Shear viscosity of a hadronic gas mixture

We discuss in detail the shear viscosity coefficient eta and the viscosity to entropy density ratio eta/s of a hadronic gas comprised of pions and nucleons. In particular, we study the effects of baryon chemical potential on eta and eta/s. We solve the relativistic quantum Boltzmann equations with binary collisions (pi pi, pi N, and NN) for a state slightly deviated from thermal equilibrium at temperature T and baryon chemical potential mu. The use of phenomenological amplitudes in the collision terms, which are constructed to reproduce experimental data, greatly helps to extend the validity region in the T-mu plane. The total viscosity coefficient eta(T,mu)=eta(pi)+eta(N) increases as a function of T and mu, indirectly reflecting energy dependences of binary cross sections. The increase in mu direction is due to enhancement of the nucleon contribution eta(N) while the pion contribution eta(pi) diminishes with increasing mu. On the other hand, due to rapid growth of entropy density, the ratio eta/s becomes a decreasing function of T and mu in a wide region of the T-mu plane. In the kinematical region we investigated T < 180 MeV, mu < 1 GeV, the smallest value of eta/s is about 0.3. Thus, it never violates the conjectured lower bound eta/s=1/4 pi similar to 0.1. The smallness of eta/s in the hadronic phase and its continuity at T similar or equal to T-c (at least for crossover at small mu) implies that the ratio will be small enough in the deconfined phase T greater than or similar to T-c. There is a nontrivial structure at low temperature and at around normal nuclear density. We examine its possible interpretation as the liquid-gas phase transition.