Langevin equation stationary solution

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. 

We have illustrated the calculation of the averages from the Langevin equation for sharp initial conditions. The solution of the Langevin equation subject to a Maxwell-Boltzmann distribution of velocities is called the stationary solution. Clearly for the stationary solution... 

One solution to this problem is that implemented in the dynamical equations (6) and (7), which couple the classical Langevin equation for the atoms in the protein-water system to the stationary Schrodinger equation for the quantum excitations (thus ensuring that only quantum eigenstates are considered). One limitation of this solution is that it is only valid when the quantum excitation responds very fast to any changes in the classical conformation, something which is assumed to be true here. 

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