Abstract
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.
Highlights
By definition, first-order phase transitions in thermodynamics feature a genuine mathematical singularity
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 general analysis of the preceding section, as a concrete model we focus on an ideal Bose gas confined in a cubic box and subject to a Dirichlet boundary condition
Summary
First-order phase transitions in thermodynamics feature a genuine mathematical singularity. Once we switch to an alternative constraint of keeping the pressure constant, we demonstrate that canonical ensembles with a finite number of physical degrees may undergo a discontinuous phase transition. We explain how canonical ensembles with a finite number of physical degrees may exhibit a discontinuous phase transition when we keep not volume but pressure constant. By numerical analysis we show that the ideal Bose gas reveals a thermodynamic instability and undergoes a first-order phase transition, if the number of particles is equal to or greater than 7616. The Appendix contains our numerical verification that ideal Bose or Boltzmann gases under periodic or Neumann boundary conditions exhibit a thermodynamic instability at low temperature near absolute zero. The earlier focus was typically on either grand canonical or microcanonical ensembles, and the computations often assumed a continuous approximation to convert discrete sums to integrals [19,20], unless an external harmonic potential sets the sum to be taken over a geometric series [21,22,23,24,25,26,27,28,29,30]
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