! STAGE POURVU !
Type of internship
M1
Duration
To be discussed
Context
Phase transitions are well characterized on regular periodic lattices, as for example the magnetic transition in the Ising model on the square lattice, exactly solved by Onsager [1]. In such cases, numerical studies use simulations on finite lattices of various sizes and determine the critical exponents using finite size scaling. In the brain, phase transitions have been proposed to explain many observed phenomena [2]. Neuron networks are then modeled by Ising spins on a nonregular graph representing brain connections. It is therefore important to determine if our knowledge about phase transitions on periodic lattices can be generalized to such nonregular graphs. The study of phase transitions on nonregular lattices has to deal with the question of the scalability of the lattice to open the possibility of finite size scaling [3].
[1] L. Onsager. Crystal statistics. I. A two-dimensional model with an order-disorder transition. Phys. Rev. 65, 117 (1944).
[2] D. Marinazzo, M. Pellicoro, G. Wu, L. Angelini, J. M. Cortés, S. Stramaglia. Information transfer and criticality in the Ising model on the human connectome. PLoS ONE 9(4): e93616 (2014).
[3] J. Viana Lopes, Y. G. Pogorelov, J. M. B. Lopes dos Santos, R. Toral. Exact solution of Ising model on a small-world network. Phys. Rev. E 70 026112 (2004).
Objective
During the internship, phase transitions will be explored on various types of nonregular graphs using Monte Carlo simulations.
Missions
- Do some literature search on the Ising model, phase transition and graph theory;
- Propose and test computer programs to generate graphs, characterize graphs, and generate graphs with given features;
- Understand the notion of finite size scaling for phase transition and specify the meaning of thermodynamic limit for nonregular graphs;
- Perform Monte Carlo simulations to investigate phase transitions on nonregular graphs.
The internship will take place on both the “Jussieu” Campus and the “Les Cordeliers” campus (if compatible with Covid restrictions).
Skills
The candidate must be able to write codes, either in Python or in C/C++.
Internship Stipend
about 600 euros per month
Contact
Laura Messio (Laboratoire de Physique Théorique de la matière Condensée, CNRS, Sorbonne Université)
laura.messio [at] lptmc.jussieu.fr
Guillaume Marrelec (Laboratoire d’imagerie biomédicale, CNRS, INSERM, Sorbonne Université)
guillaume.marrelec [at] inserm.fr