Plyukhin A V
Department of Mathematics, Saint Anselm College, Manchester, New Hampshire 03102, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2013 Nov;88(5):052115. doi: 10.1103/PhysRevE.88.052115. Epub 2013 Nov 11.
We address the problem of a microscopic derivation of the Langevin equation for a weakly relativistic Brownian particle. A noncovariant Hamiltonian model is adopted, in which the free motion of particles is described relativistically while their interaction is treated classically, i.e., by means of action-to-a-distance interaction potentials. Relativistic corrections to the classical Langevin equation emerge as nonlinear dissipation terms and originate from the nonlinear dependence of the relativistic velocity on momentum. On the other hand, similar nonlinear dissipation forces also appear as classical (nonrelativistic) corrections to the weak-coupling approximation. It is shown that these classical corrections, which are usually ignored in phenomenological models, may be of the same order of magnitude, if not larger than, relativistic ones. The interplay of relativistic corrections and classical beyond-the-weak-coupling contributions determines the sign of the leading nonlinear dissipation term in the Langevin equation and thus is qualitatively important.
我们解决了弱相对论性布朗粒子朗之万方程的微观推导问题。采用了一个非协变哈密顿模型，其中粒子的自由运动以相对论方式描述，而它们的相互作用则以经典方式处理，即通过超距相互作用势。对经典朗之万方程的相对论修正以非线性耗散项的形式出现，并且源于相对论速度对动量的非线性依赖。另一方面，类似的非线性耗散力也作为对弱耦合近似的经典（非相对论）修正出现。结果表明，这些在唯象模型中通常被忽略的经典修正，如果不比相对论修正大的话，可能具有相同的量级。相对论修正与经典的弱耦合之外的贡献之间的相互作用决定了朗之万方程中主导非线性耗散项的符号，因此在定性上很重要。
