% % Lecture presentation aid for the textbook: % % Laszlo P. Csernai: " Introduction to Relativistic Heavy Ion Collisions" % (John Wiley and Sons Ltd, Chicester, New York, Brisbane, Toronto, % Singapore, 1994; ISBN - 0-471-93420-8) % % Additional Transparencies for Lecture 4a / Chapter 4a (Not in the book) % % The following file is containing transparencies in plain LaTeX format. % Figures are usually not included, they can be pasted from the printed % textbook. Empty space is left for this purpose. Some LaTeX figures are % included in the files. References and Solutions of the assignments are % also not included! % \documentstyle[12pt]{book} % Transparencies in frame keretben \textheight=21.5cm \textwidth=19cm \oddsidemargin=10mm \evensidemargin=10mm %\voffset=-0.50cm \hoffset=-0.15cm % \newcommand{\beq}{\begin{equation}} \newcommand{\beqar}{\begin{eqnarray}} \newcommand{\eeq}[1]{\label{#1} \end{equation}} \newcommand{\eeqar}[1]{\label{#1} \end{eqnarray}} %\newcommand{\beq}{$$} %\newcommand{\eeq}{$$} \newcommand{\B}{$\bullet$ } \newcommand{\PB}{\phantom{$\bullet$} } \newcommand{\LT}{$\leadsto$ } %\renewcommand{\vec}[1]{{\bf #1}} \newcommand{\D}{\bigtriangledown} \newcommand{\etal}{{\it et al.,\/}} \newcommand{\lton}{$\raisebox{-.54ex}{$\stackrel{<}{\sim}$}$} \newcommand{\gton}{$\raisebox{-.54ex}{$\stackrel{>}{\sim}$}$} \newcommand{\gfm}{{\rm GeV/fm}^3} \newcommand{\ipint}{\int \frac{d^3 p}{p^0}} \newcommand{\dpi}{ \frac{d^3 p_i}{p_i^0}} \newcommand{\dpj}{ \frac{d^3 p_j}{p_j^0}} \newcommand{\dpk}{ \frac{d^3 p_k}{p_k^0}} \newcommand{\dpl}{ \frac{d^3 p_l}{p_l^0}} \newcommand{\dpkp}{ \frac{d^3 p'_k}{p^{'0}_k}} \newcommand{\dplp}{ \frac{d^3 p'_l}{p^{'0}_l}} \newcommand{\ip}{\int^{\infty}_{-\infty}dp} \newcommand{\pint}{\int d^3 p} \newcommand{\juttner}{ {1\over{(2 \pi \hbar)^3}} \exp \left( {{\mu - p^\mu u_\mu}\over{T}} \right) } \newcommand{\jut}{ e^{ \left( - {{ p^\mu u_\mu}\over{T}} \right) }} \newcommand{\pmum}{p^\mu u_\mu} \newcommand{\amev}{A$\cdot$MeV} \newcommand{\agev}{A$\cdot$GeV} \newcommand{\transparencyframe}[1]{ \phantom{a} \setlength{\unitlength}{1mm} \begin{picture}(10,10)(30,200) \thicklines \put(95,240){\oval(188,18)[t]} \put(95,20){\oval(188,18)[b]} \put(1,20){\line(0,1){220}} \put(189,20){\line(0,1){220}} \thinlines \put(95,240){\oval(186,16)[t]} \put(95,20){\oval(186,16)[b]} \put(2,20){\line(0,1){220}} \put(188,20){\line(0,1){220}} \put(95,8){\scriptsize\sf L.P. Csernai and Z. N\'eda (1994)} \put(10,235){\parbox[t]{170mm}{\Large\sf \parskip=0.3cm #1}}\end{picture} }%end transparencyframe % \begin{document} \pagestyle{plain} %\setlength{\unitlength}{1mm} %}%end tr-page %\newpage % transparency ===================================================== \transparencyframe{ \setcounter{page}{3800} \setcounter{chapter}{4} \setcounter{section}{2} \setcounter{subsection}{4} \vspace*{-1.2cm} \subsection{Finite QGP volume effects} \B In HI reactions QGP will have a small volume and lifetime \LT\\ - dynamics of the phase transition\\ - finite size of the QGP, have to be studied \B In a small finite system: coexisting phases around $T_c$ ---\\ - do not separate clearly \B Fluid dynamical model calculations predict 10 - 50 fm$^3$ of QGP\\ - strong first order phase transition assumed\\ - with a latent heat near to 2 GeV/fm$^3$ (optimistic SPS) \B Lattice QCD calculations are also using finite volumes!\\ - $T = 1/(N_\tau a) = \beta^{-1}$\\ - $V = (N_\sigma a)^3 $\\ \LT $ L = (N_\sigma / N_\tau) (\hbar c)/T $\\ - e.g. a lattice of $N_\sigma \times N_\tau = 48 \times 16$ \LT lattice volume $$ \Omega_{lattice} = L^3 = \left(\frac{N_\sigma (\hbar c)}{N_\tau T_c}\right)^3 \approx 61 {\rm fm}^3 , $$ - $T_c= 150 $ MeV is assumed \LT - the energy density of the system fluctuates\\ - converges to the energy density of one of the two phases\\ - the system can jump or change from one energy state to the other\\ - the probability of energy density distribution is strongly peaked }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.8cm} \B Consider a system in a canonical ensemble of volume $\Omega$,\\ - in contact with a heat-reservoir of temperature $T$\\ \LT fluctuations play an essential role\\ - the probability, $P({\bf x})$, of finding the system in a state ${\bf x}$: \beq P({\bf x})\sim \exp{(-\beta \cdot F({\bf x}))} \ , \eeq{e1} - $F({\bf x})=f({\bf x}) \cdot \Omega$ is the free-energy of the system\\ - $f({\bf x})$ is the free-energy density\\ - the system is characterized by the energy density, $e$, \LT - ${\bf x} \equiv e$. \B Bag model EOS:\\ - the eq. energy density in hadronic phase: $e_h$, - the eq. energy density QGP: $e_q$: \beq e_h(T)=\frac{\pi^2}{10 \cdot (\hbar c)^3} \cdot T^4 \ \ {\rm and} \ \ e_q(T)=\frac{\pi^2}{(\hbar c)^3}\cdot (\frac{37}{30} \cdot T^4 + \frac{34}{90} \cdot T_c^4) \eeq{e3} \B we have to do is to find the free energy curve, $f(e)$\\ - for arbitrary values of $e$\\ - not just in the well known $e_q$ and $e_h$ points: \beq f[e_q(T)]=-p_q(T)=-\frac{\pi^2}{90 \cdot (\hbar c)^3} \cdot (37 \cdot T^4 - 34 \cdot T_c^4) \ , \eeq{e4} \beq f[e_h(T)]=-p_h(T)=-\frac{\pi^2}{30 \cdot (\hbar c)^3} \cdot T^4 \eeq{e5} \B we use a parametrization\\ - emphasizes the strong first order feature of the phase transition\\ - i.e.: ``best case'', and the reality may be less favorable }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.2cm} PARAMETRIZATION OF THE FREE-ENERGY DENSITY \bigskip \B Free-energy density curve must have two minima, (in $e_h$, and $e_q$)\\ - must diverge to infinity for $e=0$ and $e\rightarrow \infty$ \setlength{\baselineskip}{10pt} {\normalsize\sf The divergence at $e=0$ leads to vanishing population of the zero energy density state. This is not exactly true for a finite system but it can be applied to systems of volume 50 fm$^3$ or larger. } \setlength{\baselineskip}{22pt} \B Parametrizing in the neighbourhood of $e_o$ (local maximum)\\ - $e_o$ is $e_q$ and $e_h$\\ \beq f(e)=f_1+\frac{K_1}{e}+K_2\cdot (e-e_o)+K_3 \cdot (e-e_o)^2 + K_4 \cdot (e-e_o)^3 \eeq{e6} \B Determination of parameters: $f_1$, $K_1$, $K_2$, $K_3$, $K_4$ and $e_o$\\ 1) - From the local maximum $f'(e_o)=0$ \LT $ K_2=K_1/e_0^2 \equiv K_1 \cdot B_2. $\\ 2) From $f'(e_h)=f'(e_q)$ we get $K_3$ and $K_4$ in terms of $K_1$ and $e_o$\\ 3) Still $\exists$ three undetermined parameters: $f_1$, $K_1$ and $e_o$ \LT\\ }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.8cm} SURFACE FREE-ENERGY OF A HADRONIC DROPLET \B Energy density profile of a spherical hadronic droplet can be obtained from: \beq -K \cdot \left(\frac{d^2}{dr^2}+\frac{2}{r} \cdot \frac{d}{dr}\right) e(r) + \frac{\partial f}{\partial e}=0 \ , \eeq{e10} - $K$: coefficient of the surface energy term in the free energy\\ - For our parametrization: \beq \frac{\partial f}{\partial e}= K_1 \cdot \frac{(e-e_q)(e-e_h)(e-e_o)} {e^2e_h^2e_q^2e_o^2} \cdot [e(e_qe_h+e_qe_o+e_he_o)+e_qe_he_o] \ . \eeq{e11} \B Near $T_c$: $R\rightarrow \infty$ \LT eq. (\ref{e10}) approximated as: \beq -K\cdot \frac{d^2e}{dr^2}+\frac{\partial f}{\partial e}=0. \eeq{e12} - Considering the origin as $x\equiv r-R$ \\ - in the neighbourhood of the surface ($x\rightarrow 0$): \beq \frac{d^2e(x)}{dx^2}+\xi_o^{-2}\cdot e(x)=0, \eeq{e13} - $\xi_o^{-2}= K_1 \cdot A_0 /K$\\ - $A_0$ can be calculated using the parametrization of $f(e)$. \B Thus the droplet profile near the surface is: \beq e(x)=\frac{1}{2}\cdot [e_h+e_q+(e_q-e_h)\tanh{(\frac{x}{\xi_o \sqrt{2}})}] +(e_o-\frac{e_h+e_q}{2})\cdot \exp{(-\frac{x^2}{2\xi_o^2})} \eeq{e16} }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.8cm} \B The surface energy of the bubble from: \beq \sigma=K\cdot \int_{-\infty}^{\infty}(\frac{de}{dx})^2 dx \eeq{e17} with the proposed energy-density profile $e(x)$ (\ref{e16}), we get \beq \sigma=\frac{K}{\xi_o}\cdot A_1, \eeq{e18} - $ A_1=\frac{(e_q-e_h)^2}{3\sqrt{2}}+\frac{(e_o-\frac{e_q+e_h}{2})^2 \sqrt{\pi}}{2} $ - Using the parameters: \beqar \sigma&=&50\: {\rm MeV/fm}^2 \nonumber\\ \xi_o&=&3\: {\rm fm} \\ T_c&=&169\: {\rm MeV} \nonumber \eeqar{e21} Thus all parameters are fixed using $\xi_o$ and (\ref{e18}) \LT \beq K_1\cdot A_o \cdot A_1 = \frac{\sigma}{\xi_o} \eeq{e23} \B The parametrized $f(e)$ can then be calculated numerically. }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.8cm} RESULTS \vspace*{17cm} Fig. 1. {\it The parametrized free energy density, $f(e)$, as a function of the energy density, $e$, for systems of three different temperatures near the critical temperature, $T_c=169$ MeV.} }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.8cm} \B The probability of finding the system in a state $e$\\ - for different $\Omega$ volumes is constructed by \beq \frac{P(e)}{P(e_{ref})}=\exp{\left(-\Omega \cdot \frac{f(e)-f(e_{ref})}{T}\right)} \eeq{e28} - $P(e_{ref})=\max\{P(e)\}$ and $e_{ref}=e_q\ \ ({\rm or} \: \: e_h)$\\ \B\B 10 fm$^3$ QGP (expected at AGS): not sufficient to see two phases\\ - Even more so, because of the large hadronic shell\\ \B\B SPS energies: 50 fm$^3$ gives already a possibility\\ - if the parameters of the EOS, $\sigma$ and $\xi$, change to the worse\\ \LT 2 separated phases can be observed at RHIC or LHC energies only. \vspace*{11cm} Fig. 2. {\it The relative probability of finding a state of a given energy density, $e$, in a system of given volume, $\Omega=10, 50$ fm$^3$, at a constant temperature, $T=T_c$.} }%end tr-page % transparency ================================================================ \end{document}