Smoluchowski equation energy derivation

The force in eqn. (208) on the particle k is F(rfe) due to the motion of all the N particles with respect to the unperturbed solvent. In Chap. 3, the Debye—Smoluchowski equation was derived from thermodynamic arguments. It was pointed out that the spatial gradient of the chemical potential at some point is the force acting at that point. If the potential energy is U, then the chemical potential is [cf. eqn. (40)]... 

The product J Eg)coQ is equal to Eg for a harmonic oscillator potential truncated at = Eg, and to 2Eg for a Morse potential with dissociation energy equal to Eg. Equation (2.41) is the low-friction limit result of Kramers. There are other methods to derive the results obtained in the previous section. One is to look for the eigenvalue with smallest positive real part of the ojjerator L defined so that dP/dt = — LP is the relevant Fokker-Planck or Smoluchowski equation. Under the usual condition of time scale separation this smallest real part is the escapie rate for a single well potential. Another way uses the concept of mean passage time. For the one-dimensional Fokker-Planck equation of the form... 

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