. The continuous energy spectrum of the system is discretized using adaptive mesh, resulting in energy levels that sample the density-of-states. By calculating time evolution of wavefunctions at sampled energies, time-dependent transport characteristics such as current and charge density distributions are obtained. Simulation results in a nanowire and a coaxially gated carbon nanotube field-effect transistor （CNTFET） are presented. Transient effects, e.g., finite rising time, are investigated in these devices.     

Nonlinear Schrödinger Equations and the Separation Property

Abstract

We investigate hierarchies of nonlinear Schrödinger equations for multiparticle systems satisfying the separation property, i.e., where product wave functions evolve by the separate evolution of each factor. Such a hierarchy defines a nonlinear derivation on tensor products of the single-particle wave-function space, and satisfies a certain homogeneity property characterized by two new universal physical constants. A canonical construction of hierarchies is derived that allows the introduction, at any particular “threshold” number of particles, of truly new physical effects absent in systems having fewer particles. In particular, if single quantum particles satisfy the usual (linear) Schrödinger equation, a system of two particles can evolve by means of a fairly simple nonlinear Schrödinger equation without violating the separation property. Examples of Galileian-invariant hierarchies are given.     

Non-equilibrium statistical mechanics in the general theory of relativity III. Collisional stellar dynamics

This paper assembles and extends earlier results to formulate a coherent theory of relativistic stellar dynamics appropriate for comparatively small systems of stars in which relativistic effects can be important. The structure of the Newtonian theory is outlined, culminating in the “collisional Boltzmann” or Fokker-Planck equation appropriate for an unconfined system of point masses. The theory of relativistic Fokker-Planck equations is then developed for general Lorentz-covariant interactions such as electromagnetism or scalar fields. The basic physical ingredients of Newtonian stellar dynamics are identified, and it is indicated how they can be reformulated relativistically. These considerations are then used to construct a relativistic Fokker-Planck equation appropriate for the evolution of a collection of point mass stars. The analysis is then generalized to allow, both Newtonianly and relativistically, for the effects of direct physical collisions between stars of finite size. By way of conclusion and illustration, the theory is applied to the study of a prototypical dense galactic nucleus which could evolve to contain a massive black hole. The paper ends by enumerating a number of tractable unsolved problems deserving of further consideration.