Brownian motion rigorous derivation

Equation (2.6) is called the Fokker-Planck equation (FPE) or forward Kolmogorov equation, because it contains time derivative of final moment of time t > to. This equation is also known as Smoluchowski equation. The second equation (2.7) is called the backward Kolmogorov equation, because it contains the time derivative of the initial moment of time to < t. These names are associated with the fact that the first equation used Fokker (1914) [44] and Planck (1917) [45] for the description of Brownian motion, but Kolmogorov [46] was the first to give rigorous mathematical argumentation for Eq. (2.6) and he was first to derive Eq. (2.7). The derivation of the FPE may be found, for example, in textbooks [2,15,17,18],... 

The excellent review of Chandrasekhar provides a detailed account of the history of the subject, to which both Smoluchowski and Einstein made fundamental contributions. It is worth mentioning the well-known paper of Kramers, who provided a rigorous derivation of the Smoluchowski equation from the complete Fokker-Planck equation of a Brownian particle in an external potential. This problem allows us to explain what we mean by a systematic version of the AEP. We can state the problem as follows. Let us consider the motion of a free Brownian particle described by the one-dimensional counterpart of Eq. (1.2),... 

The appendices contain an account of those parts of the theory of Brownian motion and linear response theory which are essential for the reader in order to achieve an understanding of relaxational phenomena in magnetic domains and in ferrofluid particles. The analogy with dielectric relaxation is emphasized throughout these appendices. Appendix D contains the rigorous derivation of Brown s equation. 

In this section, we first introduce the most frequently used definitions of static properties, which will be used throughout this chapter and in some later chapters, and then discuss stochastic processes in the motion of macromolecular chains that is. Brownian motion that leads to the well-known Fokker-Planck equation, which further reduces to the Smoluchowski equation and Langevin equation. These two equations play a very important role in describing the motion of macromolecular chains. Owing to the limited space available here, we do not present rigorous derivations of various expressions. 

---