Entropy current in two dimensional anomalous hydrodynamics and a bound on the sum of the response parameters

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The exact expression for the entropy current of a fluid in presence of two dimensional gravitational anomalies is given. To make it compatible with the second law of thermodynamics; i.e. positivity of the entropy production rate of a system (which is considered to be fundamental), we find a bound on the sum of the two response parameters (C₁ and C₂), in terms of the trace and diffeomorphism anomaly coefficients (cw and cg). The precise expression, we obtain here, is C₁ + C₂≤ 4π²cw + 8π² cg. Interestingly, when the bound is saturated corresponding to a reversible process, the result reproduces earlier findings obtained by either studying field theory on the cone or using the Israel-Hartle-Hawking boundary condition. Finally, possible physical implications and connections with the existing approaches are addressed.