Generalized Smoluchowski Equation Research Articles

Low-rank method for fast solution of generalized Smoluchowski equations

An efficient method for the fast solution of a new class of generalized Smoluchowski equations is developed. In contrast to conventional Smoluchowski equations with constant rate coefficients that describe the aggregation kinetics, the rate coefficients of the generalized Smoluchowski equations depend on time. The evolution of these coefficients is governed by another set of equations coupled with the aggregation equations. The most prominent example of such processes is the ballistic aggregation when aggregates of different sizes have different partial temperatures. The conventional approaches fail to deal efficiently with these systems when the number of equations N is large. The new method overcomes this difficulty. It exploits the low-rank approach and drastically decreases computation time: instead of O(N2) operations per time step, the new method requires only O(Nr(r+log⁡N)) operations, where r is the system rank. General techniques for an efficient solution of very large systems of ODE are discussed in the context of the new approach.

Coagulation and universal scaling limits for critical Galton–Watson processes

AbstractThe basis of this paper is the elementary observation that then-step descendant distribution of any Galton–Watson process satisfies a discrete Smoluchowski coagulation equation with multiple coalescence. Using this we obtain simple necessary and sufficient criteria for the convergence of scaling limits of critical Galton–Watson processes in terms of scaled family-size distributions and a natural notion of convergence of Lévy triples. Our results provide a clear and natural interpretation, and an alternate proof, of the fact that the Lévy jump measure of certain continuous-state branching processes (CSBPs) satisfies a generalized Smoluchowski equation. (This result was previously proved by Bertoin and Le Gall (2006).) Our analysis shows that the nonlinear scaling dynamics of CSBPs becomelinear and purely dilatationalwhen expressed in terms of the Lévy triple associated with the branching mechanism. We prove a continuity theorem for CSBPs in terms of the associated Lévy triples, and use our scaling analysis to prove the existence ofuniversalcritical Galton–Watson processes and CSBPs analogous to Doeblin's `universal laws'. Namely, these universal processes generateallpossible critical and subcritical CSBPs as subsequential scaling limits. Our convergence results rely on a natural topology for Lévy triples and a continuity theorem for Bernstein transforms (Laplace exponents) which we develop in a self-contained appendix.

Generalized Einstein relation for the mutual diffusion coefficient of a binary fluid mixture

The method employed by Einstein to derive his famous relation between the diffusion coefficient and the friction coefficient of a Brownian particle is used to derive a generalized Einstein relation for the mutual diffusion coefficient of a binary fluid mixture. The expression is compared with the one derived by de Groot and Mazur from irreversible thermodynamics and later by Batchelor for a Brownian suspension. A different result was derived by several other workers in irreversible thermodynamics. For a nearly incompressible solution, the generalized Einstein relation agrees with the expression derived by de Groot and Mazur. The two expressions also agree to first order in solute density. For a Brownian suspension, the result derived from the generalized Smoluchowski equation agrees with both expressions.

How good is the generalized Langevin equation to describe the dynamics of photo-induced electron transfer in fluid solution?

The dynamics of unimolecular photo-triggered reactions can be strongly affected by the surrounding medium for which a large number of theoretical descriptions have been used in the past. An accurate description of these reactions requires knowing the potential energy surface and the friction felt by the reactants. Most of these theories start from the Langevin equation to derive the dynamics, but there are few examples comparing it with experiments. Here we explore the applicability of a Generalized Langevin Equation (GLE) with an arbitrary potential and a non-Markovian friction. To this end, we have performed broadband fluorescence measurements with sub-picosecond time resolution of a covalently linked organic electron donor-acceptor system in solvents of changing viscosity and dielectric permittivity. In order to establish the free energy surface (FES) of the reaction, we resort to stationary electronic spectroscopy. On the other hand, the dynamics of a non-reacting substance, Coumarin 153, provide the calibrating tool for the non-Markovian friction over the FES, which is assumed to be solute independent. A simpler and computationally faster approach uses the Generalized Smoluchowski Equation (GSE), which can be derived from the GLE for pure harmonic potentials. Both approaches reproduce the measurements in most of the solvents reasonably well. At long times, some differences arise from the errors inherited from the analysis of the stationary solvatochromism and at short times from the excess excitation energy. However, whenever the dynamics become slow, the GSE shows larger deviations than the GLE, the results of which always agree qualitatively with the measured dynamics, regardless of the solvent viscosity or dielectric properties. The method applied here can be used to predict the dynamics of any other reacting system, given the FES parameters and solvent dynamics are provided. Thus no fitting parameters enter the GLE simulations, within the applicability limits found for the model in this work.

Full-time kinetics of self-assembly and disassembly in micellar solution via the generalized Smoluchowski equation with fusion and fission of surfactant aggregates.

Full-time kinetics of self-assembly and disassembly of spherical micelles with their fusion and fission in non-ionic micellar solutions has been considered in detail on the basis of direct numerical solutions of the generalized Smoluchowski equations describing the evolution of the time-dependent concentrations of molecular aggregates for every aggregation number. The cases of instant increase of the monomer concentration up or dilution of a surfactant solution below the critical micelle concentration at large initial deviations from the final equilibrium state have been studied. Different stages in assembly or disassembly of micelles have been described and compared with the results of the stepwise mechanism of monomer attachment-detachment described by the Becker-Döring kinetic equations. A relation of the full-time kinetics to micellar relaxation at small deviations from the equilibrium state has been checked.

Time-dependent Kramers escape rate in overdamped system with power-law distribution

The probability distribution of Brownian particles moving in an overdamped complex system follows the generalized Smoluchowski equation, which can be rigorously proven that the exact time-dependent solution for this equation follows Tsallis form. Time-dependent escape rate in overdamped system with power-law distributions is then established based on the flux over population theory. The stationary state escape rate in overdamped system with power-law distribution which has been obtained before based on mean first passage time theory is recovered from time-dependent escape rate as time toward infinity.

Transit probability of biomolecule in nonlinear membranal channel

Based on the generalized Smoluchowski equation that can describe the anomalous diffusion systems, the anomalous facilitated diffusion by which the molecules traverse the membrane in specific channels is studied and the expressions of generalized transit probability are obtained. Two factors of well lengths and power-law parameters on the transit probability are considered. It is shown that the nonlinear property of cell membrane leads to the non-Markov process of Brownian motion and thereby produces the subdiffusive effect. Therefore, it will be expected that our generalized expressions of transit probability are more general and more desired results for complex nonequilibrium systems.

Anomalous Spatial Diffusion and Multifractality in Optical Lattices

The transport of cold atoms in shallow optical lattices is characterized by slow, nonstationary momentum relaxation. We develop a projector operator method able to derive, in this case, a generalized Smoluchowski equation for the position variable. We show that this explicitly non-markovian equation can be written as a systematic expansion involving higher-order derivatives. We use the latter to compute arbitrary moments of the spatial distribution and analyze their multifractal properties.

Simulation of Decoherence by Averaged Semiquantum Method

We investigate the dynamics of a system coupled to an environment by averaged semiquantummethod. The theory origins from the time-dependent variational principle (TDVP) formulation and contains nondiagonal matrix elements, thus it can be applied to study dissipation, measurement and decoherence problems in the model. In the calculation, the influence of the environment governed by differential dynamical equation is incorporated using a mean field. We have performed averaged semiquantummethod for a spin-boson model, which reproduces the results from stochastic Schrodinger equation method and Hierarchical approach quite accurately. Moreover, we validate our results with noninteracting-blip approximation (NIBA) and generalized Smoluchowski equation (GSE). The problem dynamics in nonequilibrium environments has also been studied by our method. When applied to the harmonic oscillator model coupled to a heat bath with different coupling strengths and dimensionalities of the bath, we find that the loss of coherence predicted by semiquantum method is identical to the result of master equation with different initial state (Gaussian wave packet and superposed wave packets). PACS: 05.45.Mt, 03.65.Yz, 03.65.Ud

Time-Dependent Fluorescence in Nanoconfined Solvents. A Smoluchowski Equation Model Study

Time-dependent fluorescence (TDF) measurements are frequently used to probe solvation dynamics in nanoconfined solvents. It was previously found [J. Chem. Phys. 2002, 117, 6618] in simple model systems that long time scales can arise in the TDF signal due to diffusion of the solute chromophore. Here, the effects of solute diffusion after electronic excitation on the TDF signal are studied using a generalized Smoluchowski equation. In this approach, the probability density in terms of the solute radial position and a collective solvent coordinate is determined as a function of time after excitation. Model one-dimensional free energy curves with a single barrier in the solute coordinate—appropriate for relatively small, e.g., ∼2 nm diameter, cavities or pores—are used to construct the full two-dimensional free energy surface and probe different physical models. The effects of position-dependence of diffusion coefficients and solvent coordinate force constants are also considered. The results indicate that t...

Ultrafast reaction dynamics in nanoscale water droplets confined by ionic surfactants

The excited state dynamics of the dye molecule auramine O have been measured in nanoscale size water droplets stabilized by ionic surfactants using ultrafast fluorescence up-conversion. Specifically, the reaction dynamics have been measured as a function of the size of the water droplet in the range 1–10 nm and as a function of the counterion charge. The data are analysed quantitatively using a generalized Smoluchowski equation. The rate of the reaction in the reverse micelle is dramatically decreased compared to the bulk water environment. It increases as nanodroplet size increases, but never attains the bulk value. The data are not well described by a ‘core-shell’ model, i.e. by assuming a slow reaction in the interface (shell) and a bulk-like decay in the core. The effect of changing the counterion is small, with three different singly charged counterions showing essentially identical reaction dynamics, while the double charged Ca2+ counterion only slightly slows the reaction. This insensitivity to counterion is in contrast to the observations of solvation dynamics experiments and molecular dynamics simulation. The origins of these differences are discussed in terms of the solute reaction coordinate being more sensitive to different motions of the confined aqueous solvent than solvation dynamics and to more delocalised properties of the dispersed phase than are detected in simulations.

Fluctuation theory of single-file diffusion

In a one-dimensional suspension of Brownian particles, which cannot pass each other, the mean square displacement of a selected particle grows at long times with the square root of time, rather than linearly. It is shown that the coefficient of the square root, the so-called single-file mobility, can be derived from fluctuation theory, involving the velocity time scale and the fluctuation-dissipation theorem. The single-file mobility is expressed in terms of the collective diffusion coefficient and the isothermal osmotic compressibility, in agreement with the result derived earlier by Kollmann on the basis of the generalized Smoluchowski equation [M. Kollmann, Phys. Rev. Lett. 90, 180602 (2003)].

Reactions in the liquid phase

It is argued that a generalized Smoluchowski equation should be used to describe reactions in the liquid phase. The nature of the solutions to this equation is discussed. The reaction energy surface is partitioned into regions, defining the various species present, and solutions for each region are described. A set of master equations is used to calculate the rate constant. Application is made to a model situation where two structureless molecules react.

Reactive Dynamics in Confined Liquids: Interfacial Charge Effects on Ultrafast Torsional Dynamics in Water Nanodroplets

The excited-state dynamics of a reactive dye molecule, auramine O, have been studied in nanoscale water droplets stabilized by a nonionic surfactant. Spectral dynamics were measured as a function of the radius of the water nanodroplet with 50 fs time resolution using time-resolved fluorescence up-conversion method. Qualitatively, the effect of confinement is to dramatically slow the rate of the reaction compared to that of bulk water. Data were quantitatively analyzed using the one-dimensional generalized Smoluchowski equation assuming a time-dependent diffusion coefficient. The results were contrasted with our earlier analysis of auramine O in aqueous nanodroplets stabilized by the ionic surfactant AOT. The excited-state reaction is slower in the nonionic surfactant, showing that interfacial charge is not required to suppress reactions in nanoscale water droplets. The location of the dye in the heterogeneous micelle is investigated by comparing the absorption spectra of AO in the micelle with those of a water- polyethyleneglycol mixture (to mimic the surfactant head group). The results suggest that the charged dye is located in the water phase.

Reactive Dynamics in Confined Liquids: Ultrafast Torsional Dynamics of Auramine O in Nanoconfined Water in Aerosol OT Reverse Micelles

The effects of confinement on the ultrafast torsional reaction of auramine O in aqueous solution are investigated through ultrafast fluorescence up-conversion with 50 fs time resolution. The aqueous solution is confined in nanoscale water droplets by an ionic surfactant. The torsional motion is orders of magnitude slower in the confined droplets than in bulk aqueous solution. The dynamics become faster with increasing radius of the nanodroplet but never reach the bulk value, even when the radius is as large as 10 nm. Time-dependent fluorescence spectra were constructed and subsequently analyzed using a one-dimensional generalized Smoluchowski equation. An accurate description of the data was achieved using a time-dependent diffusion coefficient. This is suggested to arise because the medium friction reflects dynamics on a broad range of time scales spanning the reaction dynamics. The friction recovered suggests strongly hindered motion in the confined droplet and can be qualitatively related to solvation dynamics measured in AOT, consistent with auramine O torsional dynamics being accompanied by intramolecular charge redistribution.

Nonlinear mean field Fokker-Planck equations. Application to the chemotaxis of biological populations

We study a general class of nonlinear mean field Fokker-Planck equations in relation with an effective generalized thermodynamical (E.G.T.) formalism. We show that these equations describe several physical systems such as: chemotaxis of bacterial populations, Bose-Einstein condensation in the canonical ensemble, porous media, generalized Cahn-Hilliard equations, Kuramoto model, BMF model, Burgers equation, Smoluchowski-Poisson system for self-gravitating Brownian particles, Debye-Hückel theory of electrolytes, two-dimensional turbulence... In particular, we show that nonlinear mean field Fokker-Planck equations can provide generalized Keller-Segel models for the chemotaxis of biological populations. As an example, we introduce a new model of chemotaxis incorporating both effects of anomalous diffusion and exclusion principle (volume filling). Therefore, the notion of generalized thermodynamics can have applications for concrete physical systems. We also consider nonlinear mean field Fokker-Planck equations in phase space and show the passage from the generalized Kramers equation to the generalized Smoluchowski equation in a strong friction limit. Our formalism is simple and illustrated by several explicit examples corresponding to Boltzmann, Tsallis, Fermi-Dirac and Bose-Einstein entropies among others.

Generalized Smoluchowski equation with correlation between clusters

In this paper we compute new reaction rates of the Smoluchowski equation which takes into account correlations. The new rate K = KMF + KC is the sum of two terms. The first term is the known Smoluchowski rate with the mean-field approximation. The second takes into account a correlation between clusters. For this purpose we introduce the average path of a cluster. We relate the length of this path to the reaction rate of the Smoluchowski equation. We solve the implicit dependence between the average path and the density of clusters. We show that this correlation length is the same for all clusters. Our result depends strongly on the spatial dimension d. The mean-field term KMFi,j = (Di + Dj)(rj + ri)d−2, which vanishes for d = 1 and is valid up to logarithmic correction for d = 2, is the usual rate found with the Smoluchowski model without correlation (where ri is the radius and Di is the diffusion constant of the cluster). We compute a new rate: the correlation rate is valid for d ⩾ 1(where M(α) = ∑+∞i=1iαNi is the moment of the density of clusters and df is the fractal dimension of the cluster). The result is valid for a large class of diffusion processes and mass–radius relations. This approach confirms some analytical solutions in d = 1 found with other methods. We also show Monte Carlo simulations which illustrate some exact new solvable models.

Logotropic distributions

In all spatial dimensions d, we study the static and dynamical properties of a generalized Smoluchowski equation which describes the evolution of a gas obeying a logotropic equation of state, p = A ln ρ . A logotrope can be viewed as a limiting form of polytrope ( p = K ρ γ , γ = 1 + 1 / n ), with index γ = 0 or n = - 1 . In the language of generalized thermodynamics, it corresponds to a Tsallis distribution with index q = 0 . We solve the dynamical logotropic Smoluchowski equation in the presence of a fixed external force deriving from a quadratic potential, and for a gas of particles subjected to their mutual gravitational force. In the latter case, the collapse dynamics is studied for any negative index n, and the density scaling function is found to decay as r - α , with α = 2 n / ( n - 1 ) for n < - d / 2 , and α = 2 d / ( d + 2 ) for - d / 2 ⩽ n < 0 .

Dynamics of the colloidal suspensions

This article offers a survey on our current knowledge of the dynamics of the colloidal suspension, where each particle experiences the friction force with solvent, hydrodynamic interaction, and potential force from surrounding particles and thermodynamic force. It further contains a summary of the basic concepts about microstructures and equilibrium properties, and of analytical and numerical methods, which are relevant for the theoretical description of the suspensions. The description of the dynamics of colloidal particles, based on the generalized Smoluchowski equation, is justified for the time scale accessible in DLS experiments. The combined influence of hard sphere or electrostatic potential and solvent-mediated hydrodynamic interaction on the short-time dynamics of monodisperse suspensions is investigated in detail. A thorough study of tracer-diffusion in hard sphere and charge-stabilized suspensions is presented. Mean-square displacements and long-time tracer-diffusion coefficients are calculated with two alternative approximations, i.e., a mode-coupling scheme and a single relaxation time ansatz.

Chapman–Enskog derivation of the generalized Smoluchowski equation

We use the Chapman–Enskog method to derive the Smoluchowski equation from the Kramers equation in a high friction limit. We consider two main extensions of this problem: we take into account a uniform rotation of the background medium and we consider a generalized class of Kramers equations associated with generalized free energy functionals. We mention applications of these results to systems with long-range interactions (self-gravitating systems, 2D vortices, bacterial populations, etc.). In that case, the Smoluchowski equation is non-local. In the limit of short-range interactions, it reduces to a generalized form of the Cahn–Hilliard equation. These equations are associated with an effective generalized thermodynamical formalism.