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(β) 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(β) cannot be defined uniquely, but rather depends upon the way the measurement of E is performed.