A stochastic derivation of the multivariate master equation describing reaction diffusion systems

We derive the multivariate master equation describing reaction diffusion systems from a discrete form master equation in phase space, assuming that the elastic collisions of the chemically active substances with the inert carrier gas have relaxed. In this state of collisional equilibrium the stochastic operator modelling the displacement of the particles between spatial cells reduces to the random wall operator and the reactive collision term yields the usual birth and death operator. Correlation functions are derived and their validity is discussed.     

量子扩散通道中Wigner算符的演化规律

众所周知,量子态的演化可用与其相应的Wigner函数演化来代替.因为量子态的Wigner函数和量子态的密度矩阵一样,都包含了概率分布和相位等信息,因此对量子态的Wigner函数进行研究,可以更加快速有效地获取量子态在演化过程的重要信息.本文从经典扩散方程出发,利用密度算符的P表示,导出了量子态密度算符的扩散方程.进一步通过引入量子算符的Weyl编序记号,给出了其对应的Weyl量子化方案.另外,借助于密度算符的另一相空间表示-Wigner函数,建立了Wigner算符在扩散通道中演化方程,并给出了其Wigner算符解的形式.本文推导出了Wigner算符在量子扩散通道中的演化规律,即演化过程中任意时刻Wigner算符的形式.在此结论的基础上,讨论了相干态经过量子扩散通道的演化情况.     

Scattering of long-wavelength acoustic phonons by isotopic impurities

We consider the problem of the relaxation of an arbitrary initial distribution function of a gas of long-wave acoustic phonons scattered by isotopic impurities embedded in a crystalline medium with cubic symmetry. The spectral decomposition of the collision integral of the suitable Boltzmann-Peierls equation is obtained. The spectrum of the collision operator is purely discrete and in addition to the eigenvalue 0 consists of three other eigenvalues. Explicit analytic expressions for these eigenvalues are obtained. Within the Chapman-Enskog approximation we derive the diffusion equation for the density of phonons and obtain the explicit expression for the diffusion coefficient. The dependency of the eigenvalues of the collision operator and the diffusion coefficient on the elastic constants of the medium is studied.     

Diffusion approximations of the two dimensional master equation

The effects of rotational energy transfer on unimolecular dissociation are assessed using a two-dimensional master equation approach. The collision operator for such a system can be very large, and approximations are sought that reduce the size of the matrix that has to be stored. The density of states of the vibrational energy is large for most molecules and can be treated as virtually continuous. As a consequence, motion in vibrational energy space can be treated using a diffusion equation approach. This reduces the storage requirements greatly and increases the speed at which diagonalization occurs. Results are presented for the dissociation of ethane and methane.     

A density-corrected quantum Boltzmann equation

Binary correlations are a recognized part of the pair density operator, but the influence of binary correlations on the singlet density operator is usually not emphasized. Here free motion and binary correlations are taken as independent building blocks for the structure of the nonequilibrium singlet and pair density operators. Binary correlations are assumed to arise from the collision of twofree particles. Together with the first BBGKY equation and a retention of all terms that are second order in gas density, a generalization of the Boltzmann equation is obtained. This is an equation for thefree particle density operator rather than for the (full) singlet density operator. The form for the pressure tensor calculated from this equation reduces at equilibrium to give the correct (Beth-Uhlenbeck) second virial coefficient, in contrast to a previous quantum Boltzmann equation, which gave only part of the quantum second virial coefficient. Generalizations to include higher-order correlations and collision types are indicated.     

Multiple-relaxation-time lattice Boltzmann model for the convection and anisotropic diffusion equation

A lattice Boltzmann model with a multiple-relaxation-time (MRT) collision operator for the convection–diffusion equation is presented. The model uses seven discrete velocities in three dimensions (D3Q7 model). The off-diagonal components of the relaxation-time matrix, which originate from the rotation of the principal axes, enable us to take into account full anisotropy of diffusion. An asymptotic analysis of the model equation with boundary rules for the Dirichlet and Neumann-type (specified flux) conditions is carried out to show that the model is first- and second-order accurate in time and space, respectively. The results of the analysis are verified by several numerical examples. It is also shown numerically that the error of the MRT model is less sensitive to the variation of the relaxation-time coefficients than that of the classical BGK model. In addition, an alternative treatment for the Neumann-type boundary condition that improves the accuracy on a curved boundary is presented along with a numerical example of a spherical boundary.     

Non-linear Liouville and Shrödinger equations in phase space

Unitary representations of the Galilei group are studied in phase space, in order to describe classical and quantum systems. Conditions to write in general form the generator of time translation and Lagrangians in phase space are then established. In the classical case, Galilean invariance provides conditions for writing the Liouville operator and Lagrangian for non-linear systems. We analyze, as an example, a generalized kinetic equation where the collision term is local and non-linear. The quantum counter-part of such unitary representations are developed by using the Moyal (or star) product. Then a non-linear Schrödinger equation in phase space is derived and analyzed. In this case, an association with the Wigner formalism is established, which provides a physical interpretation for the formalism.     

Linear stability of the spatially homogeneous equilibria of the Vlasov-Poisson system with collisions

In this work we study the stability of the spatially homogeneous solutions of the Vlasov-Poisson system (Vlasov equilibria) when a collision term, in the form of a BGK operator with velocity-dependent collision frequency, is added to the Vlasov equation. Generalizing earlier results, obtained for the same collision model with a constant collision frequency, we find the spectrum and the eigenfunctions of the linear transport operator and derive a new linear dispersion relation for the linearized kinetic equations. Finally, we present some numerical results.     

非均匀等离子体湍流的重正化准线性理论——Misguich-Balescu理论的推广

本文把Misguich和Balescu建立的均匀等离子体湍流的重正化准线性理论推广到非均匀等离子体湍流。在弱耦合和弱非均匀性近似下,得到传播算符和平均湍性碰撞算符的明晰表达式。非均匀性除对算符中的扩散贡献、共振加宽或频移、Dupree衰减和平均分布函数在速度空间的陡度效应等作出一定的修正之外,还引出新的指数微分算符:在传播算符中是涨落电场关联函数非均匀性导致的新的速度微分算符,在平均湍性碰撞算符中是平均分布函数非均匀性导致的新的空间微分算符。平均分布函数非均匀性还使得在平均湍性碰撞算符中出现的两个互为逆运算的自由流传播算符不能抵消,从而使包含在(直接作用于平均分布函数的)传播算符中的非马尔可夫效应显露出来。 关键词：     

Diffusion Lattice Boltzmann Scheme on a Orthorhombic Lattice

We present a diffusion lattice Boltzmann (DLB) scheme which is derived from first principles. As opposed to the traditional lattice BGK schemes the DLB is valid for orthorhombic lattices and it has two eigenvalues of the collision operator. It is shown that the diffusion coefficient depends only on one eigenvalue of the collision operator. Hence, the DLB scheme can be optimized with means of the additional eigenvalue of the collision operator and with different lattice spacing along the principal axes. The properties of the DLB scheme concerning consistency, stability, and accuracy are studied with eigenmode analysis. This analysis shows that the DLB scheme is consistent with diffusion for a wide range of diffusion coefficients, it has unconditional stability, and that it has third-order accuracy. Furthermore, it is shown that accuracy is improved by setting the additional eigenvalue to zero and by densifying the lattice spacing along the direction of the density gradient.     

On the structure of the three-particle operator in generalized kinetic equations for inhomogeneous gases

It is shown that the three-particle kinetic operator for inhomogeneous gases obtained using Prigogine's method and the matrix representation of the Liouville equation introduced by Balescu is equivalent to the corresponding expression derived by Choh and Uhlenbeck using Bogolubov's method. Both theories take into account the space and time delocalization associated with finite collision time, and the resulting corrections to the asymptotic collision operator are equivalent.     

The covariant form of the Klein-Kramers equation and the associated moment equations

We provide a covariant, coordinate-free formulation of the many-dimensional Klein-Kramers equation for the phase space distribution of a Brownian particle. We construct a complete set of eigenfunctions of the collision operator adapted to the coordinate system, which involve covariant tensorial Hermite polynomials. The Klein-Kramers equation can then be reformulated as a system of coupled equations for the expansion coefficients with respect to this system. Truncation of this system of moment equations and application of a subsidiary condition yields a covariant generalization of Grad's thirteen-moment equations. As an application we give the explicit form of these equations for spherically symmetric, stationary solutions in spherical coordinates. We briefly comment on possible extensions of our treatment to slightly more complicated cases.     

On the theory of small-amplitude vibrations of curved-surface diaphragms

An equation is derived describing small-amplitude vibrations of an arbitrary curved diaphragm, whose surface is considered as a two-dimensional Riemannian space. The derivation is based on the variational principle, from which the motion equation and conservation law follow in a form invariant with respect to arbitrary transformations of coordinates on the diaphragm surface. It has been shown that the wave equation, along with the two-dimensional Laplace-Beltrami operator, includes an additional term proportional to the scalar curvature of the diaphragm surface. As an example, the equations are considered for a spherical diaphragm and a catenoid-shaped diaphragm with a minimal surface of revolution.     

Generalized nonlinear master equations for local fluctuations: Dimensionality expansions and augmented mean field theory

By application of a projection operator technique we derive a formally exact generalization of the nonlinear mean field master equation introduced recently for the study of local fluctuations in a reacting medium. Our starting point is a phenomenological cell master equation. The results of our theory are applicable to the theory of a fluctuating hydrodynamic reacting system. The mean field equation is placed on a firm theoretical foundation by showing it to be the lowest order approximation in an expansion in the dimensionality of the physical space keeping the product of the number of nearest neighbors (an increasing function of dimensionality) and the typical diffusion coefficient constant. A more accurate nonlinear master equation that allows for the correlation and fluctuations in the environment of a given volume element is derived in the form of an augmented mean field equation.Work supported in part by a grant from the National Science Foundation.     

On the description of transport phenomena

The linear response of a quantity like the electric current to a time and space dependent field can be described by Kubo's formula, and this again can be written as a resolvent. The expansion of this resolvent yields exactly a transport equation of the structure of the Boltzmann equation. Perturbation theory — the only practical way to deal with it — gives back the usual Boltzmann equation. This derivation has advantages like yielding a symmetric collision operator. Also including time and space dependence with this approach is very easy. The purpose of this paper is to show the closeness of linear response theory and kinetic equations. We also discuss merits and drawbacks of the resolvent method.     

Anomalous diffusion of inertial, weakly damped particles

The anomalous (i.e., non-Gaussian) dynamics of particles subject to a deterministic acceleration and a series of "random kicks" is studied. Based on an extension of the concept of continuous time random walks to position-velocity space, a new fractional equation of the Kramers-Fokker-Planck type is derived. The associated collision operator necessarily involves a fractional substantial derivative, representing important nonlocal couplings in time and space. For the force-free case, a closed solution is found and discussed.     

Finite size effects on transport coefficients for models of atomic wires coupled to phonons

We consider models of quasi-1-d, planar atomic wires consisting of several, laterally coupled rows of atoms, with mutually non-interacting electrons. This electronic wire system is coupled to phonons, corresponding, e.g., to some substrate. We aim at computing diffusion coefficients in dependence on the wire widths and the lateral coupling. To this end we firstly construct a numerically manageable linear collision term for the dynamics of the electronic occupation numbers by following a certain projection operator approach. By means of this collision term we set up a linear Boltzmann equation. A formula for extracting diffusion coefficients from such Boltzmann equations is given. We find in the regime of a few atomic rows and intermediate lateral coupling a significant and non-trivial dependence of the diffusion coefficient on both, the width and the lateral coupling. These results, in principle, suggest the possible applicability of such atomic wires as electronic devices, such as, e.g., switches.     

Series expansion solutions for the multi-term time and space fractional partial differential equations in two- and three-dimensions

Fractional partial differential equations with more than one fractional derivative in time describe some important physical phenomena, such as the telegraph equation, the power law wave equation, or the Szabo wave equation. In this paper, we consider two- and three-dimensional multi-term time and space fractional partial differential equations. The multi-term time-fractional derivative is defined in the Caputo sense, whose order belongs to the interval (1,2],(2,3],(3,4] or (0,m], and the space-fractional derivative is referred to as the fractional Laplacian form. We derive series expansion solutions based on a spectral representation of the Laplacian operator on a bounded region. Some applications are given for the two- and three-dimensional telegraph equation, power law wave equation and Szabo wave equation.