Boltzmann equation coherence

If the exciton-phonon coupling is sufficiently weak, the solution of the equation for the correlation function (B (t)Bm(t)B, (0)Bm (0)) is equivalent to the solution of the Boltzmann equation (11). In this coherent limit, the exciton states of... 

F. Kaiser, Boltzmann Equation Approach to Frohlich s Vibrational Model of Bose Condensation-Like Excitations of Coherent Modes in Biological Systems, Z. Naturforsch. 34a, 134 146. 

The time evolution in Eq. (7.75) is described by the time-dependent Schrodinger equation, provided the molecule is isolated from the rest of the universe. In practice, there are always perturbations from the environment, say due to inelastic collisions. The coherent sum in Eq. (7.75) will then relax to the incoherent sum of Eq. (7.74), that is, the off-diagonal interference terms will vanish and cn 2 — pn corresponding to the Boltzmann distribution. As mentioned earlier, the relaxation time depends on the pressure. In order to take advantage of coherent dynamics it is, of course, crucial that relaxation is avoided within the duration of the relevant chemical dynamics. 

The various coherent state matrix elements are easily evaluated analytically. Jezek and Makri have described a Monte Carlo procedure for calculating the FBSD expression for a correlation function in conjunction with a path integral treatment of the Boltzmann operator (45). If the operator A has a simple form, for example, if it is a linear function of position or momentum, the last integral in the above equations can also be performed analytically. 

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