Multipoint Distribution Functions Research Articles

TASEP and generalizations: method for exact solution

The explicit biorthogonalization method, developed in [arXiv:1701.00018] for continuous time TASEP, is generalized to a broad class of determinantal measures which describe the evolution of several interacting particle systems in the KPZ universality class. The method is applied to sequential and parallel update versions of each of the four variants of discrete time TASEP (with Bernoulli and geometric jumps, and with block and push dynamics) which have determinantal transition probabilities; to continuous time PushASEP; and to a version of TASEP with generalized update. In all cases, multipoint distribution functions are expressed in terms of a Fredholm determinant with an explicit kernel involving hitting times of certain random walks to a curve defined by the initial data of the system. The method is further applied to systems of interacting caterpillars, an extension of the discrete time TASEP models which generalizes sequential and parallel updates.

The pair distribution function for an array of screw dislocations

The paper addresses the problem of correlation within an array of parallel dislocations in a crystalline solid. The first two of a hierarchy of equations for the multi-point distribution functions are derived by treating the random dislocation distributions and the corresponding stress fields in an ensemble average framework. Asymptotic reasoning, applicable when dislocations are separated by small distances, provides equations that are independent of any specific kinetic law relating the velocity of a dislocation to the force acting on it. The only assumption made is that the force acting on any dislocation remains finite. The hierarchy is closed by making a standard closure approximation. For the particular case of a population of parallel screw dislocations of the same sign moving on parallel slip planes the solution for the pair distribution function is found analytically. For the dislocations having opposite signs the system of equations suggests that in ensemble average only geometrically necessary dislocations correlate, while balanced positive and negative dislocations would create dipoles or annihilate. Direct numerical simulations support this conclusion. In addition, the relation of the dislocation correlation to strain gradient theories and size effect is shown and discussed.

Multi-point distribution function for the continuous time random walk

We derive an explicit expression for the Fourier–Laplace transform of the two-point distribution functionp(x1, t1; x2, t2) of a continuous time random walk (CTRW), thus generalizing theresult of Montroll and Weiss for the single-point distribution functionp(x1, t1). The multi-point distribution function has a structure of a convolution of theMontroll–Weiss CTRW and the ageing CTRW single-point distribution functions. Thecorrelation function for the biased CTRW process is found. The random walk foundation of themulti-time–space fractional diffusion equation is investigated using the unbiased CTRW inthe continuum limit.

The problem of averaging random structures in terms of distribution functions

A new approach to solving averaging problems for micro-inhomogeneous continua, based on a restatement of the problem in terms of distribution functions, is described. Problems having a variational structure are considered. It is shown that, in terms of distribution functions, they reduce to the problem of minimizing a linear functional, having the meaning of the expectation value of the energy, in a set of distribution functions which is distinguished by an infinite number of linear constants. These constraints express certain matching conditions and contain multipoint distribution functions of the random characteristics of the medium. The constraints form an unlinked chain, the break of which at the n-th step contains only n-point distribution functions. In view of this, a sequence of approximate problems arises.

Multipoint distribution function hierarchy for compressible turbulent flow

The hierarchy of equations for the multipoint distribution functions is developed for compressible turbulent flow. The compressible equations reduce to previously published relations for incompressible fluids with constant transport under the appropriate approximations. Expressions for other types of incompressible fluids are also shown to be derivable from the compressible relations.

Turbulence theory on the basis of multipoint distribution functions

Abstract Systematic methods are developed for deriving closed approximate equations for probability densities of a turbulent velocity field at one and at two points. These methods are based on diagram techniques of non-equilibrium statistical mechanics and quantum field theory. The equations are written down in the weak coupling approximation. Arguments are made that the approximation which is graphically equivalent to the direct-interaction approximation well known in turbulence theory is rather exact when used for deriving closed equations for multipoint distribution functions. Analysis of the equations in the weak coupling approximation in the inertial range shows their compatibility with a local cascade mechanism of energy transfer in wave number space.

Multipoint distribution calculation of the isotropic turbulent energy spectrum

A solution for the dynamical behavior of the energy spectrum for a homogeneous isotropic fluid is obtained from the equations for the turbulent multipoint distribution functions. The calculated spectrum shows reasonable agreement with experimental data.

Semiempirical theory of turbulent transfer

A closed system of equations is constructed for flow of a nonisotropic character on the assumption that the mixing path length is not small compared with the characteristic dimension of the stream. It is assumed that the velocity pulsation field can be characterized by a multipoint distribution function, which satisfies the continuity equation. This enables equations to be obtained for the one-point and two-point distribution function. A series of assumptions is made concerning the nature of the forces acting on the turbulent formation (“turbule” or vortex) in the stream and concerning the correlation time of the random force with the scale and intensity of the turbulence. Assumptions are also made concerning the expression of the integral in the equation for the one-point distribution function and the expression for the correlation tensor in the isotropic case. After the moments are calculated, a system of Reynolds' equations is obtained in which approximations, usually acceptable from dimensional considerations, follow for a series of terms. Here, this is a consequence of the approximations for the forces in the equation for the distribution function. Closing the system of equations for the moments reduces to solving the equation for the distribution function. It turns out that the integral character of the transfer (diffusion of a nongradient type) is connected with taking third-order moments into account. A series of examples of flow is considered, and values of the empirical constants are determined. The system of equations obtained enables us to consider flow with strong anisotropy of turbulent transfer.

Solution for Turbulent Correlations Using Multipoint Distribution Functions

A solution for the time evolution of the velocity correlation function in homogeneous isotropic turbulence is obtained using multipoint distribution functions. The dependence of the velocity correlation function on the fluid parameters is calculated for a simple initial condition.