# Thermal effects in gravitational Hartree systems - Mathematics > Analysis of PDEs

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Abstract: We consider the non-relativistic Hartree model in the gravitational case,i.e. with attractive Coulomb-Newton interaction. For a given mass, we constructstationary states with non-zero temperature by minimizing the correspondingfree energy functional. It is proved that minimizers exist if and only if thetemperature of the system is below a certain thresholdpossibly infinite,which itself depends on the specific choice of the entropy functional. We alsoinvestigate whether the corresponding minimizers are mixed or pure quantumstates and characterize a positive critical temperature above which mixedstates appear.

Author: ** Gonca L. Aki, Jean Dolbeault CEREMADE, Christof Sparber**

Source: https://arxiv.org/