Langevin equation derivation

Kramers solution of the barrier crossing problem [45] is discussed at length in chapter A3.8 dealing with condensed-phase reaction dynamics. As the starting point to derive its simplest version one may use the Langevin equation, a stochastic differential equation for the time evolution of a slow variable, the reaction coordinate r, subject to a rapidly statistically fluctuating force F caused by microscopic solute-solvent interactions under the influence of an external force field generated by the PES F for the reaction... 

We recently received a preprint from Dellago et al. [9] that proposed an algorithm for path sampling, which is based on the Langevin equation (and is therefore in the spirit of approach (A) [8]). They further derive formulas to compute rate constants that are based on correlation functions. Their method of computing rate constants is an alternative approach to the formula for the state conditional probability derived in the present manuscript. 

This relation is as general as the Langevin equation itself, i.e., it holds for collisions of any strength. When deriving Eq. (1.23) from Eq. (1.26),... 

This conclusion can be confirmed by an alternative derivation of Eq. (1.71). According to Mori [52], Eq. (1.71) may be obtained from a generalized Langevin equation ... 

The friction coefficient is one of the essential elements in the Langevin description of Brownian motion. The derivation of the Langevin equation from the microscopic equations of motion provides a Green-Kubo expression for this transport coefficient. Its computation entails a number of subtle features. Consider a Brownian (B) particle with mass M in a bath of N solvent molecules with mass m. The generalized Langevin equation for the momentum P of the B... 

Analogous to the derivation of the effective Langevin equation for the chain in dilute solutions, we get in semidilute solutions... 

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... 

Note that the term involving a derivative of In / in Eq. (2.331) is identical to the velocity arising from the second term on the RHS of Eq. (2.286) for the transformed force bias in the traditional interpretation of the Langevin equation. The traditional interpretation of the Langevin equation yields a simple tensor transformation rule for the drift coefficient A , but also yields a contribution to Eq. (2.282) for the drift velocity that is driven by the force bias. The kinetic interpretation yields an expression for the drift velocity from which the term involving the force bias is absent, but, correspondingly, yields a nontrivial transformation mle for the overall drift coefficient. 

Ignoring direct interactions, neighbouring steps do not influence each other if the dynamics is dominated by evaporation-recondensation or by step-edge diffusion. In either of these cases, the single step results derived in Section 2 (i.e. Eq. (22) and (26)) then hold. However, if the dynamics is mediated by terrace diffusion, neighbouring steps influence each other through the diffusion field on the terraces, and a coupled set of Langevin equations must be solved, as shown below (see also [13-17]). 

In this paper we present a brief discussion and comparison of the probabilistic and dynamic approaches to the treatment of nonequilibrium phenomena in physical systems. The discussion is not intended to be complete but only illustrative. Details of many of the derivations appear elsewhere and only the results will be discussed here. We shall focus our attention on the probabilistic approach and shall emphasize its advantages and drawbacks. The main body of the paper deals with the properties of the master equation and, more cursorily, with the properties of the Langevin equation. 

In this section, we discuss briefly how the Langevin equation, which is a stochastic equation, can be derived from the molecular equations of motion. The stochastic model described by the Langevin equation has been of great use in interpreting a large number of experiments and physical systems. The stochastic model is extremely simple but, as always, its ultimate justification rests on the molecular dynamical laws. 

The plan of the article is as follows. First, we discuss the phenomenon of hydrodynamic interaction in general terms, and at the same time, we present some convenient notation. Then, we give the usual argument leading to the Fokker-Planck equation. After that we derive the Langevin equation that is formally equivalent to the Fokker-Planck equation, together with a statistical description of the fluctuating force. 

The traditional derivation of the Fokker-Planck equation (1.5) or (VIII. 1.1) is based on Kolmogorov s mathematical proof, which assumes infinitely many infinitely small jumps. In nature, however, all jumps are of some finite size. Consequently W is never a differential operator, but always of the type (V.1.1). Usually it also has a suitable expansion parameter and has the canonical form (X.2.3). If it then happens that (1.1) holds, the expansion leads to the nonlinear Fokker-Planck equation (1.5) as the lowest approximation. There is no justification for attributing a more fundamental meaning to Fokker-Planck and Langevin equations than in this approximate sense. 

A modified Langevin equation can be derived for any property 0t. In addition the memory function will be related to the autocorrelation function of the random force in this equation. These results can be extended to multivariate processes. 

The relaxation equations for the time correlation functions are derived formally by using the projection operator technique [12]. This relaxation equation has the same structure as a generalized Langevin equation. The mode coupling theory provides microscopic, albeit approximate, expressions for the wavevector- and frequency-dependent memory functions. One important aspect of the mode coupling theory is the intimate relation between the static microscopic structure of the liquid and the transport properties. In fact, even now, realistic calculations using MCT is often not possible because of the nonavailability of the static pair correlation functions for complex inter-molecular potential. 

In the usual derivations of the Klein-Kramers equation, the moments of the velocity increments, Eq. (68), are taken as expansion coefficients in the Chapman-Kolmogorov equation [9]. Generalizations of this procedure start off with the assumption of a memory integral in the Langevin equation to finally produce a Fokker-Planck equation with time-dependent coefficients [67]. We are now going to describe an alternative approach based on the Langevin equation (67) which leads to a fractional IGein-Kramers equation— that is, a temporally nonlocal behavior. 

In this equation g(t) represents the retarded effect of the frictional force, and /(f) is an external force including the random force from the solvent molecules. We see, in contrast to the simple Langevin equation with a constant friction coefficient, that the friction force at a given time t depends on all previous velocities along the trajectory. The friction force is no longer local in time and does not depend on the current velocity alone. The time-dependent friction coefficient is therefore also referred to as a memory kernel . A short-time expansion of the velocity correlation function based on the GLE gives (fcfiT/M)( 1 — (g/M)t2/(2r) + ), where r is the decay time of g(t), and it therefore does not have a discontinuous first derivative at t = 0. The discussion of the properties of the GLE is most easily accomplished by using so-called linear response theory, which forms the theoretical basis for the equation and is a powerful method that allows us to determine non-equilibrium transport coefficients from equilibrium properties of the systems. A discussion of this is, however, beyond the scope of this book. 

In view of the fact that the correlation function for the random force, as given by Eq. [16], is a Dirac 8 function, the strict Langevin equation (Eq. [15]) is not amenable to computer simulation. In order to circumvent the above difficulty, it is convenient to describe the motion of the fictitious particle by the generalized Langevin equation. The generalized Langevin equation, which can be derived from the Liouville equation (11), along with the supplementary conditions are... 

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. 

Let us now come back to the specific problem of the diffusion of a particle in an out-of-equilibrium environment. In a quasi-stationary regime, the particle velocity obeys the generalized Langevin equation (22). The generalized susceptibilities of interest are the particle mobility p(co) = Xvxi03) and the generalized friction coefficient y(co) = — (l/mm)x ( ) [the latter formula deriving from the relation (170) between y(f) and Xj> (f))- The results of linear response theory as applied to the particle velocity, namely the Kubo formula (156) and the Einstein relation (159), are not valid out-of-equilibrium. The same... 

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... 

In the absence of the external potential V, Eqs. (52) can be given a rigorous derivation from a microscopic Liouville equation (see Chapter I). We make the naive assumption that when an external potential driving the reaction coordinate is present, the two contributions (the deterministic motion resulting from the external potential and the fluctuation-dissipation process described by the standard generalized Langevin equation) can simply be added to each other. 

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