The Ordinary Langevin Equation

The statistical properties of the random force f(0 are modeled with an extreme economy of assumptions f(t) is assumed to be a stationary and Gaussian stochastic process, with zero mean (f(0 = 0), uncorrelated with the initial value v(t = 0) of the velocity fluctuations, and delta-correlated with itself, f(0f(t ) = f25(t -1 ) (i.e it is a purely random, or white, noise). The stationarity condition is in reality equivalent to the fluctuation-dissipation relation between the random and the dissipative forces in Equation 1.1, which essentially fixes the value of y. In fact, from Equation 1.1 and the assumed properties of f(t), we can derive the expression y(f)v(t) = exp [v(0)v(0) -+ ylM °, where Xg = In equilibrium, the long-time asymptotic value y/M must coincide with the equilibrium average (vv) = (k TIM)t given by the equipartition theorem (with I being the 3 X 3 Cartesian unit tensor), and this fixes the value of y to y= 

This set of assumptions on the statistical properties of f(t) determines the statistical properties of the solution v(0 of the stochastic differaitial equation in Equation 1.1, which are summarized saying that v(0 is a Gaussian stationary Markov stochastic process, that is, it is generally not delta-correlated. The specific results that follow from this simple mathanatical model regarding propo ties such as the velocity autocorrelation function, msd, and so on, are reviewed in standard statistical physics textbooks [48]. 

We will restrict our discussion to stationary states, that is, to the stationary solutions of the relaxation equation above, denoted by a , which solve the equation 

One then postulates that the fluctuations 8a(t) = a(i) - a will satisfy a linearized stochastic Langevin-type version of Equation 1.2, namely. 

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We now examine the response of a linear dissipative system to Levy fluctuations using the ordinary Langevin equation,... 

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 attempting to solve the nonlinear Langevin equation [Eq. (49)], one is immediately struck that a solution is not obtainable by ordinary mathematical techniques. Since the nonlinear variables A k+k-A-k are not known as a function of Ak (AA AA) Eq. (49) is simply not a closed equation for the variable of interest, A To circumvent this difficulty, it is necessary to start with the fluctuating form of the nonlinear Langevin equation. The fluctuating form of Eq. (49) is... 

In the traditional interpretation of the Fangevin equation for a constrained system, the overall drift velocity is insensitive to the presence or absence of hard components of the random forces, since these components are instantaneously canceled in the underlying ODF by constraint forces. This insensitivity to the presence of hard forces is obtained, however, only if both the projected divergence of the mobility and the force bias are retained in the expression for the drift velocity. The drift velocity for a kinetic interpretation of a constrained Langevin equation does not contain a force bias, and does depend on statistical properties of the hard random force components. Both Fixman and Hinch nominally considered the traditional interpretation of the Langevin equation for the Cartesian bead coordinates as a limit of an ordinary differential equation. Both authors, however, neglected the possible existence of a bias in the Cartesian random forces. As a result, both obtained a drift velocity that (after correcting the error in Fixman s expression for the pseudoforce) is actually the appropriate expression for a kinetic interpretation. 

On the other hand, the adoption of the Markov approximation, although yielding no mathematical inconsistencies, might correspond to annihilating important physical effects. Let us illustrate this important fact with a process related to the Anderson localization issue. Let us depict the Anderson localization as an ordinary fluctuation-dissipation process. Let us consider the Langevin equation... 

To discuss the idea of noise-activated reactions we begin by noting that the random forces which occur in the Langevin equation related with the process under investigation may have quite different origins. In an ordinary microscopic derivation of a Langevin equation (or the corresponding Fokker-Planck equation), the random term is interpreted as associated with... 

The dynamics simulation is limited to the atoms in the reaction zone. Atoms in the reaction region are treated by ordinary molecular dynamics and their motions are governed by Newton s equations of motion. Atoms in the buffer region, as indicated above, obey a Langevin equation of motion. Thus we have a set of simultaneous equations... 

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