It isn’t the calculus we knew: Equations built on fractional derivatives describe the anomalously slow diffusion observed in systems with a broad distribution of relaxation times.

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We provide a unified renewal approach to the problem of random search for several targets under resetting. This framework does not rely on specific properties of the search process and resetting procedure, allows for simpler derivation of known results, and leads to new ones. Concentrating on minimizing the mean hitting time, we show that resetting at a constant pace is the best possible option if resetting helps at all, and derive the equation for the optimal resetting pace. No resetting may be a better strategy if without resetting the probability of not finding a target decays with time to zero exponentially or faster. We also calculate splitting probabilities between the targets, and define the limits in which these can be manipulated by changing the resetting procedure. We moreover show that the number of moments of the hitting time distribution under resetting is not less than the sum of the numbers of moments of the resetting time distribution and the hitting time distribution without resetting.

Echo chambers and opinion polarization have been recently quantified in several sociopolitical contexts, across different social media, raising concerns for the potential impact on the spread of misinformation and the openness of debates. Despite increasing efforts, the dynamics leading to the emergence of these phenomena remain unclear. Here, we propose a model that introduces the phenomenon of radicalization, as a reinforcing mechanism driving the evolution to extreme opinions from moderate initial conditions. Empirically inspired by the dynamics of social interaction, we consider agents characterized by heterogeneous activities and homophily. We analytically characterize the transition between a global consensus and emerging radicalization dynamics in the population, as a function of social influence and the controversialness of the topic discussed. We contrast the model's behavior against empirical data of polarized debates on Twitter, qualitatively reproducing the observed relation between users' engagement and opinions, as well as opinion segregation based on the interaction network. Our findings shed light on the dynamics that may lie at the core of the emergence of echo chambers and polarization in social media.

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We consider different generalizations of the Fokker-Planck equation (FPE) devised to describe L e evy processes in potential force fields. We show that such generalizations can proceed along different lines. On one hand, L e evy statistics can emerge from the fractal temporal nature of the underlying process, i.e., a high variability in the rate of microscopic events. On the other hand, they may be a direct consequence of the scale-free spatial structure on which the process evolves. Although both forms considered lead to Boltzmann equilibrium, the relaxation patterns are quite different. As an example, generalized diffusion in a double-well potential is considered.

The continuous time random walk ͑CTRW͒ in a homogeneous velocity field and in arbitrary force fields is studied. Within the extended CTRW scheme, anomalous transport properties due to long-tailed waiting time or jump length distributions are consistently introduced. The connections with generalised diffusion equations in a potential field are discussed, these equations being of fractional order. In particular, the problems of a constant and a Hookean ͑linear͒ force, i.e., of a linear and a parabolic potential, are worked out.