30min lecture delivered by Prof. Beretta at the Junilee 40th Symposium on Mathematical Physics "Geometry & Quanta" on June 25, 2008 in Torun, Poland, held in honor of Prof. A. Kossakowski on his 70th birthday.The lecture summarizes the main mathematical features of the nonlinear dynamical equation proposed in the 1980’s by Prof. Beretta and recently "rediscovered" by various authors, to decribe the nonunitary evolution of density operators along the direction of steepest entropy ascent (that is, maximal entropy generation) compatible with energy conservation. The density operator is what describes the state of a system within quantum theory. More on this subject is available at www.quantumthermodynamics.org where several not readily available papers can be downloaded.