Abstract
We derive a set of nontrivial relations between second-order transport coefficients which follow from the second law of thermodynamics upon considering a regime close to uniform rotation of the fluid. We demonstrate that an extension of hydrodynamics by spin variable is equivalent to modifying conventional hydrodynamics by a set of second-order terms satisfying the relations we derived. We point out that a novel contribution to the heat current orthogonal to vorticity and temperature gradient reminiscent of the thermal Hall effect is constrained by the second law.
Highlights
Introduction.—Relativistic hydrodynamics [1] is an effective description, at large distance and timescales, of systems in local thermodynamic equilibrium parametrized by slowly varying profiles of four-velocity uμðxÞ, local temperature TðxÞ, and chemical potential μðxÞ for a conserved charge
Relativistic hydrodynamics has been successful in many branches of physics, in particular, in describing the dynamical evolution of the fireball created in relativistic heavy-ion collisions (RHIC) [2,3]
We show that the same constraints can be derived directly using the second law of thermodynamics, and are universal
Summary
This allows us to reorganize the naive gradient expansion in the entropy production rate and to derive a set of nontrivial constraints on certain second-order transport coefficients by applying the second law of thermodynamics. We consider spatial gradients of thermal vorticity to be of the same order as the dissipative gradients, i.e., ∂⊥ν ðβωμÞ ∼ εωε rather than εωε0, which means From this and the ideal equation of motion, one can show that ∂μωμ ∼ ωμ∂μβ ∼ εωε. This allows us to focus on the vorticity related terms arising from certain secondorder transport coefficients as the leading contributions to the entropy production rate up to order εωε0ε, while the dissipative terms from first-order transport coefficients are of the order ε2 ≪ εωε0ε, and are subleading. We remark that one could put a purely spatial gradient ΔμρΔγν∂γωνρ in place of Δμρ∂νωνρ, but this would be equivalent up to a redefinition of fb; c2g due to the ideal equations of motion and the thermodynamic relation βdp 1⁄4 −wdβ þ ndα
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