Fluctuations in Non-Equillibrium Phase Transitions: Critical Behavior
It is now well recognized that there exists a formal analogy between bifurcations in non-equilibrium dynamical systems and equilibrium phase transitions, hence the term non-equilibrium phase transitions now commonly used . The equivalent of a first-order phase transition can be seen in model systems  as well as in experiments, for instance in chemical systems  when bistability is observed: two stable stationary states coexist for the same value of a bifurcation parameter. A second-order transition can be defined in the same situations when, by changing some control parameter or constraint, this bistability vanishes at a critical point as shown in an example later. Also a supercritical Hopf bifurcation can be considered as another example where the amplitude of the limit cycle is taken as an order parameter. This analogy is described only within the framework of a purely deterministic approach. Now, all that we know about equilibrium phase transitions tells us that fluctuations in the state of the system play a very important role, to such an extent that a phase transition can be viewed essentially as a fluctuation phenomenon. A similar feature is suggested by the evolution equations of a dynamical system near a bifurcation point where an eigenvalue of the linear stability matrix goes to zero. In other words, some relaxation time goes to infinity. However, whereas for equilibrium phase transitions the very definition of the system includes a microscopic level having particular properties resulting from equilibrium requirements from which we can in principle deduct some description of fluctuations, no the contrary for a dynamical system evolving near a bifurcation point, we have no such possibility.
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