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Register a new userHysteresis in $eta/s$ for QFTs dual to spherical black holes
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We define and compute the (analogue) shear viscosity to entropy density ratio $tildeeta/s$ for the QFTs dual to spherical AdS black holes both in Einstein and Gauss-Bonnet gravity in five spacetime dimensions. Although in this case, owing to the lack of translational symmetry of the background, $tildeeta$ does not have the usual hydrodynamic meaning, it can be still interpreted as the rate of entropy production due to a strain. At large and small temperatures, it is found that $tildeeta/s$ is a monotonic increasing function of the temperature. In particular, at large temperatures it approaches a constant value, whereas, at small temperatures, when the black hole has a regular, stable extremal limit, $tildeeta/s$ goes to zero with scaling law behaviour. Whenever the phase diagram of the black hole has a Van der Waals-like behaviour, i.e. it is characterised by the presence of two stable states (small and large black holes) connected by a meta-stable region (intermediate black holes), the system evolution must occur through the meta-stable region and temperature-dependent hysteresis of $tildeeta/s$ is generated by non-equilibrium thermodynamics.
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Using the rules of the AdS/CFT correspondence, we compute the spherical analogue of the shear viscosity, defined in terms of the retarded Green function for the stress-energy tensor for QFTs dual to five-dimensional charged black holes of general relativity with a negative cosmological constant. We show that the ratio between this quantity and the entropy density, $tildeeta/s$, exhibits a temperature-dependent hysteresis. We argue that this hysteretic behaviour can be explained by the Van der Waals-like character of charged black holes, considered as thermodynamical systems. Under the critical charge, hysteresis emerges owing to the presence of two stable states (small and large black holes) connected by a meta-stable region (intermediate black holes). A potential barrier prevents the equilibrium path between the two stable states; the system evolution must occur through the meta-stable region, and a path-dependence of $tildeeta/s$ is generated.
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