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


 * $$\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}$$

where $$\theta$$  is the step function (função degrau). Since $$\frac{\nu_j}{s_j}=V_j$$ is the volume of the  $$j$$-th SPH particle, namely, the time-like component of freeze-out surface  $$\sigma_{j\mu}$$  in the proper frame where the fluid is static, we have  $$\frac{\nu_j}{s_j}={\sigma}_{j\mu}u_j^{\mu}$$, this is because the right hand side is a Lorentz scalar. $$n_{j\mu}$$ 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 $$u_{\mu}, p^{\mu}, n_{\mu},\frac{\nu_j n_{j\mu}}{s_j|n_{j\rho}u_j^{\rho}|}$$ ) 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 $$dp^3$$  in terms of the coordinates we use in output, namely,  $$\left\lbrace p_{\bot} ,\eta, \phi \right\rbrace$$ . This is because, we need  $$\phi$$  to calculate flows ( $$v_2,v_1$$ ) and they are functions of  $$p_{\bot}$$  as well as  $$\eta$$.

Since


 * $$\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}$$

we have


 * $$\begin{align}

&|p|=p_{\bot}\cosh \eta \\ &p_L=p_{\bot}\sinh \eta \end{align}$$

and


 * $$\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}$$

Therefore


 * $$\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}$$

or


 * $$\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}$$


 * On the other hand, through a very similar procedure, we have


 * $$\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}$$

This expression is quite well known in literature.