Superstatistics and temperature fluctuations

Superstatistics Beck and Cohen (2003) 1]] is a formalism aimed at describing statistical properties of a generic extensive quantity E in complex out-of-equilibrium systems in terms of a linear superposition of equilibrium canonical distributions. The weight function $P(\beta )$ is argued to be provided by the statistics of the intensive thermodynamic quantity β conjugate to E Beck (2011) 14]], and therefore is expected to be determined by the spatiotemporal dynamics alone of the system under consideration. In this paper, recalling a previous work Beck (2006) 21]], I show by examples that, in some cases fulfilling all the conditions for the superstatistics formalism to be applicable, $P(\beta )$ cannot be defined uniquely, but rather depends upon the way the measurement of E is performed.