Brownian motion, the Langevin equation

So the rate constant ks may be found as the ratio between j and na. This expression for the rate constant is equivalent to a one-dimensional version of Eq. (5.49) when the Boltzmann distribution is replaced by P(r, v). It is, however, important to notice that Eq. (11.2) allows for recrossings across the barrier, since the lower integration limit is —oo and not 0 as in the previous presentations of transition-state theory where the possibility of recrossings was neglected. 

We note that P(r, v t) has the dimension s m-2, so j has the dimension s-1, na is dimensionless, and ks has the dimension s 1. If both na and j are multiplied by the number of reactant molecules, we will get, respectively, the number of molecules in the a-well and the flux of reactants across the transition-state barrier. 

Since probabilistic dynamics is central to an understanding of Kramers theory for the influence of solvents on the rate constant, we shall first summarize some of the essential features in such a description. 

The problem of Brownian motion relates to the motion of a heavy colloidal particle immersed in a fluid made up of light particles. In Fig. 11.1.1 the trajectory of a Brownian particle is shown. The coordinates of a particle with diameter 2 /xm moving in water are observed every 30 s for 135 min. At the very first step in the argument one renounces an exact deterministic description of the motion and replaces it with a probabilistic description. 

Let us consider a Brownian particle of mass M immersed in a fluid. Microscopically, the laws of hydrodynamics would tell us that during its motion the particle 

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