Onsager relations and hydrodynamic balance equations
  In this paper we clarify the role of heat flux in the hydrodynamic balance equations, facilitating the formulation of an Onsager relation within the framework of this theory. Previously thought to be unobtainable from the present form of the theory [X.L. Lei, J. Cai, and L.M. Xie, Phys. Rev. B {\bf 38},1529 (1988)], our verification of the Onsager relation for linear particle and heat flux currents driven by electric fields and temperature gradients resolves a puzzling issue of long standing. Our results show that, for any temperature, when electron density is sufficiently high, the linear predictions of balance equation theory exactly satisfy the Onsager relation. The condition of high density is consonant with the requirement of strong electron-electron interactions for the validity of the Lei-Ting balance equations. Our results support the validity of this theory for a weakly nonuniform system. We also discuss a possible method of extending this theory to a system further removed from thermal equilibrium. 
M.W. Wu, H.L. Cui, and N.J.M. Horing