Dissipative or Just Nonextensive Hydrodynamics? -nonextensive/dissipative Correspondence

Recently there is renewed interest in dissipative hydrodynamical models [1, 2, 3, 4, 5] prompted by the success of perfect hydrodynamics in describing RHIC data [6] and by recent calculations of transport coefficients of strongly interacting quark-gluon system using the AdS/CFT correspondence [7]. The question is whether dissipative hydrodynamics (d-hydrodynamics) is really needed and how it should be implemented regarding problems with its formulation like ambiguities in the form equations used [3], unphysical instability of the equilibrium state in the first order theory [4] or loss of causality in the first order equation approach [5]. We would like to propose a novel view on the dissipative hydrodynamics based on nonextensive formulation of the ideal hydrodynamical model, which we call the q-hydrodynamics [8]. When applied to ideal q-fluid it can be solved exactly, in way similar to the usual ideal hydrodynamics. However, it contains additional terms which can be interpreted as due to dissipative effects expressed by the nonextensivity parameter q a single parameter here. Therefore, the following nonextensive/dissipative correspondence (NexDC) emerges: ideal q-fluid is apparently equivalent to some viscous fluid with its transport coefficients being (implicit) functions of parameter q. This parameter combines information about all possible intrinsic fluctuations and correlations existing in the collision process (in particular in the QGP being formed). Referring to [8] for more information on q-statistics it is enough to say here that it is based on (indexed by q and nonextensive, see left panel of Fig. 1) Tsallis rather than BoltzmannGibbs (BG) entropy to which it converges for q → 1. Characteristic feature here is appearance of q-exponentials, expq(−X) = [1 − (1 − q)X ] 1/(1−q) → exp(−X) for q → 1. Among other things q− 1 measures scaled variance of the corresponding intensive quantities like, for example, temperature T , or the amount of nonvanishing in the hydrodynamical limit correlations (see left panel of Fig. 1) [8]). Although q-hydrodynamics does not fully solve the problems of d-hydrodynamics, nevertheless it allows us to extend the usual perfect fluid approach (using only one new parameter q) well behind its usual limits toward the regions reserved for dissipative approach only. In this note we can only explain main points of our proposition leaving interested reader to [8] and references there for details. Our idea is visualized in Fig. 1. Left panel shows that in the usual hydrodynamics there is some spacial scale Lhyd, such that volume L 3 hyd contains enough particles composing our fluid. However, in case when there are some fluctuations and/or correlations in the system characterized by some typical correlation length l and when of l > Lhyd, taking the usual limit Lhyd → 0 removes the explicit dependence on the scale Lhyd but the correlation length l leaves its imprint as parameter q and one has to use nonextensive entropy (one can argue that in this case