Self-consistent approximations in terms of fully dressed propagators provide a simple expression for the entropy of an ultrarelativistic plasma, which isolates the contribution of the elementary excitations as a leading contribution.Further approximations, whose validity is checked on a soluble model involving a scalar field, allow us to calculate the entropy of the QCD plasma. We obtain an accurate description of lattice data for purely gluonic QCD, down to temperatures of about twice the transition temperature.

We propose a gauge-invariant and manifestly UV finite resummation of the physics of hard thermal or dense loops ͑HTL-HDL͒ in the thermodynamics of the quark-gluon plasma. The starting point is a simple, effectively one-loop expression for the entropy or the quark density which is derived from the fully self-consistent two-loop skeleton approximation to the free energy, but subject to further approximations, whose quality is tested in a scalar toy model. In contrast with the direct HTL-HDL resummation of the one-loop free energy, in our approach both the leading-order ͑LO͒ and the next-to-leading order ͑NLO͒ effects of interactions are correctly reproduced and arise from kinematical regimes where the HTL-HDL are justifiable approximations. The LO effects are entirely due to the ͑asymptotic͒ thermal masses of the hard particles. The NLO ones receive contributions both from soft excitations, as described by the HTL-HDL propagators, and from corrections to the dispersion relation of the hard excitations, as given by HTL-HDL perturbation theory. The numerical evaluations of our final expressions show very good agreement with lattice data for zero-density QCD, for temperatures above twice the transition temperature.

We study the conductivity and shear viscosity tensors of a strongly coupled N = 4 super-YangMills plasma which is kept anisotropic by a θ parameter that depends linearly on one of the spatial dimensions. Its holographic dual is given by an anisotropic axion-dilaton-gravity background and has recently been proposed by Mateos and Trancanelli as a model for the pre-equilibrium stage of quark-gluon plasma in heavy-ion collisions. By applying the membrane paradigm which we also check by numerical evaluation of Kubo formula and lowest lying quasinormal modes, we find that the shear viscosity purely transverse to the direction of anisotropy saturates the holographic viscosity bound, whereas longitudinal shear viscosities are smaller, providing the first such example not involving higher-derivative theories of gravity and, more importantly, with fully known gaugegravity correspondence.PACS numbers: 11.25. Tq, 11.10Wx, 12.38.Mh Introduction. Hydrodynamic simulations of heavyion collisions suggest [1] that the produced quark-gluon plasma is behaving like an almost perfect fluid with a ratio of shear viscosity over entropy density not far from the famous result /4π associated with the membrane paradigm of black holes [2] and which holographic gaugegravity duality maps to the corresponding quantity of maximally supersymmetric Yang-Mills theory in the limit of infinite color number and infinite 't Hooft coupling [3,4]. This value has been conjectured to form the lower bound for any realistic matter [5]. It was found to be saturated universally [6,7] in dual theories involving an isotropic horizon described by Einstein gravity. Values above this bound are obtained when corrections due to finite coupling strength are included [8], but it has been shown that values violating the bound can arise in higherderivate gravities [9], although so far no complete gaugegravity correspondence has been established for finite violations.

Non-Abelian plasma instabilities may be responsible for the fast apparent quark-gluon thermaliza-tion in relativistic heavy-ion collisions if their exponential growth is not hindered by nonlinearities. We study numerically the real-time evolution of instabilities in an anisotropic non-Abelian plasma with an SU(2) gauge group in the hard-loop approximation. We find exponential growth of non-Abelian plasma instabilities both in the linear and in the strongly nonlinear regime, with only a brief phase of subexponential behavior in between. In this Letter we present first results on the real-time evolution of non-Abelian plasma instabilities due to momentum-space anisotropies in the underlying quark and gluon distribution functions [1, 2, 3, 4, 5, 6, 7] in the nonlinear hard-loop approximation. Such anisotropies are generated during the natural expansion of the matter created during a heavy-ion collision and the resulting instabilities may be responsible for the fast apparent thermalization [8], which seems to be faster than can be accounted for by perturbative scattering processes [9, 10, 11, 12]. This type of plasma instability is the analogue of the electromagnetic Weibel instability which causes soft gauge (magnetic) fields to become nonper-turbatively large. Eventually this leads to large-angle scattering of hard particles [13], thereby rapidly accelerating the isotropization and subsequent thermalization of an Abelian plasma with a temperature anisotropy. However , in the non-Abelian case it is conceivable that the intrinsic nonlinearities could cause the instabilities to stop growing before they have large effects on hard particles and therefore reduce their efficacy in isotropizing a non-Abelian plasma. The regime where the backreaction of collective fields on the hard particles is still weak but where the self-interaction of the former may already be strongly nonlin-ear is governed by a "hard-loop" effective action which has been derived in Ref. [6] for arbitrary momentum-space anisotropies [20]. We discretize this effective action in a local auxiliary-field formulation, keeping its full dynamical nonlinearity and nonlocality. This is then applied to initial conditions that allow for an effectively 1+1-dimensional lattice simulation, extending a previous numerical study [7] that used a static and linear approximation to the hard-loop effective action. Discretized Hard-Loop Dynamics.-At weak gauge coupling g, there is a separation of scales in hard mo-menta |p| = p 0 of (ultrarelativistic) plasma constituents, and soft momenta ∼ g|p| pertaining to collective dynamics. The effective field theory for the soft modes that is generated by integrating out the hard plasma modes at one-loop order and in the approximation that the amplitudes of the soft gauge fields obey A µ ≪ |p|/g is that of gauge-covariant collisionless Boltzmann-Vlasov equations [14]. In equilibrium, the corresponding (nonlocal) effective action is the so-called hard-thermal-loop effective action [15] which has a simple generalization to plasmas w...

We calculate anomaly induced conductivities from a holographic gauge theory model using Kubo formulas, making a clear conceptual distinction between thermodynamic state variables such as chemical potentials and external background fields. This allows us to pinpoint ambiguities in previous holographic calculations of the chiral magnetic conductivity. We also calculate the corresponding anomalous current three-point functions in special kinematic regimes. We compare the holographic results to weak coupling calculations using both dimensional regularization and cutoff regularization. In order to reproduce the weak coupling results it is necessary to allow for singular holographic gauge field configurations when a chiral chemical potential is introduced for a chiral charge defined through a gauge invariant but non-conserved chiral density. We argue that this is appropriate for actually addressing charge separation due to the chiral magnetic effect.