Convergence and non-convergence to Bose-Einstein condensation
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| Format: | Preprint |
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2025
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| author | Cai, Shuzhe Lu, Xuguang |
| author_facet | Cai, Shuzhe Lu, Xuguang |
| contents | The paper is a continuation of our previous work on the strong convergence to equilibrium for the spatially homogeneous Boltzmann equation for Bose-Einstein particles for isotropic solutions at low temperature. Here we study the influence of the particle interaction potentials on the convergence to Bose-Einstein condensation (BEC). Consider two cases of certain potentials that are such that the corresponding scattering cross sections are bounded and 1) have a lower bound ${\rm const.}\min\{1, |{\bf v-v}_*|^{2η}\}$ with ${\rm const}.>0, 0\le η<1$, and 2) have an upper bound ${\rm const.}\min\{1, |{\bf v-v}_*|^{2η}\}$ with $η\ge 1$. For the first case, the long time convergence to BEC i.e. $\lim\limits_{t\to\infty}F_t(\{0\})=F_{\rm be}(\{0\})$ is proved for a class of initial data having very low temperature and thus it holds the strong convergence to equilibrium. For the second case we show that if initially $F_0(\{0\})=0$, then $ F_t(\{0\})=0$ for all $t\ge 0$ and thus there is no convergence to BEC hence no strong convergence to equilibrium. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_10073 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Convergence and non-convergence to Bose-Einstein condensation Cai, Shuzhe Lu, Xuguang Analysis of PDEs The paper is a continuation of our previous work on the strong convergence to equilibrium for the spatially homogeneous Boltzmann equation for Bose-Einstein particles for isotropic solutions at low temperature. Here we study the influence of the particle interaction potentials on the convergence to Bose-Einstein condensation (BEC). Consider two cases of certain potentials that are such that the corresponding scattering cross sections are bounded and 1) have a lower bound ${\rm const.}\min\{1, |{\bf v-v}_*|^{2η}\}$ with ${\rm const}.>0, 0\le η<1$, and 2) have an upper bound ${\rm const.}\min\{1, |{\bf v-v}_*|^{2η}\}$ with $η\ge 1$. For the first case, the long time convergence to BEC i.e. $\lim\limits_{t\to\infty}F_t(\{0\})=F_{\rm be}(\{0\})$ is proved for a class of initial data having very low temperature and thus it holds the strong convergence to equilibrium. For the second case we show that if initially $F_0(\{0\})=0$, then $ F_t(\{0\})=0$ for all $t\ge 0$ and thus there is no convergence to BEC hence no strong convergence to equilibrium. |
| title | Convergence and non-convergence to Bose-Einstein condensation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2501.10073 |