Usually, the shear stress in a fluid increases with the applied shear rate. However, in some complex fluids such as wormlike micelles (some surfactant solutions), the shear stress is almost constant over a wide range of applied shear rates . This striking feature of the surfactant solutions is hypothetically explained by the scenario that the bulk under flow is spontaneously separated into two regions, the fast shear rate region and slow one, named shear banding, and the volume fraction of the banded regions changes with the applied shear rate, just like in the case of liquid-vapor phase transitions. This scenario is examined by many experi-ments and reaches some consensus . However, why shear banding, which is a quite non equilibrium phe-nomenon, is similar to the thermodynamic phase transitions, is still unknown. In this talk, I theoretically demonstrate similarity between shear banding and thermodynamic phase transitions by deriving a scalar potential that corresponds to the free energy in the system, from the Johnson-Segalman model which is a mechanical constitutive equation that has no thermodynamic potential . In addition, we experimentally demonstrate that shear banding also appears in a solution of biological molecules, an actin solution . I dis-cuss the role of shear banding in biological systems.
 H. Rehage and H. Hoffmann, Mol. Phys. 74, 933 (1991).
 J. F. Berret, in Molecular Gels, edited by R. G. Weiss and P. Terech (Springer, Dordrecht, 2006), Chap. 19.
 K. Sato, X. F. Yuan, and T. Kawakatsu, Eur. Phys. J. E 31, 135-144 (2010).
 I. Kunita, K. Sato, Y. Tanaka, Y. Takikawa, H. Orihara, and T. Nakagaki, PRL 109, 248303 (2012).