Thermodynamic instability and first-order phase transition in an ideal Bose gas

We conduct a rigorous investigation into the thermodynamic instability of an ideal Bose gas confined in a cubic box, without assuming a thermodynamic limit or a continuous approximation. Based on the exact expression of the canonical partition function, we perform numerical computations up to ${10}^6$ particles. We report that if the number of particles is equal to or greater than a certain critical value, which turns out to be $7616$, the ideal Bose gas subject to the Dirichlet boundary condition reveals a thermodynamic instability. Accordingly, we demonstrate that a system consisting of a finite number of particles can exhibit a discontinuous phase transition that features a genuine mathematical singularity, provided we keep not volume but pressure constant. The specific number, $7616,$ can be regarded as a characteristic number of a “cube,” which is the geometric shape of the box.

Figure 1

Constant-volume curves. (a) Specific heat per particle at constant volume, $C_V/k_B$. (b) The occupancy ratio of the ground state, $\langle N_0\rangle /N$. (c) and (d) The thermodynamic instability indicator, $\varphi =-(1/N)\beta V^2\partial {}_{V}P$. In (a)–(c), the red, orange, green, blue lines are respectively for $N=1$, $N=10$, $N={10}^2$, ${10}^4$, while (d) is for $N=7614,7615,7616,7617$. The case of $N=1$ (red) also corresponds to the ideal Boltzmann gas. All the quantities are dimensionless.

Figure 2

Constant-pressure curves. (a) The dimensionless volume ${\upsilon }_P$ vs the dimensionless temperature ${\tau }_P$. (b) ${\tau }_V$ vs ${\tau }_P.$ (c) $\langle N_0\rangle /N$ vs ${\tau }_P$. (d) The dimensionless energy ${\epsilon }_P$ vs ${\tau }_P$. (e) and (f) Specific heat per particle under constant pressure $C_P/k_B$ vs ${\tau }_P$. Note that (a)–(e) are for $N=1$ (red) (i.e., ideal Boltzmann gas) or $N=10$ (orange), ${10}^2$ (green), ${10}^4$ (blue). The small boxes magnify the zigzag segments of $N={10}^4$ (blue), while (f) magnifies (e) for $N={10}^4$ (blue). The dotted pink lines denote supercooling and superheating points.

Figure 3

Thermodynamic instability due to zero mode. (a) $\varphi$ vs ${\tau }_V$ under the Neumann boundary condition. (b) $\varphi$ vs ${\tau }_V$ under the periodic boundary condition.