Dissipative structures, broken symmetry, and the theory of equilibrium phase transitions

The claim that nonlinear instabilities leading to "dissipative structures" far from thermodynamic equilibrium are analogous to equilibrium phase transitions is investigated. As a representative model, the spin wave instability occurring in a driven ferromagnetic sample is studied in arbitrary dimension. Broken symmetry of the type occurring in equilibrium phase transitions is not found in any dimension, nor does there appear to be an upper critical dimension beyond which mean field theory is correct. The Bénard instability and "Brusselator," which are known to disobey mean field theory in three dimensions, are also studied in higher dimensions; it is found here that although broken symmetry may occur in higher dimensions, again no upper critical dimension exists. Finally, we speculate under what conditions a "dissipative structure" may exhibit true broken symmetry, and a consequent generalized rigidity, in three dimensions.