Knudsen气体在随机管中的输运扩散系数
Transport diffusion coefficient for a Knudsen gas in a random tube
摘要：
我们考虑输运扩散在一个随机台球在一个随机管是细长的方向上的第一个坐标（管轴）。在随机管内，这是静态的和遍历的，不相互作用的粒子以恒定的速度直线运动。在撞击管壁时，根据余弦定律，它们被随机反射：出射方向的密度与该方向和法向量之间的角度的余弦成比例。稳态输运研究通过引入一个开放的管段如下：我们切出一个大的有限段的管段边界垂直于管轴。通过段边界离开这一片的粒子从系统中消失。通过在段的一个边界处的粒子的静止注入，保持具有非零静止粒子流的稳定状态。我们证明：（i）在热力学极限的一个无限的开放段内的粗粒密度分布是线性的，和（ii）的传输扩散系数从固定电流和有效边界密度梯度的比率等于扩散系数的标记粒子在一个无限管。因此，我们证明了菲克定律和相当一般的粗糙（随机）管的输运扩散和自扩散系数的平等。
英文摘要：
We consider transport diffusion in a stochastic billiard in a random tube which is elongated in the direction of the first coordinate (the tube axis). Inside the random tube, which is stationary and ergodic, non-interacting particles move straight with constant speed. Upon hitting the tube walls, they are reflected randomly, according to the cosine law: the density of the outgoing direction is proportional to the cosine of the angle between this direction and the normal vector. Steady state transport is studied by introducing an open tube segment as follows: We cut out a large finite segment of the tube with segment boundaries perpendicular to the tube axis. Particles which leave this piece through the segment boundaries disappear from the system. Through stationary injection of particles at one boundary of the segment a steady state with non-vanishing stationary particle current is maintained. We prove (i) that in the thermodynamic limit of an infinite open piece the coarse-grained density profile inside the segment is linear, and (ii) that the transport diffusion coefficient obtained from the ratio of stationary current and effective boundary density gradient equals the diffusion coefficient of a tagged particle in an infinite tube. Thus we prove Fick's law and equality of transport diffusion and self-diffusion coefficients for quite generic rough (random) tubes.
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