Langevin forces

Since the stochastic Langevin force mimics collisions among solvent molecules and the biomolecule (the solute), the characteristic vibrational frequencies of a molecule in vacuum are dampened. In particular, the low-frequency vibrational modes are overdamped, and various correlation functions are smoothed (see Case [35] for a review and further references). The magnitude of such disturbances with respect to Newtonian behavior depends on 7, as can be seen from Fig. 8 showing computed spectral densities of the protein BPTI for three 7 values. Overall, this effect can certainly alter the dynamics of a system, and it remains to study these consequences in connection with biomolecular dynamics. 

A completely unambiguous formulation of the Langevin equation may, however, be constructed by relating the random Langevin forces to random functions /m(f) of the type used in our discussion of the Stratonovich SDE, by taking... 

In order to find the thermal fluctuations due to heat motion in R, one replaces Q by a fluctuating quantity q and adds a rapidly fluctuating Langevin force K(t),... 

It is no longer possible to describe fluctuations by the simple device of adding a Langevin force... 

Exercise. A rod-like molecule rotates freely in a plane, but is subjected to a Langevin force due to the surroundings ... 

E is an electric field and V the interaction potential. When Ll,L2 are independent Langevin forces, write the corresponding Fokker-Planck equation. Find the equilibrium solution and determine and r2. ... 

Add a Langevin force with the properties (i)-(iv) stipulated above. 

We begin with an innocuous case. Consider a pendulum suspended in air and consequently subject to damping accompanied by a Langevin force. This force is, of course, the same as the one in equation (1.1) for the Brownian particle, because the collisions of the air molecules are the same. They depend on the instantaneous value of V, but they are insensitive to the fact that there is a mechanical force acting on the particle as well. Hence for small amplitudes the motion is governed by the linear equation (1.10). For larger amplitudes the equation becomes nonlinear ... 

Exercise. The damping of a Brownian particle is described in terms of the energy by E = -2yE. Construct a Langevin force which gives the correct fluctuations in the Stratonovich interpretation. Find another Langevin force that is correct in the Ito interpretation. 

We consider a process A(t) which shares with the Langevin force L(t) the properties (1.2) and (1.3) but not the Gaussian property (3.1). The higher cumulants of A do not vanish but they are supposed to be delta-correlated... 

Langevin-like because A(t) is not a Langevin force as defined in 1. 

Show that this leads formally to the same result (2.11). Notice, however, that L is not a true Langevin force, because its stochastic properties depend on the particular solution of (2.11a). 

The Langevin force obeys the following equations specifying its average, which is zero, and its ACF, which is a delta function ... 

As a consequence, by using Eq. (M.10), that is, the fact that the average of the Langevin force is zero, the ACF (M.15) simplifies into... 

F(f),F(f) and F°(f) Different Langevin forces dealing with the Brownian H-bond bridge and having different dimensions. 

In Eq. (22), the Langevin force F(t) may be considered as a Gaussian stationary random process of zero mean with correlation function given by Eq. (20). 

The Ohmic model memory kernel admits an infinitely short memory limit y(t) = 2y5(t), which is obtained by taking the limit a>c —> oo in the memory kernel y(t) = yG)ce c [this amounts to the use of the dissipation model as defined by Eq. (23) for any value of ]. Note that the corresponding limit must also be taken in the Langevin force correlation function (29). In this limit, Eq. (22) reduces to the nonretarded Langevin equation ... 

Adding to the Langevin force in Eq. (73) a nonrandom force proportional to 8(t — t1), one gets the expression of the displacement response function ... 

As stated above, the Langevin force F(t) can be viewed as corresponding to a stationary random process. Clearly, the same is true of the solution v(f) of the generalized Langevin equation (22), an equation which is valid once the limit ti —> —oo has been taken. Thus, Fourier analysis and the Wiener-Khintchine theorem can be used to obtain the velocity correlation function, which only depends on the observation time Cvv(t, t2) = Cvv(t —12). As in the classical case, the velocity does not age. 

For 8=1, the noise spectral density is a constant (white noise), at least in the angular frequency range co oo, the Langevin force F(t) is delta-correlated, and the Langevin equation is nonretarded. The white noise case corresponds to Ohmic friction. The cases 0 < 8 < 1 and 8 > 1 are known respectively as the sub-Ohmic and super-Ohmic models. Here we will assume that 0 < 8 < 2, for reasons to be developed below [28,49-51]. 

Equations for other operators can be obtained from the symmetry and by Hermitian conjugation. Operators of Langevin forces L r ensure the validity of... 

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