Density Functional Approach Based on Numerically Obtained Bridge Functional

The Ornstein Zernike equation is solved with the Rogers Young approximation for bulk hard sphere fluidand Lennard-Jones fluid for several state points. Then the resulted bulk fluid radial distribution function combinedwith the test particle method is employed to determine numerically the function relationship of bridge functional as afunction of indirect correlation function. It is found that all of the calculated points from different phase space statepoints for a same type of fluid collapse onto a same smooth curve. Then the numerically obtained curve is used tosubstitute the analytic expression of the bridge functional as a function of indirect correlation function required in themethodology [J. Chem. Phys. 112 (2000) 8079] to deterrnine the density distribution of non-uniform hard spherefluid and Lennard Jones fluid. The good agreement of theoretical predictions with the computer simulation data isobtained. The present numerical procedure incorporates the knowledge of bulk fluid radial distribution function intothe constructing of the density functional approximation and makes the original methodology more accurate and moreflexible for various interaction potential fluid.     

On the relationship between continuous time random walks and the non-equilibrium Ornstein-Zernike equation

An exact non-equilibrium Ornstein-Zernike (OZ) equation is derived for lattice fluid systems whose time development is given by a generalized master equation. The derivation is based on a generalization of the Montroll-Weiss continuous-time random walk on a lattice, and on their relationship with master equation solutions. Time dependent direct and total correlation functions are defined in terms of the generating functions for the probability densities of the random walker, such that, in the infinite time limit the equilibrium OZ equation is recovered. A perturbative analysis of the time dependent OZ equation is shown to be formally analogous to the perturbation of the Bloch equation in quantum field theory. Analytic results are obtained, under the mean spherical approximation, for the time dependent total correlation function for a one-dimensional lattice fluid with exponential attraction.     

A New Uniform Phase Bridge Functional: Test and Its Application to Non-uniform Phase Fluid

A new bridge functional as a function of indirect correlation function was proposed, which was basedon analysis on the asymptotic behavior of the Ornstein-Zernike (OZ) equation system and a series expansion whoserenormalization resulted in an adjustable parameter determined by the thermodynamics consistency condition. Theproposed bridge functional was tested by applying it to bulk hard sphere and hard core Yukawa fluid for the predictionof structure and thermodynamics properties based on the OZ equation. As an application, the present bridge functionalwas employed for non-uniform fluid of the above two kinds by means of the density functional theory methodology, theresulting density distribution profiles were in good agreement with the available computer simulation data.     

Observation of quantum particles on a large space-time scale

A quantum particle observed on a sufficiently large space-time scale can be described by means of classical particle trajectories. The joint distribution for large-scale multiple-time position and momentum measurements on a nonrelativistic quantum particle moving freely inR ^v is given by straight-line trajectories with probabilities determined by the initial momentum-space wavefunction. For large-scale toroidal and rectangular regions the trajectories are geodesics. In a uniform gravitational field the trajectories are parabolas. A quantum counting process on free particles is also considered and shown to converge in the large-space-time limit to a classical counting process for particles with straight-line trajectories. If the quantum particle interacts weakly with its environment, the classical particle trajectories may undergo random jumps. In the random potential model considered here, the quantum particle evolves according to a reversible unitary one-parameter group describing elastic scattering off static randomly distributed impurities (a quantum Lorentz gas). In the large-space-time weak-coupling limit a classical stochastic process is obtained with probability one and describes a classical particle moving with constant speed in straight lines between random jumps in direction. The process depends only on the ensemble value of the covariance of the random field and not on the sample field. The probability density in phase space associated with the classical stochastic process satisfies the linear Boltzmann equation for the classical Lorentz gas, which, in the limith0, goes over to the linear Landau equation. Our study of the quantum Lorentz gas is based on a perturbative expansion and, as in other studies of this system, the series can be controlled only for small values of the rescaled time and for Gaussian random fields. The discussion of classical particle trajectories for nonrelativistic particles on a macroscopic spacetime scale applies also to relativistic particles. The problem of the spatial localization of a relativistic particle is avoided by observing the particle on a sufficiently large space-time scale.     

Statistical Mechanics Approach for Uniform and Non-uniform Fluid with Hard Core and Interaction Tail

One recently proposed self-consistent hard sphere bridge functional was combined with an exponential function exp(-cr) and a re-normalized indirect correlation function to construct the bridge function for fluid with hard core and interaction tail. In the present approach, the adjustable parameter α was determined by the thermodynamic consistency realized on the compressibility modulus, the re-normalization of the indirect correlation function was realized by a modified Mayer function with the interaction potential replaced by the perturbative part of the interaction potential. As an example, the present bridge function was combined with the Ornstein-Zernike (OZ) equation to predict structure and thermodynamics properties in very good agreement with the simulation data available for Lennard-Jones (L J). Based on the universality principle of the free energy density functional and the test particle trick, the numerical solution of the OZ equation was employed to construct the first order direct correlation function of the non-uniform fluid as a functional of the density distribution by means of the indirect correlation function. In the framework of the density functional theory, the numerically obtained functional predicted the density distribution of LJ fluid confined in two planar hard walls that is in good agreement with the simulation data.     

Dynamics of a charged test particle in a hard rod fluid

The motion of a charged hard rod, accelerated by a constant and uniform external field, in a fluid of mechanically identical neutral particles is studied. The system, initially at rest, is excited through collisions with the accelerated particle. A class of initial configurations is found for which recollisions between the charged rod and the excitation caused by it (a moving particle) never occur. The evolution of the velocity distribution of the test particle is analyzed in this case. The possibility of obtaining from microscopic dynamics a kinetic equation is discussed. The dependence of the current on the external field is shown to agree with that predicted by the Boltzmann equation.     

Augmented Kierlik-Rosinberg Fundamental Measure Functional andExtension of Fundamental Measure Functional to Inhomogeneous Non-hard Sphere Fluids

From point of view of weighted density procedure, it isguessed that a Percus-Yevick (PY) compressibility excess free energydensity, appearing in the Kierlik--Rosinberg type fundamentalmeasure functional (KR-FMF) and expressed in terms of scaledparticle variables, can be substituted by a corresponding expressiondictated by a more accurate Mansoori-Carnahan-Starling-Leland(MCSL) equation of state, while retaining the original weightingfunctions; it is numerically indicated that the resultantundesirable non-self-consistency between the PY type weightingfunction and MCSL type excess free energy density had no bad effecton the performance of the resultant augmented KR-FMF which, on theone hand, preserves the exact low-density limit of the originalKR-FMF and holds a high degree of pressure self-consistency, on theother hand, improves significantly, as expected, the predictions ofdensity profile of hard sphere fluid at single hard wall contactlocation and its vicinity, and of the bulk hard sphere second orderdirect correlation function (DCF), obtained from functionaldifferentiation. The FMF is made applicable to inhomogeneousnon-hard sphere fluids by supplementing a functional perturbationexpansion approximation truncated at the lowest order with summationof higher order terms beyond the lowest term calculated by the FMFfor an effective hard sphere fluid; the resultant extended FMF onlyneeds second order DCF and pressure of the fluid considered atcoexistence state as inputs, consequently is applicable whether theconsidered temperature is above critical point or below criticalpoint. The extended MCSL-augmented KR-FMF is found to be endowedwith an excellent performance for predictions of density profile andsurface tension by comparing the present predictions of these twoquantities with available computer simulation data for inhomogeneoushard core attractive Yukawa fluid and Lennard-Jones fluid.     

Metastable molecular fluid hydrogen at high pressures

Warm dense hydrogen is studied in the region of fluid–fluid phase transition within the framework of the density functional theory. We report a procedure of obtaining metastable states and calculate the equation of state. Metastable states are diagnosed by pair correlation functions and values of conductivity. We obtain a strong overlapping through the density of metastable and equilibrium branches of pressure isotherms. This indicates the plasma nature of the phase transition.     

扩展桥密度泛函近似于非匀一非硬球流体

为了将非匀一硬球流体的桥密度泛函近似扩展到非匀一非硬球流体,提出了一个理论方案．所得的LJ流体的密度泛函近似计算简单,精确．特别是密度泛函近似仅仅需要共存体相流体的二阶直接相关函数作为输入,因而可以应用于超临界与亚临界温度．所提出的理论方案可以认为是热力学理论的非匀一对应物．     

Bridge density functional approximation for non-uniform hard core repulsive Yukawa fluid

In this work, a bridge density functional approximation （BDFA） （J. Chem. Phys. 112, 8079 （2000）） for a nonuniform hard-sphere fluid is extended to a non-uniform hard-core repulsive Yukawa （HCRY） fluid. It is found that the choice of a bulk bridge functional approximation is crucial for both a uniform HCRY fluid and a non-uniform HCRY fluid. A new bridge functional approximation is proposed, which can accurately predict the radial distribution function of the bulk HCRY fluid. With the new bridge functional approximation and its associated bulk second order direct correlation function as input, the BDFA can be used to well calculate the density profile of the HCRY fluid subjected to the influence of varying external fields, and the theoretical predictions are in good agreement with the corresponding simulation data. The calculated results indicate that the present BDFA captures quantitatively the phenomena such as the coexistence of solid-like high density phase and low density gas phase, and the adsorption properties of the HCRY fluid, which qualitatively differ from those of the fluids combining both hard-core repulsion and an attractive tail.     

SU(3) Local Gauge Field Theory as Effective Dynamics of Composite Gluons

The effective dynamics of quarks is described by a nonperturbatively regularized NJL model equation with canonical quantization and probability interpretation. The quantum theory of this model is formulated in functional space and the gluons are considered as relativistic bound states of colored quark-antiquark pairs. Their wave functions are calculated as eigenstates of hardcore equations, and their effective dynamics is derived by weak mapping in functional space. This leads to the phenomenological SU(3) gauge invariant gluon equations in functional formulation, i.e., the local gauge symmetry is a dynamical effect resulting from the dynamics of the quark model.     

Integral quadratic functionals defined in a normal Markov two-dimensional field and their statistics

We consider the problem of distribution of an integral quadratic functional defined in a stationary two-dimensional random normal Markov field. The generating function of the random-value probability distribution of the functional is found and an approximate expression for the probability distribution is obtained. The effect of the parameters of the two-dimensional field on the statistical properties of the distribution of the considered functional is analyzed. A generalization of the solution to the case of a multidimensional stationary normal Markov field is proposed. Kharkov State Polytechnical University, Kharkov, Ukraine Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 43, No. 6, pp. 562–572, June, 2000.     

Functional integrals in the statistical theory of turbulence and the burgers model equation

The functional integrals appearing in master equations for turbulent flow of an incompressible fluid and for the Burgers model equation are treated. A possible way is described to define the integration properly and related problems are discussed. For the simple example of the Burgers model equation of turbulence some results are presented.     

Ensemble mechanics for the random-forced Navier-Stokes flow

It is shown that the so-called statistical theory of turbulence, which can be exactly incorporated by the Hopf functional equation, is imperfect in that it fails to ensure the irreversible approach to a unique ultimate steady state of turbulence (for a steady boundary condition) expected from observation; and that this imperfection is removed if a stochastic random-force term is added into the Navier-Stokes equation. The ensemble mechanics for the random-forced Navier-Stokes flow is formulated by taking into account the natural random force, which has usually been neglected in the Navier-Stokes equation.     

On stochastic complex beam-beam interaction models with Gaussian colored noise

This paper is to continue our study on complex beam-beam interaction models in particle accelerators with random excitations Y. Xu, W. Xu, G.M. Mahmoud, On a complex beam-beam interaction model with random forcing [Physica A 336 (2004) 347-360]. The random noise is taken as the form of exponentially correlated Gaussian colored noise, and the transition probability density function is obtained in terms of a perturbation expansion of the parameter. Then the method of stochastic averaging based on perturbation technique is used to derive a Fokker-Planck equation for the transition probability density function. The solvability condition and the general transforms using the method of characteristics are proposed to obtain the approximate expressions of probability density function to order ε.Also the exact stationary probability density and the first and second moments of the amplitude are obtained, and one can find when the correlation time equals to zero, the result is identical to that derived from the Stratonovich-Khasminskii theorem for the same model under a broad-band excitation in our previous work.     

On the diffusion and clustering of a magnetic field in random velocity fields

We have derived an equation for the probability density of the magnetic energy in a random Gaussian, delta-correlated in time, divergent velocity field in the absence of molecular diffusion effects. Basic statistical characteristics of the energy have been calculated using this equation. Based on the ideas of statistical topography, we have studied the processes of magnetic field amplification in space and, in particular, the conditions for the formation of a cluster structure. These phenomena are coherent, occur with a probability equal to unity, and, hence, manifest themselves almost in all individual realizations of the process. The clustering effect is demonstrated with an exact solution for the magnetic field dynamics for the simplest model of a random divergent velocity field.     

Integro-differential equation for joint probability density in phase space associated with continuous-time random walk

We derive an integro-differential equation for the joint probability density function in phase space associated with the continuous-time random walk, with generic waiting time probability density function and external force. This equation permits us to investigate whole diffusion processes covering initial-, intermediate-, and long-time ranges, which can distinguish the evolution details for systems having the same behavior in the long-time limit with different initial- and intermediate-time behaviors. Moreover, we obtained analytic solutions for probability density functions both in velocity and phase spaces, and interesting dynamic behaviors are discovered.