Bose–Einstein Condensates

A Bose–Einstein condensate (BEC) is an interesting phase of matter that may occur for indistinguishable, integer spin particles, called Bosons. The BEC is a quantum mechanical phenomenon where the concept of a “ground state” is particularly important. For some incredibly small temperature we argue that the number of “excited” particles, particles with more energy than those in the ground state, is fixed and finite then every other boson in a gas must be in the ground state, which has no upper bound. Concisely, if below a certain temperature functionally all particles in a bosonic system have then least amount of energy allowable then we have a new physical state, a BEC.

The Polylogarithm

At this point it becomes convenient to introduce a special function known as the polylogarithm.

This is the integral representation of the polylog, note that it also contains the gamma function, another special function. The intricacies of this equation may be of interest to some readers but in general we only need to note two things here. That integrals of this form are ubiquitous in the study of Bose gases and that for when x=1 the polylog is only finite valued if r>1. Note in Fig.1 that for “r” larger than 1 a clear endpoint can be seen on the plot when x=1, when this is less than 1 such an end is actually never reached. From here on I will repeatedly refer to the polylog Li_r(x) but please keep in mind that this is just standing in for the above integral, identifying this function is not changing anything it just makes things more concise. 

Fig.1 : Plot of Polylogarithm for select values of “r”. Note that the first two curves are unbounded. Made in Mathematica.

Maths and Physics

One of the most simple BEC systems is an ideal, non-relativistic Bose-gas in three dimensions. In this case the energy of the particles is proportional to the momentum squared, ε∼p². To examine this system we will use the “Grand Canonical Ensemble”, this is a formulation in Statistical Physics where we allow the energy and number of particles in our system to vary. This method allows us to separate the ground and excited states from each other. Furthermore, the average number of particles in the in the excited energy levels can then be found to be proportional to the polylog, ⟨N_ex⟩∝Li_3/2(z), with z being a parameter called fugacity. We then let ourselves move to the “Canonical Ensemble” where the total number of particles in our system is fixed. In a Bose system fugacity is bounded between 0 and 1, when below some temperature z=1, this may be called the critical temperature. Following our discussion about the polylog we can now immediately see the there is an upper bound on the number of particles in an excited state. If we have a fixed number of particles, and a limit on the number of excited particles then we must infer that every other particle must be in the ground state. Importantly, the ground state can now contain any number of particles, when the ground state is densely filled due to the bound on the excited states we have a BEC.

We will now generalise this system for d-spatial dimensions and for the relation ε∼p^m.

⟨N_ex⟩ ∝ Li_d/m (z)

We can now make a range of physical predictions, using the properties of the polylogarithm. Notably, while a BEC may occur for a non relativistic particle in three dimensions, it will not occur in two or one dimensions. These predictions may seem surprising but they do hold. Here our mathematical description of a system gives a very quick criteria for a real physical phenomenon.

The versatility of this approach can then be seen by letting m=1, this energy-momentum relation can be identified to match an ultra-relativistic limit. It is clear to see now that an ultra-relativistic gas may undergo BEC for d>1, meaning that a two dimensional space may now exhibit BEC. When I first calculated this I was astounded, can this result even be experimentally tested? It can be! We may identify photons as bosons, and of course photons are relativistic massless particles. In Fig.2b the occurrence of BEC in a photon gas can be seen, characterised by the intense central point.

(a) Confinement of Photons

(b) BEC for Photons

Fig.2 : Occurrence of BEC in photon gas confined to two dimensions. [1]

A Conclusion

I hope that I have effectively conveyed my own fascination with this result. The fact that the phenomenon of BEC can be shown to depend entirely on the dimension of space and the relationship between momentum and energy is very cool to begin with. That this result follows from the convergence, or lack of convergence, of a single integral/polylogarithm function puts it on another level. It is, of course, an understatement to say that maths and physics are deeply connected, I certainly cannot imagine physics without calculus, but there is something surprising about this particular result. Most of the time when mathematically solving a problem in physics it seems like we are painting a picture of something that will happen anyway but in this case our maths breaks, diverges, if BEC does not occur. Logically there is no real difference here, emotionally I see the two situations as worlds apart. It is as if the physical world is being determined by the maths, when maths is all our own creation. Or maybe we’re just very good at making models. 

References

[1]    J. Klaers, J. Schmitt, F. Vewinger, and M. Weitz, “Bose-Einstein condensation of photons in an optical microcavity,” Nature, Nov. 2010. doi:https://doi.org/10.1038/nature09567