PHATRAMAS · Mathematical theory of phase transitions: modeling, analysis and simulations

Mathematical theory of phase transitions: modelling, analysis and simulations

(1) Systems out of equilibrium We have studied phase transitions in non-equilibrium stationary states as when a density gradient is imposed by the boundary conditions (Fick's law). The results have been reported in the paper: 'Fourier law, phase transitions and the stationary Stefan problem' (A. De Masi, E. Presutti, D. Tsagkarogiannis), Arch. Rat. Mech. and Analysis, DOI: 10.1007/s00205-011-0423-1. In the above paper, we have introduced the notion of current reservoirs as they appear in a mesoscopic description of the system. The goal is to implement the notion at the microscopic level. With this motivation we have studied the symmetric simple exclusion process in one dimension with boundary conditions which simulate a current reservoir. The results have been reported in the paper: 'Current Reservoirs in the simple exclusion process' (A. De Masi, E. Presutti, D. Tsagkarogiannis, M. E. Vares), submitted. (See http://arxiv.org/abs/1104.3445 online for further details.) In the above work we are assuming propagation of chaos. The proof of this main technical tool has been reported separately: 'Truncated correlations in the stirring process with births and deaths' (A. De Masi, E. Presutti, D. Tsagkarogiannis, M. E. Vares), submitted. (See http://arxiv.org/abs/1104.3447 online for further details). Having established the law of large numbers for the aforementioned stochastic process we are currently studying the relation between the unique equilibrium probability of the process and the unique stationary solution of the limiting equation. Then, the next step is to derive the large deviation functional and prove a fluctuation-dissipation type of theorem in the thermodynamic limit. (2) Phase transitions for systems with long- and short-range interactions We are studying an extension of some previous work (J. L. Lebowitz, A. E. Mazel, E. Presutti, 'Liquid- Vapour phase transitions for system with finite range interactions', J. Stat. Phys. 94 (1999), 955 - 1025.) on phase transitions in the continuum where we add a small short range repulsive interaction. On physical grounds, we expect that this extra interaction may be responsible for the appearance at small temperatures of crystalline structure while, due to its smallness, it should not affect the vapour-liquid transition observed in the earlier work. We have preliminary results on this latter issue while the former seems too hard to be dealt with directly and some simplifying hypotheses on the short range interaction will be presumably added. Nevertheless, we have started a new collaboration to study the phase diagram of continuum particle systems in a joint limit where both temperature and density vanish. A main step in the treatment of the microscopic Hamiltonian is to construct a coarse-grained Hamiltonian with multi-canonical constraints as required by the Lebowitz-Penrose theory. For this, the introduction of the small short range interaction creates an extra complication which is directly related to the question of validity of the cluster expansion in the canonical ensemble which may also resolve the question of the finite volume corrections to the free energy. The former has been studied in collaboration with Elena Pulvirenti (PhD student of Prof. Presutti) and has been reported in the following paper: 'Cluster expansion in the canonical ensemble' (E. Pulvirenti and D. Tsagkarogiannis), submitted. (See http://arxiv.org/abs/1105.1022 online for further details). (3) Coarse-grained algorithms for the simulation of polymeric chains Working with off-lattice systems as in the previous project, we have developed suitable cluster expansion methods for the computation of the partition function of such systems. Apart from the theoretical use of this method it can also serve as a computational tool for designing efficient coarse-grained algorithms for the computation of correlation functions of systems like polymer chains. More specifically, as a starting point we are studying the case of methane (in collaboration with V. Harmandaris and M. Katsoulakis).

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