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A general approach for treating the spatially extended stochastic
systems with the nonlinear damping and correlations between
additive and multiplicative noises is developed. Within the
modified cumulant expansion method, we derive an effective
Fokker--Planck equation with stationary solutions that describe
the character of the ordered state. We find that the fluctuation
cross--correlations lead to a symmetry breaking of the
distribution function even in the case of zero--dimensional
system. In general case, continuous, discontinuous and reentrant
noise induced phase transitions take place. It appears that the
cross--correlations play the role of bias field which can induce a chain of phase transitions of different nature. Within the mean
field approach, we give an intuitive explanation of the system
behavior by an effective potential of the thermodynamic type. This potential is written in the form of an expansion with coefficients defined by the temperature, intensity of spatial coupling, auto- and cross--correlation times and intensities of both additive and multiplicative noises.
The above formalism is applied to investigate statistical properties of three-component synergetic system of Lorenz type allowing to describe a picture of phase transitions in self-consistent manner. The model is considered in the framework of adiabatic elimination procedure. Such a kind of assumption allows to represent a behavior of the system described by slow variables (order parameter and conjugate field) where additive noises are transformed to multiplicative ones. It is shown that in the case of slow variation of the order parameter the system undergoes reentrant phase transition with a variation of additive noise autocorrelation time. In the case of both additive and multiplicative noises being uncorrelated the system becomes ordered in standard manner. If above noises are correlated then the picture of phase transition is more complicated as introduced above. |
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