Right now Kazimierz Rzążewski from the Center for Theoretical Physics at the Polish Academy of Sciences is giving a talk on “Bose statistics and classical fields”.
Abstract: Statistical properties of quantum systems are the heart of quantum statistical physics. Probability distributions of Bose-Einstein condensate are well understood for an ideal gas. In the presence of interactions only crude approximations are available. In this talk I will argue that now we have a powerful computational tool to study the statistics of weakly interacting Bose gas which is based on the so-called classical field approximation.
For a 3d ideal gas of bosonic atoms trapped in a 3d harmonic oscillator potential, the fraction of atoms in the ground state goes like
for T below a certain critical value, and 0 above that.
The grand canonical ensemble, where we assume the number of particles in our gas and its total energy are both variable, is a dubious method for Bose-Einstein condensates, because there’s no contact with a particle reservoir. The canonical ensemble is also fishy, where we assume the particle number is fixed both the total energy is variable, is also fishy. Why? Because there’s not contact with a heat reservoir, either. The microcanonical ensemble, where the energy and number of particles are both fixed, is closest to experimental reality.
We see this when we compute the fluctuations of the number of particles in the ground state. For the grand canonical ensemble, the standard deviation of the number of particles in the ground state becomes infinite at temperature below a certain value!
The fun starts when we move from the ideal gas to a weakly interacting gas. Most papers here consider particles trapped in a box, not in a harmonic oscillator — and they use the Bogoliubov approximation, which is exactly soluble for a box. This approximation involves a quadratic Hamiltonian that’s a sum of terms, one for each mode in the box. To set up this equation we need to use the Bogoliubov-deGennes equations.
As the temperature goes up, the Bogoliubov approximation breaks down… so we need a new approach.
Here is Rzążewski’s approach. A gas of bosons is described by a quantum field. But we can approximate the long-wavelength part of this quantum field by a classical field. Of course the basic idea here is not new. In our study of electromagnetism — this is what lets us approximate the quantum electromagnetic field by a classical field obeying the classical Maxwell equations. But the new part is setting up a theory that keeps some of the virtues of the quantum description, while approximating it with a classical one at low frequencies (i.e., large distance scales).
So: for modes below the cutoff we describe the system using annihilation and creation operators; for each mode above the cutoff we have 2d classical phase space. But: how to put in a nice ‘cutoff’ where we make the transition from the quantum field to the classical field?
Testing this problem on an exactly soluble model is a good idea: for example, the 1-dimensional ideal gas!
It turns out that by choosing the cutoff in an optimal way, the approximation is very good — not just for the 1d ideal gas, but also the 3d case, in both a harmonic potential and in a box. There is an analytic form for this optimal cutoff.
But more significant is the nonideal gas, where the particles repel each other. Here it’s easiest to start with the 1d case of a gas trapped in a harmonic oscillator potential. Now it’s more complicated. But we can simulate it numerically using the Metropolis algorithm!
We can also study ‘quasicondensates‘, where the coherence length is shorter than the size of the box, or the size of the cloud of atoms. (For example, in 2 dimensions, at temperatures above the Berezinskii-Kosterlitz-Thouless transition, there are lots of vortices in the gas, so the phase of the gas is nearly uniform only in small patches.)
• E. Witkowska, M. Gajda, and K. Rzążewski,
Bose statistics and classical fields, Phys. Rev. A 79 (2009), 033631.
• E. Witkowska, M. Gajda, and K. Rzążewski,
Monte Carlo method, classical fields and Bose statistics,
Opt. Comm. 283 (2010), 671-675.
• Z. Idziaszek, L. Zawitkowski, M. Gajda, and K. Rzążewski, Fluctuations of weakly interacting Bose-Einstein condensate, Europhysics Lett. 86 (2009), 10002.
As usual, I’d love it if an expert came along and explained anything more about these ideas. For example, I’m pretty vague about how exactly the Metropolis algorithm is used here.