The Langevin Equation by W. T. Coffey, Yu. P. Kalmykov, J. T. Waldron

By W. T. Coffey, Yu. P. Kalmykov, J. T. Waldron

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The continuity equation [Eq. 7) at which is the specialised form of the Fokker-Planck equation known as the Smoluchowski equation, which describes approximately the evolution o f / i n configuration space. (For a detailed account of Smoluchowski's method of derivation of this equation, which was based on a specific detailed kinetic model, namely, collisions of hard spheres, see Mazo [100]). 7) was first [5] given by Einstein in 1905 for the special case of V=0. In general direct justification of inclusion of the thermal agitation by adding the diffusive term of Eq.

Now since the volume element Av is arbitrary, Eq. 8) at which is the continuity equation of fluid mechanics. Now f .. 9) ox dp ox op ydx dp j and, using Eqs. 10) dx dp dx y dp dp\ dx from the equality of the mixed second order partial derivatives. Furthermore, x^-+p^= u-gmdp ox op and the continuity Eq. 11) ^ + ugrad/7 = ^ = 0. 12) is known as the Liouville equation, which for N particles moving in three dimensions (so that we have a 6N dimensional 32 The Langevin Equation phase space or 127V dimensional if the rotational degrees of freedom are added) is: dp 3JV dH dp dH dp = 0.

5) will also be used in these sections. , x (t) are realisations. In the remainder of the book unless it is evident from the context, we shall always use £, (t) to denote a random (stochastic) variable. 3 The Langevin Equation The theory of the Brownian movement as formulated by Einstein [2] and Smoluchowski [9] although in agreement with experiment seemed far removed from the Newtonian dynamics of particles [1] as it appeared to rely entirely on the concept of the underlying probability density distribution of Brownian particles and the Fokker-Planck equation for the time evolution of that distribution.

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