Course 1: Bose-Einstein Condensates in Atomic Gases: Simple Theoretical Results

1.1 1925: Einstein’s prediction for the ideal Bose gas Einstein considered N non-interacting bosonic and non-relativistic particles in a cubic box of volume L3 with periodic boundary conditions. In the thermodynamic limit, defined as $$ N,L \to \infty {\text{ }}with{\text{ }}\frac{N} {{L^3 }} = \rho = constant, $$ (1.1) a phase transition occurs at a temperature Tc defined by: $$ \rho \lambda _{{\text{dB}}}^3 (T_c ) = \zeta (3/2) = 2.612... $$ (1.2) > where we have defined the thermal de Broglie wavelength of the gas as function of the temperature T: $$ \lambda _{{\text{dB}}} (T) = \left( {\frac{{2\pi \hbar ^2 }} {{mk_B T}}} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} $$ (1.3) and where \( \zeta (\alpha ) = \sum\nolimits_{k = 1}^\infty {1/k^\alpha } \) is the Riemann Zeta function.