Derivation Notes on Particle Distribution Function in SPheRIO
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.