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The diffusive effects of the fluctuations must be balanced by the dissipative effects of the drag. A version of this relation was used by Einstein in his study of Brownian motion to relate the diffusion coefficient ???? to the drag force experienced by the particle, thereby obtaining an explicit relation between the mean-squared displacement and Boltzmann’s constant (cf. Eq. 13 on page 7). The fluctuation-dissipation relation was later used by Nyquist and Johnson in the study of thermal noise in a resistor.

With ????1 < ????2 < ????3 , integration over ????2 gives, ∫∞ ???? (????1 , ????3 ; ????1 , ????3 ) = ???? (????1 , ????1 ) ???? (????3 , ????3 ∣????2 , ????2 ) ???? (????2 , ????2 ∣????1 , ????1 ) ????????2 , −∞ and, using ???? (????1 , ????3 ; ????1 , ????3 ) = ???? (????3 , ????3 ∣????1 , ????1 ) ⋅ ???? (????1 , ????1 ) (Eq. 1). It is a functional equation relating all conditional probability densities ???? (???????? , ???????? ∣???????? , ???????? ) for a Markov process, where the time ordering in the integrand is essential. The converse is also true: if ???? (????1 , ????1 ) and ???? (????2 , ????2 ∣????1 , ????1 ) obey the consistency condition, Eqs.

1 , . . , ???????? ; ????1 , . . , ???????? ) = ???? ∂ ???? (????1 , . . , ???????? ; ????1 , . . , ???????? ) . ∂????1 . . 6) Note that the ????????ℎ -order distribution function determines all lower-order distribution functions, and in fact, the ????????ℎ -order distribution function completely determines the stochastic process. Remarks: 1. The distribution function Eq. 5 is not arbitrary, but must satisfy the following, ∙ Symmetry condition. For every permutation (????1 , ????2 , . . , ???????? ) of (1, 2, .

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