Superstatistics in nanoscale electrochemical systems

Stochastic electrochemical reaction steps on nanosized electrodes are non-Markovian when externally driven by an applied voltage. We show that, compared to the Markovian case (when external driving is absent), nanoscale electrochemical systems obey a superstatistics characterized by a superposition of Tsallis' q indices. The distribution of Tsallis' q indices along stochastic trajectories can be calculated from the electrochemical master equation and normal distributions from Boltzmann-Gibbs thermostatistics are recovered in the thermodynamic limit (the infinite electrode size limit). Although on the nanoscale the external control makes intricate correlations between the microstates, in the superstatistical frame one can still address the microstates as if they were uncorrelated. The resulting superstatistical entropic form is additive in this frame and Tsallis' indices have on the time-average values hqi ≤ 1, which is, indeed, an example of a superstatistical system where no ad hoc distribution has to be assumed for the fluctuations; rather, the distribution is directly calculated from a mesoscopic master equation without freely adjustable parameters.