Towards Thermodynamics of Spatiotemporal Chaos

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Abstract
Thermodynamics of spatiotemporal chaos is discussed, with the use of coupled map lattices. Structural stability of fully developed spatiotemporal chaos (FDSTC) is confirmed. The stability is sustained by the destruction of all windows through spatiotemporal intermittency and supertransients. In FDSTC, spatial and temporal correlations are found to decay exponentially, as are measured by mutual information. The existence of finite correlation length assures the density of thermodynamic quantifiers. In FDSTC, it is possible to obtain the thermodynamical densities only from measurement within a subspace. On this purpose, subspace Lyapunov spectra are introduced. Only from data of the dynamics within a subspace, thermodynamic quantifiers such as Kolmogorov-Sinai entropy density, Lyapunov dimension density, and scaled Lyapunov spectra are estimated. Application of this approach to the diagnosis of experimental data of spatially extended systems is proposed. To study the fluctuation of Lyapunov spectrum, sub-spacetime Lyapunov exponent is introduced. It provides a way to distinguish chaotic region from ordered region in spacetime. Distribution of sub-spacetime Lyapunov exponents is calculated which characterizes the change of pattern dynamics clearly. As a theoretical approach to spatiotemporal chaos, self-consistent Perron-Frobenius operator is introduced. The invariant measure in a subspace is obtained as the fixed point function for this operator. Some applications to spatiotemporal intermittency transitions and pattern dynamics are presented.

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