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In our Novi Sad conference paper (1999) we described Dehn type surgeries of the famous Gieseking (1912) hyperbolic ideal simplex manifold $\mathcal{S}$, leading to compact fundamental domain $\mathcal{S}(k)$, $k = 2, 3, \dots$ with singularity geodesics of rotation order $k$, but as later turned out with cone angle $2(k-1)/k$. We computed also the volume of $S(k)$, tending to zero if $k$ goes to infinity. That time we naively thought that we obtained orbifolds with the above surprising property. As the reviewer of Math. Rev., Kevin P. Scannell (MR1770996 (2001g:57030)) rightly remarked, "this is in conflict with the well-known theorem of D. A. Kazhdan and G. A. Margulis (1968) and with the work of Thurston, describing the geometric convergence of orbifolds under large Dehn fillings". In this paper we refresh our previous publication. Correctly, we obtained cone manifolds (for $k > 2$), as A. D. Mednykh and V. S. Petrov (2006) kindly pointed out. We complete our discussion and derive the above cone manifold series (Gies.1 and Gies.2) in two geometrically equivalent form, by the half turn symmetry of any ideal simplex. Moreover we obtain a second orbifold series (Gies.3 and 4), tending to the regular ideal simplex as the original Gieseking manifold., Comment: 19 pages, 9 figures

We study anisotropic fluid dynamics derived from the Boltzmann equation based on a particular choice for the anisotropic distribution function within a boost-invariant expansion of the fluid in one spatial dimension. In order to close the conservation equations we need to choose an additional moment of the Boltzmann equation. We discuss the influence of this choice of closure on the time evolution of fluid-dynamical variables and search for the best agreement to the solution of the Boltzmann equation in the relaxation-time approximation., Comment: 4 pages, 4 figures, proceedings for Quark Matter 2017

Fluid-dynamical equations of motion can be derived from the Boltzmann equation in terms of an expansion around a single-particle distribution function which is in local thermodynamical equilibrium, i.e., isotropic in momentum space in the rest frame of a fluid element. However, in situations where the single-particle distribution function is highly anisotropic in momentum space, such as the initial stage of heavy-ion collisions at relativistic energies, such an expansion is bound to break down. Nevertheless, one can still derive a fluid-dynamical theory, called anisotropic dissipative fluid dynamics, in terms of an expansion around a single-particle distribution function, $\hat{f}_{0\bf k}$, which incorporates (at least parts of) the momentum anisotropy via a suitable parametrization. We construct such an expansion in terms of polynomials in energy and momentum in the direction of the anisotropy and of irreducible tensors in the two-dimensional momentum subspace orthogonal to both the fluid velocity and the direction of the anisotropy. From the Boltzmann equation we then derive the set of equations of motion for the irreducible moments of the deviation of the single-particle distribution function from $\hat{f}_{0\bf k}$. Truncating this set via the 14-moment approximation, we obtain the equations of motion of anisotropic dissipative fluid dynamics., Comment: 52 pages; Abstract slightly changed; new references and new paragraph added

We consider the optimal approximation of certain quantum states of a harmonic oscillator with the superposition of a finite number of coherent states in phase space placed either on an ellipse or on a certain lattice. These scenarios are currently experimentally feasible. The parameters of the ellipse and the lattice and the coefficients of the constituent coherent states are optimized numerically, via a genetic algorithm, in order to obtain the best approximation. It is found that for certain quantum states the obtained approximation is better than the ones known from the literature thus far., Comment: 11 pages, 12 figures

In Denicol et al., Phys. Rev. D 85, 114047 (2012), the equations of motion of relativistic dissipative fluid dynamics were derived from the relativistic Boltzmann equation. These equations contain a multitude of terms of second order in Knudsen number, in inverse Reynolds number, or their product. Terms of second order in Knudsen number give rise to non-hyperbolic (and thus acausal) behavior and must be neglected in (numerical) solutions of relativistic dissipative fluid dynamics. The coefficients of the terms which are of the order of the product of Knudsen and inverse Reynolds numbers have been explicitly computed in the above reference, in the limit of a massless Boltzmann gas. Terms of second order in inverse Reynolds number arise from the collision term in the Boltzmann equation, upon expansion to second order in deviations from the single-particle distribution function in local thermodynamical equilibrium. In this work, we compute these second-order terms for a massless Boltzmann gas with constant scattering cross section. Consequently, we assess their relative importance in comparison to the terms which are of the order of the product of Knudsen and inverse Reynolds numbers., Comment: 34 pages; corrections for Appendix I, and new references added

Israel-Stewart theory is a causal, stable formulation of relativistic dissipative fluid dynamics. This theory has been shown to give a decent description of the dynamical behavior of a relativistic fluid in cases where shear stress becomes important. In principle, it should also be applicable to situations where heat flow becomes important. However, it has been shown that there are cases where Israel-Stewart theory cannot reproduce phenomena associated with heat flow. In this paper, we derive a relativistic dissipative fluid-dynamical theory from kinetic theory which provides a good description of all dissipative phenomena, including heat flow. We explicitly demonstrate this by comparing this theory with numerical solutions of the relativistic Boltzmann equation., Comment: 13 pages, 5 figures

We review the traditional derivation of the fluid-dynamical equations from kinetic theory according to Israel and Stewart. We show that their procedure to close the fluid-dynamical equations of motion is not unique. Their approach contains two approximations, the first being the so-called 14-moment approximation to truncate the single-particle distribution function. The second consists in the choice of equations of motion for the dissipative currents. Israel and Stewart used the second moment of the Boltzmann equation, but this is not the only possible choice. In fact, there are infinitely many moments of the Boltzmann equation which can serve as equations of motion for the dissipative currents. All resulting equations of motion have the same form, but the transport coefficients are different in each case., Comment: 15 pages, 3 figures, typos fixed and discussions added; EPJA: Topical issue on "Relativistic Hydro- and Thermodynamics"

Experimental particle spectra can be successfully described by power-law tailed energy distributions characteristic to canonical equilibrium distributions associated to R\'enyi's or Tsallis' entropy formula - over a wide range of energies, colliding system sizes, and produced hadron sorts. In order to derive its evolution one needs a corresponding dynamical description of the system which results in such final state observables. The equations of relativistic fluid dynamics are obtained from a non-extensive Boltzmann equation consistent with Tsallis' non-extensive $q$-entropy formula. The transport coefficients like shear viscosity, bulk viscosity, and heat conductivity are evaluate based on a linearized collision integral., Comment: 11 pages, corrected typos and more discussions added; EPJA: Topical issue on "Relativistic Hydro- and Thermodynamics"

We study the influence of a temperature-dependent shear viscosity over entropy density ratio $\eta/s$, different shear relaxation times $\tau_\pi$, as well as different initial conditions on the transverse momentum spectra of charged hadrons and identified particles. We investigate the azimuthal flow asymmetries as a function of both collision energy and centrality. The elliptic flow coefficient turns out to be dominated by the hadronic viscosity at RHIC energies. Only at higher collision energies the impact of the viscosity in the QGP phase is visible in the flow asymmetries. Nevertheless, the shear viscosity near the QCD transition region has the largest impact on the collective flow of the system. We also find that the centrality dependence of the elliptic flow is sensitive to the temperature dependence of $\eta/s$., Comment: 13 pages, 20 figures

In this work we present a general derivation of relativistic fluid dynamics from the Boltzmann equation using the method of moments. The main difference between our approach and the traditional 14-moment approximation is that we will not close the fluid-dynamical equations of motion by truncating the expansion of the distribution function. Instead, we keep all terms in the moment expansion. The reduction of the degrees of freedom is done by identifying the microscopic time scales of the Boltzmann equation and considering only the slowest ones. In addition, the equations of motion for the dissipative quantities are truncated according to a systematic power-counting scheme in Knudsen and inverse Reynolds number. We conclude that the equations of motion can be closed in terms of only 14 dynamical variables, as long as we only keep terms of second order in Knudsen and/or inverse Reynolds number. We show that, even though the equations of motion are closed in terms of these 14 fields, the transport coefficients carry information about all the moments of the distribution function. In this way, we can show that the particle-diffusion and shear-viscosity coefficients agree with the values given by the Chapman-Enskog expansion., Comment: 27 pages

We derive equations for fluid dynamics from a non-extensive Boltzmann transport equation consistent with Tsallis' non-extensive entropy formula. We evaluate transport coefficients employing the relaxation time approximation and investigate non-extensive effects in leading order dissipative phenomena at relativistic energies, like heat conductivity, shear and bulk viscosity., Comment: 9 pages, 5 figures. Some small corrections in the text and in the first figure caption; accepted for publication in Physical Review C

We derive the equations of second order dissipative fluid dynamics from the relativistic Boltzmann equation following the method of W. Israel and J. M. Stewart. We present a frame independent calculation of all first- and second-order terms and their coefficients using a linearised collision integral. Therefore, we restore all terms that were previously neglected in the original papers of W. Israel and J. M. Stewart., Comment: 8 pages, added references and corrected typos; Proceedings for HCBM 2010: International Workshop on Hot and Cold Baryonic Matter 2010, 15-20 August 2010, Budapest, Hungary

Employing a microscopic transport model we investigate the evolution of high energetic jets moving through a viscous medium. For the scenario of an unstoppable jet we observe a clearly strong collective behavior for a low dissipative system $\eta/s \approx 0.005$, leading to the observation of cone-like structures. Increasing the dissipation of the system to $\eta/s \approx 0.32$ the Mach Cone structure vanishes. Furthermore, we investigate jet-associated particle correlations. A double-peak structure, as observed in experimental data, is even for low-dissipative systems not supported, because of the large influence of the head shock., Comment: 4 pages, 3 figures, to appear in the conference proceedings of Hot Quarks 2010

We solve the relativistic Riemann problem in viscous matter using the relativistic Boltzmann equation and the relativistic causal dissipative fluid-dynamical approach of Israel and Stewart. Comparisons between these two approaches clarify and point out the regime of validity of second-order fluid dynamics in relativistic shock phenomena. The transition from ideal to viscous shocks is demonstrated by varying the shear viscosity to entropy density ratio $\eta/s$. We also find that a good agreement between these two approaches requires a Knudsen number $Kn < 1/2$., Comment: Version as published in PRC 82, 024910 (2010); 16 pages, 16 figures, typos corrected

To investigate the formation and the propagation of relativistic shock waves in viscous gluon matter we solve the relativistic Riemann problem using a microscopic parton cascade. We demonstrate the transition from ideal to viscous shock waves by varying the shear viscosity to entropy density ratio $\eta/s$. We show that an $\eta/s$ ratio larger than 0.2 prevents the development of well-defined shock waves on time scales typical for ultrarelativistic heavy-ion collisions. These findings are confirmed by viscous hydrodynamic calculations., Comment: 4 pages, 3 figures - To appear in the conference proceedings for Quark Matter 2009, March 30 - April 4, Knoxville, Tennessee

We present numerical methods to solve the Israel-Stewart (IS) equations of causal relativistic dissipative fluid dynamics with bulk and shear viscosities. We then test these methods studying the Riemann problem in (1+1)-- and (2+1)-dimensional geometry. The numerical schemes investigated here are applicable to realistic (3+1)--dimensional modeling of a relativistic dissipative fluid., Comment: 21 pages, 4 figures

We solve the relativistic Riemann problem in viscous gluon matter employing a microscopic parton cascade. We demonstrate the transition from ideal to viscous shock waves by varying the shear viscosity to entropy density ratio $\eta/s$ from zero to infinity. We show that an $\eta/s$ ratio larger than 0.2 prevents the development of well-defined shock waves on timescales typical for ultrarelativistic heavy-ion collisions. Comparisons with viscous hydrodynamic calculations confirm our findings., Comment: Version as published in PRL 103, 032301 (2009). 4 pages, 4 figures

A novel higher order theory of relaxation of heat and viscosity is proposed based on corrections to the traditional treatment of the relativistic energy density. In the framework of generalized Bjorken scaling solution to accelerating longitudinal flow we point out that the energy flux can be consequently set to zero in the stationary case, independently of the choice of a specific local rest frame, like the Landau-Lifshitz or Eckart one. We investigate and compare several cooling and re-heating scenarios for the Quark Gluon Plasma (QGP) within this approach., Comment: 13 pages, 4 figures

Entropy production due to shear viscosity during the continuous freeze-out of a longitudinally expanding dissipative fluid is addressed. Assuming the validity of the fluid dynamical description during the continuous removal of interacting matter we estimated a small entropy production as function of the freeze-out duration and the ratio of dissipative to non-dissipative quantities in case of a relativistic massless pion fluid., Comment: 5 pages, 3 figures

We study entropy production in the early stage of high-energy heavy-ion collisions due to shear viscosity. We employ the second-order theory of Israel-Stewart with two different stress relaxation times, as appropriate for strong coupling or for a Boltzmann gas, respectively, and compare the hydrodynamic evolution. Based on present knowledge of initial particle production, we argue that entropy production is tightly constrained. We derive new limits on the shear viscosity to entropy density ratio $\eta/s$, independent from elliptic flow effects, and determine the corresponding Reynolds number. Furthermore, we show that for a given entropy production bound, that the initial time $\tau_0$ for hydrodynamics is correlated to the viscosity. The conjectured lower bound for $\eta/s$ provides a lower limit for $\tau_0$., Comment: 10 pages, 10 figures, Text updated, references added, two new figures added, normalization in Fig.1 corrected, final version to be published in PRC

The freeze out of the expanding systems, created in relativistic heavy ion collisions, will be discussed. We combine kinetic freeze out equations with Bjorken type system expansion into a unified model. Such a model is a more physical generalization of the earlier simplified non-expanding freeze out models. We shall see that the basic freeze out features, pointed out in the earlier works, are not smeared out by the expansion., Comment: 6 pages, 3 figures. Presented at the 19th International Conference On Ultrarelativistic Nucleus-Nucleus Collisions: Quark Matter 2006 (QM 2006), Shanghai, China, November 14-20, 2006

The freeze out (FO) of the expanding systems, created in relativistic heavy ion collisions, is discussed. We start with kinetic FO model, which realizes complete physical FO in a layer of given thickness, and then combine our gradual FO equations with Bjorken type system expansion into a unified model. We shall see that the basic FO features, pointed out in the earlier works, are not smeared out by the expansion., Comment: 3 pages, 2 figures

The freeze out of a massive nucleon gas through a finite layer with time-like normal is studied. The impact of in-medium nucleon mass shift on the freeze out process is investigated. A considerable modification of the thermodynamical variables temperature, flow-velocity, energy density and particle density has been found. Due to the nucleon mass shift the freeze out particle distribution functions are changed noticeably in comparison with evaluations, which use vacuum nucleon mass., Comment: submitted to Physical Review C

The freeze out of particles from a layer of finite thickness is discussed in a phenomenological kinetic model. The proposed model, based on the Modified Boltzman Transport Equation, is Lorentz invariant and can be applied equally well for the freeze out layers with space-like and time-like normal vectors. It leads to non-equilibrated post freeze out distributions. The dependence of the resulting distribution on the thickness of the layer is presented and discussed for a space-like freeze out scenario., Comment: Minor corrections to improve the presentation. 4 pages, 2 figures, to appear in the Proceedings of "Quark Matter 2005", August 4-9, 2005, Budapest, Hungary

We develop and analyze a covariant FO probability valid for a finite space-time layer., Comment: Proceedings of "Quark Matter 2005", 4 pages, 3 figures, with corrections

We study Freeze Out process in high energy heavy ion reaction. The description of the process is based on the Boltzmann Transport Equation (BTE). We point out the basic limitations of the BTE approach and introduce Modified BTE. The Freeze Out dynamics is presented in the 4-dimensional space-time in a layer of finite thickness, and we employ Modified BTE for the realistic Freeze Out description., Comment: 9 pages, 2 figures

The problem of Freeze Out (FO) in relativistic heavy ion reactions is addressed. We develop and analyze an idealized one-dimensional model of FO in a finite layer, based on the covariant FO probability. The resulting post FO phase-space distributions are discussed for different FO probabilities and layer thicknesses., Comment: 16 pages, 19 figures, changed content, references added

The Freeze Out (FO) problem is addressed for a covariant FO probability and a finite FO layer with a time-like normal vector continuing the line of studies introduced in Ref. [1]. The resulting post FO momentum distribution functions are presented and discussed. We show that in general the post FO distributions are non-thermal and asymmetric distributions even for time-like FO situations., Comment: 10 pages, 12 figures, major rewrite with changed content, corrected typos and new references added

Recently several works have appeared in the literature in which authors try to describe Freeze Out (FO) in energetic heavy ion collisions based on the Boltzmann Transport Equation (BTE). The aim of this work is to point out the limitations of the BTE, when applied for the modeling of FO or other very fast process, and to propose the way how the BTE approach can be generalized for such a processes., Comment: Presented at the 18th Nuclear Physics Division Conference of the EPS "Phase transitions in strongly interacting matter", Prague, Czech Republic, August 23-29, 2004, 4 pages

Fluid dynamical calculations with QGP showed a softening of the directed flow while with hadronic matter this effect is absent. On the other hand, we indicated that a third flow component shows up in the reaction plane as an enhanced emission, which is orthogonal to the directed flow. This is not shadowed by the deflected projectile and target, and shows up at measurable rapidities, $y_cm = 1-2$. To study the formation of this effect initial stages of relativistic heavy ion collisions are studied. An effective string rope model is presented for heavy ion collisions at RHIC energies. Our model takes into account baryon recoil for both target and projectile, arising from the acceleration of partons in an effective field. The typical field strength (string tension) for RHIC energies is about 5-12 GeV/fm, what allows us to talk about "string ropes". The results show that QGP forms a tilted disk, such that the direction of the largest pressure gradient stays in the reaction plane, but deviates from both the beam and the usual transverse flow directions. The produced initial state can be used as an initial condition for further hydrodynamical calculations. Such initial conditions lead to the creation of third flow component. Recent $v_1$ measurements are promising that this effect can be used as a diagnostic tool of the QGP. Collective flow is sensitive to the early stages of system evolution. To study the sensitivity of the flow signal, we have calculated flow harmonics from a Blast Wave model, a tilted, ellipsoidally expanding source. We studied recent experimental techniques used for calculation of the $v_n$ Fourier coefficients and pointed out a few possible problems connected to these techniques, which may impair the sensitivity of flow analysis., Comment: 15 pages, 4 figures, talk presented at the "Hadron-RAMP 2004" meeting, March 28 - April 3, 2004, Angra dos Reis (Rio de Janeiro), Brasil

Recently several works have appeared in the literature that addressed the problem of Freeze Out in energetic heavy ion reaction and aimed for a description based on the Boltzmann Transport Equation (BTE). In this paper we develop a dynamical Freeze-Out description, starting from the BTE, pointing out the basic limitations of the BTE approach, and the points where the BTE approach should be modified., Comment: More explanations about the Modified Boltzmann Transport Equation are added. 5 pages, 1 figure. Please note that there is a more detailed paper on the Modified Boltzmann Transport Equation published in the ArXiv: hep-ph/0502185

Earlier fluid dynamical calculations with QGP show a softening of the directed flow while with hadronic matter this effect is absent. On the other hand, we indicated that a third flow component shows up in the reaction plane as an enhanced emission, which is orthogonal to the directed flow. This is not shadowed by the deflected projectile and target, and shows up at measurable rapidities, $y_cm = 1-2$. To study the formation of this effect initial stages of relativistic heavy ion collisions are studied. An effective string rope model is presented for heavy ion collisions at RHIC energies. Our model takes into account baryon recoil for both target and projectile, arising from the acceleration of partons in an effective field. The typical field strength (string tension) for RHIC energies is about 5-12 GeV/fm, what allows us to talk about "string ropes". The results show that QGP forms a tilted disk, such that the direction of the largest pressure gradient stays in the reaction plane, but deviates from both the beam and the usual transverse flow directions. The produced initial state can be used as an initial condition for further hydrodynamical calculations. Such initial conditions lead to the creation of third flow component. Recent $v_1$ measurements are promising that this effect can be used as a diagnostic tool of the QGP.

Modelling Quark-Gluon Plasma formation and decay in high energy heavy ion reactions is presented in a framework of a multi-module setup. The collective features, governing the equlibrated fluid dynamical stages of the model are emphasized. Flow effects formed from the initial conditions are discussed. Particular attention is given to the improvement of the final hadronization and freeze-out part of the reaction which has strong effects on the observables., Comment: The misprint in the surname of one of the co-authers was corrected. 16 pages, 4 figures, presented at the NATO Advances Study Institute "Structure and Dynamics of Elementary Matter", September 22 - October 2, 2003, Kemer, Turkey