Herrmann Joachim
Max Born Institute, Max Born Strasse 2a, D12489 Berlin, Germany.
Phys Rev E Stat Nonlin Soft Matter Phys. 2009 Nov;80(5 Pt 1):051110. doi: 10.1103/PhysRevE.80.051110. Epub 2009 Nov 11.
The Markovian diffusion theory is generalized within the framework of the special theory of relativity. Since the velocity space in relativity is a hyperboloid, the mathematical stochastic calculus on Riemanian manifolds can be applied but adopted here to the velocity space. A generalized Langevin equation in the fiber space of position, velocity, and orthonormal velocity frames is defined from which the generalized relativistic Kramers equation in the phase space in external force fields is derived. The obtained diffusion equation is invariant under Lorentz transformations and its stationary solution is given by the Jüttner distribution. Besides, a nonstationary analytical solution is derived for the example of force-free relativistic diffusion.
马尔可夫扩散理论在狭义相对论框架内得到推广。由于相对论中的速度空间是一个双曲面，黎曼流形上的数学随机微积分可以应用于此，但这里是应用于速度空间。在位置、速度和正交速度框架的纤维空间中定义了一个广义朗之万方程，由此推导出外力场中相空间的广义相对论克莱默斯方程。所得到的扩散方程在洛伦兹变换下是不变的，其定态解由尤特纳分布给出。此外，针对无外力相对论扩散的例子推导了一个非定态解析解。
