在崎岖的能量景观中扩散的异常维数依赖性：一维有多病态？
Anomalous dimensionality dependence of diffusion in a rugged energy landscape : How pathological is one dimension ?
摘要：
具有空间分布的最大值和最小值的崎岖（或粗糙）能量景观（REL）经常用于物理、化学和生物学的应用（酶动力学、蛋白质折叠、无序固体中的扩散、有机半导体中的输运、随机自旋系统中的弛豫、过冷液体和玻璃）。有时系统需要被建模为高维的随机游走器（如在蛋白质折叠/解折叠中），其中维度可以是不同氨基酸残基之间的距离（如在HP-36的解折叠中）。然而，大多数对这些现象的理论研究仍然采用一维描述。尽管预测在粗糙（或崎岖）的能量景观（REL）中，一维（1d）中的扩散预测与任何更高维度的扩散在病理上不同，遇到破碎遍历性的机会增加（Stein和Newman，2012）。我们探讨了维数依赖的扩散REL进行有效的介质近似为基础的分析计算，并与现有的计算机模拟结果进行比较。我们发现，在中等水平的坚固性（假设有一个高斯分布），扩散是明确定义的，有效扩散系数的值取决于维数和变化（增加）的几个因素（~5-10），从1d到2d。相比之下，后续过渡（如2d到3d和3d到4d等）的变化要温和得多，只有10-20%的数量级。当坚固性是由随机陷阱与指数分布的障碍高度，均方位移是次扩散（一个众所周知的结果），但MSD的增长是由不同的指数在一个和更高的维度。增长的指数是在更高的维度比在1d更大。
英文摘要：
Rugged (or, rough) energy landscape (REL) with spatially distributed maxima and minima are often employed in applications of physics, chemistry and biology (enzyme kinetics, protein folding, diffusion in disordered solids, transport in organic semiconductors, relaxation in random spin systems, in supercooled liquids and glasses). Sometimes the system needs to be modeled as a random walker in high dimensions (like in protein folding/unfolding) where dimensions could be the distances between different amino acid residues (as in unfolding of HP-36). Nevertheless, most of the theoretical studies of these phenomena still employ a one dimensional description. This is despite the prediction that in a rough (or, rugged) energy landscape (REL), diffusion in one dimension (1d) is predicted to be pathologically different from any higher dimension with the increased chance of encountering broken ergodicity (Stein and Newman, 2012). We explore the dimensionality dependent diffusion on REL by carrying out an effective medium approximation based analytical calculations and compare them with the available computer simulation results. We find that at intermediate level of ruggedness (assumed to have a Gaussian distribution), where diffusion is well-defined, the value of the effective diffusion coefficient depends on dimensionality and changes (increases) by several factors (~5-10) in going from 1d to 2d. In contrast, the changes in subsequent transitions (like 2d to 3d and 3d to 4d and so on) are far more modest, of the order of 10-20% only. When ruggedness is given by random traps with an exponential distribution of barrier heights, the mean square displacement is sub-diffusive (a well-known result), but the growth of MSD is described by different exponents in one and higher dimensions. The exponent of growth is larger in higher dimensions than in 1d.
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