Langevin-diffusion equation

The density functional theory for classical(equilibrium) statistical mechanics is generalized to deal with various dynamical processes associated with density fluctuations in liquids and solutions. This is effected by deriving a Langevin-diffusion equation for the density field. As applications of our theory we consider density fluctuations in both supercooled liquids and molecular liquids and transport coefficients. 

Constraints may be introduced either into the classical mechanical equations of motion (i.e., Newton s or Hamilton s equations, or the corresponding inertial Langevin equations), which attempt to resolve the ballistic motion observed over short time scales, or into a theory of Brownian motion, which describes only the diffusive motion observed over longer time scales. We focus here on the latter case, in which constraints are introduced directly into the theory of Brownian motion, as described by either a diffusion equation or an inertialess stochastic differential equation. Although the analysis given here is phrased in quite general terms, it is motivated primarily by the use of constrained mechanical models to describe the dynamics of polymers in solution, for which the slowest internal motions are accurately described by a purely diffusive dynamical model. 

D only assumes a constant value over times long compared with rc( 10rc), such that (u(O)u(f)), the velocity autocorrelation function, is nearly zero. The diffusion equation is not valid over times < 10rc(i.e. a few picoseconds at least). A better approach would be to use a generalised Langevin equation with a friction coefficient which has a memory and such that the velocity autocorrelation takes 0.5ps to decay to insignificant levels (see Chap. 11) [453]. 

By contrast, when both the reactive solute molecules are of a size similar to or smaller than the solvent molecules, reaction cannot be described satisfactorily by Langevin, Fokker—Planck or diffusion equation analysis. Recently, theories of chemical reaction in solution have been developed by several groups. Those of Kapral and co-workers [37, 285, 286] use the kinetic theory of liquids to treat solute and solvent molecules as hard spheres, but on an equal basis (see Chap. 12). While this approach in its simplest approximation leads to an identical result to that of Smoluchowski, it is relatively straightforward to include more details of molecular motion. Furthermore, re-encounter events can be discussed very much more satisfactorily because the motion of both reactants and also the surrounding solvent is followed. An unreactive collision between reactant molecules necessarily leads to a correlation in the motion of both reactants. Even after collision with solvent molecules, some correlation of motion between reactants remains. Subsequent encounters between reactants are more or less probable than predicted by a random walk model (loss of correlation on each jump) and so reaction rates may be expected to depart from those predicted by the Smoluchowski analysis. Furthermore, such analysis based on the kinetic theory of liquids leads to both an easy incorporation of competitive effects (see Sect. 2.3 and Chap. 9, Sect. 5) and back reaction (see Sect. 3.3). Cukier et al. have found that to include hydrodynamic repulsion in a kinetic theory analysis is a much more difficult task [454]. 

Having discussed some aspects of motion of molecules in liquids, it is now more interesting to return to the classical hydrodynamic description of the motion of the molecules of a liquid. In this section, a brief account of the assumption inherent in the Langevin equation are presented and the assumptions required to reduce this to the diffusion equation are discussed. 

In a stochastic approach, one replaces the difficult mechanical equations by stochastic equations, such as a diffusion equation, Langevin equation, master equation, or Fokker-Planck equations.5 These stochastic equations have fewer variables and are generally much easier to solve than the mechanical equations, One then hopes that the stochastic equations include the significant aspects of the physical equations of motion, so that their solutions will display the relevant features of the physical motion. 

Local motions which occur in macromolecular systems can be probed from the diffusion process of small molecules in concentrated polymeric solutions. The translational diffusion is detected from NMR over a time scale which may vary from about 1 to 100 ms. Such a time interval corresponds to a very large number of elementary collisions and a long random path consequently, details about mechanisms of molecular jump are not disclosed from this NMR approach. However, the dynamical behaviour of small solvent molecules, immersed in a polymer melt and observed over a long time interval, permits the determination of characteristic parameters of the diffusion process. Applying the Langevin s equation, the self-diffusion coefficient Ds is defined as... 

Langevin form, and through a diffusion equation, for not too small an auto-correlation function which is in turn of simple form only for... 

In Section 2 the DFT is briefly reviewed and in Section 3 a Langevin-diffusion(L-D) equation for the density field n r,t) is presented. In Section 4 we consider, as applications of the TD-DFT, (A) density fluctuations in liquids and solutions and (B)mass flow around a fixed petrticle to calculate transport coefficients of liquids. Section 5 contains some remarks and summary of this paper. 

We now present a Langevin-diffusion(L-D) equation for the density field n(r,t) and the corresponding... 

Next, Fig. 5 shows a typical center of mass trajectory for the case of a full chaotic internal phase space. The eyecatching new feature is that the motion is no more restricted to some bounded volume of phase space. The trajectory of the CM motion of the hydrogen atom in the plane perpendicular to the magnetic field now closely resembles the random motion of a Brownian particle. In fact, the underlying equation of motion at Eq. (35) for the CM motion is a Langevin-type equation without friction. The corresponding stochastic Lan-gevin force is replaced by our intrinsic chaotic force — e B x r). A main characteristic of random Brownian motion is the diffusion law, i.e. the linear dependence of the travelled mean-square distance on time. We have plotted in Fig. 6 for our case of a chaotic force for 500 CM trajectories the mean-square distance as a function of time. Within statistical accuracy the plot shows a linear dependence. The mean square distance

of the CM after time t, therefore, obeys the diffusion equation... 

Finally, the fractional calculus was used to construct fractional diffusion equations. One such equation, in particular, models the evolution of the Levy nestable probability density describing Levy diffusion, another mechanism for generating anomalous diffusion. It was shown that this probability density satisfies the scaling relation [Eq. (35)] with the Levy index a such that 8 = 1 /a. The dynamics of a Levy diffusion process, using a Langevin equation, were also considered. The probability density for a simple dissipative process being driven by Levy noise is also Levy but with a change in parameters. This is a possible alternate model of the fluctuations in the interbeat intervals for the human heart shown to be Levy stable over a decade ago [25],... 

This form is useful in the generalization of Eq. (485) to fractional diffusion. The investigation of the diffusion equation (485) began when Louis Bachellier (Jules Poincare s student) wrote his thesis in 1900. It was called The Theory of Speculations and was devoted to the evolution of the stock market. Many of the most famous scientists have contributed to our knowledge of diffusion processes, amongst them Einstein, Langevin, Smoluchowski, Fokker, Planck, Levy, and others. 

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