On the equivalence of thermodynamics ensembles for flexible polymer chains

Although the problem of the ensembles equivalence for flexible polymers has aroused considerable interest, there is not an overall consensus on this topic. In this work, we present a theoretical investigation on the asymptotic equivalence of two ensembles for single flexible polymer chains (without confinement effects, i.e. fluctuating in the entire space): the first is the Gibbs (or isotensional) ensemble with one end-terminal of the chain tethered to a given point and the other subjected to an applied force; the other ensemble is the Helmholtz (or isometric) one characterized by both terminals tethered to fixed points. The equivalence property is rigorously proved for a class of potentials characterized by a continuous pairing interaction between neighboring monomers. To approach the problem we adopted an original analytical formalism based on the stationary phase technique and on the exact determination of the eigenvalues sign of the Hessian matrix of the phase function. To give some examples of application, the general result is successively applied to freely-jointed chains, to flexible polymers with extensible bonds and to chains with domains that exhibit conformational transitions between two stable states.