% % 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) % % Transparencies for Lecture 5 / Chapter 5 % % 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{\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: Inroduction to Relativistic Heavy Ion Collisions (Wiley, 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{chapter}{4} \vspace*{-1.9cm} \chapter{Relativistic Fluid Dynamics} \B Frequently used for heavy ion reactions at all energies.\\ - Transparent and analytic or quasi analytic solutions.\\ - Input is the equation of state (EOS). \B Idealized continuum description assuming local equilibrium.\\ - Microscopic numerical models account better for fluctuations and finite particle effects. \section{Energy domains, stopping power} Heavy ions colliding at ultra-relativistic energies:\\ \B at lower energies the stopping power: sufficient to stop the nuclear matter \LT nearly equilibrated hot and dense system in the C.M. \B at very high energies the baryons can penetrate through each other initially: development of the central zones and the side (fragmentation) regions is different. }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-1.5cm} \subsection{Stopping energy region} 1) \B BEVALAC energies:\\ - Central, highly excited zone with high pressure develops \LT collective flow\\ - Mean free path is not negligibly small at these energies but it is still smaller than the size of the system \LT \\ - Fluid dynamical effects are still observable. 2) \B Increasing energies:\\ - nucleon nucleon cross section becomes more and more forward peaked \LT \\ - Longer mean free path and more transparency. \LT \\ - Two fluid and three fluid models. 3) \B QCD plasma threshold:\\ - degrees of freedom increase rapidly \LT \\ - increased stopping power.\\ \B Stopping power studies, (based on p+A 20-100 GeV):\\ - heavy target nucleus slows down the incident proton by $\Delta y \approx 2.5 $. \LT\\ - Stopping for the heaviest nuclei up to about 15-35 GeV/nucleon Lab. beam energy. \\ - One-fluid description is OK, or two-fluid dynamics to allow for a partial transparency at the initial stages. }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-1.5cm} \subsection{Transparent reactions, mid rapidity region} \B Energies above E=30-100 GeV/nucleon stopping not anticipated. \\ - Even in central hits: baryons/valence quarks slow down only slightly.\\ - At the initial impact target and projectile nucleons interact;\\ - (int.: represented by gluon or chromoelectric fields lasts ...)\\ - Leading valence quarks (carrying the baryon chg.) propagate further.\\ - Large central energy density - fields representing this interaction.\\ - Later fields neutralize: pion and other hadron pairs (uniform in $y$) \LT \B Energy density is also "uniform" in space time. - Described more easily than the other regions\\ - Simple and imperative theoretical reaction description\\ \vskip 7.8truecm %\caption[New] Figure 5.1 {\it Space-time picture of an ultra-relativistic heavy ion collision. The projectile and the target and later their valence quarks propagate close to the light-cone. The thermalized matter in the mid-rapidity region changes its state according to the proper time counted from the event of collision. } %\label{f4.1} %\end{figure} }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ %\vspace*{-1.0cm} \subsection{Transparent reactions, fragmentation region} - The most involved description of the so called\\ - Based on p+p and p+A experiments \LT \\ \B Baryons after collision will be in a rapidity range near their original rapidity.\\ \B \LT Net baryon charge: distributed in two peaks, these are the so called {\bf fragmentation regions}. - However, target nucleons still accelerated by the projectile\\ \ \ \ \ and vice versa \LT \\ Increase in the energy density: sufficient to reach the QGP threshold }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-1.0cm} \section{Perfect fluid dynamics} - Chapret 3: Conservation laws of a continuum can be expressed in a differential form via the energy-momentum tensor $\sim$ FD. - Validity of equations is wider! \\ \B Dense liquids can also be described by these equations. The equations of Perfect Fluid Dynamics are the conservation laws \beq N^\mu ,_\mu = 0 \ \ \ \ {\rm or} \ \ \ \partial_\mu (n u^\mu) = 0, \eeq{a4.36} and \beq T^{\mu\nu}_{,\mu} = 0 \ \ \ \ {\rm or} \ \ \ \partial_\mu (T^{\mu\nu}) = 0. \eeq{a4.37} Using $u^\mu = (\gamma, \gamma \vec{v}), \ $ $ w = e +P,\ $ $ T^{ik} = w \gamma^2 v_i v_k + P \delta_{ik}, \ $\\ $ T^{0i} = -T_{0i} = w \gamma^2 v_i, \ $ $ T^{00} = T_{00} = (e + P v^2) \gamma^2 $ ,\\ and introducing the apparent density \beq {\cal N} \equiv n \gamma = {\sf n} , \eeq{a4.37c} and the momentum current density and apparent energy density: \beq \vec{\cal M} \equiv T^{0i} = w \gamma^2 \vec{v}, \eeq{a4.38a} \beq {\cal E} \equiv T^{00} = (e+P \vec{v}\,^2) \gamma^2 , \eeq{a4.38b} the equations of fluid dynamics take the more familiar form. }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.8cm} The continuity equation \beq (\partial_t + \vec{v} \, {\rm grad} ) {\cal N} = - {\cal N} {\rm div}\vec{v}. \eeq{a4.38} The energy and momentum conservation will take the form \beqar (\partial_t + \vec{v} \, {\rm grad} ) \vec{\cal M} & = & - \vec{\cal M} ({\rm div}\vec{v}) - {\rm grad} P , \\ (\partial_t + \vec{v} \, {\rm grad} ) {\cal E} & = & - {\cal E} {\rm div}\vec{v} - {\rm div} (P \vec{v}). \eeqar{a4.39} \B Last two equations: Euler equation of fluid dynamics, and Energy conservation. \B ${\cal N}$, ${\cal E}$, $\vec{\cal M}$ not directly from EOS, but solve algebraic equations (5.3-5), to obtain the thermodynamical quantities.\\ - Equations of fluid dynamics are not complete without an EOS.\\ \B Viscous fluid dynamics is seldom used in relativistic physics. {\small\sf\baselineskip=0pt Due to the fact that there are still questions around the proper relativistic generalization of viscous fluid dynamics [10]. It was shown that the usual relativistic generalizations of viscous fluid dynamics may lead to unstable solutions. Dissipative effects are, nevertheless, important as many non- relativistic calculations indicate. There exist a few relativistic viscous calculations which can be viewed upon as approximations.} }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-1.0cm} \section{Numerical solutions} - First solved numerically in 1975 in Los Alamos [11-13] for a heavy ion collision in full complexity and 3-dim. - By now several $\exists$ research groups: 3-dimensional relativistic fluid dynamical codes. - Considerable computing power and extensive numerical work in developing proper numerical methods. }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-1.0cm} \subsection{Equation of state} - The nuclear Equation of State (EOS) was discussed in the previous chapter. - Here just the most basic information: \B In the {\bf transport theory} the pressure is that of an ideal gas\\ - (because we neglected all interactions among the particles): \LT (?) - FD valid only under the assumptions: local equilibrium everywhere, (J\"uttner distribution). Consequently, our EOS: classical relativistic ideal gas. \B Restriction is not necessary: FD can be derived in many different ways,\\ - (only energy, momentum and mass conservation). \LT \B EOS should satisfy all thermodynamical requirements.\\ \B EOS can be more realistic than a simple ideal gas.\\ }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.8cm} \B Basic features of the nuclear EOS: - Stable ground state (T=0) at the normal nuclear density, ${\sf n}_0$ = 0.145-0.17 fm$^{-3}$ with -8 MeV/nucleon energy. \\ - At higher temperatures the specific energy is higher. \\ - The curvature of the specific energy curve is characterized by the {\bf nuclear compressibility} \\. %\begin{figure}[hbtp] %\begin{center} \setlength{\unitlength}{0.45mm} \begin{picture}(200,190)(-80,0) \put(5,10){\vector(1,0){185}} \put(8,60){\line(1,0){180}} \put(8,20){\line(1,0){4}} \put(10,5){\vector(0,1){165}} \put(192,2){\sf n} \put(80,120){\sf T$>$0} \put(165,68){\sf T=0} \put(72,2){${\sf n}_0$} \put(2,182){$\varepsilon [MeV]$} \put(2, 63){$0$} \put(-15, 23){$-8$} \end{picture} \setlength{\unitlength}{1mm} %\end{center} %\caption[New] Figure 5.2 {\it The nuclear EOS: the specific energy as function of the nuclear density. The parameter is the temperature, $T$.} %\label{f4.2} %\end{figure} - EOS represented via the specific energy, $\varepsilon$: $$ \varepsilon(n,T) = m_0 + \varepsilon_0(n) + \varepsilon_T(n,T), $$ where $\varepsilon_0(n) \equiv \varepsilon(n,T=0)$, and $m_0$ is he rest mass.\\ - This separation of the thermal, $\varepsilon_T$, and compressional, $\varepsilon_0$, energy is unique, but (!) \\ - $\varepsilon(n,T)$ is not a thermodynamical potential! }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-1.9cm} \subsection{Flow characteristics from numerical solutions} \vspace*{-0.6cm} Examples from numerical results of fluid dynamical calculations.\\ Heavy ion collisions by relativistic fluid dynamical model (Los Alamos): %\begin{figure}[hbtp] \vskip 14truecm %\caption[F:AH77-f.2-f.3] Figure 5.3a {\it Change of density configuration in a relativistic heavy ion collision. The darkness illustrates the density of nuclear matter. (Each dot represents a so called ``marker particle'' of PIC method. If several marker particles are exactly above each other we see only one dot as their projection.) One marker particle represents a fraction of baryon charge, typically 1/100-1/500. The most apparent feature is the relatively sharp shock front at the initial compression. From ref. [12]} %\label{f4.3} %\end{figure} }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{21cm} Figure 5.3b {\it continued} }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.8cm} Fig. 5.4 shows the results of a Soviet group: %\begin{figure}[hbtp] \vskip 12truecm %\caption[F:RR81-f.??] Figure 5.4 {\it The maximum of the nuclear density as a function of time from a numerical calculation for the reaction: $3.6\ GeV/nucl. \ \ C + Pb$ at different impact parameters. The parameter of the curves is the impact parameter, denoted by $\rho$ in units of $R_t+R_p$, \ $b=\rho(R_p+R_t)$. Curve 1: $\rho=0.1$, Curve 2: $\rho=0.3$, Curve 3: $\rho=0.5$, Curve 4: $\rho=0.7$, Curve 5: $\rho=0.9$. From [17]} %\label{f4.4} %\end{figure} \B Increase of energy density is larger than the increase of baryon density,\\ - due to the compressional and thermal energy. - In ideal gas: No compressional energy \LT very {\it soft}.\\ }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.8cm} The time dependence of the energy density reachable in a HI collision\\ calculated at the University of Frankfurt. \B Large projectile and target mass is more important to reach high energy density for as many nucleons as possible.\\ \B Increasing beam energy is secondary but it also results in an increase of energy density. %\begin{figure}[hbtp] \vskip 11.5truecm %\caption[F:ReG87-f] Figure 5.5 {\it The number of nucleons that exceed a given energy density, $e_{crit}$, at a given time, $t$, in a heavy ion collision. The energy density of the ground state nuclear matter is $e_0 = E/V = n_0 \varepsilon_0(n_0) \approx 0.15 GeV/fm^3$ so that $e_{crit} = 2 GeV/fm^3$ is already more than ten times larger than that of the normal nuclear matter. $A$ is the number of nucleons that are in the region where $e>e_{crit}$ at different times $t$. From [19] } %\label{f4.5} %\end{figure} }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.8cm} \B Sharp fronts of widths of a few Fermi.\\ - Speed of the incoming projectile and the flow velocity are {\em supersonic}.\\ - Sound speed in (g. s.) nuclear matter is about $v_{sound} \approx 0.2 c$.\\ \B {\em In the Eulerian fluid dynamics}: Sharp fronts = discontinuity\\ - For (``infinitely sharp'') discontinuities: Rankine-Hugoniot-Taub equations [20, 56]\\ \B {\em Physically} shock waves are not infinitely sharp.\\ - Dissipative processes (viscosity, heat conductivity, reaction rates, incomplete equilibrium) \LT finite width \\ \B Final state is, however, identical \\ - Determined by the energy, momentum and baryon conservation.\\ - Dissipative processes \LT Internal structure of the front\\ $\exists$ problem: Eulerian numerical codes $\sim$ shock fronts of finite width.\\ - Numerical method \LT so called {\em numerical viscosity}\\ - The final state is, however, correct\\ - Only the internal structure (and width) of the front depends on the method\\ }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-1.5cm} \subsection{Conclusions} \begin{description} \item[1] Final states of compression fronts: Taub equations\\ - Final state depends only on the Equation of State of the nuclear matter\\ - Compression phenomena include shocks, detonations, deflagrations, etc.\\ \item[2] Realistic descriptions: dissipation studied in two ways: \begin{quote} (2a) Small deviations from eq. \LT expansion around $f^{eq.}$\\ - {\em Chapman-Enskog expansion} \LT {\em Navier-Stokes} equations \end{quote} \begin{quote} (2b) Large deviations: other approximations:\\ - Multi component fluid dynamics, Cascade models, Mean field theories with cascading particles like, Vlasov-Uehling-Uhlenbeck (VUU), Boltz\-mann-Uehling-Uhlenbeck (BUU), Landau-Vlasov (LV) or Molecular Dynamical models. \end{quote} \item[3] Perfect relativistic fluid dynamics: {\bf no} entropy increase, adiabatic.\\ - Entropy: generated in discontinuities (shock fronts) only. \end{description} }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-1.5cm} \section{Numerical methods} $\exists$ Two basic approaches: {\bf Eulerian} and {\bf Lagrangian} methods\\ \B Eulerian method: fix computational computational grid in the space\\ - Fluid flows across these grid cells\\ - Used if flow around some object (ship, pipeline, channel)\\ - Practical for incompressible fluids\\ \B Lagrangian method: reference frame is fixed to the fluid\\ - The fluid does not flow in or out of the cells\\ - Cells change their shape, position, volume if the fluid is compressible\\ - For big changes in compression, no fixed objects\\ - Not for: 3-dim., large shears, turbulent flow\\ \ \ \ \ because the cells neighbors are not the same\\ \B In relativistic HI physics two large scale solution methods are used:\\ - Particle in Cell method\\ - Flux Corrected Transport method. Here demonstrated following Maruhn [22]. }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-1.5cm} \subsection{The Particle in Cell (PIC) method} Developed at Los Alamos by Amsden and Harlow [23,24] - Eulerian with a Lagrangian admixture\\ - via inclusion `marker particles' (MP) that flow through an Eulerian-grid\\ - $N$, $\vec{M}$, and $E$ are defined on this space-fixed grid\\ - Marker particles serve the exchange of these between neighboring cells\\[0.7ex] \B Divided into two calculational phases:\\ (i) Updating $\vec{M}$ and $E$ without transport between cells: \beq \frac{\partial \vec{M}}{\partial t} = - \ \nabla p, \ \ \ \frac{\partial E}{\partial t} = - \ \nabla (p\ \vec{v}). \eeq{m-148} (ii) Momentum and energy content of cells distributed evenly among the marker particles in the cell\\ (iii) MPs then assigned a velocity by interpolating the velocities of the neighboring grid points to their position\\ - then MPs moved using that velocity.\\ - if an MP crosses cell boundary, its momentum and energy content are passed to the new cell\\ \B The transport is "quantized" in the method: \\ - the number of particles in each cell determines in which increments mass, energy, and momentum is exchanged\\ - Constraining in the final stages of the reaction when densities are low\\ Precise formulation is given by Harlow, Amsden and Nix [21] }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-1.5cm} \subsection{The Flux Corrected Transport algorithm (FCT)} Developed originally by Boris and Book [25-27].\\ - Based on: any one-dimensional conservation equation of type (like FD) \beq \frac{\partial n}{\partial t} + \frac{\partial (n \cdot v)}{\partial x} = S \ . \eeq{m-149} - 3-dim.: one after the other axes "time step splitting". \\ \B Lagr. transport: moving density, $n$, according to the flow velocity\\ - Reinterpolating the result onto the space-fixed Eulerian grid \LT \\ - Equations for the density, $n$, after the motion: \beq \bar{n}_j = \frac{1}{2} Q_j^{+2} (n_{j+1} - n_j) + \frac{1}{2} Q_j^{-2} (n_{j-1} - n_j) + + (Q_j^+ + Q_j^-) n_j + \Delta t \ S_j , \eeq{m-150} \beq {\sf with} \ \ \ Q_j^+ = \epsilon_j^- / (\epsilon_{j+1}^+ - \epsilon_j^-), \ \ \ \ Q_j^- = 1 - Q_{j-1}^+ , \ \ \ {\sf and} \eeq{m-151} \beq \epsilon_j^\pm = \frac{1}{2} \pm v_j \Delta t / \Delta x . \eeq{m-152} - These finite difference equations \LT strong diffusion\\ - Easily seen for the simple case $v = S = 0$: \beq \bar{n}_j = n_j + \frac{1}{8} ( n_{j+1} - 2 n_j + n_{j-1}) . \eeq{m-153} - Dangerous for numerical accuracy: smears out all structures\\ - Prevents instability near shock fronts\\ \B FCT: remove the diffusion at safe points, using an anti-diffusion step: \beq \tilde{n}_j = \bar{n}_j - \frac{1}{8} ( \bar{n}_{j+1} - 2 \bar{n}_j + \bar{n}_{j-1}) . \eeq{m-154} - but cut off the correction for new extrema in the profile\\ - Physical extrema can still be created in the first diffusive step\\ }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-1.5cm} \section{Simple analytic solutions --- Shock waves} \noindent \B Some important basic solutions:\\ - Shock, detonation and deflagration waves, Bjorken's fluid dynamical model, approximate spherical expansion model, Landau's fluid dynamical model, ... \\ - Central density profile %\begin{figure}[hbtp] %\noindent %\begin{center} \setlength{\unitlength}{0.4mm} \begin{picture}(370,216)(20,0) %$fig61.pit 2 \thinlines \put(26.29,36.76){\vector(1,0){412.38}} \put(223.50,27.33){\vector(0,1){146.43}} \thicklines \put( 69.02,65.97){\oval(50.40,58.69)[br]} \put(125.26,65.97){\oval(62.08,16.96)[tl]} \put(124.90,88.59){\oval(61.35,28.27)[br]} \put(221.31,89.06){\oval(131.47,29.22)[t]} \put(314.43,89.53){\oval(54.78,30.16)[bl]} \put(315.16,65.97){\oval(53.32,16.96)[tr]} \put(358.62,65.97){\oval(33.60,58.69)[bl]} \thinlines \put(167.99,30.16){\line(0,1){108.53}} \put(144.62,30.16){\line(0,1){108.95}} \put(275.36,29.22){\line(0,1){108.00}} \put(299.46,29.22){\line(0,1){107.25}} \put(328.68,47.12){\vector(-1,0){18.41}} \put(105.18,47.12){\vector(1,0){27.75}} \put(236.65,173.57){\makebox(0,0)[tl]{n}} \put(143.89,161.16){\makebox(0,0)[tl]{Shock}} \put(273.17,161.16){\makebox(0,0)[tl]{Shock}} \put(143.89,151.73){\makebox(0,0)[tl]{front}} \put(273.17,151.73){\makebox(0,0)[tl]{front}} \put(176.75,85.76){\makebox(0,0)[tl]{$v = 0$}} \put(144.62,132.88){\vector(-1,0){11.36}} \put(299.46,132.88){\vector(1,0){11.54}} \put(44.55,181.11){\makebox(0,0)[tl]{C.M. frame}} \put(436.77,49.01){\makebox(0,0)[tl]{z}} \put(96.41,25.45){\makebox(0,0)[tl]{Projectile}} \put(305.30,25.45){\makebox(0,0)[tl]{Target}} \put(448.75,11.31){\line(0,1){188.93}} \put(448.75,200.24){\line(-1,0){431.35}} \put(17.40,200.24){\line(0,-1){188.93}} \put(17.53,11.31){\line(1,0){431.35}} \end{picture} \setlength{\unitlength}{1mm} %\end{center} %\caption[New] Figure 5.6 {\it Schematic density profile of two colliding nuclei along the central beam axis, $z$. In supersonic impact the density increase happens in shock fronts. The matter ia at rest in the middle. } %\label{f4.6} %\end{figure} }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.8cm} The shock can be approximated by a one dimensional problem %\begin{figure}[hbtp] %\noindent %\begin{center} \setlength{\unitlength}{0.5mm} \begin{picture}(400,200)(-30,0)%$fig62.pit 3 \thinlines \put(34.33,27.33){\vector(0,1){146.43}} \put(24.10,36.76){\vector(1,0){221.84}} \thicklines \put(32.14,89.06){\oval(131.47,29.22)[tr]} \put(125.26,89.53){\oval(54.78,30.16)[bl]} \put(110.29,74.45){\line(1,0){130.89}} \thinlines \put(86.19,29.22){\line(0,1){108.00}} \put(110.29,29.22){\line(0,1){108.00}} \put(139.50,47.12){\vector(-1,0){18.41}} \put(110.29,132.88){\vector(1,0){11.54}} \put(47.48,173.57){\makebox(0,0)[tl]{n}} \put(84.00,161.16){\makebox(0,0)[tl]{Shock}} \put(84.00,151.73){\makebox(0,0)[tl]{front}} \put(45.28,85.76){\makebox(0,0)[tl]{$v = 0 $}} \put(247.60,49.01){\makebox(0,0)[tl]{z}} \put(146.81,49.95){\makebox(0,0)[tl]{$v_1 > v_s$}} \put(124.17,63.14){\makebox(0,0)[tl]{"1"}} \put(60.62,63.14){\makebox(0,0)[tl]{"2"}} \put(153.38,160.37){\makebox(0,0)[tl]{C.M frame}} \put(259.58, 11.31){\line(-1,0){246.43}} \put(259.58,190.24){\line(-1,0){246.43}} \put( 13.15,11.31){\line(0,1){178.93}} \put(259.58,11.31){\line(0,1){178.93}} \end{picture} \setlength{\unitlength}{1mm} %\end{center} %\caption[New] Figure 5.7 {\it One of the shock fronts presented in the previous Figure in the c.m. frame } %\label{f4.7} %\end{figure} \noindent In this figure the local sound velocity is denoted by $v_s$. }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.8cm} Shock front viewed from the front's reference frame,\\ - where the front is at rest and the matter flows across %\begin{figure}[hbtp] %\noindent %\begin{center} \setlength{\unitlength}{0.5mm} \begin{picture}(400,250)(-30,0)%$fig62a.pit 4 \thinlines \put(24.10,36.76){\vector(1,0){221.84}} \thicklines \put(32.14,89.06){\oval(131.47,29.22)[tr]} \put(125.26,89.53){\oval(54.78,30.16)[bl]} \put(110.29,74.45){\line(1,0){130.89}} \thinlines \put(86.19,29.22){\line(0,1){108.00}} \put(110.29,29.22){\line(0,1){107.25}} \put(84.00,161.16){\makebox(0,0)[tl]{Shock}} \put(84.00,151.73){\makebox(0,0)[tl]{front}} \put(247.60,49.01){\makebox(0,0)[tl]{z}} \put(30.00,49.01){\makebox(0,0)[tl]{$v_2 < v_s$ }} \put(146.81,49.95){\makebox(0,0)[tl]{$v_1 > v_s$}} \put(124.17,63.14){\makebox(0,0)[tl]{"1"}} \put(60.62,63.14){\makebox(0,0)[tl]{"2"}} \put(95.00,128.00){$V_w$} \put(153.38,175.37){\makebox(0,0)[tl]{Shock frame}} \put(259.58, 11.31){\line(-1,0){246.43}} \put(259.58,190.24){\line(-1,0){246.43}} \put(259.58,11.31){\line(0,1){178.93}} \put(13.15, 11.31){\line(0,1){178.93}} \put(81.07,81.05){\vector(-1,0){11.10}} \put(139.50,47.12){\vector(-1,0){25.86}} \end{picture} \setlength{\unitlength}{1mm} %\end{center} %\caption[New] Figure 5.8 {\it The same shock front presented in the previous Figure but in the frame of the shock front. In this frame the shock front is kept at the same location, matter is flowing in and out but with different velocities } %\label{f4.8} %\end{figure} \B Stationary front \LT energy, particle number and momentum are constant\\ - in a comoving box \LT \\ - Incoming and outgoing ${\cal N, E}$, $\vec{\cal M}$ are equal\\ Denote two sides: "1" and "2"; Difference of a quantity, $Q$,:\\ - $[Q] = Q_2 - Q_1$ (e.g. $[v] = v_2 - v_1 $ in the fronts frame) }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.8cm} \B Eulerian FD: sharp surface, flow is discontinuous\\ - Unit normal vector of the surface of disc. is $\Lambda^\mu$, in the space-time. It can be space-like or time-like: \beq \Lambda^\mu \Lambda_\mu = \left\{ \begin{array}{ll} +1 &:\ time-like \ \ surface \\ -1 &:\ space-like \ \ surface \end{array} \right. . \eeq{6.0} Energy and momentum flow are identical on the two sides: \beq [ T^{\mu\nu} \Lambda_\nu ] = 0, \eeq{6.1} and the particle number should also be conserved: \beq [ N^\mu \Lambda_\mu ] = 0. \eeq{6.2} Eqs. (5.17-18) are the relativistic Rankine-Hugoniot equations\\ - first by A. Taub [20] (space-like only; extended to timelike in [56])\\ \B Space like shocks propagate with $v_{shock} < 1$, \LT \\ - The points of the front are in causal connection\\ - Space-like fronts can be transformed into their local rest frame\\ \B Time-like discontinuities can be transformed into local rest frame where the matter at a time $t$ goes over a sudden transition everywhere (e.g. phase transition).\\ - $v_{shock} > 1$ possible: matter or inf. does not move with the front\\ - Caused by the initial conditions, not by neighboring fluid elements\\ - Example is very slow homogeneous compression or heating:\\ \phantom{- }\ Temperature may exceed a critical value at the same time\\ \phantom{- }\ and phase transition may take place\\ }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-2.1cm} %\begin{figure}[hbtp] %\noindent %\begin{minipage}[t]{7.5cm} \setlength{\unitlength}{0.55mm} %{\unitlength 0.9pt \begin{picture}(200,200)(-30,0)%$fig63a.pit 0 \thinlines \put(25.56,18.85){\vector(1,0){173.40}} \put(26.29,18.85){\vector(0,1){150.90}} \thicklines \put(26.29,18.85){\vector(1,1){151.11}} \put(196.48,30.16){\makebox(0,0)[tl]{x}} \put(34.33,168.70){\makebox(0,0)[tl]{t}} \put(34.33,77.28){\makebox(0,0)[tl]{"1"}} \put(157.03,134.77){\makebox(0,0)[tl]{"2"}} \put(75.96,170.58){\makebox(0,0)[tl]{Space-like}} \put(75.96,161.16){\makebox(0,0)[tl]{surface}} \put(86.92,49.01){\vector(4,1){14.26}} \put(109.56,67.86){\makebox(0,0)[tl]{$\Lambda^\mu$}} \put(134.39,119.69){\makebox(0,0)[tl]{Light}} \put(134.39,105.27){\makebox(0,0)[tl]{cone}} \end{picture} %}%endunitlength %\end{minipage} \ %\begin{minipage}[t]{7.5cm} %{\unitlength 0.9pt \begin{picture}(200,200)(-30,0)%$fig63b.pit 0 \thinlines \put(25.56,18.85){\vector(1,0){173.40}} \put(26.29,18.85){\vector(0,1){150.90}} \thicklines \put(26.29,18.85){\vector(1,1){151.11}} \put(196.48,30.16){\makebox(0,0)[tl]{x}} \put(34.33,168.70){\makebox(0,0)[tl]{t}} \put(34.33,77.28){\makebox(0,0)[tl]{"1"}} \put(157.03,134.77){\makebox(0,0)[tl]{"2"}} \put(154.84,75.40){\makebox(0,0)[tl]{Time-like}} \put(154.84,65.03){\makebox(0,0)[tl]{surface}} \put(42.36,147.02){\makebox(0,0)[tl]{$\Lambda^\mu$}} \put(134.39,119.69){\makebox(0,0)[tl]{Light}} \put(134.39,105.27){\makebox(0,0)[tl]{cone}} \put(42.36,110.27){\vector(1,2){7.12}} \end{picture} %}%endunitlength \setlength{\unitlength}{1mm} %\end{minipage} %\caption[New] Figure 5.9 {\it Space-like (a) and time-like (b) surfaces of discontinuity} %\label{f4.9} %\end{figure} }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{6cm} %\begin{figure}[hbtp] %\begin{center} \setlength{\unitlength}{0.7mm} \begin{picture}(400,175)%$fig64.pit 0 \thinlines \put(25.56,18.85){\vector(1,0){173.40}} \put(26.29,18.85){\vector(0,1){150.90}} \put(196.48,30.16){\makebox(0,0)[tl]{x}} \put(34.33,168.70){\makebox(0,0)[tl]{t}} \put(34.33,77.28){\makebox(0,0)[tl]{"1"}} \put(157.03,134.77){\makebox(0,0)[tl]{"2"}} \put(199.40,75.40){\makebox(0,0)[tl]{Spacelike}} \put(199.40,65.03){\makebox(0,0)[tl]{surface}} \put(69.39,158.33){\makebox(0,0)[tl]{surface}} \put(68.66,168.70){\makebox(0,0)[tl]{Timelike}} \put(111.02,121.58){\line(1,1){22.53}} \put(133.66,121.58){\line(-1,1){23.96}} \end{picture} \setlength{\unitlength}{1mm} %\end{center} %\caption[New] Figure 5.10 {\it Smooth change from spacelike to timelike detonation} %\label{f4.10} %\end{figure} }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-1.5cm} \subsection{Taub adiabat for finite particle densities} \B Eliminate the velocity from eqs. (5.17-18) \LT scalar equation\\ - Connecting thermodynamical quantities on two sides of the shock\\ - Called the {\em shock or detonation adiabat}\\ - Outside shock fronts - no dissipation \LT Landau's def. = Eckart's def.\\ - Note: if no conserved particles: number, derivation is not applicable\\ - Still: Taub adiabat can be derived, see later - Use Eckart's def. ($N^\mu = n u^\mu$) - Define particle current across the surface: \beq j \equiv N^\mu \Lambda_\mu . \eeq{6.3} - $j$ is invariant scalar, same value on both sides of the shock!\\ - $j \equiv n u^\mu \Lambda_\mu $: no dissipation outside the front\\ In the LR of the front: \beq \Lambda_{\mu \ LR} = \left\{ \begin{array}{ll} (1,\vec{0}) &:\ time-like\ \ surface \\ (0,\vec{e}) &:\ space-like\ \ surface \end{array} \right. , \eeq{6.3a} where $\vec{e}$ is a unit 3-vector. Using $ u^\mu = (\gamma, \gamma \vec{v})$: \beq j_{LR} = \left\{ \begin{array}{ll} n \gamma &:\ time-like\ \ surface \\ n \gamma (\vec{v} \vec{e}) = n \gamma v_\perp &:\ space-like\ \ surface \end{array} \right. , \eeq{6.3b} where $v_\perp$ is the component of velocity normal to the front.\\ Since $j$ is invariant scalar $ j = j_{LR} $, \LT \beq [j] = 0, \eeq{6.4} i.e. $ j$ is constant across the front!! }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.8cm} Eq. (5.17) is a 4-vector equation\\ - To connect thermodynamical (invariant scalar) quantities,\\ - we need two scalar equations projected from (5.17): \begin{itemize} \item 1 Parallel projection to the surface \item 2 Orthogonal projection to the surface \end{itemize} PARALLEL PROJECTION\\ \B Component orthogonal to surface, (parallel to the normal): $$ [ T^{\mu\nu} \Lambda_\nu ] = 0 \ \ \leadsto [ T^{\mu\nu} \Lambda_\nu \Lambda_\mu ] = 0 $$ $$ \ \leadsto [ w u^\mu u^\nu \Lambda_\mu \Lambda_\nu - p \Lambda^\mu \Lambda_\mu ] = 0. $$ - Using (5.19): $j = n u^\mu \Lambda_\mu, \ ( \ \leadsto \ u^\mu \Lambda_\mu = {j \over n})$,\\ - Inserting this into the equation above $$ \left[ {w \over {n^2}} j^2 - P \ \Lambda^\mu \Lambda_\mu \right] = 0, $$ $$ \left[ {w \over {n^2}} \right] j^2 - \left[ P \right] (\Lambda^\mu \Lambda_\mu) = 0. $$ \LT the equation of the {\em Rayleigh-line}: \beq j^2 = {{ [P] ( \Lambda^\mu \Lambda_\mu )} \over { [X] }}, \eeq{6.5} where $X \equiv {w \over{ n^2}}$, is the {\em generalized specific volume}.\\ \B In non-relativistic limit: - $X \longrightarrow m_0 V_{spec.}$\\ - $[P,X]$ plane corresponds to the $[P,V]$ plane \\ \B The Rayleigh line is a straight line on the $[P,X]$ plane: - locus of possible final states, $P_2, X_2$'s, for given initial state ``1''\\ - slope: given by the current across the front $j$. \\ - In heavy ion reactions: $j$ increases with beam energy }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.8cm} ORTHOGONAL PROJECTION\\ \B Construct orthogonal projection using the projector: $$ \Delta^{\mu\nu}_{\Lambda} = g^{\mu\nu} - {{\Lambda^\mu \Lambda^\nu } \over {( \Lambda^\mu \Lambda_\mu )}} $$ 1) we get a 4-vector tangent to the plane: $ G^\tau \equiv T_{\mu\nu} \Lambda^\mu \Delta^{\nu\tau}$,\\ 2) then a scalar equation by taking the norm: $$ [G^\tau G_\tau] = \ [ T_{\mu\nu} \Lambda^\mu \Delta^{\nu\tau} \ \ T^{\sigma\omega} \Lambda_\sigma \Delta_{\omega\tau} ] = 0. $$ \B Calculate 1): $$ G^\mu = (w u_\tau u_\nu \Lambda^\nu - P \Lambda_\tau ) \left( g^{\tau\mu} - {{\Lambda^\tau \Lambda^\mu }\over{(\Lambda^\sigma \Lambda_\sigma)}} \right) = $$ $$ = w (u_\nu \Lambda^\nu) u^\mu - w ( u_\tau \Lambda^\tau) (u_\nu \Lambda^\nu) {{ \Lambda^\mu}\over{(\Lambda_\sigma \Lambda^\sigma)}} - P \Lambda^\mu + P {{(\Lambda_\tau \Lambda^\tau)}\over{(\Lambda_\sigma \Lambda^\sigma)}} \Lambda^\mu = $$ $$ = {w \over n} j u^\mu - {w \over {n^2}} j^2 {{\Lambda^\mu } \over{(\Lambda_\sigma \Lambda^\sigma)}}. $$ \B 2) Then the scalar equation $[G^\mu G_\mu] = 0$:\\ $$ \left[ {{ w^2} \over {n^2}} j^2 u^\mu u_\mu - {{ w^2} \over {n^4}} j^4 {{\Lambda^\mu \Lambda_\mu} \over{(\Lambda_\sigma \Lambda^\sigma)^2}} \right] = 0, \ \leadsto $$ $$ \left[ {{ w^2} \over {n^2}} \right] - \left[ X^2 \right] j^2 (\Lambda^\mu \Lambda_\mu) = 0. $$ - This leads to another scalar equation for the current: \beq j^2 = {{ [ wX ] } \over { [X^2] (\Lambda^\mu \Lambda_\mu)}}. \eeq{6.6} }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.8cm} - Comparing the two projections, i.e. eqs. (5.23) and (5.24) $$ {{ [P] ( \Lambda^\mu \Lambda_\mu )} \over { [X] }} = {{ [ wX ] } \over { [X^2] (\Lambda^\mu \Lambda_\mu)}} \ \ \ \ \leadsto \ \ \ \ {{ [P] ( \Lambda^\mu \Lambda_\mu )^2} \over { [X] }} = {{ [ wX ] } \over { [X] (X_2 + X_1)}}, $$ \B \LT The equation of {\em Taub adiabat}, shock adiabat or relativistic Rankine-Hugoniot adiabat: \beq [P] = {{ [ wX ] } \over {(X_2 + X_1)}}. \eeq{6.7} - This is an eq. defining a curve in the $[P,X]$ plane for a given EOS\\ - Locus of the possible final states, $P_2, X_2$'s \\ - Depends on the EOS and on the initial state $"1"$ %\begin{figure}[hbtp] %\begin{center} \setlength{\unitlength}{0.5mm} \begin{picture}(400,175)(-50,0) %$fig65.pit 1 \thinlines \put(25.56,18.85){\vector(1,0){173.40}} \put(26.29,18.85){\vector(0,1){150.90}} \put(196.48,30.16){\makebox(0,0)[tl]{X}} \put(83.26,49.95){\line(1,0){117.30}} \put(111.02,32.99){\line(0,1){108.84}} \put(89.11,43.35){\makebox(0,0)[tl]{"1"}} \put(34.33,168.70){\makebox(0,0)[tl]{P}} \put(111.75,159.27){\makebox(0,0)[tl]{Detonation}} \put(123.44,148.91){\makebox(0,0)[tl]{adiabat}} \put(186.98,62.00){\makebox(0,0)[tl]{$j=0$}} \put(80.00,123.46){\makebox(0,0)[tl]{$j=\infty$}} \put(116.86,35.81){\makebox(0,0)[tl]{Shock adiabat}} \end{picture} \setlength{\unitlength}{1mm} %\end{center} %\caption[New] Figure 5.11 {\it Shock and Detonation adiabat in the [P,X] plane} %\label{f4.11} %\end{figure} }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.8cm} - Shock- and Poisson adiabats are similar; - But shock adiabat depends both \\ --- on the equation of state of the matter on the side "2" and\\ --- on the initial state "1"\\ \B If the initial "1" and final "2" states have the same EOS\\ - we have a normal shock, sh. adiabat goes through the initial point "1"\\ - at this point the Poisson adiabat and the shock adiabat are parallel:\\ $$ -\left. {{\partial P}\over{\partial X}}\right|_{"1"} = ( n_1 \gamma_s v_s )^2 , $$ i.e. "infinitely weak" shock wave and the sound wave are identical. \\ Weakest shock wave propagates with the speed of sound, $u_s = \gamma_s v_s$\\ %\begin{figure}[hbtp] %\begin{center} \setlength{\unitlength}{0.4mm} \begin{picture}(400,175)(-70,0) %$fig67.pit 1 \thinlines \put(25.56,18.85){\vector(1,0){173.40}} \put(26.29,18.85){\vector(0,1){150.90}} \put(196.48,30.16){\makebox(0,0)[tl]{X}} \put(83.26,49.95){\line(1,0){117.30}} \put(111.02,32.99){\line(0,1){108.84}} \put(89.11,43.35){\makebox(0,0)[tl]{"1"}} \put(34.33,168.70){\makebox(0,0)[tl]{P}} \put(60.62,129.11){\makebox(0,0)[tl]{"2"}} \put(111.02,49.95){\line(-3,4){51.00}} \put(148.27,117.81){\vector(-2,-1){61.28}} \thicklines \put(147.54,155.50){\line(0,-1){78.92}} \put(155.57,156.45){\makebox(0,0)[tl]{Rayleigh}} \put(155.57,146.08){\makebox(0,0)[tl]{line}} \put(155.57,117.81){\makebox(0,0)[tl]{slope $= -j^2$}} \put(155.57,101.78){\makebox(0,0)[tl]{${{[P]}\over{[X]}}=-j^2$}} \put(36.52,32.04){\makebox(0,0)[tl]{compression}} \put(126.36,32.04){\makebox(0,0)[tl]{expansion}} \end{picture} \setlength{\unitlength}{1mm} %\end{center} %\caption[New] Figure 5.12 {\it The slope of the chord and the current across the shock } %\label{f4.12} %\end{figure} %\vfill }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-1.2cm} \B "Strength of a shock" $\Leftrightarrow$ $j$\\ - $j$, or beam energy given: it selects (usually) one final state "2"\\ \B The relative speed of matter on the two sides of the front with respect to each other (see assignment 6a): $$ v_{12} = \frac{v_1 - v_2}{1-v_1 v_2/c^2} = \sqrt{ {{ (P_2-P_1) (e_2-e_1) }\over{ (e_1+P_2) (e_2+P_1)}} } . $$ - Symm. HI reactions, {\bf A+A}, c.m. beam energy is $E_{c.m.}= m(\gamma_{12}-1)$\\ - In HI collisions the initial state, "1", is the ground state of matter\\ - With a given EOS (5.25) can be solved: %\begin{figure}[hbtp] \vspace*{10truecm} %Insert figure 6.8 [3] from ref. [3] %\caption[F:CB79p-f.??] Figure 5.13 {\it Shock velocity, $v_{S}$, compression, $n/n_0$, temperature of the shocked matter, $kT$, and the width of the front, $\Delta l$, calculated analytically with two different Equations of state: stiff --- full lines, soft --- dashed lines. Points are taken from a 1-dimensional numerical, viscous Relativistic Fluid Dynamical calculation. From [31] } %\label{f4.13} %\end{figure} }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-1.5cm} \subsection{Relativistic detonations} - Also called: Deflagration, Slow combustion, or Condensation waves\\ \B EOS before and after the front may be different\\ - due to some chemical reaction, rearrangement, structural change, phase transition etc. \LT \\ - Taub adiabat does not go through the initial point "1" in the $[P,X]$ plane\\ - We have two possibilities in this case: \begin{itemize} \item a) if exotherm change: it goes above "1" \item b) if endotherm change: it goes below "1" \end{itemize} }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{2.5cm} %\begin{figure}[hbtp] %\begin{center} \setlength{\unitlength}{0.75mm} \begin{picture}(400,175)(20,0)%$fig68.pit 2 \thinlines \put(25.56,18.85){\vector(1,0){173.40}} \put(26.29,18.85){\vector(0,1){150.90}} \put(196.48,30.16){\makebox(0,0)[tl]{X}} \put(111.02,32.99){\line(0,1){108.84}} \put(116.13,83.88){\makebox(0,0)[tl]{"1"}} \put(34.33,168.70){\makebox(0,0)[tl]{P}} \put(59.16,89.53){\line(1,0){139.21}} \put(43.09,48.06){\makebox(0,0)[tl]{endotherm}} \put(159.23,157.39){\makebox(0,0)[tl]{exotherm}} \end{picture} \setlength{\unitlength}{1mm} %\end{center} %\caption[F:BC85-f.4] Figure 5.14 {\it Exotherm and endotherm discontinuities. From [32]} %\label{f4.14} %\end{figure} }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.5cm} MINIMUM ENERGY TO REACH A NEW PHASE:\\ - Have an exotherm reaction, and reach the final state, "2" from state, "1"\\ - Usually by compression ($X$ decreases, $X_2 < X_1$)\\ - or in expansion ($X$ increases, $X_2 > X_1$)\\ \B These two cases are called {\bf detonation} and {\bf deflagration}, respectively\\ - Deflagrations are also called slow combustion %\begin{figure}[hbtp] %\begin{center} \setlength{\unitlength}{0.5mm} \begin{picture}(400,175)(-30,0)%$fig69.pit 2 \thinlines \put(25.56,18.85){\vector(1,0){173.40}} \put(26.29,18.85){\vector(0,1){150.90}} \put(196.48,30.16){\makebox(0,0)[tl]{X}} \put(83.26,49.95){\line(1,0){117.30}} \put(111.02,32.99){\line(0,1){108.84}} \put(89.11,43.35){\makebox(0,0)[tl]{"1"}} \put(34.33,168.70){\makebox(0,0)[tl]{P}} \put(60.62,129.11){\makebox(0,0)[tl]{CJ}} \put(111.02,49.95){\line(-4,5){74.93}} \put(57.70,115.92){\line(-4,-3){7.26}} \put(140,95){\makebox(0,0)[tl]{Taub adiabat of the new phase}} \put(140,45){\makebox(0,0)[tl]{Taub adiabat of the initial phase}} \end{picture} \setlength{\unitlength}{1mm} %\end{center} %\caption[New] Figure 5.15 {\it Minimum energy detonation} %\label{f4.15} %\end{figure} }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.8cm} - The least steep chord corresponding to a detonation from state "1" is tangent to the Taub adiabat\\ - This is a Rayleigh-line touching the Taub adiabat at point "CJ".\\ \B "CJ" is called the {\em Chapman-Jouguet point}\\ - The slope of this Rayleigh line is: $$ j^2_{min} = - {{P_2 -P_1}\over {X_2 - X_1}} = (n_{CJ}\ \gamma_{CJs}\ v_{CJs})^2 $$ - In a weaker shock the new phase is never reached. \vspace*{12.5truecm} Figure 5.16 {\it Detonation adiabat for QGP EOS. Slopes and sound speeds. Instead of "1" and "2" here the initial state is denoted by "0" and the final state by "1" ! From [32]} }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.8cm} STABILITY \B The final state is not always stable\\ - Study of stability of shock fronts is quite involved\\ - Stability of fast detonations dominated by radiation is even more\\ - Complicated further by relativistic detonations, where discontinuities across timelike surfaces become possible\\ \B Stability: the velocity of matter leaving the front, $v_2$, should be smaller than the local sound velocity, $v_{2s}$, in that shocked matter $ v_2 < v_{2s}$\\ - at final point "1" the Poisson adiabat is steeper than the Rayleigh line\\ - all points above the Chapman-Jouguet point satisfy this relation \LT these points represent mechanically stable final states\\ - in the final state (at point "1" in Fig. 5.16) the local weak shock adiabat (corresponding to point "1") and the Poisson adiabat are parallel\\ - both having a slope related to the sound velocity as: $ -(u_{1s} n_1)^2$\\ \B Let us list systematically the different possibilities. \noindent - a minimum current is necessary to have a shock or detonation front\\ - shocks: speed of incoming matter should exceed the speed of sound\\ - detonations: it should exceed the speed of sound at the CJ point. }%end tr-page \newpage % =============================================================== \transparencyframe{ \vspace*{2cm} {\normalsize\sf %\begin{figure}[hbtp] \begin{minipage}[b]{5.5cm} {\unitlength 0.7pt \noindent \begin{picture}(200,170)%$fig611.pit 3 \thinlines \put(25.56,18.85){\vector(1,0){173.40}} \put(26.29,18.85){\vector(0,1){150.90}} \put(196.48,30.16){\makebox(0,0)[tl]{X}} \put(83.26,49.95){\line(1,0){117.30}} \put(111.02,32.99){\line(0,1){108.84}} \put(89.11,43.35){\makebox(0,0)[tl]{"1"}} \put(34.33,169.64){\makebox(0,0)[tl]{P}} \put(42.36,104.61){\makebox(0,0)[tl]{CJ}} \put(111.02,49.95){\line(-3,4){80.64}} \end{picture} }%endunitlength \end{minipage} \ \begin{minipage}[b]{9cm} {\Large\sf Detonation with $j_{min}$. This is the smallest current which leads to a detonation. The final state is a CJ point: $ j = n_{CJ} u_{CJs}$} \medskip\hrule \vspace*{14pt} \end{minipage} \begin{minipage}[b]{5.5cm} {\unitlength 0.7pt \noindent \begin{picture}(200,170)%$fig611.pit 3 \thinlines \put(25.56,18.85){\vector(1,0){173.40}} \put(26.29,18.85){\vector(0,1){150.90}} \put(196.48,30.16){\makebox(0,0)[tl]{X}} \put(83.26,49.95){\line(1,0){117.30}} \put(111.02,32.99){\line(0,1){108.84}} \put(89.11,43.35){\makebox(0,0)[tl]{"1"}} \put(34.33,169.64){\makebox(0,0)[tl]{P}} \put(42.36,104.61){\makebox(0,0)[tl]{CJ}} \put(57.70,115.92){\line(-4,-3){7.26}} \put(50.40,117.81){\line(5,-6){7.77}} \put(111.02,49.95){\line(-3,4){80.64}} \put(41.63,142.31){\line(-4,-3){4.47}} \put(80.34,90.47){\line(-5,-4){4.56}} \put(50.40,147.02){\makebox(0,0)[tl]{A}} \put(83.26,98.96){\makebox(0,0)[tl]{B}} \put(115.0,92.06){\makebox(0,0)[tl]{C}} \end{picture} }%endunitlength \end{minipage} \ \begin{minipage}[b]{9cm} {\Large\sf Detonation with $j>j_{min}$. There are two solutions: A and B, A is stable, at B the outgoing matter propagates faster than the local sound velocity, so the shock is unstable. However, if strong heat transfer is present due to radiation, final states in the section CJ-C are also realizable. This can be realized in nuclear bombs, rocket engines and in heavy ion collisions.} \medskip\hrule \vspace*{14pt} \end{minipage} }%end-normalsizesf }%end-transparency \newpage % ============================================================= \transparencyframe{ \vspace*{1.5cm} {\normalsize\sf \begin{minipage}[b]{5.5cm} {\unitlength 0.7pt \noindent \begin{picture}(200,170)%$fig612.pit 4 \thinlines \put(25.56,18.85){\vector(1,0){173.40}} \put(26.29,18.85){\vector(0,1){150.90}} \put(196.48,30.16){\makebox(0,0)[tl]{X}} \put(83.26,49.95){\line(1,0){117.30}} \put(111.02,32.99){\line(0,1){108.84}} \put(89.11,43.35){\makebox(0,0)[tl]{"1"}} \put(34.33,169.64){\makebox(0,0)[tl]{P}} \thicklines \put(107.37,84.82){\line(1,0){7.58}} \put(116.86,93.30){\makebox(0,0)[tl]{C}} \put(97.14,152.68){\makebox(0,0)[tl]{$j=\infty$}} \end{picture} }%endunitlength \end{minipage} \ \begin{minipage}[b]{9cm} {\Large\sf At point C the front propagates with the velocity of light: $j \rightarrow \infty $. This is the absolute boundary of propagating shocks due to causality.} \medskip\hrule \vspace*{14pt} \end{minipage} \begin{minipage}[b]{5.5cm} {\unitlength 0.7pt \noindent \begin{picture}(200,170)%$fig613.pit 4 \thinlines \put(25.56,18.85){\vector(1,0){173.40}} \put(26.29,18.85){\vector(0,1){150.90}} \put(196.48,30.16){\makebox(0,0)[tl]{X}} \put(83.26,49.95){\line(1,0){117.30}} \put(111.02,32.99){\line(0,1){108.84}} \put(89.11,43.35){\makebox(0,0)[tl]{"1"}} \put(34.33,169.64){\makebox(0,0)[tl]{P}} \put(111.02,49.01){\line(1,2){19.86}} \put(130.01,100.84){\makebox(0,0)[tl]{D}} \end{picture} }%endunitlength \end{minipage} \ \begin{minipage}[b]{9cm} {\Large\sf Time-like detonations can only be caused by the special initial conditions. Otherwise the points on the front are not in causal connection to each other.} \medskip\hrule \vspace*{14pt} \end{minipage} \begin{minipage}[b]{5.5cm} {\unitlength 0.7pt \noindent \begin{picture}(200,170)%$fig614.pit 4 \thinlines \put(25.56,18.85){\vector(1,0){173.40}} \put(26.29,18.85){\vector(0,1){150.90}} \put(196.48,31.10){\makebox(0,0)[tl]{X}} \put(60.62,112.15){\makebox(0,0)[tl]{"1"}} \put(34.33,169.64){\makebox(0,0)[tl]{P}} \put(81.80,32.99){\line(0,1){115.17}} \put(43.09,117.81){\line(1,0){169.93}} \put(113.21,120.63){\line(0,-1){5.62}} \put(113.21,135.06){\makebox(0,0)[tl]{$j=0$}} \put(81.80,117.81){\line(4,-1){52.33}} \put(159.41,91.94){\line(3,-1){27.02}} \put(169.45,81.05){\line(2,5){5.42}} \put(174.56,74.45){\makebox(0,0)[tl]{CJ}} \put(148.27,111.21){\makebox(0,0)[tl]{E}} \put(90.57,60.32){\makebox(0,0)[tl]{expansion}} \put(90.57,44.95){\makebox(0,0)[tl]{deflagration}} \put(90.57,65.03){\vector(1,0){23.42}} \end{picture} }%endunitlength \end{minipage} \ \begin{minipage}[b]{9cm} {\Large\sf In slow combustion or deflagrations both $v_1$ and $v_2$ are subsonic. Thus the stable final point "E" should lie between the Chapman-Jouguet point and the crossing by the $j=0$ line. If the reaction rate does not limit the current across the front, the deflagration will propagate with sound velocity with respect to the matter behind the front, "2". This speed maximizes the entropy production.} \medskip\hrule \vspace*{14pt} \end{minipage} %\caption[New] }%end-normalsizesf Figure 5.17 {\it Detonations with different currents across the front } %\label{f4.16a} %\end{figure} }%end-transparency \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-1.5cm} \subsection{Detonations to QGP} In HI collisions:\\ - the phase transition to QCD plasma is endotherm, but\\ - the transition back to the nuclear-matter in expansion is exotherm\\ \B Latter: deflagration or slow combustion, hadronization of QGP \bigskip\hrule \bigskip \begin{quotation} {\large\sf Aside: \\ - Recall ideal Bose and Fermi gases: - If we neglect the baryon density, then the pressure and the energy density depends on the temperature $T$ only\\ - Good QGP approximation at high temperatures and low densities\\ - Energy density of an ideal photon gas (ideal massless boson gas) is (c.f. eq. (60.16) in [33]) \beq e_\gamma = g_\gamma {{\pi^2}\over{30 (\hbar c)^3}} T^4, \eeq{6.e23} where $g_\gamma = 2$ (degeneracy of photons, two helicity states)\\ - Energy density of ideal fermion gas at high temperatures, (rest mass is negligible compared to the temperature, c.f. eq. (104.4) in [33]): \beq e^+ = g^+ {7 \over 4} {{\pi^2}\over{30 (\hbar c)^3}} T^4, \eeq{6.e24} where $g^+ = 1$ is the degeneracy of positrons\\ - Electrons have the same energy density. The pressure is one third of the energy density $P = {1\over 3}e$ }%end-largesf \end{quotation} \medskip\hrule \bigskip }%end-transparency \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.8cm} \noindent \B The equation of state of QCD plasma in zeroth order of the perturbation theory can be given based on the formulae above\\ - The plasma contains\\ $2 \times [N_c^2-1]$ gluons (bosons), ($N_c=3$ is the number of colors), and\\ $2 \times N_c N_f$ quarks (fermions), ($N_f$ is the number of flavors, $N_f = 2-4$)\\ - E.g. for 2 flavors and 3 colors: $2\times [(N_c^2-1) + {7 \over 4} N_c N_f] = 37$,\\ so the pressure of the QGP at vanishing baryon chemical potential is\\ $$ P_q = {{37 }\over{90}} \pi^2 T^4 / (\hbar c)^3 $$ \B QGP exists in the {\em perturbative vacuum} and not in the physical vacuum\\ - The energy density of the perturbative vacuum is higher,\\ - its pressure is lower than in physical vacuum (MIT bag model)\\ \B This correction is taken into account via a bag constant $B = \Lambda_B^4 / (\hbar c)^3$\\ - Typical bag constant is $B \approx 0.4 $GeV/fm$^3$, i.e. $\Lambda_B = 235 $MeV QGP PRESSURE: \beq P_q=\left( {{37}\over{90}} \pi^2 T^4 + {{1}\over{9}} \mu_B^2 T^2 + {{1}\over{162 \pi^2}}\mu_B^4-\Lambda_B^4 \right){1\over{(\hbar c)^3}}, \eeq{6a1} $\mu_B$ is the baryon chemical potential\\ (can be expressed in terms of the quark chemical potential as $\mu_B = 3 \mu_q$.) }%end-transparency \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.8cm} Other thermodynamical quantities of the QGP EOS are: \beq e_q=\left( {{37}\over{30}} \pi^2 T^4 + {{1}\over{3}} \mu_B^2 T^2 + {{1}\over{54 \pi^2}}\mu_B^4+\Lambda_B^4 \right){1\over{(\hbar c)^3}}, \eeq{6a2} \beq w_q=\left( {{74}\over{45}} \pi^2 T^4 + {{4}\over{9}} \mu_B^2 T^2 + {{2}\over{81 \pi^2}}\mu_B^4\right){1\over{(\hbar c)^3}}, \eeq{6a3} \beq n_q={2\over 9}\left( \mu_B T^2 + {{1}\over{9 \pi^2}}\mu_B^3\right){1\over{(\hbar c)^3}}, \eeq{6a4} \beq s_q=\left( {{74}\over{45}} \pi^2 T^3 + {{2}\over{9}} \mu_B^2 T \right){1\over{(\hbar c)^3}}, \eeq{6a5} where $n_q$ is the {\em baryon number} density in the QGP!\\ \B Now we observe that the Equation of State is rather simple: \beq P_q = {{e_q}\over 3} - {4 \over 3} B \ = \ {w_q \over 4} - B , \eeq{6a6} so it follows that $ w_q = 4(P_q+B)$. {\large\sf Dropping the index "2" and assuming that the initial state is the ground state nuclear matter ($P_1=P_0=0$ and $X_0 \approx 6$GeV/fm$^3$)}, the \\ \B Taub adiabat: $$ P = {{wX-w_0 X_0}\over {X+X_0}}. $$ "0" is representing our initial state \LT \beq (P+{4\over 3}B) (X-{1 \over 3} X_0) = {1 \over 3} X_0 ( w_0 - {4 \over 3} B ). \eeq{6a7} \B This is a hyperbola with its center at $(-{4 \over 3} B, \ {1\over 3} X_0)$ on the [P,X] plane. }%end-transparency \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.8cm} Depending on $B$ or $\Lambda_B$ the parameter of the hyperbola\\ $ Q= {1\over 3} X_0 ( w_0 - {4 \over 3} B )$ can be positive or negative. The parameter changes sign when $B={3\over 4} w_0$, or in other form, when $\Lambda_B=167$MeV. \vspace*{12truecm} %\caption[F:BC85-f.1] Figure 5.18 {\it Taub adiabats of the QGP initiated from normal ground state nuclear matter depending on the bag constant $B$ or $\Lambda_B$. From [32] } %\label{f4.18} %\end{figure} If $\Lambda_B$ is smaller than 127 MeV the reaction is exotherm, else endotherm. Usual value $\Lambda_B \approx 200$MeV. }%end-transparency \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.8cm} \B It seems that there is no QGP threshold for this bag constant\\ (Intersection $\exists$ between Rayleigh line and Taub adiabat even at $j=0$)\\ - QGP could be reached in a zero energy detonation front\\ - This does not seem to be reasonable at all!!\\ For any $j = n_0 \gamma_0 v_0 = n_2 \gamma_2 v_2$, we can get a solution $$ j^2 = - {{P - 0}\over{X-X_0}}. $$ \LT $X = X_0 - P/(j^2)$. Using now the Taub adiabat we get a quadratic equation for the pressure of the final state \beq P_q = {1\over 3} \left[ (X_0 j^2 - 2B) \pm \sqrt{ \left( 2B-X_0j^2\right)^2 + 3 X_0j^2 (4B- w_0) } \ \right]. \eeq{6a9} - This is a seemingly perfect solution for any current $j$\\ \B We can solve the puzzle by calculating all other thermodynamical quantities like the temperature, etc. (Assignment [6.a]) }%end-transparency \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-1.5cm} \subsection{Detonations in baryon free plasma} We assumed that the baryon density is large enough to define a baryon current\\ If this is not the case: we still can use the formalism [28,29]\\ \B 1) Calculate the {\em normal projection} of the energy current across the front: $$ [T^{\mu\nu} \Lambda_\mu \Lambda_\nu ] = 0. $$ Substituting the expression of $T^{\mu\nu} = w u^\mu u^\nu - P g^{\mu\nu} $, it follows that: \beq [ w (u^\mu \Lambda_\mu )^2 ] = [ P ] ( \Lambda^\mu \Lambda_\mu ) . \eeq{4i10} \B 2) Similarly the orthogonal projection to $\Lambda^\mu$ can be calculated by using the projector $ \Delta^{\mu\nu} = g^{\mu\nu} - {{\Lambda^\mu \Lambda^\nu}\over{ (\Lambda^\xi \Lambda_\xi )}}. $ The resulting projection $G^\mu $ is parallel to the surface: $$ G^\mu = \Delta^{\mu\nu} T_{\xi\nu} \Lambda^\xi = \hspace*{6cm} $$ $$ = w (u^\nu \Lambda_\nu ) u^\mu - w (u^\nu \Lambda_\nu )^2 {{\Lambda^\mu }\over{(\Lambda^\xi \Lambda_\xi)}} - P \Lambda^\mu + P \Lambda^\mu {{ (\Lambda^\xi \Lambda_\xi ) }\over{ (\Lambda^\mu \Lambda_\mu )}} = $$ $$ = w (u^\nu \Lambda_\nu )u^\mu - w (u^\nu \Lambda_\nu )^2 {{\Lambda^\mu }\over{ (\Lambda^\nu \Lambda_\nu )}} . $$ }%end-transparency \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.8cm} \B The length of this vector is an invariant scalar, ($G^\mu G_\mu$), so instead of $[G^\mu ] = 0$, we can use $[G^\mu G_\mu ] = 0$, which leads to $$ [ \ \ w^2 (u^\mu \Lambda_\mu )^2 + w^2 (u^\mu \Lambda_\mu )^4 \left((\Lambda^\xi \Lambda_\xi )- {{2}\over{(\Lambda^\nu \Lambda_\nu )}}\right) \ \ ] =0\ . $$ Multiplying this with $(\Lambda^\mu \Lambda_\mu )$ we get: \beq [w^2 (u^\mu \Lambda_\mu )^2 ] (\Lambda^\nu \Lambda_\nu ) = [ w^2 (u^\mu \Lambda_\mu )^4 ] \ . \eeq{4i11} \B Let us now introduce the quantity [29] $$ x \equiv {{w (u^\mu \Lambda_\mu )^2 }\over{ ( w_0 (u_0^\mu \Lambda_\mu )^2)}} , $$ and insert it into eqs. (5.36-37) \beq w_0 (u_0^\mu \Lambda_\mu )^2 [ x ] = [ P ] (\Lambda^\mu \Lambda_\mu ) , \eeq{4i12} \beq w_0 (u_0^\mu \Lambda_\mu )^2 [wx] (\Lambda^\xi \Lambda_\xi ) = w_0^2 (u_0^\mu \Lambda_\mu )^4 [x^2] , \eeq{4i13} }%end-transparency \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.8cm} We can express $w_0 (u_0^\mu \Lambda_\mu )^2$ from both (5.38) and (5.39) \LT $$ [P] / [x] = [wx] / [x^2]. $$ $\Lambda^\mu$ falls out \LT applicable both for spacelike and timelike surfaces(!): \beq (P_1 - P_0) (x_0 + x_1) = ( w_1 x_1 - w_0 x_0) . \eeq{4i14} \B This is the equation of the shock adiabat. {\large\sf It depends on the initial state and EOS. On the $[P,x]$ plane the final state "1" lies on this curve} Furthermore: \beq w_0 (u_0^\mu \Lambda_\mu )^2 = (\Lambda^\mu \Lambda_\mu ) [P] / [x] . \eeq{4i15} - Equivalent of Rayleigh line\\ - Contains the information about the spacelike or timelike nature of the surface. \\ - The tangent of the chord is positive for timelike surfaces (bulk transitions) and negative for propagating fronts.\\[1ex] If there is a conserved charge, $n$, the corresponding conserved current is \beq j = n u^\mu \Lambda_\mu , \eeq{4i16} which is also an invariant scalar and has the same value $j$ on both sides\\ Then $X$ can be introduced: $ X = w/n^2$. This is related to $x$ by $$ x = X_0 $$ Using this variable, from (5.40-41) \beq [P](X_1 + X_0) = [wX], \eeq{4i17} \beq j^2 = (\Lambda^\mu \Lambda_\mu )[P]/[X]. \eeq{4i18} Eq. (5.43) now contains only thermodynamical quantities of the initial and final states. }%end-transparency \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-1.5cm} \subsection{Deflagrations from QGP (*)} \B Widely investigated area of Quark-Gluon Plasma research.\\ - Baryon free plasma: ultra-relativistic energies at mid-rapidity\\ - Baryon rich plasma: fragmentation regions and at lower `stopping' energies\\ -Both cases: rarefaction discontinuity or deflagration wave (exotherm!!)\\ \B $\exists$ Standard theoretical treatment (Danielewicz and Ruuskanen [29])\\ - Shock adiabat (5.40) and Rayleigh line (5.41) corresponding to a constant enthalpy current $w u^\mu \Lambda_\mu$ or $w_1u_1^2=w_0u_0^2$ are already defined.\\ \B The Poisson adiabat for $n=0$:\\ (we cannot require the constancy of specific entropy! $\sigma=s/n$.\\ - If the incoming entropy flow is \\ $s_0 u_0$ or $s_0 u_0^\mu \Lambda_\mu$, the outgoing entropy flow\\ $s_1 u_1$ or $s_1 u_1^\mu \Lambda_\mu$ should be greater or equal! \LT \\ \beq s_1 u_1^\mu \Lambda_\mu \ \ > \ \ s_0 u_0^\mu \Lambda_\mu . \eeq{4i29} }%end-transparency \newpage % transparency ===================================================== \transparencyframe{ \vspace*{0.8cm} Eliminate the four velocities from this eq. using the definition \\ $x \equiv {{w(u^\mu \Lambda_\mu )^2}\over{w_0(u_0^\mu \Lambda_\mu)^2}}$ \LT \beq {{s_1^2}\over{ w_1}} x_1 \ \ge \ {{s_0^2}\over{w_0}} x_0 . \eeq{4i30} - In case of equality, (5.46) is the Poisson adiabat. - When $n = 0$, we have $w=Ts$ and so the Poisson adiabat becomes: \beq {{s_1}\over{T_1}} x_1 = {{s_0}\over{T_0}} x_0 . \eeq{4i31} - For a given EOS Shock and Poisson adiabats cross each other at the initial point. - Shock transitions with $P_1>P_0$ satisfy the entropy increase law\\ - The slope of this Poisson adiabat is related to the sound speed as [29] \beq w u^2_{sound} = -x \left. \left({{\partial P}\over{\partial x}}\right)\right|_{Poisson} . \eeq{4i32} - At the initial point the Poisson adiabat and the shock adiabat are tangent to each other }%end-transparency \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.8cm} QGP EOS FOR $N = 0$: TAUB (SHOCK) AND POISSON ADIABATS\\ - Shock adiabat for the mixed phase is $p=p_{cr.}= $const.\\ - The two phases yield two hyperbolas.\\ - The QGP hyperbola is much steeper\\ - appears as a vertical line on Figures below\\ \vspace*{8truecm} Figure 5.19 {\it Deflagration adiabat (lower thick line) and Poisson adiabat (upper thin line) corresponding to an initial state "0". The OA section of the deflagration adiabat is in the plasma phase, AC is in the mixed phase and below C is in the hadron phase. CJ is the Chapman-Jouguet point. The dashed lines are the continuations of the plasma adiabats representing a supercooled plasma and the continuation of the hadron adiabat representing a superheated hadronic phase. The Poisson adiabat lies everywhere above the shock adiabat. This means that the entropy on the shock adiabat is always lower than the initial entropy. The development of a deflagration front is impossible if $T_0 \gg T_c$ or $p_0 \gg p_c$. From [29]} }%end-transparency \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-1.0cm} \B Difficulty at rarefaction fronts:\\ initial state is not well defined and constant. \B In the $n \gg 0$ case more initial possibilities \LT \\ rarefaction studies mainly for baryon free plasma. \B In a highly excited plasma, $T>>T_{cr.}$ the hadronization via rarefaction discontinuity is not possible \LT plasma cools and expands. \vspace*{9truecm} Figure 5.20 {\it The same as the previous Figure with a different initial condition of a smaller initial pressure. The Poisson adiabat (thin line) intersects the shock adiabat at A and B. In the AB section of the shock adiabat the final state has a larger entropy than the initial state at "0". The entropy increase is maximum at CJ. Spontaneously developing rarefaction fronts will propagate with the sound speed, maximizing the entropy increase. The current corresponding to this process can be obtained by using the "0"-CJ chord (dashed-dotted line). If the finite reaction time limits the speed of hadronization the final state will be on the A - CJ section of the shock adiabat. From [29] } }%end-transparency \newpage % transparency ===================================================== \transparencyframe{ \vspace*{12truecm} Figure 5.21 {\it The same as the previous figure with an initial state "0" in the mixed phase. Final states on the A-CJ section of the shock adiabat are realizable. The entropy increase is smaller than in the previous case. From [29]} }%end-transparency \newpage % transparency ===================================================== \transparencyframe{ \vspace*{12truecm} Figure 5.22 {\it The initial state is now in the supercooled (metastable) plasma phase. The entropy increase is bigger than in the previous cases. The maximum of the entropy production is at CJ. Slow deflagrations yield final states laying on the B-CJ section of the adiabat. At B the enthalpy flow across the front vanishes. From [29]} }%end-transparency \newpage % transparency ===================================================== \transparencyframe{ \vspace*{0.8cm} \B When the plasma becomes cold \LT rarefaction front may develop\\ - Initially the current across the discontinuity is, however, small \LT\\ - The matter cools further in plasma phase \LT\\ - Supercooled metastable state develops\\ - Then timelike deflagration will become possible and\\ - the matter can undergo a rapid hadronization, (Fig. 5.22).\\ \B Although final states above B are usually considered unphysical [29],\\ - we have seen in the general derivation (sect. 5.5.4, eq. (5.41)) \\ - that final states on the B-E section of the curve can also be realized in timelike deflagrations [56].\\ \B In time-like deflagrations:\\ - In LR this transition is an instantaneous bulk phase transition\\ - In LR flow may exist and break across the surface\\ - In Laboratory frame it does not happen everywhere at the same time. }%end-transparency \newpage % transparency ===================================================== \transparencyframe{ \vspace*{8truecm} Figure 5.23 {\it Space-time picture of the evolution of QCD-plasma in the mid- rapidity region. The expanding plasma is surrounded by a spacelike and a timelike surface. The plasma temperature monotonically decreases with time according to the calculation. With realistic initial and boundary conditions the time-like surface is not necessarily at $t=const.$ From [52]} \B The necessity to complete the hadronization process with a timelike deflagration was indicated by the results of several works [44, 52, 53] \B Earlier works did not realize possibility of time-like deflagrations \LT \\ - Conclusion (false): hadronization across the plasma surface is too slow \B Consequently: formation of hadronic bubbles, and their expansion [43] \LT observable density and rapidity fluctuations. \B Spontaneous hadronic bubble formation in volume; percolation \LT \\ - can be approximated by change on an effective timelike surface [56]\\ - Two hadronization pictures: modeling essentially the same process. }%end-transparency \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.8cm} \section{Assignment 5} \begin{description} \item[5.a] Show that in perfect relativistic fluid dynamics the entropy increase in continuous flow is zero. \item[5.b] Calculate the critical temperature and critical pressure for a phase transition between ideal massless hadronic matter (pion gas) and ideal quark gluon plasma. Both are described by Stefan- Boltzmann EOS: $$ P_\pi = {{g_\pi \pi^2}\over{90}} T^4, \ \ \ e_\pi = 3P_\pi, \ \ \ g_\pi=3, $$ $$ P_q = {{g_q \pi^2}\over{90}} T^4 -B, \ \ \ e_q = 3P_q+4B, \ \ \ g_q=37, $$ where B is the "Bag constant". At the critical point $P_\pi=P_q=P_{cr}$, and $T_\pi=T_q=T_{cr}$ (These are Gibb's criteria!). Plot the $P(e)$ and $P(T)$ functions for the different phases and for the phase mixture. \item[5.c] Calculate the latent heat of this phase transition (per unit volume) in terms of the bag constant $ B$, and of the critical pressure $P_{cr}$. \item[5.d] Calculate the energy density and pressure for a distribution $$ f(x,p) = \sum^{N}_{i=1} f^{Juttner}(x,p;\mu_i,T_i,V_i^\mu), $$ in terms of parameters $\mu_i,T_i,V_i^\mu$. For help see ref. [57]. \end{description} }%end tr-page % transparency ================================================================ \end{document}