Accelerating and retarding anomalous diffusion: A Bernstein function approach

We have discovered here a duality relation between infinitely divisible subordinators which can produce both retarding and accelerating anomalous diffusion in the framework of the special Bernstein function approach. As a consequence, we show that conjugate pairs of Bernstein functions taken as Laplace exponents can produce in a natural way both retarding and accelerating anomalous diffusion (either subdiffusion or superdiffusion). This provides a unified way to control the dynamics of complex biological processes leading to transient anomalous diffusion in single-particle tracking experiments. Moreover, this permits one to explain better the relaxation diagram positioning two different power laws of relaxation, including the celebrated Havriliak-Negami law.