By Thomas H. Stix

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**Additional resources for Stix, Waves in plasmas**

**Sample text**

28) is in fact at its finite maximum at T = TBEC . If the number of particles is fixed, this means that at T < TBEC some of the singleparticle states must be occupied by a finite fraction of the total number of particles to make up for the difference. This singular contribution has been missed by approximating the discrete sum over wavevectors with the integral, and needs to be added separately. As the occupation number is a decreasing function of energy, the macroscopically occupied state below TBEC is the ground state at k = 0.

This singular contribution has been missed by approximating the discrete sum over wavevectors with the integral, and needs to be added separately. As the occupation number is a decreasing function of energy, the macroscopically occupied state below TBEC is the ground state at k = 0. This is the phenomenon of Bose–Einstein condensation. The Bose–Einstein condensation temperature TBEC is therefore determined by the density of particles and Eq. 29), and the number of particles in the condensate N0 for T < TBEC is N0 =1− N T TBEC 3/2 .

32) where the average | (r , τ )|2 is to be self-consistently computed. This is the so-called “Hartree approximation”. 33) in the partition function for the effective non-interacting system, as in Eq. 23). 34) where we introduced again an infinitesimal η. The Bose–Einstein condensation now occurs not at μ = 0 but at μ ˜ = 0, which corresponds to μ = μc (T ) = λ 2 k2 dk 2mkB T e −1 (2π )d −1 . 35) For d > 2 the integral over the wavevectors is finite. Rescaling the wavevectors with temperature, we see that at low temperatures the transition line behaves d like μc (T ) ∝ T 2 .