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q^{\uparrow}_j (t) = \sum_i^{n_i} f_{ij} q^{\downarrow}_i(t) + - \tau_j \cdot \frac{ d q^{\uparrow}_jn}{ dt }= \sum_m f_{nm} \cdot q^{\downarrow}_m(t) - \beta_jn \cdot \frac{d p_jn}{dt} |
This equation can be integrated explicitly:
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