Distribution of the Radius of Gyration for an ISAW
We aim at calculating an explicit expression for the finite-size probability distribution of the radius of gyration $R$ of an Interacting Self-Avoiding Walk (ISAW), as a function of chain length $N$ and monomer-monomer interaction energy $\varepsilon$. We first derive the explicit free energy expression for a non-interacting Self-Avoiding Walk, introducing a new natural scale variable $t = ρ^g$ expressed as a power of the density $ρ$. Then, thanks to a cumulant expansion approach introduced by Lhuillier, Victor and coworkers, we extend it to the interacting case, capturing both the coil-globule transition and finite-size corrections to scaling, including entropic and surface-energy contributions. The radius of gyration distribution determined by the new free energy expression is then compared, using Bayesian inference to estimate the model parameters, to results from extensive Monte Carlo simulations of three-dimensional ISAWs, showing excellent agreement except very close to the $Θ$-point, where finite chain lengths limit the accessible scaling regime. This validates our resulting free energy expression and allows us, as an application, to construct a phase diagram of the coil-globule transition.