WIAS Preprint No. 1544, (2010)

Thermal effects in gravitational Hartree systems


  • Aki, Gonca L.
  • Dolbeault, Jean
  • Sparber, Christof

2010 Mathematics Subject Classification

  • 35Q40 49J40 82B10 47G20 85A15

2008 Physics and Astronomy Classification Scheme

  • 02.30.Jr 02.30.Xx 05.70.Jk 04.40.-b 64.10.+h


  • Gravitation, Hartree energy, entropy, ground states, free energy, Casimir functional, pure states, mixed states


We consider the non-relativistic Hartree model in the gravitational case, i.e. with attractive Coulomb-Newton interaction. For a given mass $M>0$, we construct stationary states with non-zero temperature $T$ by minimizing the corresponding free energy functional. It is proved that minimizers exist if and only if the temperature of the system is below a certain threshold $T^*>0$ (possibly infinite), which itself depends on the specific choice of the entropy functional. We also investigate whether the corresponding minimizers are mixed or pure quantum states and characterize a critical temperature $T_c in (0, T^*)$ above which mixed states appear.

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