We investigate the microphysics of contrail formation behind commercial aircraft by means of the particle-based LCM (Lagrangian Cloud Module) box model. We extend the original LCM to cover the basic pathway of contrail formation on soot particles being activated into liquid droplets that soon after freeze into ice crystals. In our particle-based microphysical approach, simulation particles are used to represent different particle types (soot, droplets, ice crystals) and properties (mass/radius, number). The box model is applied in two frameworks. In the classical framework, we prescribe the dilution along one average trajectory in a single box model run. In the second framework, we perform a large ensemble of box model runs using 25 000 different trajectories inside an expanding exhaust jet as simulated by the LES (large-eddy simulation) model FLUDILES. In the ensemble runs, we see a strong radial dependence of the temperature and relative humidity evolution. Droplet formation on soot particles happens first near the plume edge and a few tenths of a second later in the plume centre. Averaging over the ensemble runs, the number of formed droplets and ice crystals increases more smoothly over time than for the single box model run with the average dilution. Consistent with previous studies, contrail ice crystal number varies strongly with atmospheric parameters like temperature and relative humidity near the contrail formation threshold. Close to this threshold, the apparent ice number emission index (product of freezing fraction and soot number emission index) strongly depends on the geometric-mean dry core radius and the hygroscopicity parameter of soot particles. The freezing fraction of soot particles slightly decreases with increasing soot particle number, particularly for higher soot number emissions. This weakens the increase of the apparent ice number emission index with rising soot number emission index. Comparison with box model results of a recent contrail formation study by Lewellen (2020) (using similar microphysics) shows a later onset of our contrail formation due to a weaker prescribed plume dilution. If we use the same dilution data, our evolution and Lewellen's evolution in contrail ice nucleation show an excellent agreement cross-verifying both microphysics implementations. This means that differences in contrail properties mainly result from different representations of the plume mixing and not from the microphysical modelling. Using an ensemble mean framework instead of a single trajectory does not necessarily lead to an improved scientific outcome. Contrail ice crystal numbers tend to be overestimated since the interaction between the different trajectories is not considered. The presented aerosol and microphysics scheme describing contrail formation is of intermediate complexity and thus suited to be incorporated in an LES model for 3D contrail formation studies explicitly simulating the jet expansion. Our box model results will help interpret the upcoming, more complex 3D results.