Journal of Statistical Physics

, Volume 175, Issue 2, pp 289–350 | Cite as

The Spatially Homogeneous Boltzmann Equation for Bose–Einstein Particles: Rate of Strong Convergence to Equilibrium

  • Shuzhe Cai
  • Xuguang LuEmail author


The paper is a continuation of our previous work on the spatially homogeneous Boltzmann equation for Bose–Einstein particles with quantum collision kernel that includes the hard sphere model. Solutions \(F_t\) under consideration that conserve the mass, momentum, and energy and converge at least weakly to equilibrium \(F_{\mathrm{be}}\) as \(t\rightarrow \infty \) have been proven to exist at least for isotropic initial data that have positive entropy, and \(F_t\) have to be Borel measures for the case of low temperature. The new progress is as follows: we prove that the long time convergence of \(F_t(\{0\})\) to the Bose–Einstein condensation \(F_{\mathrm{be}}(\{0\})\) holds for all isotropic initial data \(F_0\) satisfying the low temperature condition. This immediately implies the long time strong convergence to equilibrium. We also obtain an algebraic rate of the strong convergence for arbitrary temperature. Our proofs are based on entropy control, positive lower bound of entropy, Villani’s inequality for entropy dissipation, a suitable time-dependent convex combination between the solution and a fixed positive function (in order to deal with logarithm terms), the convex-positivity of the cubic collision integral, and an iteration technique for obtaining a positive lower bound of condensation.


Bose–Einstein particles Entropy Strong convergence Equilibrium Low temperature Condensation 



We are grateful to a referee for helpful suggestions on the presentation of the paper. This work was supported by National Natural Science Foundation of China Grant No.11771236.


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Authors and Affiliations

  1. 1.Department of Mathematical SciencesTsinghua UniversityBeijingPeople’s Republic of China

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