Chemin auto-évitant

Résumé

droite|vignette|200x200px|Chemin auto-évitant dans un réseau carré, hamiltonien dans un carré de côté 14, donc de longueur 15²-1. vignette|Chemin auto-évitant dans un réseau carré aboutissant à un cul-de-sac. vignette|Idem dans un réseau hexagonal. En mathématiques, un chemin auto-évitant (CAE), ou marche auto-évitante, est un chemin dans un réseau ne passant jamais par le même sommet ; lorsqu'il est fermé, on parle de polygone auto-évitant (PAE). Pour le graphe infini associé au réseau, les notions de CAE et de PAE correspondent à celles de chaînes et de cycles. Les CAE ont été introduits en 1948 par le chimiste Paul Flory (prix Nobel 1974) dans le but de modéliser le comportement des polymères plongés dans un solvant. Depuis cette introduction, des physico-chimistes ont proposé de nombreuses conjectures qui ont été confirmées par des simulations numériques, mais peu d'entre elles ont été démontrées mathématiquement. Un CAE de longueur infinie a une structure fractale. On démontre q

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Entropy in the Coil-to-Globule Transition of Macromolecules

Carlo Maffi

The coil to globule transition is a fundamental phenomenon in the physics of macro-molecules by reason of the multiplicity of arrangements of their conformation. Such conformational freedom is the main source of entropy in the molecule and is the main opponent to the transition towards the compact state, since a system tends to the state of maximum entropy. This phenomenon is captured by very simple models, such as the ensemble of Interacting Self avoiding Walks on the lattice. This model shows that the coil to globule transition belongs to the universality class of continuous transition called Θ point. Starting from a critical inspection of the definition of the interacting walks model, we introduce a refinement aiming to represent more precisely the entropy sourced from the local fluctuations of the molecule around its equilibrium conformations; this contribution is absent in the standard model which includes only the entropy generated by the multiplicity of the global conformations. Through the study of the vibrational properties of the model and an exhaustive numerical simulation of the phase transition we show that the new model presents a transition which is different in character and belongs to a distinct universality class. In particular, in 3 dimensions the model shows a discontinuous transition; thanks to this, the model presents a common framework underlying the physics of both homo-polymers and proteins. Considering the results obtained in 2 and 3 dimensions we identify the convexity of the vibrational entropy as the responsible of the new class of transitions. These results and a revision of experimental measures of viscosity bring into question the standard description of the homo-polymer transition.

EPFL2012

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In this work we investigate force-induced desorption of linear polymers in good solvents in non-homogeneous environment, by applying the model of self-avoiding walk on two- and three-dimensional fractal lattices, obtained as generalization of the Sierpinski gasket fractal. For each of these lattices one of its boundaries represents an adsorbing wall, whereas along one of the fractal edges, not lying in the adsorbing wall, an external force acts on the self-avoiding walk. The hierarchical nature of the lattices under study enables an exact real-space renormalization group treatment, which yields the phase diagram of polymer critical behavior. We show that for this model there is no low-temperature reentrance in the cases of two-dimensional lattices, whereas in all studied three-dimensional cases the force-temperature dependance is reentrant. We also find that in all cases the force-induced desorption transition is of first order.

2011

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Two-dimensional O(n) models and logarithmic CFTs

Bernardo Zan

We study O(n)-symmetric two-dimensional conformal field theories (CFTs) for a continuous range of n below two. These CFTs describe the fixed point behavior of self-avoiding loops. There is a pair of known fixed points connected by an RG flow. When n is equal to two, which corresponds to the Kosterlitz-Thouless critical theory, the fixed points collide. We find that for n generic these CFTs are logarithmic and contain negative norm states; in particular, the O(n) currents belong to a staggered logarithmic multiplet. Using a conformal bootstrap approach we trace how the negative norm states decouple at n = 2, restoring unitarity. The IR fixed point possesses a local relevant operator, singlet under all known global symmetries of the CFT, and, nevertheless, it can be reached by an RG flow without tuning. Besides, we observe logarithmic correlators in the closely related Potts model.

SPRINGER2020

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