The multidimensional Fokker-Planck equation

The above analysis was done for a single particle moving in one dimension, but can be extended to higher dimensions using the same procedure. The starting point is the multidimensional analog of Eq. (8.141), given by 

Problem 8.14. Show that the second of Eqs (8.148) is indeed the correct relationship between fluctuation and dissipation for such a system. 

Note that the form of Eq. (8.148) expresses the possibility that the different degrees of freedom Xj are coupled to each other not only via their interaction potential V, but in principle also through their mutual coupling to the environment. 

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