Langevin equation and Brownian motion

The above equation is called the Langevin equation, which is an extension of Newton s law to include random forces. The velocity of the particle, u(r), as the solution of the above equation is also random, due to the randomness of the force r (t). Since F (t) is expected to be highly fluctuating in the timescales of 

In view of the random nature of F and hence u, a solution of Equation 6.31 is implied only in some average sense. In fact, a formal solution of Equation 6.31 can be readily obtained but it is only useful for computing average properties of u based on the average properties of the random force. As an example of the simplest situation and at the same time of common occurrence in experiments, let us assume that the random force F(t) fluctuates independently of u(t) about zero average and is correlated essentially locally in time. 

Since Equation 6.33 relates the correlation function of the fluctuating noise F (t) at temperature T with the friction coefficient C associated with the frictional dissipation of energy at the particle-solvent interface, it is referred to as the fluctuation-dissipation theorem. Therefore, the random force and the friction coefficient are not arbitrary but must satisfy Equation 6.33 at a fixed temperature of the system. 

The velocity correlation functions and averages related to the displacement of the particle can be derived by constmcting these quantities from Equation 6.31, and then averaging the resulting expressions with the use of Equations 6.32 and 6.33. In the absence of any external forces ((ext = 0), the velocity correlation 

Since t is a measure of over how long u(t) and u(t ) are correlated to each other, it is called the correlation time of the particle. The mean square displacement of the center of mass. Remit), of the particle follows from Equation 6.35 as 

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