Abstract
We investigate the time-dependent perturbations of strongly coupled mathcal{N} = 4 SYM theory at finite temperature and finite chemical potential with a second order phase transition. This theory is modelled by a top-down Einstein-Maxwell-dilaton description which is a consistent truncation of the dimensional reduction of type IIB string theory on AdS5×S5. We focus on spin-1 and spin-2 sectors of perturbations and compute the linearized hydrodynamic transport coefficients up to the third order in gradient expansion. We also determine the radius of convergence of the hydrodynamic mode in spin-1 sector and the lowest non-hydrodynamic modes in spin-2 sector. Analytically, we find that all the hydrodynamic quantities have the same critical exponent near the critical point θ = frac{1}{2} . Moreover, we propose a relation between symmetry enhancement of the underlying theory and vanishing of the only third order hydrodynamic transport coefficient θ1, which appears in the shear dispersion relation of a conformal theory on a flat background.
Highlights
super YangMills (SYM) theory at finite temperature and finite chemical potential with a second order phase transition
We propose a relation between symmetry enhancement of the underlying theory and vanishing of the only third order hydrodynamic transport coefficient θ1, which appears in the shear dispersion relation of a conformal theory on a flat background
After reviewing general aspects of hydrodynamics as a gradient expansion in section 2, Green’s function and linear response theory in section 3, we focused on a specific example from section 4 onwards, namely the 4-dimensional N = 4 SYM theory at finite temperat√ure and finite chemical potential which poses a second order phase transition at μ = πT / 2
Summary
We will review the basic principles of the relativistic hydrodynamics of the boundary theory. The existence of the hydrodynamic equations is due to the conservation laws which are related to the continuous symmetries of the underlying theory. Where T μν is the energy-momentum tensor corresponding to the space-time symmetries and Jμ is a conserved current corresponding to a local U(1) symmetry of the underlying theory. The corresponding third order transport coefficients λ(i3) are not uniquely defined but for a given underlying theory there exist at most 20 of them. In the case of relativistic kinetic theory 14 third-order transport coefficients were determined in [58] by studying evolution equation for shear stress tensor. We present the basic ingredients and methods to derive the transport coefficients for N = 4 SYM theory at finite temperature and finite chemical potential by using the gauge/gravity duality.
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