The phase-space density of stationary nonequilibrium particle systems is known to be a multifractal object with an information dimension smaller than the phase-space dimension. The rate of heat flowing through the system, divided by the Boltzmann constant and the kinetic temperature, is equal to the sum of the Lyapunov exponents. The reduction in dimensionality is determined from the spectrum of Lyapunov exponents. We show here that also many-body systems in nonequilibrium states with stochastic thermostats can be found that have similar properties and support fractal structures in phase space. We study two two-dimensional examples: first, color conductivity for a system of hard disks, which are thermostated by a stochastic map which affects the momenta of randomly chosen particles; second, color conductivity of a system of soft disks which are subjected to a stochastic force and perform Brownian motion. Full Lyapunov spectra were computed for both models, and the information dimensions of their underlying attractors determined.
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