$^4$ He超流相变中涡旋环的频率动力学
Finite-frequency dynamics of vortex loops at the $^4$He superfluid phase transition
摘要：
$^4$ He超流相变的有限频率动力学可以用热激励涡环对振荡流场的响应来表示。关键参数是环的Hausdorff分形维数$d_H$，其影响动力学，因为环上的摩擦力与环的总周长$P$成比例，其随$P \sim a^{d_H}$变化，其中$a$是环直径。在频率$ω$下求解回路响应的3D Fokker-Planck方程，得到在$T_λ$下变化为$ω^{1/(d_H -1)}$的超流体密度。由于动态指数$z$被确定为$z = d_H-1$，所以这种随$ω$的幂律变化与Fisher、Fisher和Huse发现的标度形式一致。Flory标度的自避免循环给出了一个分形维数的空间维数$d$为$d_H = (d+2)/2$，产生$z = d/2 = 3/2$为d = 3，在完全符合动态标度。
英文摘要：
The finite-frequency dynamics of the $^4$He superfluid phase transition can be formulated in terms of the response of thermally excited vortex loops to an oscillating flow field. The key parameter is the Hausdorff fractal dimension $d_H$ of the loops, which affects the dynamics because the frictional force on a loop is proportional to the total perimeter $P$ of the loop, which varies as $P \sim a^{d_H}$ where $a$ is the loop diameter. Solving the 3D Fokker-Planck equation for the loop response at frequency $ω$ yields a superfluid density which varies at $T_λ$ as $ω^{1/(d_H -1)}$. This power-law variation with $ω$ agrees with the scaling form found by Fisher, Fisher, and Huse, since the dynamic exponent $z$ is identified as $z = d_H-1$. Flory scaling for the self-avoiding loops gives a fractal dimension in terms of the space dimension $d$ as $d_H = (d+2)/2$, yielding $z = d/2 = 3/2$ for d = 3, in complete agreement with dynamic scaling.
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