An analytical model has been developed to describe Monte Carlo simulation results for the non-equilibrium incorporation of dopants during crystallization. These Monte Carlo simulation results, which are presented in the J. Crystal Growth, this issue. doi: 10.1016/j.jcrysgro.2004.07.074. as well as in earlier publications, have successfully reproduced experimental results on solute trapping. The analytical model presented here can also be used to describe extended solubility and other phenomena associated with rapid first-order phase transformations. The underlying physical picture, based on insights gained from Monte Carlo computer simulations, is that the interface can transfer, between species, the chemical potential differences which are driving the transformation. The analytical model starts with a set of rate equations, one for the crystallization of each component, based on standard reaction rate theory. The new equations describing growth reduce to the standard growth rate expression for a pure material and for growth near equilibrium, and they reduce to the usual quasi-equilibrium equations for alloy crystallization. They incorporate a transition from the independent crystallization of each component near equilibrium, which is governed by the differences in the individual chemical potentials between the two phases, to a cooperative growth mode far from equilibrium, which depends on the difference in free energy between the two phases of the alloy. The transition depends on a dimensionless parameter, beta, which is the ratio of the distance an atom can diffuse to the distance the interface moves, during the time it takes an atom to join the crystal. This parameter embodies the relationship between the growth and diffusion parameters found in the Monte Carlo simulations (J. Crystal Growth, this issue. doi: 10.1016/j.jcrysgro.2004.07.074.), and was found earlier by Temkin (J. Crystal Growth 5 (1969) 193; Phys. Crystallogr 17 (1972) 405) in his analyses of alloy crystallization. The analytical model is compared with experimental results and, in the J. Crystal Growth, this issue. doi: 10.1016/j.jcrysgro.2004.07.074., with Monte Carlo computer simulations. This comparison requires a single fitting constant which also provides a reasonable fit to the available experimental data. (C) 2004 Elsevier B.V. All rights reserved.