  A novel mathematical treatment is proposed for computing the time evolution
of dynamic nuclear polarization processes in the low temperature thermal mixing
regime. Without assuming any a priori analytical form for the electron
polarization, our approach provides a quantitative picture of the steady state
that recovers the well known Borghini prediction based on thermodynamics
arguments, as long as the electrons-nuclei transition rates are fast compared
to the other relevant time scales. Substantially different final polarization
levels are achieved instead when the latter assumption is relaxed in the
presence of a nuclear leakage term, even though very weak, suggesting a
possible explanation for the deviation between the measured steady state
polarizations and the Borghini prediction. The proposed methodology also allows
to calculate nuclear polarization and relaxation times, once specified the
electrons/nuclei concentration ratio and the typical rates of the microscopic
processes involving the two spin species. Numerical results are shown to
account for the manifold dynamical behaviours of typical DNP samples.
