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0〈N(0)N(t)〉dt. This is just Eq. (3.27) with approximation (3.39) for the vibrational heat capacity of the solute taken into account. Of course, at first sight this analogy should not be very close because particle momentum is a vector whereas energy is a scalar quantity. However, the vibrational energy is a sum of energies of vibrational modes, and the external force capacity can be split into a sum of normal mode contributions, too. If all the modes relax independently of each other, the expression for the relaxation time of a particular mode will comprise only the respective normal mode capacity as in the case of relaxation of a particular momentum projection. This peculiarity makes the analogy between the Langevin equations for particle momentum and for vibrational energy very close and explains at an intuitive level of understanding the appearance of the number of vibrational degrees of freedom in the denominator of Eqs. (3.27), (3.29), and (3.38).
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