Statistical Model Showing Synchronization-Desynchronization Transitions

6 Statistical Model Showing Synchronization-Desynchronization Transitions 

The phase-transition-like phenomena expected from system (5.5.15) seem to be far more difficult to treat than the usual thermodynamic phase transitions. This is simply because we do not know what is the steady solution, if any, of (5.5.15). In view of the fact that exact analytical results are seldom available even if the equilibrum distribution is known, we shall have to resort to some drastic assumptions about the interaction in order to obtain some specific results. In this connection, a Husimi-Temperly-type model 

In contrast, for interactions of infinitely long range like (5.6.1), it is possible to get a single equation for n involving no higher correlation functions. Note that the same idea does not, unfortunately, work for the deterministic system of Sect. 5.4. The derivation of the equation for n is made easier by making use of the identities 

By making a partial integration, and using the properties (5.6.3a, b), this equation becomes 

Consequently, we obtain a simple nonlinear diffusion equation, 

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