Research Paper Notes on Flow Fluctuations and Non-flow Analysis

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Research Paper Notes on Fluctuation and Non-flow Analysis

本文档除了包括推导,疑惑外,做读书重点的记录

参考文献

  • nucl-th/0607009 Eccentricity fluctuations and elliptic flow at RHIC
  • arXiv:0708.0800 Elliptic flow in the Gaussian model of eccentricity fluctuations
  • arXiv:0904.2315 Effect of flow fluctuations and nonflow on elliptic flow methods
  • arXiv:0907.4664 Effects of flow fluctuations and partial thermalization on v4
  • arXiv:1209.2323 Eliminating experimental bias in anisotropic-flow measurementsof high-energy nuclear collisions

nucl-th/0607009 Eccentricity fluctuations and elliptic flow at RHIC

Eq.(11)注意到平均值的方差(variance)实际上随着测量次数的增加而减少,因为按定义

<math>\begin{align}

Var{(x_i-\bar{x})}=\sigma^2 \end{align}</math>

<math>\begin{align}

Var\left({\frac{\sum_{i=1}^N(x_i-\bar{x})}{N}} \right)=\frac{\sum Var{(x_i-\bar{x})}}{N^2}=\frac{N\sigma^2}{N^2}=\frac{\sigma^2}{N} \end{align}</math> 其中利用了每次独立测量的偏差是独立的,所以他们的方差的和等于和的方差.由此得知,平均偏离<math> \sqrt{\sigma} </math>随着测量次数而减少.如果体系由<math> N </math>个核子构成,那么如果一个物理量由所有核子的平均值决定,这个物理量的平均值的涨落随着核子数目的增加而减少.

Eq.(12)右边来自类似<math> <x^2\cos^2\phi>-<x^2\sin^2\phi> </math>

arXiv:0708.0800 Elliptic flow in the Gaussian model of eccentricity fluctuations

arXiv:0904.2315 Effect of flow fluctuations and nonflow on elliptic flow methods

Eq.(3)我们知道,当存在流的时候,粒子之间存在连接关联.这是流的定义是

<math>\begin{align}

v_n\equiv<\cos[n(\phi-\Psi_r)]>=\frac{1}{N}\sum_{i=1}^N\cos[n(\phi_i-\Psi_r)] \sim\frac{1}{N}\sum_{i=1}^N\cos(n\phi_i)=\frac{Q}{N} \end{align}</math> 文中涉及的是不同的上下文,当不存在任何关联的时候,<math> v_n=\bar{Q}=0 </math>,由于这时平均值为零,我们只能讨论由于涨落导致的流

<math>\begin{align}

Var(Q)=Var\sum_{i=1}^N\cos(n\phi_i)=N\times Var[\cos(n\phi_i)]=N\sigma^2\sim N \end{align}</math> 这时候<math> Q </math>的大小与<math> \sqrt{N} </math>成正比.

Eq.(11) 这个表达式是把测量值<math>v</math>在平均值<math><v></math>附近泰勒展开<math>f(v)=f((v-<v>)+<v>)</math>,然后去平均.注意到一次项为<math>f(<v>)(v-<v>)</math>,取平均后为零,故而没有贡献.如果我们仅仅考虑到两次项为止,即得.

值得指出当<math>f(v)=v^2</math>时对应常见的标准偏差的关系式,这时可以直接推导.

Eq.(13) 因为仅仅考虑到两次项,我们可以在计算中随意的添加一些高阶的项,如下

<math>\begin{align}

&f(v)=v^4\\ &<v^4>=<v>^4+\frac{\sigma_v^2}{2}4\times3<v>^2+\dots=<v>^4+\frac{\sigma_v^2}{2}4\times3<v>^2+\sigma_v^4\\ &<v^2>=<v>^2+\sigma_v^2\\ &(2<v^2>^2-<v^4>)^{1/2}=[<v>^4-2\sigma_v^2<v>^2+\sigma_v^4]^{1/2}=<v>^2-\sigma_v^2 \end{align}</math>

arXiv:0907.4664 Effects of flow fluctuations and partial thermalization on v4

arXiv:1209.2323 Eliminating experimental bias in anisotropic-flow measurementsof high-energy nuclear collisions