A superdiffusive random walk model with exponentially decaying memory is reported. This seems to be a self-contradictory statement, since it is well known that random walks with exponentially decaying temporal correlations can be approximated arbitrarily well by Markov processes and that central limit theorems prohibit superdiffusion for Markovian walks with finite variance of step sizes. The solution to the apparent paradox is that the model is genuinely non-Markovian, due to a time-dependent decay constant associated with the exponential behavior.

We perform large-scale simulations of a two-dimensional restricted height conserved stochastic sandpile, focusing on particle diffusion and mobility, and spatial correlations. Quasistationary (QS) simulations yield the critical particle density to high precision [p c = 0.7112687(2)], and show that the diffusion constant scales in the same manner as the activity density, as found previously in the one-dimensional case. Short-time scaling is characterized by subdiffusive behavior (mean-square displacement ∼ t γ with γ < 1), which is easily understood as a consequence of the initial decay of activity, ρ(t) ∼ t −δ , with γ = 1 − δ. We verify that at criticality, the activity-activity correlation function/ , as expected at an absorbing-state phase transition.

We investigate a system governed by a fractional diffusion equation with an integro-differential boundary condition on the surface. This condition can be connected with several processes such as adsorption and/ or desorption or chemical reactions due to the presence of active sites on the surface. The solutions are obtained by using the Green function approach and show a rich class of behaviors, which can be related to anomalous diffusion.

The thermodynamics for systems of non-interacting bosons with multifractal energy spectrum is considered. The critical attractors of one-dimensional generalized logistic and circular maps are used to generate multifractal bounded spectra with well defined scaling exponents. The specific heat is then calculated for both cases of conserved and non-conserved particle number, showing a power-law behaviour which is modulated by log-periodic oscillations when the energy spectrum is not dense. The occurrence of Bose–Einstein condensation for systems with conserved particle number, at which the specific heat is discontinuous, is also analyzed.

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