The Modified Langevin Equation

The second approach starts from the modified Langevin equation Eq. (37) and uses the equivalence of the Kramers theory to the multi-dimensional TST. It has been established by numerical comparison that there is agreement between the two approaches. 

Kubo calls this relationship the Second Fluctuation Dissipation Theorem. For its proof it should be noted that the modified Langevin equation can be written as... 

The phenomenological Langevin Eqs. (227) and (228) are only applicable to a very restricted class of physical processes. In particular, they are only valid when the stochastic forces and torques have infinitely short correlation times, i.e., their autocorrelation functions are proportional to Dirac delta functions. As was shown in the previous section, these restrictions can be removed by a suitable generalization of these Langevin equations. As we saw in the particular case of the velocity, the modified Langevin equation is... 

If an N-unit chain is immersed in a viscous medium, then the movement of any monomer i can be described by the modified Langevin equation ... 

Fig. 265. Cu2 l-4-HO CeH., CHjCH(NHj)COO 4. Temperature dependence of Xm sttd VXm- Curves fitted according to the modified Langevin equation...

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. 

Interestingly, due to the linearity of the generalized Langevin equation (22), the same effective temperature T,eff(( ) can consistently be used in the modified Nyquist formula linking the noise spectral density C/ /- ([Pg.313]

Onsager model with generalized mechanical behaviour. It is clear from the last four equations that if the true y obeys a modified Langevin equation with memory function 2D(t), the essential transform... 

If we make the less restrictive assumption that it is a linear function of the preceding velodties we obtain a modified Langevin equation of the form... 

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... 

As before, (Av) and ((Av)2) may be determined from the Langevin equation. It differs slightly from the Langevin equation in Eq. (11.5), since the motion takes place in an external potential U(r), as sketched in Fig. 11.0.2. This is the potential from the gas phase modified by the interactions from the solvent molecules. 

Let x t) be the temporal variable of the excitable system which is also subject to external noise. A corresponding differential equation for the time evolution of x t) is called a Langevin-equation and includes random parts. We specify to the situation where randonmess is added linearly to modify the time derivative of x t), i.e. 

Fig. 10 KMC prediction of the bimolecular recombination rate constant symbols) for bulk heterojimctions with 4 nm (black) and 35 nm (redIgrey) domains. Squares correspond to no energetic disorder, while triangles assume Gaus sian-distributed energetic disorder with a = 15 meV. The solid line shows the prediction of the Langevin equation. Modified from [12]. American Physical Society...

The Langevin equation must be modified in order to take into account the hydrodynamic... 

These modified equations of motion are solved numerically using a Verlet type of algorithm which includes the effect due to the presence of Langevin forces99 102 (see Chapt. IV.G). 

Buckingham and Pople (1955) have modified the Langevin-Born equations for electric birefringence to allow for a field-dependent polarizability such as (38) ... 

Wolf and Villain (WV) [8], Villain, Lai, and Das Sanna (VLD) [9] and others [10]. The rate equations for the film growth resulting from these models are modified versions of the Langevin stochastic equation [10] for the variation with time of the interface height. 

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