Tonks-Girardeau Gas

I had the chance to kick back and read a few papers over spring break, including a number of experimental biophysics and atomic/molecule/optical (AMO) physics papers for well-roundedness. Some dealt with the Tonks-Girardeau gas, something I first encountered in my liquid crystals class last semester (I’m taking a few of the equations below from my notes from that class). Among other things, this system is interesting because it was one of the first exactly-solvable systems in statistical mechanics, of which there still remain very few (the Ising model in 1D/2D being another). From a soft condensed-matter physics point of view, the Tonks gas serves as a good starting point for exploring the notion of excluded volume, a surprisingly important concept (for example, leading to ideas like the depletion attraction and entropic attraction/organization - see this book for a nice introduction.)

Working at the research laboratory of the General Electric company, John Bradshaw Taylor and Irvin Langmuir spent a good deal of time in the early 30’s developing very precise means of determining the number of caesium atoms adsorbed on tungsten and using these to study the properties of such monoatomic films. Motivated by this, Lewi Tonks derived the equation of state not just for a two-dimensional gas, but for a one-dimensional and three-dimensional gas as well, ignoring the nature of the forces between them and treating the simplest case of hard elastic spheres. (Twenty-four years later, Marvin Girardeau established a rigorous one-to-one correspondence between this system and a 1D system of spinless fermions. This ‘fermionization’ helps explain why such a system of bosons doesn’t undergo condensation.)

For example, one can calculate the equation of state for a dilute gas of strongly-interacting impenetrable bosons confined to the one dimensional space quite simply - since we’re treating the particles as hard spheres of radius , the associated potential is given by for and for . Then the -sphere partition function is just:

(There’s a constant prefactor involving and what Kittel & Kroemer call the quantum concentration, but it’s not important here since we’ll be taking derivatives.) A slick way of solving this is by changing variables to :

and since and , where in this case the ‘volume’ , one finds that

It is interesting to note that including a simplistic two-body attractive interaction regains the van der Waals equation of state exactly (see for example section 4.9 of Mattis’ somewhat misleadingly-titled book on statistical mechanics). This can also be extended to three dimensions by expanding the expression for P and considering the virial expansion, and a lot of interesting concepts, like the depletion interaction, fall out very nicely (see the first chapter of the soft condensed-matter physics book I mentioned above).

Another soft-matter application of the Tonks gas that I came across a while back is this paper by Tom Chou at UCLA. He’s developed an exact one-dimensional theory of histone adsorption and wrapping on DNA by considering each histone as a Tonks gas particle, which I think is pretty neat; after all, histones are pretty much hard spheres confined to a line of sorts - as wikipedia puts it, they “act as spools around which DNA winds”.

(This model can be theoretically extended in other ways, too. For example, Takahashi added in the effect of an arbitrary bounded interaction potential between going from the zero potential to the infinite hard-core potential; see another book by Mattis for more on this.)

But the Tonks gas is important in other areas of physics, as well - in particular, in the study of atomic gases at low temperature and particle density. During the summer of 2004, two groups published papers detailing their experiments using ultracold rubidium-87 atoms trapped using optical lattices in which they observed a transition to the strongly-correlated 1D Tonks gas regime (although they verified this in different ways: the Penn State group studied the energy and size of their system while the European group looked at the momentum distribution, both comparing their results to relevant theoretical predictions). And the research has been continuing ever since: for example, the Penn State group has extended these measurements to the study of a quantum Newton’s cradle, and other groups have studied 1D Bose gases as both Mott insulators and Luttinger liquids (the latter being a particularly nice connection to carbon nanotubes).