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

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 ($\bar{C}_1$ and $\bar{C}_2$), in terms of the trace and diffeomorphism anomaly coefficients ($c_w$ and $c_g$). The precise expression, we obtain here, is $\bar{C}_1 + \bar{C}_2\leq 4\pi^2c_w + 8\pi^2 c_g$. 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 {\it Israel-Hartle-Hawking} boundary condition. Finally, possible physical implications and connections with the existing approaches are addressed.