% Pages 1-3 and 17-25 are ready (4-16 are not!) Aug. 14, 1994 L.Cs. % % 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 9 / Chapter 9 % % 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}{{\sf 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} % transparency =============================================================== %\setlength{\unitlength}{1mm} %}%end tr-page %\newpage % transparency ====================================== p. 1 \transparencyframe{ \setcounter{chapter}{8} \vspace*{-1.8cm} \chapter{Direct Solutions of Kinetic Theory} \B Heavy Ion collisions: \\ - \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ a higly dynamical, non-equilibrium system.\\ - \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ not in thermal equilibrium globally, \&\\ - \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ at most of the time not even locally.\\ \B The solutions we have presented so far:\\ - \ \ \ \ \ \ \ \ were based on perfect fluid dynamics, assuming local eq.\\ Usually this approximation is valid only at later stages. \B We search for the real physical distribution function, $f$,\\ - \ \ \ \ for for all particles\\ - \ \ \ \ not only for the single particle distribution function\\ - \ \ \ \ but for the two and more particle distribution functions also. }%end tr-page \newpage % transparency =========================================== p. 2 \transparencyframe{ \vspace*{0.8cm} \B $\exists$ different approaches to find these non-eq. distributions:\\ - \ \ \ \ the Chapman-Enskog method, or\\ - \ \ \ \ its 1st order correction the viscous fluid dynamics,\\ - \ \ \ \ two-, three-, or more component fluid dynamics,\\ - \ \ \ \ nuclear cascade models,\\ - \ \ \ \ nuclear cascade models with a mean field, like the\\ -- \ \ \ \ \ \ \ \ Boltzmann-Uehling-Uhlenbeck (BUU),\\ -- \ \ \ \ \ \ \ \ Vlasov-Uehling-Uhlenbeck (VUU) or \\ -- \ \ \ \ \ \ \ \ Landau-Vlasov (LV) approaches,\\ - \ \ \ \ molecular dynamics, considering each interaction separately,\\ -- \ \ \ \ \ \ \ \ like the Quantum Molecular Dynamics (QMD) approach.\\ - \ \ \ \ Microscopic simulations aimed for higher energies\\ - \ \ \ \ where multiparticle production is dominant:\\ -- \ \ \ \ \ \ \ \ string or flux tube models\\ -- \ \ \ \ \ \ \ \ RQMD, QGSM, VENUS, FRITIOF, HIJET, ATTILA, DPM, ...\\ -- \ \ \ \ \ \ \ \ parton cascade models, PCM, ... \setlength{\baselineskip}{9pt} {\large\sf To present all these models and methods is exceeding the scope of an itroductory book. We will discuss only a few of these approaches and give a superficial insight into some of these models, which will enable us just to understand the basic results and constraints of the models.} }%end tr-page \newpage % transparency ======================================= p. 3 \transparencyframe{ \vspace*{-0.8cm} {THE CHAPMAN-ENSKOG METHOD:} \B Start from a perfect FD solution, the local eq. distribution, $f^{(0)}(x,p)$. \\ Then search for corrections, step by step.\\ \B The 1st order corrected distribution function, \ \ \ \ \ \ \ \ \ \ \ $f^{(1)}(x,p)$,\\ - \ is assumed to be only slightly different from the local eq. solution:\\ - \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $f^{(1)}(x,p) = f^{(0)}(x,p)+g(x,p)$. \\ This $f^{(1)}$ should satisfy the Boltzmann Transport Equation (BTE) $$ p^\mu \partial_\mu f^{(1)} \equiv D f^{(1)} = C \left[f f^{(1)}_1 f^{(1)}_2 \right] \ . $$ Since $g\ll f^{(0)}$ and for $f^{(0)}$ the collision integral, $C \left[f f^{(0)}_1 f^{(0)}_2 \right]=0$\\ \LT \ \ for $f^{(1)}$ the BTE takes the following form $$ D f^{(0)} = C \left[f f^{(0)}_1 f^{(1)}_2 \right] = C \left[f f^{(0)}_1 g_2 \right] \ . $$ From this eq. we can determine $g(x,p)$. \B Continue with the next correction, $f^{(2)}=f^{(1)}+h$:\\ - \ \ \ by inserting, $f^{(2)}$ back to the BTE we can evaluate $h$ too [1,2]\\ - \ \ \ \ \ \ \ \ \ \ This is schematically the Chapman Enskog expansion. \setlength{\baselineskip}{12pt} {\large\sf \B It is usefull but the solutions even after many steps of iteration will not deviate essentially or qualitatively from the local equilibrium solution.\\ - \ \ \ For example the initial two peaked distribution (sect. 3.4) in a HI collision cannot be obtained this way.\\ \phantom{K}\\ \B The IMPORTANCE of this method is:\\ \LT the viscous fluid dynamics and the transport coefficients at 1st order.\\ \B The non-relativistic derivation of viscous fluid dynamics is straightforward [1,3]. \\ \B The relativistic counterpart[2] is quite involved. } }%end tr-page \newpage % transparency ======================================= p. 4 \transparencyframe{ \vspace*{-0.8cm} \section{Viscous Fluid Dynamics} \B Viscous fluid dynamics is seldom used in relativistic physics.\\ --- Due to questions around the proper relativistic generalization [4]\\ --- Usual relativistic generalizations may lead to unstable solutions. \B Dissipative effects are, nevertheless, important\\ \B $\exists$ Relativistic viscous FD: $\sim$ APPROXIMATIONS. \B Rel. dissipative FD: new terms in the stress-energy-momentum tensor\\ - - - and to the baryon current:\\ (see sect. 2.3.1-2,) \begin{equation} T^{\mu \nu} = -Pg^{\mu \nu} + (P+e)u^{\mu}u^{\nu} + T^{\mu \nu \ (1)}, \end{equation} \begin{equation} N^{\mu} = nu^{\mu} + N^{\mu\ (1)}. \end{equation} \B No dissipation: $T^{\mu\nu\ (1)}$ \& $N^{\mu\ (1)}$ are both zero,\\ - - - and $u^{\mu}$ is the flow four-velocity of the matter. \B When dissipation is present one has a choice of defining $u^{\mu}$\\ - - to be the velocity of baryon flow or the velocity of energy flow.\\ These are the Eckart and Landau-Lifshitz approaches, respectively. }%end tr-page \newpage % transparency ======================================= p. 5 \transparencyframe{ \vspace*{-0.8cm} \B In the Eckart approach $N^{\mu\ (1)}$ is zero by definition.\\ - - - Then $T^{\mu\nu\ (1)}$ is a linear combination of a shear tensor, a projection tensor on the hyperplane normal to $u^{\mu}$, and a heat flow vector, with coefficients being the shear viscosity $\eta$, the bulk viscosity $\zeta$, and heat conductivity $\lambda$, respectively. \B In the Landau-Lifshitz approach $T^{\mu\nu\ (1)}$ does not have the term involving the thermal conductivity;\\ - - - rather, $N^{\mu\ (1)}$ is nonzero and is in fact proportional to $\lambda$. - These two approaches are completely equivalent.\\ \B Small/no baryon number \LT Eckart approach is indeterminate. \LT\\ \B $u^{\mu}$ must be the velocity of energy flow. \B \LT Thermal conductivity is not defined\\ because there is no net baryon density to define a frame of reference with respect to which energy can be conducted. }%end tr-page \newpage % transparency ========================================== p. 6 \transparencyframe{ \vspace*{-0.8cm} \subsection{Entropy production} Let us use Landau's definition of the flow velocity.\\ (Perf.rel.FD: divergence of entropy 4-current vanishes, Assign.5a) \B Both heat conductivity and shear and bulk viscosity\LT dissipation \beq \partial_\iota \left(s u^\iota - \frac{\mu}{T} N^{\iota\ (1)} \right) = - N^{\iota\ (1)} \partial_\iota \left(\frac{\mu}{T}\right) - \frac{1}{T} \Pi_\nu^\iota \ \partial_\iota u^\nu \ , \eeq{la.5} where\\ $\mu$ is the chemical potential,\\ $N^{\nu\ (1)} = - {n \over w} I_q^\nu$ is the particle flux density 4-vector\\ - - (with respect to the flow four-velocity), \&\\ $\Pi_{\mu \nu}$ is the stress tensor. \B The particle flux density takes the form \beq N^{\iota\ (1)} = - {n \over w} I_q^\iota = - \frac{\lambda}{c} \left( \frac{nT}{w}\right)^2 \left[ \partial^\iota \left(\frac{\mu}{T}\right) - u^\iota u^\nu \partial_\nu \left(\frac{\mu}{T}\right) \right] \ , \eeq{la.7} arising from the requirement that the entropy should not decrease [5].\\ \B From the same requirement the stress tensor is $$ \Pi_{\mu \nu} = - \eta \left( \partial_\mu u_\nu + \partial_\nu u_\mu - u_\mu u^\iota \partial_\iota u_\nu - u_\nu u^\iota \partial_\iota u_\mu \right) $$ \beq - ( \xi - \frac{2}{3} \eta ) (\partial_\iota u^\iota) \left(g_{\mu \nu} - u_\mu u_\nu\right) \ , \eeq{la.6} where $\xi$ and $\eta$ are the coefficients of bulk and shear viscosity. }%end tr-page \newpage % transparency ========================================== p. 7 \transparencyframe{ \vspace*{-0.8cm} \B Dissipative rel. FD: self consistent set of partial diff. eqs.\\ - Rel. generalization of the Navier-Stokes equations of FD.\\ \B\B These eqs. are parabolic type \& can be acausal [4,7,8] !!! \B \LT severe limitation on the application of dissipative FD. \setlength{\baselineskip}{9pt}{\normalsize\sf If, however, the flow structure is of a length scale much larger than the mean free path, this precludes propagation velocities faster than the thermal velocities of particles.[9] This is also necessary due to the requirement that dissipative perturbations should be small, compared to other dynamical processes. } \setlength{\baselineskip}{19pt}{\Large\sf \B Consequently, we may use the relativistic theory without any danger,\\ - - for slow, non-relativistic process,\\ - - like the droplet fromation in relativistic matter [10]. } }%end tr-page \newpage % transparency ======================================== p. 8 \transparencyframe{ \vspace*{-0.6cm} ASIDE \bigskip\hrule\bigskip \setlength{\baselineskip}{9pt} {\normalsize\sf Viscous rel. FD can be formulated similar to the nonrelativistic theory.\\ For compact notation we can introduce the operator of derivation orthogonal to the flow \beq \D^\iota \equiv \Delta^{\iota \nu} \partial_\nu = \partial^\iota - u^\iota u^\nu \partial_\nu \ , \eeq{la.511} where $\Delta^{\iota\nu} = g^{\iota\nu} - u^\iota u^\nu$ is the projector in the direction orthogonal to the flow.\\ (These have the properties: $\D^0_{(LR)} = 0$, and $\D^i_{(LR)} = $ $ -\D_{i\ (LR)} = $ $ - \partial_i$,\\ which means that it is purely spatial in the local rest frame (LR)\\ $u^\mu \D_\mu = 0$,\ $\Delta^{\mu\nu}_{(LR)} = \Delta_{\mu\nu (LR)} = $ diag(0,-1,-1,-1), \ $\Delta^\mu_{\nu\ (LR)} $ diag(0,1,1,1) and $\Delta^\mu_\mu= 3 $.\\ These opearators have the following properties:\\ $\Delta^{\mu\nu} u_\nu = 0$, \ $\Delta^{\mu\nu} \Delta_{\nu\sigma} = \Delta^\mu_\sigma$ and $\D_\mu u^\mu = \partial_\mu u^\mu$.)\\ Using these projectors the stress tensor, eq. (\ref{la.6}), can be written in the form \beq \Pi^{\mu \nu} = - \eta (\D^\mu u^\nu + \D^\nu u^\mu ) - \left( \xi - \frac{2}{3} \eta \right) (\D_\iota u^\iota) \Delta^{\mu\nu} \eeq{la.63} Using the thermodynamical identity \beq d\left(\frac{\mu}{T}\right) = - \left(\frac{w}{nT^2}\right) dT + \left(\frac{1}{nT}\right) dP \ , \eeq{la.10} the particle flux takes the form \beq N^{\iota\ (1)} = \frac{\lambda}{c} n \left( \frac{1}{w}\right) \left[ \D^\iota T - \frac{T}{w} \D^\iota P \right]\ , \eeq{la.71} where $\lambda$ is the coefficient of thermal conductivity.\\ $N^{\nu\ (1)}$ tends to zero if the baryon density $n$ tends to zero.\\ Using the thermodynamical identity (\ref{la.10}) the dissipation given by eq. (\ref{la.5}) takes the form $$ \partial_\iota \left(s u^\iota - {\mu \over T} N^{\iota\ (1)} \right) = - \frac{\lambda}{c} \left\{ \frac{1}{T^2} \partial_\iota T \D^\iota T + \frac{1}{w^2} \partial_\iota P \D^\iota P \right. $$ \beq \left. - \frac{1}{wT} \left[ \partial_\iota T \D^\iota P + \partial_\iota P \D^\iota T \right]\right\} - \frac{1}{T} \Pi_\nu^\iota \ \partial_\iota u^\nu \ . \eeq{la.51} } }%end tr-page \newpage % transparency ==================================== p. 9 \transparencyframe{ \vspace*{-0.8cm} \setlength{\baselineskip}{9pt} {\normalsize\sf For baryon free matter from (\ref{la.10}) it follows that $ (1/T) \D_\iota T = (1/w) \D_\iota p$.\\ Using this for transformations on the right hand side of eq. (\ref{la.51}), the first term vanishes.\\ I.e., the heat conductivity alone does not lead to dissipation in baryon free matter, and obviously $N^{\mu\ (1)} = - {n \over w} I^\mu_q \rightarrow 0$. Thus, for baryon free matter the equation for the entropy production, (\ref{la.51}), takes the form \beq \partial_\iota \left(s u^\iota \right) = \frac{1}{T} \eta {1 \over 2} [ \D^\mu u_\nu + \D_\nu u^\mu - {2 \over 3} \Delta^\mu_\nu (\partial_\sigma u^\sigma)]^2 + \xi (\partial_\sigma u^\sigma)^2 \ , \eeq{la.513} so that the dissipation is proportional to $\eta$ and $\xi$. Using the relations $\D^\nu = \D_\mu \Delta^{\mu\nu}$ and $\Delta^{\mu\nu} \D_\nu = \D^\mu$, the four divergence of the the stress tensor takes the form $$ \D_\mu \Pi^{\mu\nu} = \Delta_\mu^\sigma \partial_\sigma \Pi^{\mu\nu} = - \eta ( \D_\mu \D^\mu u^\nu ) - \left(\xi +\frac{1}{3} \eta\right) \D_\mu \Delta^{\mu\nu} (\D_\iota u^\iota) \ . $$ In the shearless limit, $\D_\mu \D^\mu u^\nu = \D^\nu (\D_\mu u^\mu)$, this reduces to \beq \D_\mu \Pi^{\mu\nu} = - \left(\xi + \frac{4}{3} \eta\right) \D_\mu \Delta^{\mu\nu} (\D_\iota u^\iota)\ \ \ \propto\ \ \ \partial_\mu \Pi^{\mu\nu} = - \left(\xi + \frac{4}{3} \eta\right) \partial_\mu \Delta^{\mu\nu} (\partial_\iota u^\iota) \ . \eeq{la.523} Comparing this to the four divergence of the energy-momentum tensor of a perfect fluid \beq \partial_\mu T^{\mu\nu} = \partial_\mu (e u^\mu u^\nu) + \partial_\mu \Delta^{\mu\nu} P \ , \eeq{la.533} we can see that in the equations of motion, in the shearless limit the only change is that the\\ \B\B pressure is replaced by \beq P \rightarrow P^* = P + \pi \ , \eeq{r.1} where the correction to the pressure in the one dimensional case relevant to us, $\pi$, is \beq \pi = - \left(\xi + \frac{4}{3} \eta \right) (\partial_\mu u^\mu )\ , \eeq{r.2} where the unit of $\xi, \ \eta$ is [MeV fm$^{-2}$ c$^{-1}$]. }%endfootnotesize \bigskip\hrule\bigskip }%end tr-page \newpage % transparency ========================================= p. 10 \transparencyframe{ \vspace*{-0.8cm} \B The contribution to the stress-energy-momentum tensor\\ - due to dissipation is $$ T_{\mu\nu\ (1)} = - \eta \left( \partial_\mu u_\nu + \partial_\nu u_\mu - u_\mu u^\iota \partial_\iota u_\nu - u_\nu u^\iota \partial_\iota u_\mu \right) $$ \begin{equation} - ( \zeta - \frac{2}{3} \eta ) (\partial_\iota u^\iota) \left(g_{\mu \nu} - u_\mu u_\nu\right) \ . \end{equation} \B The entropy should not decrease\\ \LT Shear and bulk viscosity must be positive.\\ - - The divergence of the entropy current in the absence of baryons is \begin{equation} \partial_{\mu}(su^{\mu}) = -\frac{1}{T} T^{\mu\ (1)}_{\nu} \partial_{\mu}u^{\nu} \ , \end{equation} from which one can calculate the total entropy change of the system. \B This eq. is a consequence of the eqs. of motion:\\ - - the conservation of energy and momentum, $\partial_{\mu}T^{\mu \nu} = 0$.\\ \B In general this description is applicable if the flow 3-velocity is small compared to the speed of light.\\ \B The relativistic treatment is required because\\ - - the pressure is comparable to the energy density, and\\ - - to define LR for baryonless matter }%end tr-page \newpage % transparency ======================================== p. 11 \transparencyframe{ \vspace*{-1.3cm} \subsection{Shock front profiles} The entropy production can be evaluated in the shearless limit (\ref{r.2}) following the standard derivation given in sect 127 of ref. [5]. Introducing the notation \beq \dot{a} \equiv u^\mu \partial_\mu a \equiv u^\mu a,_\mu \eeq{def1} the entropy production is [11]: \beq \dot{\sigma} = \left(\xi + \frac{4}{3} \eta \right) {1 \over {nT}} \left( \frac{\dot{n}}{n} \right)^2 \ = \eta_{eff} {1 \over {nT}} \left( \frac{\dot{n}}{n} \right)^2 \ . \eeq{ent3} \B \LT simple estimate for the width of the compression shock fronts\\ - - If the width of the front is $L_{sf}$, the derivatives can be estimated as \beq \dot{\sigma} = \frac{\Delta \sigma}{L_{sf}} \ , \ \ \ \ \ \ \ \dot{n} = \frac{\Delta n}{L_{sf}} \ . \eeq{e12.22} - $\Delta \sigma$ and $\Delta n$ are fixed by the Rankine-Hugoniot-Taub relations,\\ - \LT these do not depend on the structure or thickness of the front. Thus eq. (\ref{ent3}) yields \beq \frac{\Delta \sigma}{L_{sf}} \sim \bar{\eta}_{eff} \frac{1}{\bar{n}\bar{T}} \frac{\Delta n^2}{L_{sf}^2 \bar{n}^2}\ . \eeq{e12.23} \B Width of the front is proportional to the average effective viscosity \beq L_{sf} \ \sim \ \ \bar{\eta}_{eff} \ . \eeq{e12.24} \B The width of the shock front is important:\\ - FD considerations for the initial stages of the reaction hold\\ - only if the system is considerably larger than the width of the front. \B Estimates for intermediate energy HI reactions are $L_{sf} \approx 0.5-3$fm. }%end tr-page \newpage % transparency ===================================== p. 12 \transparencyframe{ \vspace*{-0.8cm} The viscosity can be calculated in kinetic theory,\\ - - for dilute gases it is evaluated in standard textbooks [1,3].\\ The viscosity is proportional with $\sqrt{T}$ for dilute gases.\\ The viscosity of the quark gluon plasma will be discussed in section 10.4.\\ Estimates for low energy nuclear matter $\eta = 6-20 $MeV fm$^{-2}$ c$^{-1}$. \B For sub-QGP densities and temperatures the transport coefficients were evaluated by Danielewicz [12] (see Fig. 9.1). \\ \B The viscosity diverges for low temperatures and high densities.\\ - - caused by the Pauli priciple which limits the possible collisions. \vspace*{12.2truecm} Figure 9.1 {\it The temperature and density dependence of the nuclear matter transport coefficients. The dotted lines denote the Chapman-Enskog results with the effective cross section $\sigma = 30$mb. From ref. [12] } }%end tr-page \newpage % transparency ======================================== p. 13 \transparencyframe{ \vspace*{-0.8cm} \noindent \B Pauli principle \LT wide shock front in low energy reactions\\ \B \LT FD approach is not valid in this energy domain.\\ \B The shock front is sufficiently sharp already at around 400 A MeV\\ - - and above: \vspace*{13truecm} Figure 9.2 {\it Shock front profiles. Restframe density, $n$, temperature, $T$, and pressure, $P$, as a function of the distance, $z$, in the shock-wave frame. The shock wave corresponds to $E_{Lab} = 400$A MeV. Curves A and B are calculated using coefficients from Fig. 9.1, curves C with $\eta = 18.6 \ (1+T/20)^{1/2}$ MeV/fm$^2$c and curves D with $\eta = 6$ MeV/fm$^2$c. Curve A includes the effect of heat conductivity also the others do not. From ref. [12] } }%end tr-page \newpage % transparency ========================================== p. 14 \transparencyframe{ \vspace*{-0.1cm} \B Large scale numerical calculations are mostly in the perfect FD:\\ - - However, yield shock waves with similar thickness\\ - - This is due to the so called numerical viscosity in these models. \B Important effects of viscosity on FD scaling were shown in Chpt. 8.\\ - These viscous effects are apparent in the observed transverse flow.\\ - - (The theoretical values for the viscosity were taken from Fig. 9.1) \B The good agreement between theory and experiment indicates\\ - that HI reactions \LT\\ - accurate information both on the EOS and on the transport propeties. }%end tr-page \newpage % transparency ========================================== p. 15 \transparencyframe{ \vspace*{-1.8cm} \section{Multi Component Fluid Dynamics} \B The initial distribution in HI collisions is far from a thermal eq.\\ - Fermi distribution both in the target and the projectile, \\ - Separated from each other by the beam momentum,\\ - - which is much larger than the Fermi momentum at rel. energies. \B Basic idea behind the multi-fluid models is\\ - - to start with two initially independent fluid components\\ - - these then interact with each other during the HI collision, and\\ - - either get modified or populate other fluid components\\ - - - - like a central thermalized nucleon fluid. \B Possible to consider different particle species as fluid components.\\ (2-fluid dynamics is standard theory to describe electron-ion plasma.) \B There is a large number of such approaches in the literature [13-17]. \B Start from the BTE,\\ - if components are identical particles, like nucleons, then $$ f(x,p) = \sum_{k=1}^{N_{comp}} f_k(x,p_k) \ . $$ Each component is then described by a separate BTE $$ D\ f_k = \sum_j C_{kj} \left[f f_k f_j \right] \ . $$ The sum of these equations returns the full BTE. \B \LT Conservation laws for each component (sect. 3.6) \\ $\sim$ \ \ \ FD type equations (sect. 3.9).\\ \B Collision integrals will \LT couplings terms: drag terms. }%end tr-page \newpage % transparency ===================================== p. 16 \transparencyframe{ \vspace*{-0.5cm} \B Drag terms among the fluid components: From kinetic theory\\ - simplifying assumptions are frequently applied \B E.g. in ref. [14] each collision among two different components \\ \LT particles belonging to the third thermalized central component.\\ - Thus the FD equations have a source term in the continuity eq.\\ - - - for the 3rd component and \\ - two corresponding loss or drain terms\\ - - - in the 1st (projectile) and 2nd (target) component. \B The particles scattering into the 3rd thermal component\\ - - carry all their energy and momentum with them to the 3rd component.\\ \B \LT Target and projectile components will have the same specific energy\\ - - but the temperature of the 3rd, thermal component will increase.\\ \B \LT gradual thermalization and local thermal equilibrium\\ - - - achieved in roughly 8fm/c after the impact }%end tr-page \newpage % transparency ===================================== p. 17 \transparencyframe{ \vspace*{11.8truecm} Figure 9.3 {\it The development of the momentum distribution of the central region in the beam direction and in the ortogonal direction. The speed of thermalization is similar to the one obtained in cascade and kinetic models. The arrows indicate the initial target and projectile velocity. The given percent values indicate the degree of thermalization; $/$, in the region $\Delta z$ around the c.m. From ref. [14] } \B In multi-fluid models shock waves also develop,\\ - their thickness is of the order of 1-3fm.\\ \B If the nucleon-nucleon c.s. is reduced to $\sigma = 2 mb$,\LT\\ - - the model becomes transparent, a majority of particles will continue their propagation after the collision in the same direction and only a small third component will develop. }%end tr-page \newpage % transparency ===================================== p. 18 \transparencyframe{ \vspace*{-0.8cm} \B Disadvantage of the multi-component fluid dynamical approach is\\ - - that the components are not always forming a dilute gas,\\ - - consequently the coupled fluid dynamical equations\\ - - \ \ \ \ \ cannot be derived in a rigorous way from the BTE. \B Particlularly problematic to use more fluid dynamics\\ - - to describe a phase transition. --- The fluid components are generally overlapping in space\\ - - \ \ \ with different partial flow velocities and other char. parameters.\\ - - \ \ \ ?? How can one find a unique well defined description\\ - - \ \ \ in a more fluid model \\ - - \ \ \ for a system undergoing a phase transition. --- Just imagine one fluid component is in one phase\\ and another component,\\ - \ \ \ \ \ \ \ \ \ overlapping with it but flowing in a different direction,\\ is in the other phase or in the mixed phase! }%end tr-page \newpage % transparency ========================================== p. 19 \transparencyframe{ \vspace*{-0.8cm} \section{Solutions on microscopic level} GENERAL REMARKS \B Most straightforward way of solving the Boltzmann Transport Eq.:\\ - to simulate the particle propagation and collisions on a computer.\\ \B This Monte-Carlo method of solution can be performed in many ways\\ -- most usual to represent each particle numerically:\\ -\ \ \ \ easy and straightforward to implement the collision term,\\ -\ \ \ \ cross sections, etc. \B One simulation \LT very fluctuating one-ptcl. distribution, $f(x,p)$,\\ - but, calculate a large number of simulations for the same collision, \&\\ - the average of this ensemble of simulated events:\\ \B \LT a smooth distribution. \\ - Most frequently a statistical ensemble of $10^3-10^4$ events is realized.\\ - The statistical accuracy of the experiments may be reached. (Exceptions: exotic particles, with a multiplicity of 10$^{-4}$ or 10$^{-5}$ per collision. To reproduce these cross sections with satisfactory accuracy is quite expensive numerically.) }%end tr-page \newpage % transparency ========================================== p. 20 \transparencyframe{ \vspace*{-0.8cm} \B Most important advantage of microscopic models:\\ - \LT simulated physical events closely similar to the experimental ones.\\ -- \ \ \ \ \ \ \ \ \ \ called also EVENT GENERATORS\\ \B The final set of emerging particles in a numerical simulation\\ -- \ \ can be analysed exatly the same way as it is done in experiments. \\ All measurables can be evaluated in a straightforward way. The numerical codes doing such experimental evaluation are frequently the same and they are used both for measured-, as well as for simulated data. \B Obviously these microscopic models are\\ - the most frequently used, and \\ - the most popular in the study of heavy ions\\ - they provide immediate explanation to many measured effects \\ - \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \-based on microscopic assumptions only. \B The realization of all possible microscopic processes\\ - is an extensive task,\\ - is straightforward and\\ - can be performed safely.\\ (These codes are not facing problems with numerical instabilities, etc. unlike many of the numerical fluid dynamical models.) \B To provide a deeper macroscopic insight into the reaction mechanism\\ - \ \ \ \ \ \ \ \ \ \ \ \ \ and into the collective behaviour of the matter\\ -\B needs, nevertheles, a serious extra effort:\\ - \ \ \ \ \ \ \ \ \ \ \ \ \ \ Macroscopic parameters, e.g. $n,\ e,\ T$,\\ - \ \ \ \ \ \ \ \ \ \ \ \ \ \ can be evaluated using coarse graining. \bigskip \setlength{\baselineskip}{11pt} {\large\sf The calculation of all thermodynamical quantities, like the local entropy density, is not an easy task any more. Thus to evaluate e.g. the underlying EOS of such a model is possible but usually so difficult that it is hardly ever done. } }%end tr-page \newpage % transparency ========================================= p. 21 \transparencyframe{ \vspace*{-1.2cm} \subsection{Intranuclear cascade models} \B The Intra-Nuclear Cascade (INC) model:\\ -\ \ \ originally introduced for high energy (GeV) p+A collisions\\ -\ \ \ more than 40 years ago. \B It followed numerically the path of the proton in the phase space,\\ - followed all its collisions with the nucleons of the target\\ - a collision is simulated the following way:\\ --\ \ \ if the proton along its path approaches a nucleon\\ --\ \ \ \ \ \ \ closer than some minimum distance, $d=\sigma(\sqrt{s})$,\\ --\ \ \ the proton will scatter, and\\ --\ \ \ the final state is randomly generated from $\sigma(\sqrt{s})$,\\ --\ \ \ the total energy and momentum is conserved\\ --\ \ \ the outgoing particles propagate along straight line trajectories. \bigskip \B Inelastic processes, e.g. $N+N \rightleftharpoons N+\Delta$ or $\Delta \rightleftharpoons \pi + N$,\\ can be included. \B At low energies the Pauli priciple is also taken account\\ --\ \ \ usually by randomly {\bf Pauli-blocking} a collision\\ --\ \ \ \ \ \ \ according to the probability $[1-f(x,p)]$,\\ --\ \ \ \ \ \ \ evaluated at the final state of the outgoing particles. \B The Nucleus - Nucleus INC is a superposition of the p+A models.\\ - the collisions of all projectile (and target) nucleons are described. }%end tr-page \newpage % transparency ============================================ p. 22 \transparencyframe{ \vspace*{-0.8cm} \B Initial state: \\ --\ \ \ \ random positions for the nucleons according to $n(r)$, and\\ --\ \ \ \ random momenta according to the Fermi distribution\\ --\ \ \ \ (These momenta are then boosted by the beam velocity\\ --\ \ \ \ in the frame of calculation.) \B To avoid that the target and projectile fly apart before their impact\\ - the random Fermi momenta are {\bf frozen in} until their 1st collision,\\ - i.e. they propagate only according to the beam velocity.\\ - The first collision unfreezes the random Fermi motion. \B The proper treatment of SECONDARY Collisions is also important\\ - two nucleons within the projectile nucleus or the target nucleus\\ - are not allowed to collide until their first collision,\\ - * - but if a nucleon has already collided, such secondary nucleons\\ - - - are allowed to collide on any nucleon in the system.\\ (Earlier INC models did not always allow all collisions,\\ \phantom{-} \ \ \ \ limiting therefore the possibility to achieve thermalization.) \bigskip \setlength{\baselineskip}{11pt} {\large\sf It turns out, however, that the INC is not a solution of the BTE in a strict sense, but rather of an other set of kinetic equations. For details see ref. [18] and references therein. \bigskip Many high energy string Monte-Carlo models are based on INC models. } }%end tr-page \newpage % transparency ========================================= p. 23 \transparencyframe{ \vspace*{-1.2cm} \subsection{Mean field models: BUU - VUU - BN - LV} \B The BTE was introduced in Chapter 3 without an external field.\\ In the non-relativistic theory it is usual to include on the left hand side of the BTE the effect of an external force [3] $$ (\partial_t + \vec{v} \nabla_x - \frac{\vec{F}}{m} \nabla_p ) f_1(x,p) = C[f_1 f_2]\ \ {\sf or} $$ $$ (\partial_t + \vec{v} \nabla_x - \frac{\nabla_x U}{m} \nabla_p ) f_1(x,p) = C[f_1 f_2] \ . $$ \B The nucleus is self-bound, kept together by a mean field potential $U$.\\ \LT Nucleons are then bound in this self-consistent mean field potential,\\ - and propagate along curved trajectories, determined by the mean field. \B The nucleons were artificially kept together initially in the INC model,\\ - furthermore, they did not feel the repulsive effect of nearby nucleons\\ - even at large nucleon densities.\\ --\ \ \ \ \ \ \ To avoid these problems: introduce an ``external'' potential. \B Although, the mean field potential is calculated in low energy NP\\ - in most cases the mean field is taken as a simple function of $n_B(x)$. \B The baryon density, $n_B(x)$, is calculated from the average density\\ --\ \ \ \ \ of the whole calculational ensemble.\\ --\ \ \ \ \ This provides a smooth potential surface\\ --\ \ \ \ \ responsible for binding and collective repulsion. }%end tr-page \newpage % transparency ============================================== p. 24 \transparencyframe{ \vspace*{-0.8cm} The mean field potential is able to keep the nuclei together.\\ --\ Freezing in the momentum distribution before the collision\\ --\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ is not necessary any more.\\ \LT In this model the initial states of nuclei can be dynamically stable nuclear configurations. \B The introduction of mean field led to a revolution\\ --\ \ in the capacity of the microscopic models.\\ --\ \ \ \ \ \ E.g.: INC models were not able to describe accurately\\ --\ \ \ \ \ \ the observed collective transverse flow in HI collisions. \B The introduction of mean field,\\ - which is repulsive at high avarage baryon densities\\ -- \ \ \ reproduced the data without any difficulty. \B The mean field potential has a certain degree of freedom,\\ - for example the potential may be {\bf momentum dependent}. \\ --\ \ \ Considering such a momentum dependence,\\ --\ \ \ \ \ \ the same transverse flow experiment\\ --\ \ \ \ \ \ can be described with a much softer potential. }%end tr-page \newpage % transparency ============================================= p. 25 \transparencyframe{ \vspace*{-0.8cm} \B Contrary to the overwhelming success of these microscopic models with mean field there are some DRAWBACKS of the model which should be mentioned:\\ i) * One is a theoretical problem\\ --\ \ \ \ of separating the nucleon nucleon interaction\\ --\ \ \ \ \ \ \ \ \ \ \ to a mean field and a cross section part.\\ --\ \ \ \ In a dense matter this is not a trivial task and\\ --\ \ \ \ \ \ \ \ \ \ \ obviously the nucleon-nucleon cross section \\ --\ \ \ \ \ \ \ \ \ \ \ must not be the same as in the vacuum. ii) * The other problem is related to the ensemble averaging:\\ \LT the mean field is smooth and density fluctuations are decreased\\ \LT large discrepancies compared to observed nuclear fragment formation. --\ \ \ The smooth mean field does not permit the formation\\ --\ \ \ of small selfbound clusters or nuclear fragments.\\ --\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ This was clearly contrary to experiments. \bigskip Thus these models are manifestly applicable for single particle observables, but phenomena like fragment formation or multifragmentation cannot be calculated in a one particle theory. }%end tr-page \newpage % transparency ============================================== p. 26 \transparencyframe{ \vspace*{-0.8cm} \subsection{Models of Molecular Dynamics} To solve the previous problem\\ -\ \ \ \ \ we need to consider N-particle correlations and\\ -\ \ \ \ \ N-particle distributions explicitely.\\ \B This is done in the {\bf classical equations of motion} approach or\\ -\ \ in molecular dynamics (MD) approaches or\\ -\ \ in the Quantum Molecular Dynamics (QMD) approach [20-22]. \B Here we follow each particle separately and\\ -\ \ calculate their interactions with all other particles one by one.\\ -\ \ This is a much larger computation than in the previous approaches,\\ -\ \ but it is necessary to maintain the N-paricle correlations. \B In the purely classical approaches quantum features are missing. \\ -\ \ The theoretical problem of constructing a nucleon-nucleon interaction\\ -\ \ which reproduces both the scattering and the ground state properties\\ -\ \ is also a nontrivial problem [19]. \B The QMD is essentially a classical equation of motion approach\\ -\ but it includes a few quantum effects:\\ --\ \ \ \ \ \ e.g. random scatterings, Pauli blocking of scatterings and\\ --\ \ \ \ \ \ stochastic particle decays. These models are quite succesful in reproducing multifragmentation experiments, but similarly to the previous microscopic models it requires a considerable extra effort to extract coolective macroscopic information from the model applicable for the dense and hot matter. }%end tr-page \newpage % transparency ======================================== p. 27 \transparencyframe{ \vspace*{-0.8cm} \section{Assignment 9} \begin{description} \item[9.a] Assume that we have two coexisting phases in a fluid, $i=1, 2$, described by volume fractions, $\lambda_i = V_i/V_{tot}$, or equivalently by particle number fractions, $\alpha_i = N_i/N_{tot}$, so that $\lambda_i = \alpha_i n/n_i$, where $ n_i = N_i/V_i$. Let us assume that the system is thermalized but the Gibbs criteria are not necessarily fulfilled. The energy momentum tensor for the whole system can be given as $$ T^{\mu\nu} = \sum_i \lambda_i T^{\mu\nu}_i \ . $$ Calculate the entropy production as a function of the change of $\alpha_i$ and $\lambda_i$. \end{description} }%end tr-page % transparency ================================================================ \end{document}