Derivation Notes on Particle Distribution Function in SPheRIO

Page last edited 4,465 days ago
From LaTeX Notes Series
Jump to navigation Jump to search

SPheRIO Implementation of One Particle Distribution Directly in Terms of Freeze-out Surface Information

Cooper-Frye Formulae

Particle distribution in terms of SPH particle degree of freedom reads

<math>\begin{align}

E\frac{d^3N}{dp^3}=\sum_j \frac{\nu_j n_{j\mu}p^{\mu}}{s_j|n_{j\rho}u_j^{\rho}|}\theta(n_{j\delta}p^{\delta})f(u_{j\omega}p^{\omega}) \end{align}</math>

where <math>\theta</math> is the step function (função degrau). Since <math>\frac{\nu_j}{s_j}=V_j</math> is the volume of the <math>j</math>-th SPH particle, namely, the time-like component of freeze-out surface <math>\sigma_{j\mu}</math> in the proper frame where the fluid is static, we have <math>\frac{\nu_j}{s_j}={\sigma}_{j\mu}u_j^{\mu}</math>, this is because the right hand side is a Lorentz scalar. <math>n_{j\mu}</math> is not necessarily normalized in the code, since its magnitude cancels in numerator and denominator.

SPH implementation

  • In the code, all the quantities (such as <math>u_{\mu}, p^{\mu}, n_{\mu},\frac{\nu_j n_{j\mu}}{s_j|n_{j\rho}u_j^{\rho}|}</math> ) are well defined.
  • In weigpr() integral with respect to solid angle was carried out, while in prodis() approximation in angular part was made.
  • The new implementation consists of two steps: 1) Save all the information on freeze-out surface (frozen-out SPH particles) into a file 2) Calculate the flows by directly accessing the files.
  • The only thing left to do is to express <math>dp^3</math> in terms of the coordinates we use in output, namely, <math>\left\lbrace p_{\bot} ,\eta, \phi \right\rbrace</math> . This is because, we need <math>\phi</math> to calculate flows ( <math>v_2,v_1</math> ) and they are functions of <math>p_{\bot}</math> as well as <math>\eta</math>.

Since

<math>\begin{align}

\eta=\frac{1}{2}\ln\frac{|p|+p_L}{|p|-p_L}=-\ln\left[\tan\left(\frac{\theta}{2}\right)\right] \end{align}</math>

we have

<math>\begin{align}

&|p|=p_{\bot}\cosh \eta \\ &p_L=p_{\bot}\sinh \eta \end{align}</math>

and

<math>\begin{align}

&d\eta=-\frac{d\theta}{\sin \theta} \\ &\sin \theta = \frac{1}{\cosh\eta} \\ &\frac{\partial (|p|,\theta,\phi)}{\partial (p_{\bot},\eta,\phi)}= \begin{bmatrix} \frac{\partial |p|}{\partial p_{\bot}} & \frac{\partial |p|}{\partial \eta} & \frac{\partial |p|}{\partial \phi} \\ 0 & \frac{\partial \theta}{\partial \eta} & 0 \\ 0 & 0 & 1 \\ \end{bmatrix} =\begin{bmatrix} \cosh \eta & \frac{\partial |p|}{\partial \eta} & 0 \\ 0 & -\sin \theta & 0 \\ 0 & 0 & 1 \\ \end{bmatrix} =-\cosh \eta \sin \theta \end{align}</math>

Therefore

<math>\begin{align}

dp^3=|p|^2\sin\theta \begin{vmatrix} \frac{\partial (|p|,\theta,\phi)}{\partial (p_{\bot},\eta,\phi)} \end{vmatrix}dp_{\bot}d\eta d\phi =p_{\bot}^2 \cosh^3 \eta \sin^2 \theta dp_{\bot}d\eta d\phi = p_{\bot}^2 \cosh\eta dp_{\bot}d\eta d\phi \end{align}</math>

or

<math>\begin{align}

\frac{d^3N}{dp_{\bot}d\eta d\phi} = \sum_j \frac{\nu_j n_{j\mu}p^{\mu}}{s_j|n_{j\rho}u_j^{\rho}|}\theta(n_{j\delta}p^{\delta})f(u_{j\omega}p^{\omega}) \frac{ p_{\bot}^2 \cosh \eta}{(p_{\bot}^2 \cosh ^2 \eta+m^2)^{1/2}} \end{align}</math>

  • On the other hand, through a very similar procedure, we have
<math>\begin{align}

&y=\frac{1}{2}\ln\frac{E+p_L}{E-p_L} \\ &E=m_{\bot}\cosh y \\ &p_L=m_{\bot}\sinh y \\ &dp^3=\sqrt{m^2+p_{\bot}^2}\cosh y p_{\bot}dp_{\bot}dy d\phi \\ &\frac{d^3N}{dp_{\bot}dy d\phi} = \sum_j \frac{\nu_j n_{j\mu}p^{\mu}}{s_j|n_{j\rho}u_j^{\rho}|}\theta(n_{j\delta}p^{\delta})f(u_{j\omega}p^{\omega})p_{\bot} \end{align}</math>

This expression is quite well known in literature.