The Landau equation is a kinetic equation based on the weak coupling approximation of the interaction between the particles. In the framework of dry active matter this new kinetic equation relies on the weak coupling approximation of both the alignment strength and the magnitude of the angular noise, instead of the hypothesis of diluteness. Therefore, it is a kinetic equation bridging between the Boltzmann (Bertin et al 2006 Phys. Rev. E 74 022101), and the Smoluchowski (Baskaran et al 2010 J. Stat. Mech. P04019) approximations, and allowing analytical descriptions at moderate densities. The form of the equation presents non-linear and density dependent diffusions and advections fully derived by the microscopic equations of motions. Finally, implementing the BGL procedure (Peshkov et al 2014 Eur. Phys. J. Spec. Top. 223 1315–44), the parameters of the Toner–Tu equations are derived showing the appearance of linearly stable homogeneous ordered solutions and mimicking the results obtained from the Boltzmann approach.

兰道方程是基于颗粒之间相互作用的弱耦合近似的动力学方程。在干燥的活性物质的框架中，这个新的动力学方程式依赖于对准强度和角噪声幅度的弱耦合近似，而不是稀释度的假设。因此，这是在玻尔兹曼（Bertin等人， 2006 Phys。Rev. E 74 022101）和Smoluchowski（Baskaran等人， 2010 J. Stat。Mech。Acad。Sci.USA，95：1593）之间的动力学方程式。P04019）近似值，并允许在中等密度下进行分析描述。该方程式的形式呈现出非线性的和与密度有关的扩散和对流，这些扩散和对流是由微观运动方程式完全推导出来的。最后，在实施BGL程序（Peshkov等人2014欧元。物理学。J.规格。顶部。 223 1315年至1344年），将调色剂涂方程的参数，导出表示线性稳定均匀的外观有序的解决方案和模拟的结果从玻尔兹曼方法获得。