Integral Equations for Memory Functions Involving Projection Operators

Kinetic equations for the phase„space-time correlation functions contain memory functions that involve projection operators. It is shown that these memory functions can be represented by integral equations involving only real-time correlation functions, thereby eliminating the projection operators completely in the kinetic description of correlation functions. The weak-coupling and density expansions of the memory functions have been obtained through these integral equations.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/86138/1/PhysRevA.7.182-AZA.pd

Fluctuation Analysis in Simple Fluids

A classical analysis of time correlations in simple fluids based on the generalized Langevin equation is presented. Formulas for the current-current correlations are developed explicitly in a region of frequencies (__1013 sec-1) and wave numbers (k_108 cm-1) which are explored in typical slow-neutron-scattering measurements. Where applicable, comparisons are made with the results of the numerical calculations of Rahman in argonlike liquids, and good agreement is generally found. The analysis is based on a hydrodynamic description of fluids involving frequency- and wavelength-dependent transport parameters. The frequency and wavelength dependence of shear and longitudinal viscosities are given explicitly for argon-like liquidsPeer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/86142/1/PhysRevA.2.962-AZA.pd

Derivation of Kinetic Equations from the Generalized Langevin Equation

The projection operator techniques of Zwanzig and Mori are used to obtain a generalized Langevin equation describing the time evolution of the fluctuation of the microscopic phase density _g(x_,p_,t)_g(x_,p_,t)-_g(x_,p_,t)_for a classical many-particle system. This equation is then used to develop an exact kinetic equation for the time-correlation function _g(x_,p_,0)_g(x__,p__,t) [which is the generalization of the Van Hove time-dependent pair correlation function G(r_,t)]. In the lowest order of approximation, this kinetic description reduces to the Vlasov-like equation which has been used to study neutron scattering from liquids. A less restrictive approximation is obtained by utilizing weak-coupling perturbation theory to yield a generalized Fokker-Planck equation for the time-correlation function. Other possible approximation schemes are also discussed.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/86140/1/PhysRev.188.479-AZA-JJD.pd

A closed‐form, free‐energy functional for a binary polymer mixture

A new, closed‐form, free‐energy functional is derived for a binary polymer mixture. When the free‐energy functional is expanded in series form around the mean concentration, the leading term in the expansion is the usual Flory–Huggins free energy. The Fourier transform of the coefficients of this expansion are approximate vertex functions Γ(n). A useful and tractable form for Γ(n) is obtained for all n which only depends on the magnitudes of the n wave vectors. It is shown that Γ(2) is exact and Γ(3) and Γ(4) reduce to the correct limiting values in the small and large wave vector limits.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/70577/2/JCPSA6-88-12-7847-1.pd

The Calculation of Current Correlations in Classical Fluids via Modeled Kinetic Equations

The transverse and longitudinal current correlations in a simple classical fluid were originally calculated from a modeled kinetic equation using an approximate solution valid for only small values of wave number k. These correlation functions have been recalculated using the exact solution of this kinetic equation for arbitrary values of k.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/86141/1/PhysRevA.2.1097-JJD-AZA.pd

Calculation of Current Correlations in Classical Fluids via Modeled Kinetic Equations

In an earlier paper, the generalized Langevin approach of Mori and Zwanzig has been applied to generate an exact kinetic equation describing the time correlation of fluctuations in the microscopic phase density for a classical many-body system. In the present work, this kinetic equation is modeled using a single-relaxation-time form for the damping kernel, and the solution of this equation is then studied for large and small values of k and _. The modeled kinetic equation is used to calculate transverse and lingitudinal current-current correlations in simple classical liquids, and the results are compared with the molecular-dynamics calculations of Rahman for argonlike systems.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/86143/1/PhysRevA.1.905-JJD-AZA.pd

Nonlinear kinetics of spinodal decomposition, and dissolution of inhomogeneities formed by spinodal decomposition in polymer blends

Nonlinear kinetics of both spinodal decomposition at early stages, and the dissolution of homogeneities formed during spinodal decomposition, is studied. Variation of the scattering intensity during a complete cycle consisting of a step temperature change from T1 in the one‐phase region to T2 in the two‐phase region, a period of spinodal decomposition followed by a temperature drop from T2 back to T1, and the subsequent relaxation to the original equilibrium state, is investigated at various wavenumbers. Step temperature changes within one‐phase region are also investigated.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/113684/1/19920620107_ftp.pd

On the dynamics of polyelectrolyte solutions

A general formalism to study the dynamics of polyelectrolyte solutions is presented. We show in particular that the Berne–Pecora equations for charged pointlike particles are obtained by neglecting the memory function and using the Debye–Huckel potential with the linear approximation exp(−U/kBT)≂1−U/kBT. We generalize Berne–Pecora results by introducing the effect of hydrodynamic interaction. Our calculations show a plasmon mode which corresponds to a nonzero frequency at zero scattering angle.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/70342/2/JCPSA6-80-6-2762-1.pd

Diffusion of charged particles in tokamak‐like stochastic magnetic and electric fields

In this paper the diffusion of guiding centers induced by stochastic magnetic and electric field fluctuations, with both time and space dependence, is analyzed for the case of tokamak plasmas. General experimental results on tokamak fluctuations are used to derive guiding‐center equations that properly describe the particle motion. These equations assume uniform average magnetic and electric fields with random stationary Gaussian fluctuations that constitute a homogeneous and cylindrically symmetric turbulence. By applying Novikov’s theorem, a Fokker–Planck equation for the probability distribution function is derived and an expression for the guiding‐center diffusion coefficient is obtained. This coefficient not only contains the standard terms due to the stochastic wandering of the magnetic lines and the stochastic electric drift, but also new terms due to the stochastic curvature and ∇B drifts. The form of these terms is shown explicitly in terms of the correlation functions of the fields.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/70770/2/PFBPEI-4-12-3935-1.pd