  In earlier work we provided the first evidence that the collapse, or
coil-globule, transition of an isolated polymer in solution can be seen in a
four-dimensional model. Here we investigate, via Monte Carlo simulations, the
canonical lattice model of polymer collapse, namely interacting self-avoiding
walks, to show that it not only has a distinct collapse transition at finite
temperature but that for any finite polymer length this collapse has many
characteristics of a rounded first-order phase transition. However, we also
show that there exists a `$\theta$-point' where the polymer behaves in a simple
Gaussian manner (which is a critical state), to which these finite-size
transition temperatures approach as the polymer length is increased. The
resolution of these seemingly incompatible conclusions involves the argument
that the first-order-like rounded transition is scaled away in the
thermodynamic limit to leave a mean-field second-order transition. Essentially
this happens because the finite-size \emph{shift} of the transition is
asymptotically much larger than the \emph{width} of the pseudo-transition and
the latent heat decays to zero (algebraically) with polymer length. This
scenario can be inferred from the application of the theory of Lifshitz,
Grosberg and Khokhlov (based upon the framework of Lifshitz) to four
dimensions: the conclusions of which were written down some time ago by
Khokhlov. In fact it is precisely above the upper critical dimension, which is
3 for this problem, that the theory of Lifshitz may be quantitatively
applicable to polymer collapse.
