Esler J G, Ashbee T L, McDonald N R
Department of Mathematics, University College London, London, United Kingdom.
Phys Rev E Stat Nonlin Soft Matter Phys. 2013 Jul;88(1):012109. doi: 10.1103/PhysRevE.88.012109. Epub 2013 Jul 9.
The statistics of a neutral point-vortex gas in an arbitrary two-dimensional simply connected and bounded container are investigated in the framework of the microcanonical ensemble, following the cumulant expansion method of Pointin and Lundgren [Phys. Fluids 19, 1459 (1976)]. The equation for vorticity fluctuations, obtained when a thermodynamic scaling limit is taken, is solved explicitly. The solution depends on an infinite sequence of negative "domain inverse temperatures," determined by the domain shape, which are obtained from solutions of a "vorticity mode" eigenvalue problem. An explicit expression for the thermodynamic curve relating inverse temperature and energy is found and is shown to depend on the geometry and not on the scale of the domain. Explicit formulas are then obtained for the time variance of the projection of the vorticity field onto each vorticity mode. The results are verified by two methods. First, for a chosen single-parameter family of domains, direct sampling of the microcanonical ensemble is used to demonstrate the accuracy of the formula for the thermodynamic curve. Second, direct numerical simulations are used to verify the formulas for the variance of the projections of the vorticity field, with convincing results.
在微正则系综的框架下，遵循波廷和伦德格伦[《物理流体》19, 1459 (1976)]的累积量展开方法，研究了任意二维单连通有界容器中中性点涡气体的统计特性。当取热力学标度极限时得到的涡度涨落方程被明确求解。该解依赖于由区域形状决定的无穷序列负“区域逆温度”，这些负“区域逆温度”由一个“涡度模式”本征值问题的解得到。找到了关于逆温度和能量的热力学曲线的显式表达式，并表明它依赖于几何形状而非区域的尺度。然后得到了涡度场在每个涡度模式上投影的时间方差的显式公式。结果通过两种方法得到验证。首先，对于选定的单参数区域族，使用微正则系综的直接采样来证明热力学曲线公式的准确性。其次，使用直接数值模拟来验证涡度场投影方差的公式，结果令人信服。
