# System of $N$ bosons

What precaution needs to be observed in writing down an expression for the total number of bosons $N$ valid at low temperatures?

• You mean in second quantization? – Giuseppe Mar 30 '18 at 11:23

For a gas of $N$ bosons, the chemical potential $\mu_b$ is negative at high temperatures. As the temperature decreases towards a critical temperature $T_c$, $|\mu_b|$ also decreases until it reaches zero, and then remains stuck at zero for lower temperatures. The average number of bosons in the ground state is: $$N_0=\frac{1}{\exp\Big(\frac{|\mu_b|}{k_B T}\Big)-1}$$ Since the argument of the exponential is very small as $T\rightarrow T_c^+$ because $|\mu_b|\rightarrow 0$, you can taylor expand the exponential to first order to get: $$N_0=\frac{k_B T}{|\mu_b|}$$
In the case of $T\rightarrow T_c^-$, since the expression for $N_0$ would diverge, it is convenient to find an expression for the number of bosons which is not in the ground state, where $\mu_b=0$. $$N-N_0=\int_0^{\infty}\frac{g(\epsilon)d\epsilon}{\exp\Big(\frac{\epsilon}{k_B T}\Big)-1}$$ where $g(\epsilon)$ is the density of energy states. This was a specific example for the Bose-Einstein condensation, where you can see that the ground state is populated by a macroscopic number of particles. It is the most relevant example when considering a gas of bosons at low temperatures.