Coppex François, Droz Michel, Trizac Emmanuel
Department of Theoretical Physics, University of Genève, CH-1211 Genève 4, Switzerland.
Phys Rev E Stat Nonlin Soft Matter Phys. 2005 Aug;72(2 Pt 1):021105. doi: 10.1103/PhysRevE.72.021105. Epub 2005 Aug 23.
The hydrodynamic description of probabilistic ballistic annihilation, for which no conservation laws hold, is an intricate problem with hard spherelike dynamics for which no exact solution exists. We consequently focus on simplified approaches, the Maxwell and very-hard-particle (VHP) models, which allows us to compute analytically upper and lower bounds for several quantities. The purpose is to test the possibility of describing such a far from equilibrium dynamics with simplified kinetic models. The motivation is also in turn to assess the relevance of some singular features appearing within the original model and the approximations invoked to study it. The scaling exponents are first obtained from the (simplified) Boltzmann equation, and are confronted against direct Monte Carlo simulations. Then, the Chapman-Enskog method is used to obtain constitutive relations and transport coefficients. The corresponding Navier-Stokes equations for the hydrodynamic fields are derived for both Maxwell and VHP models. We finally perform a linear stability analysis around the homogeneous solution, which illustrates the importance of dissipation in the possible development of spatial inhomogeneities.
概率性弹道湮灭的流体动力学描述不存在守恒定律，是一个复杂的问题，具有硬球状动力学，不存在精确解。因此，我们专注于简化方法，即麦克斯韦模型和超硬粒子（VHP）模型，这使我们能够通过解析计算几个量的上下界。目的是测试用简化动力学模型描述这种远离平衡态动力学的可能性。其动机还在于评估原始模型中出现的一些奇异特征以及为研究它而采用的近似的相关性。首先从（简化的）玻尔兹曼方程获得标度指数，并与直接蒙特卡罗模拟进行对比。然后，使用查普曼 - 恩斯科格方法获得本构关系和输运系数。针对麦克斯韦模型和VHP模型，推导了流体动力学场的相应纳维 - 斯托克斯方程。我们最后围绕均匀解进行线性稳定性分析，这说明了耗散在空间不均匀性可能发展中的重要性。
