Diffusion in multilayer media: transient behavior of the lateral diffusion coefficient

A general formalism for treating lateral diffusion in a multilayer medium is developed. The formalism is based on the relation between the lateral diffusion and the distribution of the cumulative residence time, which the diffusing particle spends in different layers. We exploit this fact to derive general expressions which give the global and local time-dependent diffusion coefficients in terms of the average cumulative times spent by the particle in different layers and the probabilities of finding the particle in different layers, respectively. These expressions are used to generalize two recently obtained results: (a) A solution for the short-time behavior of the lateral diffusion coefficient in two layers separated by a permeable membrane obtained by a perturbation theory is extended to the entire range of time. (b) A solution for the time-dependent diffusion coefficient of a ligand, which repeatedly dissociates and rebinds to sites on a planar surface, obtained under the assumption that the medium above the surface is infinite, is generalized to allow for the medium layer of finite thickness. For the latter problem we derive an expression for the Fourier-Laplace transform of the propagator in terms of the double Laplace transform of the probability density of the cumulative residence time spent by the ligand in the medium layer.     

Diffusion in the presence of periodically spaced permeable membranes

The diffusion of molecules in biological tissues and some other microheterogeneous systems is affected by the presence of permeable barriers. This leads to the slowdown of diffusion at long times as compared to barrier-free diffusion. At short times the effect of barriers is weak. In consequence, the diffusion coefficient D(t) decreases as a function of time. We derive an exact solution for the Laplace transform of D(t) for diffusion in a space separated into layers by equally spaced, parallel identical planes of arbitrary permeability. Additionally, we give an approximation to D(t) which is reasonably accurate over the whole range of the partition permeability from zero (the case of isolated layers) to infinity (the case of no barriers).     

Efficient computation of the first passage time distribution of the generalized master equation by steady-state relaxation

The generalized master equation or the equivalent continuous time random walk equations can be used to compute the macroscopic first passage time distribution (FPTD) of a complex stochastic system from short-term microscopic simulation data. The computation of the mean first passage time and additional low-order FPTD moments can be simplified by directly relating the FPTD moment generating function to the moments of the local FPTD matrix. This relationship can be physically interpreted in terms of steady-state relaxation, an extension of steady-state flow. Moreover, it is amenable to a statistical error analysis that can be used to significantly increase computational efficiency. The efficiency improvement can be extended to the FPTD itself by modelling it using a gamma distribution or rational function approximation to its Laplace transform.     

Segmental distribution properties of a polymer chain near an interacting barrier

The exact distribution of segments for self-avoiding walks of lengths N=4–14 bonds in the presence of an interacting barrier on the diamond lattice has been obtained by the method of direct enumeration. Behavior for the infinite chain was estimated and compared with Rubin's results for the normal random walk. It is shown that the onset of a well defined transition for the self-avoiding walk coincides with the location predicted for the normal random walk. It was found that for a self-avoiding walk a plot of θ₂ (the fraction of segments in level z) versus the interaction parameter φ is shifted to the right (higher φ), for all z, as compared with a similar plot for the normal random walk. Conditional probabilities for a self-avoiding walk having its t th segment in level z (when φ=0) are reported.     

Two-particle continuous-time random walks and binary reactions in disordered media

The relation between unary-(trappihg) and binary (mutual annihilation) reactions in disordered systems is studied in the framework of the continuous-time random walk. It is found that if the waiting-time distribution of the walk has infinite moments, a time-independent binary rate constant may exist even though a unary one does not.     

Exact expressions of mean first-passage times and splitting probabilities for random walks in bounded rectangular domains

We give exact and explicit expressions of mean first-passage times for random walks in a rectangular domain in both cases of reflecting boundary conditions and periodic boundary conditions. The situations with one or two absorbing targets are considered.     

On describing the steady absorption of brownian particles by a restricted random walk

This paper is concerned with idealizing brownian motion as a random walk, using the diffusion equation, and finding the boundary condition at an absorbing surface - all with an eye towards chemical kinetics. Three models of random walk (due to Smoluchowski, Fermi, and Lorentz) are considered, and it is concluded that the lorentzian model is the most appropriate.     

Diagrammatic kinetic theory for a lattice model of a liquid. I. Theory

We present a diagrammatic formalism for the time correlation functions of density fluctuations for an excluded volume lattice gas on a simple d-dimensional hypercubic lattice. We consider a multicomponent system in which particles of different species can have different transition rates. Our theoretical approach uses a Hilbert space formalism for the time dependent dynamical variables of a stochastic process that satisfies the detailed balance condition. We construct a Liouville matrix consistent with the dynamics of the model to calculate both the equation of motion for multipoint densities in configuration space and the interactions in the diagrammatic theory. A Boley basis of fluctuation vectors for the Hilbert space is used to develop two formally exact diagrammatic series for the time correlation functions. These theoretical techniques are generalizations of methods previously used for spin systems and atomic liquids, and they are generalizable to more complex lattice models of liquids such as a lattice gas with attractive interactions or polymer models. We use our formalism to construct approximate kinetic theories for the van Hove correlation and self-correlation function. The most simple approximation is the mean field approximation, which is exact for the van Hove correlation function of a one component system but an approximation for the self-correlation function. We use our first diagrammatic series to derive a two site multiple scattering approximation that gives a simple analytic expression for the spatial Fourier transform of the self-correlation function. We employ our second diagrammatic series to derive a simple mode coupling type approximation that provides a system of equations that can be solved for the self-correlation function.     

On the fluctuation theorem for the dissipation function and its connection with response theory

Recently, there has been considerable interest in the fluctuation theorem (FT). The Evans-Searles FT shows how time reversible microscopic dynamics leads to irreversible macroscopic behavior as the system size or observation time increases. We show that the argument of this FT, the dissipation function, plays a central role in nonlinear response theory and derive the dissipation theorem, giving exact relations for nonlinear response of classical N-body systems that are more widely applicable than previous expressions. These expressions should be verifiable experimentally. When linearized they reduce to the well-known Green-Kubo expressions for linear response.     

Theory for a wavelet analysis of electrochemical noise in the laplace space with use of two operational frequencies

The image of a random process in the Laplace space may be viewed as resulting from use of a oneway continuous wavelet transform with an exponential as the basic function, i.e. as resulting from the application of the Laplace wavelet. If the Laplace-wavelet variance of an electrochemical noise allows one to determine the Laplace transform of a time-correlation function, i.e. a factual operational spectral density of the noise, then the covariance of two Laplace waveletes of an electrochemical noise, each of which corresponds to its own operational frequency, allows one to verify a local consistency of the initial experimental noise data. The Laplace waveletes are applied rather widely. In fact, any stationary random process and stationary random time sequence can be described with operational noise spectra. Dedicated to the ninetieth anniversary of Ya.M. Kolotyrkin’s birth.     

Theory of Laplace Analysis of Non-Gaussian Noise

An algorithm for directly calculating third-order noise operation spectra, which includes no evaluation of the third-order correlation function as a preliminary stage, is found. For the Ershler–Randles circuit, an expression is found, which links bispectra of the equilibrium electrode potential fluctuation determined in the imaginary and real axes of the Laplace plane. Advantages of using the Laplace space in studies of the fine non-Gaussian structure of random time series are discussed.     

A new time evolving Gaussian series representation of the imaginary time propagator

Frantsuzov and Mandelshtam [J. Chem. Phys. 121, 9247 (2004)] have recently demonstrated that a time evolving Gaussian approximation (TEGA) to the imaginary time propagator exp(-betaH) is useful for numerical computations of anharmonically coupled systems with many degrees of freedom. In this paper we derive a new exact series representation for the imaginary time propagator whose leading order term is the TEGA. One can thus use the TEGA not only as an approximation but also to obtain the exact imaginary time propagator. We also show how the TEGA may be generalized to provide a family of TEGA's. Finally, we find that the equations of motion governing the evolution of the center and width of the Gaussian may be thought of as introducing a quantum friction term to the classical evolution equations.     

Exact solution of the excited-state geminate A*+B <==> C*+D reaction with two different lifetimes and quenching

The authors obtain, in the Laplace transform space, the exact analytic solution for the Green function and survival probabilities for the excited-state diffusion-influenced reversible geminate reaction, A*+B <==> C*+D, with two different lifetimes and in the presence of an added quenching process. This extends a previous investigation by Popov and Agmon [J. Chem. Phys. 117, 5770 (2002)] of the ground-state reaction without quenching. The long-time asymptotic behavior of the survival probabilities is obtained in the time domain. It is found to be different from the equal-lifetime case. This paper also provides a useful short-time approximation for the kinetics.     

Error analysis of molecular dynamics and fractal time approximants from a combinatorial perspective

Trotter's theorem forms the theoretical basis of most modern molecular dynamics. In essence this theorem states that a time displacement operator (a Lie operator) constructed by exponentiating a sum of noncommuting operators can be approximated by a product of single operators provided the time interval is "very small." In theory "very small" implies infinitesimally small (at which point the approximate product becomes exact), while in practical analysis a finite time interval is divided into several small subintervals or steps. It follows, therefore, that the larger the number of steps the better the approximation to the exact time displacement operator. The question therefore arises: How many steps are sufficient? For bounded operators, standard theorems are available to provide the answer. In this paper we show that a very simple combinatorial formula can be derived which allows the computation of the global differences (as a function of the number of steps) between the Taylor coefficients of the exact time displacement operator and an approximate one constructed by using a finite number of steps. The formula holds for both bounded and nonbounded operators and shows, quantitatively, what is qualitatively expected-that the error decreases with increasing number of steps. Furthermore, the formula applies irrespective of the complexity of the system, boundary conditions, or the thermodynamic ensemble employed for averaging the initial conditions. The analysis yields explicit expressions for the Taylor coefficients which are then used to compute the errors. In the case of the algorithmically based practical numerical simulations in which fixed, albeit small, steps are repeatedly applied, the rise in the number of steps does not reduce the size of the steps but increases the total time of interest. The combinatorial formula shows that, here, the errors diverge. Furthermore, this work can be used to supplement other efforts such as the use of shadow Hamiltonians where the truncation of the series expansion of the latter will produce errors in the higher order propagator moments.     

Symmetric random walk on a regular lattice with an elastic barrier: diffusion equation and the boundary condition

We examine here the symmetric random-walk model in which a particle can jump only to adjacent sites on a regular lattice, and discuss, in the light of the works of Chandrasekhar, Goodrich, van Kampen, and Oppenheim, the reduction of the problem of random walk in the presence of an elastic barner to a boundary value problem.     

The statistical theory of linear capillary chromatography with uniform stationary phase

Based on the random walk model and probability theory, general relations between the moments of column residence time and the moments of step sojourn time and step displacement are established. And starting from the mass-balances principle of solute molecules in the mobile and stationary phases, the moments of step sojourn time and step displacement are derived and expressed in terms of the basic parameters. Substituting the step moments into the general relations, the moments of column residence time are then obtained. The expression of retention time is completely identical to the well-known, the expressions of second moment or HETP unite and generalize various expressions of stochastic theory and mass balance theory, and the third and forth moments are given in more exact form.     

Stochastic dynamics of complexation reaction in the limit of small numbers

We study stochastic dynamics of the non-linear bimolecular reaction A + B?AB. These reactions are common in several bio-molecular systems such as binding, complexation, protein multimerization to name a few. We use master equation to compute the full distribution of several stochastic equilibrium properties such as number of complexes formed (N(c)), equilibrium constant (K). We provide exact analytical and simpler approximate expression for equilibrium fluctuation quantities to quickly estimate the amount of noise as a function of reactant molecules and rates. We construct the phase diagram for a fluctuational quantity f, defined as the ratio of standard deviation to average (f=√(ΔN(c))(2)/N(c)), as a function of different number of reactant molecules and reaction rates. One of the striking result is, it is possible to have f as high as 45% or higher in significant regions of the phase diagram even when number of reactants involved are around 20-40, typical in biology. Our finding indicates studying averages alone using mass action law needs careful scrutiny. We also outline possible application of our findings in gene expression. Furthermore, we compute average and fluctuation properties of time dependent quantities and derive equations of motion for different moments such as N(c)(t) and N(c)(t)(2). While mean-field mass action law fails to reproduce the exact time dependence, approximate solutions of coupled equations of motions for different moments, capturing fluctuation, is in good agreement with exact results. This may be a way to compute time development of averages and fluctuations in such non-linear systems where mass action law breaks down. Moreover, for this reaction, we outline connection to variational principle of maximum caliber and other more traditional approaches such as chemical Langevin equation. We derive noise statistics for the equivalent Langevin equation and show possible departure from Gaussian white noise. We believe quantitative estimates of phase diagrams for noise, time dependent quantities, and simple analytical expression for equilibrium quantities will be particularly useful to guide experiments involving such non-linear reactions with small numbers of reactants that are often encountered in biology.     

Efficient electron dynamics with the planewave-based real-time time-dependent density functional theory: absorption spectra, vibronic electronic spectra, and coupled electron-nucleus dynamics

The electron dynamics with complex third-order Suzuki-Trotter propagator (ST(3)) has been implemented into a planewave (PW) based density functional theory program, and several applications including linear absorption spectra and coupled electron-nucleus dynamics have been calculated. Since the ST(3) reduces the number of Fourier transforms to less than half compared to the fourth-order Suzuki-Trotter propagator (ST(4)), more than twice faster calculations are possible by exploiting the ST(3). We analyzed numerical errors of both the ST(3) and the ST(4) in the presence∕absence of an external field for several molecules such as Al(2), N(2), and C(2)H(4). We obtained that the ST(3) gives the same order of numerical errors (10(-5) Ry after 100 fs) as the ST(4). Also, the time evolution of dipole moments, hence the absorption spectrum, is equivalent for both ST(3) and ST(4). As applications, the linear absorption spectrum for an ethylene molecule was studied. From the density difference analysis, we showed that the absorption peaks at 6.10 eV and 7.65 eV correspond to the π → 4a(g) and π → π* excitation bands, respectively. We also investigated the molecular vibrational effect to the absorption spectra of an ethylene molecule and the dynamics of a hydrogen molecule after the σ → σ* transition by formulating coupled electron-nucleus dynamics within the Ehrenfest regime. The trajectory of nuclei follows the excited state potential energy curve exactly.