To account for frequently documented low-temperature deviations from Arrhenius rate law, the proposed expansion of inverse activation energy against inverse temperature is shown to yield a first order linearizing parameter which is formally correlated with Tsallis non-extensive classical statistical mechanics. Its sign provides a heuristic criterion, especially appealing in biochemistry, for assigning deviations as due either: (i) to quantum mechanical under-barrier tunneling, or (ii) to 'classical' collective phenomena. For (i), an explicit relationship is here derived in terms of barrier features. Case (ii) typically occurs in enzymatic or heterogeneous catalysis, in membrane mediated processes and in those controlled by diffusion or by transport in general. (C) 2013 Elsevier B. V. All rights reserved.