Nonlinear extensions of Schr\"odinger-von Neumann quantum dynamics: a   list of conditions for compatibility with thermodynamics
  We propose a list of conditions that consistency with thermodynamics imposes on linear and nonlinear generalizations of standard unitary quantum mechanics that assume a set of true quantum states without the restriction $\rho^2=\rho$ even for strictly isolated systems and that are to be considered in experimental tests of the existence of intrinsic (spontaneous) decoherence at the microscopic level. 
Gian Paolo Beretta