Derivation of the Langevin equation from a microscopic model

5 Derivation of the Langevin equation from a microscopic model 

The stochastic equation of motion (8.13) was introduced as a phenomenological model based on the combination of experience and intuition. We shall now attempt to derive such an equation from first principles, namely starting from the Newton 

The interaction that appears in (8.48) contains, in addition to a linear coupling term x cjqj, also a compensating term /(Imjcoj ) that has the effect 

Equation (8.50) is an inhomogeneous differential equation for (z), whose solution can be written as 

The function y (Z), which is mathematically identical to the variable A of Section 6.5.1, represents a stochastic force that acts on the system coordinate X. Its stochastic nature stems from the lack of information about qjo and qjQ. All we know about these quantities is that, since the thermal bath is assumed to remain in equilibrium throughout the process, they should be sampled from 

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