Abstract
We determine analytically the dependence of the approach to thermal equilibrium of strongly coupled plasmas on the breaking of scale invariance. The theories we consider are the holographic duals to Einstein gravity coupled to a scalar with an exponential potential. The coefficient in the exponent, $X$, is the parameter that controls the deviation from the conformally invariant case. For these models we obtain analytic solutions for the plasma expansion in the late-time limit, under the assumption of boost-invariance, and we determine the scaling behaviour of the energy density, pressure, and temperature as a function of time. We find that the temperature decays as a function of proper time as $T\sim \tau^{-s/4}$ with $s$ determined in terms of the non-conformality parameter $X$ as $s=4(1-4X^2)/3$. This agrees with the result of Janik and Peschanski, $s=4/3$, for the conformal plasmas and generalizes it to non-conformal plasmas with $X\neq 0$. We also consider more realistic potentials where the exponential is supplemented by power-law terms. Even though in this case we cannot have exact solutions, we are able under certain assumptions to determine the scaling of the energy, that receives logarithmic corrections.
Highlights
In the linear response regime, the holographic prescription boils down to solving the linearized equations for perturbations of the metric or other fields around a given background
We study the dependence of the approach to thermal equilibrium of strongly coupled plasmas on the breaking of scale invariance
The coefficient in the exponent, X, is the parameter that controls the deviation from the conformally invariant case. For these models we obtain analytic solutions for the plasma expansion in the late-time limit, under the assumption of boost-invariance, and we determine the scaling behaviour of the energy density, pressure, and temperature as a function of time, which is found to agree with the hydrodynamical expectation
Summary
Let us review the picture of the boost-invariant flow advocated by Bjorken [2]. It is convenient to introduce the pseudo-rapidity and the proper-time as t = τ cosh(y) x1 = τ sinh(y). We outline the idea of [1] that we will follow closely in this paper They start by considering the most general Ansatz for a bulk metric in AdS5 consistent with the symmetries of the Bjorken flow; this has the form ds. The Einstein equations reduce to a set of coupled non-linear differential equations that can be solved by the following change of variables: a(v) = A(v) − 2m(v) , b(v) = A(v) + (2s − 2)m(v) , c(v) = A(v) + (2 − s)m(v) Such solutions correspond to a boundary energy density behaving as ∼ τ −s, and there is a solution for generic s, the form of the solution shows a potential singularity at v4 = 1/∆(s). The subleading corrections in 1/τ contain informations about the deviation from perfect fluid, in particular the viscosity coefficients [11], which we will not consider here
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