The study of strongly out-of-equilibrium states and their time evolution towards thermalization is critical to the understanding of an ever widening range of physical processes. We present a numerical method that for the first time allows for the numerical solution of the most difficult part of the time-dependent Boltzmann equation: the full scattering term. Any number of bands (and quasiparticles) with arbitrary dispersion, any number of high order scattering channels (we show here four legs scatterings: electron–electron scattering) can be treated far from equilibrium. No assumptions are done on the population and all the Pauli-blocking factors are included in the phase-space term of the scattering. The method can be straightforwardly interfaced to a deterministic solver for the transport. Finally and most critically, the method conserves to machine precision the particle number, momentum and energy for any resolution, making the computation of the time evolution till complete thermalization possible.

We apply this approach to two examples, a metal and a semiconductor, undergoing thermalization from a strongly out-of-equilibrium laser excitation. These two cases, which are in literature treated hitherto under a number of approximations, can be addressed free from those approximations and straightforwardly within the same numerical method.

对强烈失衡状态及其向热化的时间演化的研究对于理解不断扩大的物理过程至关重要。我们提出了一种数值方法，该方法首次允许对与时间相关的Boltzmann方程中最困难的部分：完整的散射项进行数值求解。任意数量的具有任意色散的谱带（和准粒子），任意数量的高阶散射通道（在此我们显示四个分支散射：电子-电子散射）都可以远离平衡处理。没有对总体做任何假设，并且所有的Pauli阻挡因子都包含在散射的相空间项中。该方法可以直接与用于运输的确定性求解器接口。最后也是最关键的一点

我们将这种方法应用于两个示例（金属和半导体），这些示例由于强烈的不平衡激光激发而发生热化。迄今为止，这两种情况在文献中都以许多近似值进行了处理，可以从这些近似值中解脱出来，并且可以在相同的数值方法中直接解决。