The generalised Langevin equation

In the previous section, the phenomenological description of Brownian motion was presented. The Langevin analysis leads to a velocity autocorrelation function which decays exponentially with time. This is characteristic of a Markovian process, as Doobs has shown (see ref. 490). Since it is known heyond question that the velocity autocorrelation function is far from such an exponential function, the effect that the solvent structure has on the progress of a chemical reaction cannot be assessed very reliably by means of phenomenological Langevin description. Since the velocity of a solute is correlated with its velocity a while before, a description which fails to consider solute and solvent velocities can hardly be satisfactory. Necessarily, the analysis requires a modification of the Langevin or Fokker—Plank description. In this section, some comments are made on this new and exciting area of research. 

The simplicity of the Langevin description commends itself strongly. A simple extension of the Langevin equations allows any velocity autocorrelation function to be described. By writing 

By taking the Laplace transform of eqn. (290) and then multiplying throughout by the initial velocity u(0) and finally taking an ensemble average, the Laplace transformed velocity autocorrelation function is [490] 

Consequently, the time-dependent function coefficient can be calculated. 

---

Adelman [530] and Stillman and Freed [531] have discussed the reduction of the generalised Langevin equation to a generalised Fokker— Planck equation, which provides a description of the probability that a molecule has a velocity u at a position r at a time t, given certain initial conditions (see Sect. 3.2.). The generalised Fokker—Planck equation has important differences by comparison with the (Markovian) Fokker— Planck equation (287). However, it has not proved so convenient a vehicle for studies of chemical reactions in solution as the generalised Langevin equation (290). 

D only assumes a constant value over times long compared with rc( 10rc), such that (u(O)u(f)), the velocity autocorrelation function, is nearly zero. The diffusion equation is not valid over times < 10rc(i.e. a few picoseconds at least). A better approach would be to use a generalised Langevin equation with a friction coefficient which has a memory and such that the velocity autocorrelation takes 0.5ps to decay to insignificant levels (see Chap. 11) [453]. 

As far as the linear generalised Langevin equation of a single macromolecule is concerned, general speculations allow us to write down the form of the final results before the calculations are carried out. The requirements of covariance... 

---