What a Difference a Term Makes

The recent post on the AMO competition for perfect fluidity offers a wonderful opportunity to compare and contrast the techniques of condensed matter physics versus those of relativistic heavy ion physics. The experimental techniques, while vastly different, each push the state-of-the-art in their respective disciplines. But determining the all important viscosity to entropy density ratio η/s could not be more different between the two fields: in the case of the trapped atoms, an elegant yet straightforward perturbative analysis makes a direction connection between the damping rate of breathing modes and η/s. Nothing could be further from the truth in nuclear collisions- the development of causal, viscous hydrodynamics for truly relativistic systems remains very much a work in progress.

These issues are not apparent from consultation with the classic references. It’s hard to do better than the exposition found in Weinberg’s Gravitation and Cosmology, published in 1972. Weinberg begins with a perfect fluid, defined as one that appears isotropic to a co-moving observer. The equations of motion then follow directly from conservation of energy and momentum, the continuity equation for any conserved currents and the first law of thermodynamics, along with an equation of state to close the system. This is the so-called zero-th order formalism, which admits many analytic solutions such as the famous Bjorken expansion.

The introduction of “imperfection” in which the thermodynamics quantities vary appreciably on distances comparable to the mean free path will take the system out of strict thermal equilibrium and necessarily leads to dissipation. Weinberg does state this “raises certain delicate issues of principle”, then proceeds to develop a first-order expansion for weak space-time gradients, noting that this already leads to ambiguities in the definitions of the thermodynamic quantities. One of these is the definition of the co-moving four-velocity: Landau and Lifshitz take this to be the velocity of the energy transport, while in the approach of Eckart the velocity of particle transport is chosen. After stating a personal preference for the Eckart formalism, Weinberg then develops a consistent first-order formalism, taking care to insure that the 2nd law of thermodynamics is preserved.

However, hidden in this first-order formalism is a serious inconsistency. In 1985 Hiscock and Lindblom established that all first-order hydrodynamic expansions are unstable, with extraordinary rapid (10-34 sec for water at STP) exponential growth terms. It is of more than a little interest to note that their results, applied to a QGP with a value of η/s saturating the famous viscosity bound of Kovtun, Son and Starinets (KSS) gives a growth time for instabilities of order 1/(4πT), that is, as fast or faster than the thermalization time(!). In addition to this problem, first-order theory violates causality. A nice discussion of this can be found in

by Natsuume and Okamura, who simply argue that classical diffusive terms have Green’s functions ~ exp[-x2/4Dt], which implies super-luminal propagation.

Given these issues, it is perhaps surprising that a second-order expansion suffices to resolve them. Nonetheless, that seems to be the case. The “causal hydrodynamics” of Israel and Stewart was shown by Hiscock and Lindblom to be free of the instabilities found in first-order theories. An excellent description of this formalism may be found in

which is part of the highly recommended Living Reviews in Relativity archive. However, the Israel-Steward is neither unique not complete. For instance, the article notes that

Although the Israel–Stewart model resolves the problems of the first-order descriptions for near equilibrium situations, difficult issues remain to be understood for nonlinear problems… The fact that the formulation breaks down in nonlinear problems is not too surprising. After all, the basic foundation is a “Taylor expansion” in the various fields. However, it raises important questions. There are many physical situations where a reliable nonlinear model would be crucial, e.g. heavy-ion collisions and supernova core collapse. This problem requires further thought.

That “further thought” is now actively proceeding. Recent developments indicate that the progress is substantial and rapid. A not-so-obvious case-in-point is provided by

This one-page note is like a skeleton key- by following its references much of the recent progress in the field can be traced. The authors note that an oft-neglected 2nd-order term first noted by Muronga leads to a factor of 3 (!) increase in the viscous relaxation time computed in

by an impressive application of AdS/CFT technology. Note that even with the factor of 3 correction, the relaxation time computed in this strong-coupling limit is an order-of-magnitude smaller than the perturbative value. The importance of this same term is discussed in

note in particular Reference (footnote) 44 of that paper.

Much of the current rapid progress is being driven by the power of AdS/CFT methods to provide an exact description of hydrodynamic modes in the gravity dual. For example, in

the authors compute various transport coefficients by this method, and note conformal invariance requires additional terms neglected in the Israel-Stewart formalism that, while non-linear, are still second-order. However, it is not clear that second-order effects are the logical endpoint to the bookkeeping. In addressing the question

the authors argue that the scales in an expanding QGP require resummation of all “higher-order” viscosity terms, which they perform using the results of an AdS/CFT study on sound damping . Somewhat surprisingly, they find that higher-order terms contribute with both signs, leading to a reduction of the overall entropy produced as compared to that based on first-order calculations. So while it is clear that much work remains to be done, it is also clear that powerful advances are being made in creating a hydrodynamic formalism appropriate to the special case of the highly relativistic, approximately conformal QGP.

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