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q^{\uparrow} (t) =\tau^{-1} \exp(-t/\tau)  \cdot \left[ \ q^{\uparrow} (0) + \tau^{-1} \cdot  \int_0^t \exp(s/\tau) \left[ f \cdot q^{\downarrow}(s) - \gamma \frac{dp}{ds} \right] ds   \ \right]

q^{\uparrow}_n (t) =\tau_n^{-1} \exp(-t/\tau_n) \cdot \left[ \  q^{\uparrow}_n (0) + \tau_n^{-1}  \cdot \int_0^t \exp(s/\tau_n) \left[ \sum_m f_{nm} q^{\downarrow}_m(s) - \gamma_n \frac{dp_n}{ds} \right] ds \ \right]