The density of states of an ideal Bose-Einstein gas in a cubic box of volume \(V\) is $$ g(E)= \begin{cases}\alpha E^{3} & \text { if } \quad E>0 \\ 0 & \text { if } \quad E<0\end{cases} $$ where \(\alpha\) is a constant. Compute the critical temperature for Bose- Einstein condensation.

The concept of the 'density of states' (DOS) is crucial in understanding how particles are distributed among available energy levels. In the context of an ideal Bose-Einstein gas, the DOS function defines how many states are available at a certain energy level. Here, the DOS is given by: \[ g(E) = \begin{cases} \alpha E^3 & \text{if} \quad E>0 \ 0 & \text{if} \quad E<0 \end{cases} \]For positive energies, it follows a cubic relationship (E^3), which means the number of states increases rapidly with energy. The constant \text\beta is unique to the system's properties, like volume and particle mass. Understanding these states' distribution helps predict the critical temperature (T_c) for Bose-Einstein condensation when particles congregate in the lowest energy state.

The critical temperature, T_c, marks the point where Bose-Einstein condensation occurs, causing a large fraction of particles to settle into the ground state.To find T_c, we use the condition: \[ N = \int_0^{\infty} \frac{g(E)}{e^{E / (k_B T_c)} - 1} \, dE \]Here:

• N represents the number of particles
• E is the energy
• k_B is the Boltzmann constant
• \text Tc is the critical temperature.

By substituting the given DOS function and simplifying through integral transformation, we get: \[ T_c = \left(\frac{15 N}{text'alpha k_B^4 \pi^4}\right)^{1/4} \]This formula helps us calculate the exact temperature where condensation begins for a system of bosons (particles following Bose-Einstein statistics).

In solving for the critical temperature, an integral transformation is essential. The transformation process simplifies complex integrals by changing variables. Initially, we have:\[ N = \int_0^{\infty} \frac{text'alpha E^3}{e^{E / (k_B T_c)} - 1} \, dE \]Introducing a dimensionless variable x, where \[ x = E / (k_B T_c) \], changes the integral as follows:\[ N = \text'alpha (k_B T_c)^4 \int_0^{\infty} \frac{x^3}{e^x - 1} \, dx \]The integral \[ \int_0^{\infty} \frac{x^3}{e^x - 1} \, dx \] is known and equal to \frac{\pi^4}{15}. Simplifying and solving helps us derive the expression for T_c. This approach is vital in physical problems to make equations more manageable and solvable by applying known mathematical results.