Mathematical and Statistical Computing Laboratory, Division of Computational Bioscience, Center for Information Technology, National Institutes of Health, Bethesda, Maryland 20892, USA.
J Chem Phys. 2009 Dec 14;131(22):224110. doi: 10.1063/1.3271998.
Reduction of three-dimensional (3D) description of diffusion in a tube of variable cross section to an approximate one-dimensional (1D) description has been studied in detail previously only in tubes of slowly varying diameter. Here we discuss an effective 1D description in the opposite limiting case when the tube diameter changes abruptly, i.e., in a tube composed of any number of cylindrical sections of different diameters. The key step of our approach is an approximate description of the particle transitions between the wide and narrow parts of the tube as trapping by partially absorbing boundaries with appropriately chosen trapping rates. Boundary homogenization is used to determine the trapping rate for transitions from the wide part of the tube to the narrow one. This trapping rate is then used in combination with the condition of detailed balance to find the trapping rate for transitions in the opposite direction, from the narrow part of the tube to the wide one. Comparison with numerical solution of the 3D diffusion equation allows us to test the approximate 1D description and to establish the conditions of its applicability. We find that suggested 1D description works quite well when the wide part of the tube is not too short, whereas the length of the narrow part can be arbitrary. Taking advantage of this description in the problem of escape of diffusing particle from a cylindrical cavity through a cylindrical tunnel we can lift restricting assumptions accepted in earlier theories: We can consider the particle motion in the tunnel and in the cavity on an equal footing, i.e., we can relax the assumption of fast intracavity relaxation used in all earlier theories. As a consequence, the dependence of the escape kinetics on the particle initial position in the system can be analyzed. Moreover, using the 1D description we can analyze the escape kinetics at an arbitrary tunnel radius, whereas all earlier theories are based on the assumption that the tunnel is narrow.
先前仅在直径缓慢变化的管中详细研究了将具有可变横截面的管中的三维（3D）扩散描述简化为近似一维（1D）描述的问题。在这里，我们讨论了当管直径突然变化时的相反极限情况（即由不同直径的任意数量的圆柱形段组成的管）中的有效 1D 描述。我们方法的关键步骤是将粒子在管的宽部和窄部之间的跃迁近似描述为具有适当选择的捕获率的部分吸收边界的捕获。边界均匀化用于确定从管的宽部到窄部的跃迁的捕获率。然后，将该捕获率与详细平衡条件结合使用，以找到从管的窄部到宽部的相反方向的跃迁的捕获率。与 3D 扩散方程的数值解进行比较可以使我们能够测试近似的 1D 描述并确定其适用性条件。我们发现，当管的宽部不太短时，建议的 1D 描述效果很好，而窄部的长度可以任意。在扩散粒子通过圆柱形隧道从圆柱形腔中逸出的问题中利用这种描述，我们可以取消在早期理论中接受的限制假设：我们可以平等地考虑粒子在隧道和腔中的运动，即我们可以放松在所有早期理论中都使用的快速腔内弛豫的假设。结果，可以分析粒子初始位置在系统中的逃逸动力学的依赖性。此外，使用 1D 描述我们可以分析任意隧道半径处的逃逸动力学，而所有早期理论都是基于隧道狭窄的假设。
