Through cumulant expansions of the free energy and the susceptibility, a new variational procedure is proposed with the purpose of improving the standard variational method in equilibrium statistical mechanics. The procedure is tested for two types of classical anharmonic single oscillators, namely, those whose elastic potential is proportional to x " (n =1,2, . . .) and those of the type ax +bx, whose exact free energy, specific heat, and susceptibility are herein established. Although convergence problems (similar to those appearing in the asymptotic series) exist (at least for the free energy) in the limit of high perturbative orders, great improvement (typically of the order of 40) with respect to the standard variational method is obtained in all the physically meaningfulwituations, and a quite satisfactory description is provided (with a "single shot") for both limits T~O and T~oo simultaneously.

We investigate diusive phenomena governed by fractional diusion equations in the presence of a spatial dependent diusion coecient and external forces. We consider a fractional operator which enables us to analyze, in a unified way, dierent types of fractional time derivatives. This approach opens the possibility of considering new situations related to memory eects. After determining exact analytical solutions, we show that dierent diusive behaviors can be obtained, depending on the fractional time operator employed. In this framework, we also discuss the role of the anomalous diusion processes.

-The elephant walk model originally proposed by Schütz and Trimper to investigate non-Markovian processes led to the investigation of a series of other random-walk models. Of these, the best known is the Alzheimer walk model, because it was the first model shown to have amnestically induced persistence -i.e. superdiffusion caused by loss of memory. Here we study the robustness of the Alzheimer walk by adding a memoryless stochastic perturbation. Surprisingly, the solution of the perturbed model can be formally reduced to the solutions of the unperturbed model. Specifically, we give an exact solution of the perturbed model by finding a surjective mapping to the unperturbed model. Copyright c EPLA, 2012Introduction. -The theoretical foundations of the Brownian motion are well known and were laid more than one hundred years ago (see [1][2][3] and references therein). Since then, the random walk (RW), the simplest realization of the Brownian motion, has been used in the scientific literature as a prototype for modelling applications in various fields. The diffusion of the traditional RW is known as normal diffusion.Superdiffusion [4][5][6][7] is an accelerated diffusion, for which the mean squared displacement grows faster than linearly in time. Theoretical studies of anomalous diffusion are based on generalized Langevin equations [8,9], the fractional Fokker-Planck equation [10,11] and the continuous-time random-walk [12-14] approaches. Superdiffusion is possible only if the necessary and sufficient conditions of the central limit theorem for sums of random variables are not met, otherwise the behavior is always normal diffusion (e.g., Brownian motion). There are two basic mechanisms (not mutually exclusive) by which a random walker can undergo superdiffusion. The first is an infinite second moment for the random-walk step size distribution, as seen in Lévy processes [15][16][17]. The second mechanism is long-range power law correlations, i.e. long-range memory [18,19].