Construction of point vortex equilibria via Brownian ratchets
Construction of point vortex equilibria via Brownian ratchets
复制标题
通过布朗棘轮构建点涡平衡
DOI：
10.1098/rspa.2007.1832
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发表时间：
2007
期刊：
影响因子：
--
通讯作者：
G. Chamoun
中科院分区：
文献类型：
--
作者：
P. Newton;G. Chamoun
本文叙述了一种能产生平面内点涡平衡构形的理论，沿着有一个计算它们的数值方案。这个理论被表述为一个线性代数问题，其中必须找到矩阵方程的解，其中A是通过要求所有涡间距离保持固定而获得的（1/2）N（N−1）×N非正常位形矩阵，并且是N-涡强度。为了平衡的存在，A必须有一个非平凡的零空间。我们考虑A的奇异值;当它有一个或多个零奇异值时，A的零空间是非空的，并且对于某些选择的Γ存在一个平衡。通过在平面上随机放置N个点，在数值上找到新的平衡位形，这一般会产生具有空零空间的位形矩阵A。使用A的k个最小奇异值的平方和作为“棘轮”，我们“热波动”的配置，允许每个点在平面上执行随机行走，只保留那些配置，减少这个数量在下一步。因此，配置被驱动为具有零空间（A）=k>0的配置。这些收敛态不一定在它们的初始构型附近，通常它们是不对称的，并且我们经常可以将相同的初始态驱动到几个不同的平衡。还描述了一种反向棘轮方法，该方法可以产生将演化到指定平衡状态的初始条件。一旦达到收敛的最终状态，A的完全奇异值分解被用于计算A的零空间的最优基集，从而计算所有允许的Γ。奇异值的分布给出了关于每个平衡态的大小（由弗罗贝纽斯范数测量），它们彼此之间的距离（间距和密度）以及平面上随机选择的N个点的系统与具有指定秩的最近平衡配置的距离的重要信息，以及它的香农熵。
A theory capable of producing equilibrium configurations of point vortices in the plane, along with a numerical scheme to compute them, is described. The theory is formulated as a problem in linear algebra where one must find solutions to the matrix equation , where A is the (1/2)N(N−1)×N non-normal configuration matrix obtained by requiring that all intervortical distances remain fixed, and are the N-vortex strengths. For existence of an equilibrium, A must have a non-trivial nullspace. We consider the singular values of A; when this has one or more zero singular values, the nullspace of A is non-empty and an equilibrium exists for some choice of Γ. New equilibrium configurations are found numerically by randomly depositing N points in the plane, which generically gives rise to a configuration matrix A with empty nullspace. Using the sum of squares of the k smallest singular values of A as a ‘ratchet’, we ‘thermally fluctuate’ the configuration, allowing each point to execute a random walk in the plane, retaining only those configurations which reduce this quantity at the next step. The configuration is thus driven to one with nullspace (A)=k>0. These converged states are not necessarily nearby their initial configurations, typically they are asymmetric, and often we can drive the same initial state to several different equilibria. A reverse-ratchet method is also described, which can produce initial conditions that would evolve to a specified equilibrium state. Once a converged final state is achieved, the full singular value decomposition of A is used to calculate an optimal basis set for the nullspace of A and thus all allowable Γ. The distribution of the singular values gives important information on the size of each equilibrium state (as measured by Frobenius norm), their distance from each other (spacing and density) and how far a randomly chosen system of N points in the plane is from the nearest equilibrium configuration with a specified rank, as well as its Shannon entropy.
影响因子：
56.9
作者：
Abo-Shaeer, JR;Raman, C;Ketterle, W
通讯作者：
Ketterle, W