Langevin equation generalized form

In either interpretation of the Langevin equation, the form of the required pseudoforce depends on the values of the mixed components of Zpy, and thus on the statistical properties of the hard components of the random forces. The definition of a pseudoforce given here is a generalization of the metric force found by both Fixman [9] and Hinch [10]. An apparent discrepancy between the results of Fixman, who considered the case of unprojected random forces, and those of Hinch, who was able to reproducd Fixman s expression for the pseudoforce only in the case of projected random forces, is traced here to an error in Fixman s use of differential geometry. 

The dynamics of an isolated Kuhn segment chain in its bead-and-spring form is considered in a viscous medium without hydrodynamic backflow or excluded-volume effects. The treatment is based on the Langevin equation generalized for Brownian particles with internal degrees of freedom. A first, crude formalism of this sort was reported by Kargin and Slonimskii [45]. In-... 

In order to proceed now to a statistical mechanical description of the corresponding relaxation process, it is convenient to solve the equation of motion for the creation and destruction operators and cast them in a form ressembling a Generalized Langevin equation. We will only sketch the procedure. 

The present analysis builds directly on three previous analyses of SDEs for constrained systems by Fixman [9], Hinch [10], and Ottinger [11]. Fixman and Hinch both considered an interpretation of the inertialess Langevin equation as a limit of an ordinary differential equation with a finite, continuous random force. Both authors found that, to obtain the correct drift velocity and equilibrium distribution, it was necessary to supplement forces arising from derivatives of C/eff = U — kT n by an additional corrective pseudoforce, but obtained inconsistent results for the form of the required correction force. Ottinger [11] based his analysis on an Ito interpretation of SDEs for both generalized and Cartesian coordinates, and thereby obtained results that... 

In both traditional and kinetic interpretations of the Cartesian Langevin equation for a constrained system, one retains some freedom to specify the hard and mixed components of the force variance tensor Several forms for Z v have been considered in previous work, corresponding to different types of random force, which generally require the use of different corrective pseudo forces ... 

Thus we have found the general form of the quantum master equation that corresponds to the Schrodinger-Langevin equation (5.3). 

The particle trajectories can be simulated using a random force in the generalized Langevin equation that is constant during a small time step ts with values given by a Gaussian distribution. The memory function for this form of random force is (12)... 

In Section II, motivated by the fact that in typical experiments an aging system is not isolated, but coupled to an environment which acts as a source of dissipation, we recall the general features of the widely used Caldeira-Leggett model of dissipative classical or quantum systems. In this description, the system of interest is coupled linearly to an environment constituted by an infinite ensemble of harmonic oscillators in thermal equilibrium. The resulting equation of motion of the system can be derived exactly. It can be given, under suitable conditions, the form of a generalized classical or quantal Langevin equation. 

One considers a particle interacting linearly with an environment constituted by an infinite number of independent harmonic oscillators in thermal equilibrium. The particle equation of motion, which can be derived exactly, takes the form of a generalized Langevin equation, in which the memory kernel and the correlation function of the random force are assigned well-defined microscopic expressions in terms of the bath operators. 

Except for the term —mx(ti)y(t — f,), which we will discuss later, Eq. (6) is similar to a generalized Langevin equation, in which y(t) acts as a memory kernel and F(t) acts as a random force. By using, instead of y(t), the retarded memory kernel2 y(t) = 0(f)y(t), the upper integration bound of the integral in the left-hand side of Eq. (6) can be set equal to +oo. This latter equation can then be rewritten in the following equivalent form ... 

As will be seen below, the kernel y(f) decreases on a characteristic time of the order of mf. 1, the angular frequency ooc characterizing the bandwidth of the bath oscillators effectively coupled to the particle. The quantity y(t — t,) is thus negligible if (i)c(t - U) 1. Mathematically, this condition can be realized at any time t by referring the initial time f, to -oo. Then, the initial particle position becomes irrelevant in Eq. (10), which takes the form of the generalized Langevin equation, namely,... 

Here I is the unit matrix and is the frequency square matrix in the space of nonreactive modes Equation (6.1) is a generalized Langevin equation of the form used in treating the one-dimensional case in Section V, and leads to the result of Eq. (5.25) (with m = 1) for the steady-state probability distribution of the reactive mode near the barrier. In the present multidimensional treatment it is convenient to redefine the distribution according to... 

When the system dynamics depends on what occurred earlier—that is, the environment has memory—Eq. (141) is no longer adequate and the Langevin equation must be modified. The generalized Langevin equation takes this memory into account through an integral term of the form... 

However, neither of these models is adequate for describing multiffactal statistical processes as they stand. A number of investigators have recently developed multifractal random walk models to account for the multiffactal character of various physiological phenomena, and here we introduce a variant of those discussions based on the fractional calculus. The most recent generalization of the Langevin equation incorporates memory into the system s dynamics and has the simple form of Eq. (131) with the dissipation parameter set to zero ... 

The pseudo-Liouville operator does couple these doublet fields to triplet fields such as 8 abs cds involving the solvent molecules. Thus one of the simplest forms for the pair kinetic equation can be obtained by explicitly including doublet and triplet fields in the generalized Langevin equation. This procedure yields a treatment of the effects of solvent dynamics on the motion of the reactive pair that is much more sophisticated than that given in the singlet kinetic equation discussed in the preceding... 

Some features of late-stage interface dynamics are understood for model B and also for model A. We now proceed to discuss essential aspects of this interface d5mamics. Consider the Langevin equations without noise. Equation (A3.3.571 can be written in a more general form ... 

This equation indeed gives good account of the observed N/pY >-dependence of xr by applying the r T) from simulations. They attempted to justify this equation by taking an equation from Schweizer s (1989) formally exact, nonlinear generalized Langevin equation (GLE) for a flexible probe polymer in a dense entangled polymer melt (see Eq. (3.50) in Schweizer). They rewrote this same equation in the form... 

There are two forms of phenomenological equations for describing Brownian motion the Smoluchowski equation and the Langevin equation. These two equations, essentially the same, look very different in form. The Smoluchowski equation is derived from the generalization of the diffusion equation and has a clear relation to the thermodynamics of irreversible processes. In Chapters 6 and 7, its application to the elastic dumbbell model and the Rouse model to obtain the rheological constitutive equations will be discussed. In contrast, the Langevin equation, while having no direct relation to thermodynamics, can be applied to wider classes of stochastic processes. In this chapter, it will be used to obtain the time-correlation function of the end-to-end vector of a Rouse chain. 

The Langevin equation of a normal mode has the general form ... 

In this section the generalized Langevin equation (GLE) for density correlation functions for molecular liquids is derived based on the memory-function formalism and on the interaction-site representation. In contrast to the monatomic liquid case, all functions appearing in the GLE for polyatomic fluids take matrix forms. Approximation schemes are developed for the memory kernel by extending the successful frameworks for simple liquids described in Sec. 5.1. 

In this chapter we have described a theory for dynamics of polyatomic fluids based on the memory-function formalism and on the interaction-site representation of molecular liquids. Approximation schemes for memory functions appearing in the generalized Langevin equation have been developed by assuming an exponential form for memory functions and by employing the mode-coupling approach. Numerical results were presented for longitudinal current spectra of a model diatomic liquid and water, and it has been discussed how the results can be interpreted in... 

It should be emphasized that the essence of the Rouse model is in the universal nature of the modelling of the dynamics of a connected object. The central assumption in the Rouse model is that the dynamics is governed by the interactions localized along the diain. In fact, if one assumes a linear Langevin equation for R with localized interaction, one ends up with the Rouse model in the long time-scale behaviour. To see this, consider the general form of the linearized Langevin equation... 

We have now eliminated as many variables from consideration as is possible with the use of general agreements it is time to start writing the nonlinear Langevin equation for the remaining variables. To obtain the equations, we must form the product (/n +L°)0 ) The only quantity in this product that is not at least formally available is the inversion of an infinite rank matrix can be a formidable task. Thus, we first attempt to find out if X might be inverted in some simpler manner. 

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