Hierarchical maximum entropy principle for generalized superstatistical systems and Bose-Einstein condensation of light

Abstract

A principle of hierarchical entropy maximization is proposed for generalized superstatistical systems, which are characterized by the existence of three levels of dynamics. If a generalized superstatistical system comprises a set of superstatistical subsystems, each made up of a set of cells, then the Boltzmann-Gibbs-Shannon entropy should be maximized first for each cell, second for each subsystem, and finally for the whole system. Hierarchical entropy maximization naturally reflects the sufficient time-scale separation between different dynamical levels and allows one to find the distribution of both the intensive parameter and the control parameter for the corresponding superstatistics. The hierarchical maximum entropy principle is applied to fluctuations of the photon Bose-Einstein condensate in a dye microcavity. This principle provides an alternative to the master equation approach recently applied to this problem. The possibility of constructing generalized superstatistics based on a statistics different from the Boltzmann-Gibbs statistics is pointed out.

Thermalization and breakdown of thermalization in photon condensates

“…Inspired by these experiments, there has also been significant theoretical work on a variety of topics related to photon condensation. Many of these works have concentrated on the unique properties of the photon system even in thermal equilibrium [36][37][38][39]. These have included exploration of the role of the dye molecules as a reservoir for excitations, leading to grand canonical statistics [36,38], and exploring effects of the nonlinearity of coupling to dye molecules inducing effective interactions [39].…”

Spatial dynamics, thermalization, and gain clamping in a photon condensate

“…We will refer to condensation throughout this paper, but we present phenomena that can be interpreted either as lasing or BEC. Following these experiments many theoretical works [19][20][21][22][23][24][25][26][27][28][29][30][31] explored topics including equilibration, phase coherence, and photon statistics of the photon BEC, and later experiments studied photon statistics [32].…”

Bose-Einstein Condensation of Photons versus Lasing and Hanbury Brown-Twiss Measurements with a Condensate of Light

“…In the here discussed dye microcavity system, the dye molecules act both as a heat bath and a particle reservoir for the photon gas in a grand-canonical sense, with the contact between system and reservoir not being realized diffusively but by absorption and emission processes, in a spatially overlapping geometry. We have predicted grand-canonical number fluctuations for this system [18], and the obtained theory results for the photon number distribution have been confirmed [19].…”

“…This article reviews recent experiments of our group studying the transition from laser-like dynamics to Bose-Einstein condensation of photons upon variation of the thermalization rates of the photon gas to the dye medium [21]. Moreover, we describe work observing grand-canonical number statistics of the condensate emission, as understood from effective particle exchange with the reservoir of photo-excitable dye molecules [18,19,22].…”