We present a thermokinetic description of anomalous diffusion of single particles and clusters in a viscoelastic medium in terms of a non-Markovian diffusion equation involving memory functions. The scaling behavior of these functions is analyzed by considering hydrodynamics and cluster-size space random walk arguments. We explain experimental results on diffusion of Brownian particles in the cytoskeleton, in cluster-cluster aggregation, and in a suspension of micelles.

We analyze the concept of equilibrium temperature in a set of interacting argon atoms, confined in a nanostructure, a zeolite with an intricate distribution of channels through which the atoms may move. The temperature is computed following two procedures: by averaging over the kinetic energy of the particles and over the forces acting on them. It is shown that for external surfaces and for regions which do not fall under the whole pattern of potential energy distribution, smaller than a quarter of a crystal unit cell, both temperatures, kinetic and configurational, show significant differences. The configurational temperature accounts for the different interactions on the particles in the different parts of the channels which makes them move in an energetically heterogeneous environment. The kinetic temperature is practically not affected by these inhomogeneities. The observed disparity between both temperatures disappears when averages are taken over larger regions of the zeolite. The size of these regions imposes a lower limit for a consistent thermodynamic description of a small-scale systems such as nanostructured materials, catalytic cells, and nano heat-exchangers.

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