Nonlinear Fokker-Planck equations play a major role in modeling large systems of interacting particles with a proved effectiveness in describing real world phenomena ranging from classical fields such as fluids and plasma to social and biological dynamics. Their mathematical formulation has often to face with physical forces having a significant random component or with particles living in a random environment which characterization may be deduced through experimental data and leading consequently to uncertainty-dependent equilibrium states. In this work, to address the problem of effectively solving stochastic Fokker-Planck systems, we will construct a new equilibrium preserving scheme through a micro-macro approach based on stochastic Galerkin methods. The resulting numerical method, contrarily to the direct application of a stochastic Galerkin projection in the parameter space of the unknowns of the underlying Fokker-Planck model, leads to highly accurate description of the uncertainty dependent large time behavior. Several numerical tests in the context of collective behavior for social and life sciences are presented to assess the validity of the present methodology against standard ones.

翻译：非线性Fokker-Planck方程在描述大规模相互作用粒子系统中扮演重要角色，其有效性已通过从经典领域（如流体与等离子体）到社会和生物动力学等真实世界现象得到验证。该方程的数学建模常需应对具有显著随机分量的物理力，或处于随机环境中的粒子——此类环境特征可通过实验数据推断，并由此导致依赖于不确定性的平衡态。为有效求解随机Fokker-Planck系统，本研究基于随机Galerkin方法构建了一种新型保平衡微宏格式。相较于直接对底层Fokker-Planck模型未知量的参数空间应用随机Galerkin投影，本数值方法能高精度描述依赖于不确定性的长期行为。通过多个社会与生命科学中群体行为数值实验，验证了该标准方法的本方法论有效性。