Continuum Mechanics and Thermodynamics

, Volume 30, Issue 3, pp 459–483 | Cite as

Helmholtz and Gibbs ensembles, thermodynamic limit and bistability in polymer lattice models

  • Stefano Giordano
Original Article


Representing polymers by random walks on a lattice is a fruitful approach largely exploited to study configurational statistics of polymer chains and to develop efficient Monte Carlo algorithms. Nevertheless, the stretching and the folding/unfolding of polymer chains within the Gibbs (isotensional) and the Helmholtz (isometric) ensembles of the statistical mechanics have not been yet thoroughly analysed by means of the lattice methodology. This topic, motivated by the recent introduction of several single-molecule force spectroscopy techniques, is investigated in the present paper. In particular, we analyse the force–extension curves under the Gibbs and Helmholtz conditions and we give a proof of the ensembles equivalence in the thermodynamic limit for polymers represented by a standard random walk on a lattice. Then, we generalize these concepts for lattice polymers that can undergo conformational transitions or, equivalently, for chains composed of bistable or two-state elements (that can be either folded or unfolded). In this case, the isotensional condition leads to a plateau-like force–extension response, whereas the isometric condition causes a sawtooth-like force–extension curve, as predicted by numerous experiments. The equivalence of the ensembles is finally proved also for lattice polymer systems exhibiting conformational transitions.


Polymer lattice models Polymer stretching Isotensional and isometric ensembles Ensemble equivalence Lattice Green functions 


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Institute of Electronics, Microelectronics and Nanotechnology - UMR 8520, LIA LICS, University of Lille, CNRS, Centrale Lille, ISENUniversity of ValenciennesLilleFrance

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