Entropy Current in Conformal Hydrodynamics (0801.3701v2)

Abstract: In recent work (arXiv:0712.2456, arXiv:0712.2451) the energy-momentum tensor for the N=4 SYM fluid was computed up to second derivative terms using holographic methods. The aim of this note is to propose an entropy current (accurate up to second derivative terms) consistent with this energy-momentum tensor and to explicate its relation with the existing theories of relativistic hydrodynamics. In order to achieve this, we first develop a Weyl-covariant formalism which simplifies the study of conformal hydrodynamics. This naturally leads us to a proposal for the entropy current of an arbitrary conformal fluid in any spacetime (with d>3). In particular, this proposal translates into a definite expression for the entropy flux in the case of N=4 SYM fluid. We conclude this note by comparing the formalism presented here with the conventional Israel-Stewart formalism.

Entropy Current in Conformal Hydrodynamics: Insights from N = 4 SYM Theory

The paper "Entropy Current in Conformal Hydrodynamics," represented as TIFR/TH/08-05, is a robust investigation into the entropy transport phenomena within the framework of conformal hydrodynamics, intricately linked with the energy-momentum tensor derived from N = 4 supersymmetric Yang-Mills (SYM) theory. This investigation leverages holographic techniques under the AdS/CFT correspondence, particularly focusing on second-order hydrodynamics expansion terms.

Constitutive Relations and Holographic Techniques

The author, R. Loganayagam from the Tata Institute of Fundamental Research, builds upon previous works that have computed the energy-momentum tensor for N = 4 SYM fluids using holographic methods. This paper proposes an expression for the entropy current coherent with these derived tensors and discusses its connection with the Weyl-covariant formalism, facilitating the analysis of conformal hydrodynamics.

N = 4 SYM theory, due to its super-conformal properties and the duality with IIB string theory, serves as an exquisite model to explore these hydrodynamic behaviors. The duality enunciates a parallel between the thermodynamics of AdS black holes and gauge theory, which is pivotal in deriving transport coefficients such as shear viscosity within this fluidic context.

Entropy Current Proposal

An entropy current up to second derivative accuracy is articulated as a crucial component consistent with the hydrodynamic derivative expansion. This current is expressed explicitly for spacetimes with dimensions $d > 3$, and its form in N = 4 SYM fluid is notably derived. The entropy flux's definitiveness is achieved by ensuring compatibility with the governing energy-momentum dynamics and adherence to the second law of thermodynamics.

$$J_s^\lambda = 4\pi\eta u^\lambda - \frac{1}{8(\pi T)^2}\left[(\ln2 \sigma_{\mu\nu} \sigma^{\mu\nu} + \omega_{\mu\nu} \omega^{\mu\nu}) u^\lambda + 2u^\lambda (G_{\mu\nu}^\lambda + F_{\mu\nu}^\lambda) + 6 D_\omega^{\nu\lambda}\right]$$

Comparison with Traditional Formalisms

In juxtaposition to the conventional Israel-Stewart framework, which incorporates non-linear relaxation dynamics to address causality issues in relativistic dissipative hydrodynamics, the approach in this paper provides a more expansive inclusion of transport phenomena. Particularly, the expressions derived for N = 4 SYM fluid incorporate additional viscoelastic terms, indicating the existence of scenarios not fully captured by traditional linear response theories. The emphasis on conformally covariant formulations also distinguishes this work, allowing for a more detailed parametrization of hydrodynamic responses.

Implications and Future Directions

The implications of this research are profound both for theory and potential applications in high-energy physics, particularly within the domain of heavy-ion collisions where quark-gluon plasma behaves akin to a nearly perfect fluid and hence, is conjectured to exhibit traits of conformal hydrodynamics. The insights gained can refine the understanding of transport processes, especially under extreme conditions akin to those produced at facilities like RHIC and LHC.

A crucial aspect needing further exploration lies in deriving the entropy current through direct gravitational computations, potentially corroborating the proposed formulations. Additionally, extending these analyses to charged conformal fluids could unravel further fundamental aspects of various gauge theories.

In conclusion, the insights drawn from Loganayagam's categorization of the entropy current not only enrich the understanding of N = 4 SYM hydrodynamics but also pave pathways for future theoretical developments and practical investigations into the dynamics of conformal fluids in both cosmological and quantum contexts.