Research Paper Notes on Flow Fluctuations and Non-flow Analysis

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

 * Eccentricity fluctuations and elliptic flow at RHIC, nucl-th/0607009, by Rajeev S. Bhalerao and Jean-Yves Ollitrault
 * Elliptic flow in the Gaussian model of eccentricity fluctuations, arXiv:0708.0800, by Sergei A. Voloshin et al.
 * Effect of flow fluctuations and nonflow on elliptic flow methods, arXiv:0904.2315, by Jean-Yves Ollitrault, Arthur M. Poskanzer, Sergei A. Voloshin
 * Effects of flow fluctuations and partial thermalization on v(4), arXiv:0907.4664, by Clement Gombeaud, Jean-Yves Ollitrault
 * Understanding anisotropy generated by fluctuations in heavy-ion collisions, arXiv:1107.5485v2, by Rajeev S. Bhalerao, Matthew Luzum, and Jean-Yves Ollitrault
 * Eliminating experimental bias in anisotropic-flow measurements of high-energy nuclear collisions, arXiv:1209.2323, by Matthew Luzum, Jean-Yves Ollitrault
 * Non-Gaussian eccentricity fluctuations, arXiv:1604.07230v2, by Hanna Gronqvist, Jean-Paul Blaizot, and Jean-Yves Ollitrault

Eccentricity fluctuations and elliptic flow at RHIC, nucl-th/0607009, by Rajeev S. Bhalerao and Jean-Yves Ollitrault
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
 * $$\begin{align}

Var{(x_i-\bar{x})}=\sigma^2 \end{align}$$
 * $$\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}$$ 其中利用了每次独立测量的偏差是独立的,所以他们的方差的和等于和的方差.由此得知,平均偏离$$ \sqrt{\sigma} $$随着测量次数而减少.如果体系由$$ N $$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$$ - $$

Effect of flow fluctuations and nonflow on elliptic flow methods, arXiv:0904.2315, by Jean-Yves Ollitrault, Arthur M. Poskanzer, Sergei A. Voloshin
这是著名的通过多粒子关联来获得集体流涨落(标准偏差),从而获得非流贡献的讨论.

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
 * $$\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}$$ 文中涉及的是不同的上下文,当不存在任何关联的时候,$$ v_n=\bar{Q}=0 $$,由于这时平均值为零,我们只能讨论由于涨落导致的流
 * $$\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}$$ 这时候$$ Q $$的大小与$$ \sqrt{N} $$成正比.

Eq.(11) 这个表达式是把测量值$$v$$在平均值$$$$附近泰勒展开$$f(v)=f((v-)+)$$,然后去平均.注意到一次项为$$f()(v-)$$,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$$f(v)=v^2$$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
 * $$\begin{align}

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

Effects of flow fluctuations and partial thermalization on v(4), arXiv:0907.4664, by Clement Gombeaud, Jean-Yves Ollitrault
本文档除了包括推导, 疑惑 外,做 读书重点 的记录 $$ $$

Understanding anisotropy generated by fluctuations in heavy-ion collisions, arXiv:1107.5485v2, by Rajeev S. Bhalerao, Matthew Luzum, and Jean-Yves Ollitrault
本文档除了包括推导, 疑惑 外,做 读书重点 的记录 $$ $$

Eliminating experimental bias in anisotropic-flow measurements of high-energy nuclear collisions, arXiv:1209.2323, by Matthew Luzum, Jean-Yves Ollitrault
本文档除了包括推导, 疑惑 外,做 读书重点 的记录 $$ $$

Non-Gaussian eccentricity fluctuations, arXiv:1604.07230v2, by Hanna Gronqvist, Jean-Paul Blaizot, and Jean-Yves Ollitrault
本文档除了包括推导, 疑惑 外,做 读书重点 的记录 $$ $$