School of Engineering and Applied Sciences and Kavli Institute for Bionano Science and Technology, Harvard University, Cambridge, MA 02138, USA.
Proc Natl Acad Sci U S A. 2013 Jan 2;110(1):E5-14. doi: 10.1073/pnas.1211720110. Epub 2012 Dec 17.
Particles interacting with short-ranged potentials have attracted increasing interest, partly for their ability to model mesoscale systems such as colloids interacting via DNA or depletion. We consider the free-energy landscape of such systems as the range of the potential goes to zero. In this limit, the landscape is entirely defined by geometrical manifolds, plus a single control parameter. These manifolds are fundamental objects that do not depend on the details of the interaction potential and provide the starting point from which any quantity characterizing the system--equilibrium or nonequilibrium--can be computed for arbitrary potentials. To consider dynamical quantities we compute the asymptotic limit of the Fokker-Planck equation and show that it becomes restricted to the low-dimensional manifolds connected by "sticky" boundary conditions. To illustrate our theory, we compute the low-dimensional manifolds for n ≤ 8 identical particles, providing a complete description of the lowest-energy parts of the landscape including floppy modes with up to 2 internal degrees of freedom. The results can be directly tested on colloidal clusters. This limit is a unique approach for understanding energy landscapes, and our hope is that it can also provide insight into finite-range potentials.
粒子与短程势相互作用引起了越来越多的关注，部分原因是它们能够模拟胶体通过 DNA 或耗尽相互作用等介观系统。我们考虑了势的范围趋于零时的自由能景观。在这个极限下，景观完全由几何流形以及单个控制参数定义。这些流形是基本的对象，不依赖于相互作用势的细节，并为任意势下计算任何描述系统的特性——平衡或非平衡——提供了起点。为了考虑动力学量，我们计算了福克-普朗克方程的渐近极限，并表明它被限制在由“粘性”边界条件连接的低维流形上。为了说明我们的理论，我们计算了 n ≤ 8 个相同粒子的低维流形，提供了景观中最低能量部分的完整描述，包括多达 2 个内部自由度的柔软模式。结果可以直接在胶体簇上进行测试。这个极限是理解能量景观的独特方法，我们希望它也能为有限范围势提供深入的了解。
