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Abstract:

The interacting quantum Bose gas is a random ensemble of many

Brownian bridges (cycles) of various lengths with interactions between any

pair of legs of the cycles. It is one of the standard mathematical models in

which a proof for the famous Bose–Einstein condensation phase transition

is sought for. We introduce a simplified version of the model in $Z^d$ instead

of $R^d$ and with an organisation of the particles in deterministic boxes in-

stead of Brownian cycles as the marks of a reference Poisson point process.

We derive an explicit and interpretable variational formula in the thermody-

namic limit for the canonical ensemble for any value of the particle density.

In this formula, each of the microscopic particles and the macroscopic part

of the configuration are seen explicitly (if they exist); the latter receives the

interpretation of the condensate. The methods comprises a two step large-

deviation approach for marked Poisson point processes and an explicit dis-

tinction into microscopic and macroscopic marks. We discuss the conden-

sate phase transition in terms of existence of minimizer. (based on joint

works with Adams/Collevecchio (2011) and Collin/Jahnel (preprint 2022).)

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Gianmarco Bet (he/him)
Senior researcher

https://gianmarco.bet
Phone: (+39) 055 2751491

Department of Mathematics and Informatics "U. Dini"
University of Florence
Viale Morgagni, 65
50134 Firenze, Italy
Office 64
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