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A.1 q-Hamiltonian dynamics We can define a continuum of dynamics with Newtonian dynamics at one extreme and energysampling dynamics at the other via Tsallis statistics. H(x, v) =E(x)+K(v) The potential energy function, E(x), is the target distribution that we would like to sample and is defined by our problem. The kinetic energy, K, can be chosen in a variety of ways, but we consider the following class. However, we found it interesting that Newtonian and ESH kinetic energy terms could be viewed as opposite ends of a spectrum defined by Tsallis statistics. The situation mathematically resembles the thermodynamic continuum that emerges from Tsallis statistics. In that case, standard thermodynamics with Brownian motion corresponds to one extreme for q and in the other extreme so-called "anomalous diffusion" occurs [23]. Analogously, we can refer to the q-logarithmic counterpart of Newtonian momentum as anomalous momentum.