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The exact expression for the entropy current of a fluid in presence of twodimensional gravitational anomalies is given. To make it compatible with thesecond law of thermodynamics; i.e. positivity of the entropy production rate ofa system (which is considered to be fundamental), we find a bound on the sum ofthe two response parameters ($\bar{C}_1$ and $\bar{C}_2$), in terms of thetrace and diffeomorphism anomaly coefficients ($c_w$ and $c_g$). The preciseexpression, we obtain here, is $\bar{C}_1 + \bar{C}_2\leq 4\pi^2c_w + 8\pi^2c_g$. Interestingly, when the bound is saturated corresponding to a reversibleprocess, the result reproduces earlier findings obtained by either studyingfield theory on the cone or using the {\it Israel-Hartle-Hawking} boundarycondition. Finally, possible physical implications and connections with theexisting approaches are addressed.

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