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Complex nonextensive systems are expected to temporary and spatially fluctuate. In the su- perstatistics, it is assumed that locally the equilibrium state is described by the Boltzmann- Gibbs statistics, and that their global properties may be expressed by a superposition of them over some intensive physical quantity, e.g. the inverse temperature [30]-[33]. Many applications of the concept of the superstatistics have been pointed out (for a recent re- view, see [33]). It is, however, not clear how to obtain the mixing probability distribution of fluctuating parameter from first principles. This problem is currently controversial and some attempts to this direction have been proposed [34]-[37]. Previous studies in the su- perstatistics have paid attention mostly to the stationary PDF, which generally has the non-Gaussian form. However, thermodynamical quantities such as energy and entropy have not been calculated in the superstatistics, which will be studied in this paper. 

The paper is organized as follows: In Sec. II, we briefly review the OLM-MEM with the q -average [6] and the original MEM with the normal average [1, 24]. MEMs with the two average methods are applied to ideal gases and harmonic oscillators in Secs. III and IV, respectively. In Sec. V, the fluctuation of the energy and applications of the superstatistics [30]-[33] to N -body systems are studied. Sec. VI is devoted to our conclusion. 

## II. MAXIMUM-ENTROPY METHODS 

## A. OLM-MEM with the q -average 

We consider an independent N -body system whose hamiltonian is given by 

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In order to obtain the N -variate density matrix ρ$_{q}$ from the Tsallis entropy with the OLM- MEM [6], we impose the constraints given by 

with 

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