Four-dimensional polymer collapse: Pseudo-first-order transition in interacting self-avoiding walks

In an earlier work we provided the first evidence that the collapse, or coi l-globule transition of an isolated polymer in solution can be seen in a fo ur-dimensional model. Here we investigate, via Monte Carlo simulations, the canonical lattice model of polymer collapse, namely, interacting self-avoi ding 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, w e 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 arg ument that the first-order-like rounded transition is scaled away in the th ermodynamic limit to leave a mean-field second-order transition. Essentiall y this happens because the finite-size shift of the transition is asymptoti cally much larger than the width of the pseudotransition and the latent hea t decays to zero (algebraically) with polymer length. This scenario can be inferred from the application of the theory of Lifshitz, Grosberg, and Khok hlov (based upon the framework of Lifshitz) to four dimensions: the conclus ions of which were written down some time ago by Khokhlov. In fact it is pr ecisely above the upper critical dimension, which is 3 for this problem, th at the theory of Lifshitz may be quantitatively applicable to polymer colla pse.