Bound on viscosity and the generalized second law of thermodynamics

Abstract

We describe a new paradox for ideal fluids. It arises in the accretion of an ideal fluid onto a black hole, where, under suitable boundary conditions, the flow can violate the generalized second law of thermodynamics. The paradox indicates that there is in fact a lower bound to the correlation length of any real fluid, the value of which is determined by the thermodynamic properties of that fluid. We observe that the universal bound on entropy, itself suggested by the generalized second law, puts a lower bound on the correlation length of any fluid in terms of its specific entropy. With the help of a new, efficient estimate for the viscosity of liquids, we argue that this also means that viscosity is bounded from below in a way reminiscent of the conjectured Kovtun-Son-Starinets lower bound on the ratio of viscosity to entropy density. We conclude that much light may be shed on the Kovtun-Son-Starinets bound by suitable arguments based on the generalized second law.

Gravitation, thermodynamics, and the bound on viscosity

“…Currently this bound is considered a conjecture well supported for a certain class of field theories [3]. So far, all known materials satisfy the bound for the range of temperatures and pressures examined in the laboratory.…”

“…It is not clear how to obtain such a bound directly from microscopic physics [3]. Inspection of the Green-Kubo formula [5] which relates the viscosity of a fluid to its fluctuations shows no apparent connection of the viscosity and the entropy density.…”

“…This may seem as bad news for our aspirations to prove a KSS-like bound. But one need not loose heart-there is general agreement that black hole entropy reflects some aspect of the quantum theory of gravity [3].…”

“…The GSL is a unique law of physics that bridges thermodynamics and gravity [3,9]. It asserts that in any interaction of a black hole with an ordinary matter, the sum of the entropies (matter+hole) never decreases.…”

“…The minimal length scale (radius) ℓ for which a consistent thermodynamic description is available is given by ℓ min = 1/2πT [3,12].…”