自引力布朗粒子和细菌群体的爆炸时间与弛豫时间估计
Estimate of blow-up and relaxation time for self-gravitating Brownian particles and bacterial populations.
作者信息
Chavanis P-H, Sire C
机构信息
Laboratoire de Physique Théorique (UMR 5152 du CNRS), Université Paul Sabatier, 118, route de Narbonne, 31062 Toulouse Cedex 4, France.
出版信息
Phys Rev E Stat Nonlin Soft Matter Phys. 2004 Aug;70(2 Pt 2):026115. doi: 10.1103/PhysRevE.70.026115. Epub 2004 Aug 30.
We determine an exact asymptotic expression of the blow-up time t(coll) for self-gravitating Brownian particles or bacterial populations (chemotaxis) close to the critical point in d=3. We show that t(coll) = t() (eta- eta(c) )(-1/2) with t() =0.917 677 02..., where eta represents the inverse temperature (for Brownian particles) or the mass (for bacterial colonies), and eta(c) is the critical value of eta above which the system blows up. This result is in perfect agreement with the numerical solution of the Smoluchowski-Poisson system. We also determine the exact asymptotic expression of the relaxation time close to but above the critical temperature and derive a large time asymptotic expansion for the density profile exactly at the critical point.
我们确定了三维空间中自引力布朗粒子或细菌群体（趋化作用）接近临界点时爆聚时间(t_{coll})的精确渐近表达式。我们证明(t_{coll}=t^(\eta - \eta_c)^{-\frac{1}{2}})，其中(t^ = 0.91767702\cdots)，这里(\eta)表示逆温度（对于布朗粒子）或质量（对于细菌菌落），(\eta_c)是(\eta)的临界值，超过该值系统会发生爆聚。该结果与斯莫卢霍夫斯基 - 泊松系统的数值解完全一致。我们还确定了接近但高于临界温度时弛豫时间的精确渐近表达式，并推导出恰好在临界点处密度分布的长时间渐近展开式。
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