Lecturer at the University Paris Diderot (Paris 7), doing his research in the IMNC laboratory, in the modelling team. Formerly post-doc, in the team of Thomas Nattermann. Former PhD student at the LPTENS under the advice of Rémi Monasson.
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E-mail: deroulers # imnc.in2p3.fr Office: E151 Phone: +33 1 69 15 36 41 Fax: +33 1 69 15 71 96 |
It has been shown experimentally that contact interactions may influence the migration of cancer cells. Previous works have modelized this thanks to stochastic, discrete models (cellular automata) at the cell level. However, for the study of the growth of real-size tumors with several million cells, it is best to use a macroscopic model having the form of a partial differential equation (PDE) for the density of cells. The difficulty is to predict the effect, at the macroscopic scale, of contact interactions that take place at the microscopic scale. To address this, we use a multiscale approach: starting from a very simple, yet experimentally validated, microscopic model of migration with contact interactions, we derive a macroscopic model. We show that a diffusion equation arises, as is often postulated in the field of glioma modeling, but it is nonlinear because of the interactions. We give the explicit dependence of diffusivity on the cell density and on a parameter governing cell-cell interactions. We discuss in detail the conditions of validity of the approximations used in the derivation, and we compare analytic results from our PDE to numerical simulations and to some in vitro experiments. We notice that the family of microscopic models we started from includes as special cases some kinetically constrained models that were introduced for the study of the physics of glasses, supercooled liquids, and jamming systems.
Motivated by recent experiments on nanowires and carbon nanotubes, we study theoretically the effect of strong, point-like impurities on the linear electrical resistance R of finite length quantum wires. Charge transport is limited by Coulomb blockade and cotunneling. ln R is slowly self-averaging and non Gaussian. Its distribution is Gumbel with finite-size corrections which we compute. At low temperatures, the distribution is similar to the variable range hopping (VRH) behaviour found long ago in doped semiconductors. We show that a result by Raikh and Ruzin does not apply. The finite-size corrections decay with the length L like 1/ln L. At higher temperatures, this regime is replaced by new laws and the shape of the finite-size corrections changes strongly: if the electrons interact weakly, the corrections vanish already for wires with a few tens impurities.
The probability Psuccess(α, N) that stochastic greedy algorithms successfully solve the random SATisfiability problem is studied as a function of the ratio α of constraints per variable and the number N of variables. These algorithms assign variables according to the unit-propagation (UP) rule in presence of constraints involving a unique variable (1-clauses), to some heuristic (H) prescription otherwise. In the infinite N limit, Psuccess vanishes at some critical ratio α_H which depends on the heuristic H. We show that the critical behaviour is determined by the UP rule only. In the case where only constraints with 2 and 3 variables are present, we give the phase diagram and identify two universality classes: the power law class, where Psuccess[αH (1+ε N-1/3), N] ∼ A(ε)/Nγ; the stretched exponential class, where Psuccess[αH (1+ε N-1/3), N] ∼ exp[-N1/6 Φ(ε)]. Which class is selected depends on the characteristic parameters of input data. The critical exponent γ is universal and calculated; the scaling functions A and Φ weakly depend on the heuristic H and are obtained from the solutions of reaction-diffusion equations for 1-clauses. Computation of some non-universal corrections allows us to match numerical results with good precision. The critical behaviour for constraints with >3 variables is given. Our results are interpreted in terms of dynamical graph percolation and we argue that they should apply to more general situations where UP is used.
The probability P(α, N) that search algorithms for random Satisfiability problems successfully find a solution is studied as a function of the ratio α of constraints per variable and the number N of variables. P is shown to be finite if α lies below an algorithm-dependent threshold α_A, and exponentially small in N above. The critical behaviour is universal for all algorithms based on the widely-used unitary propagation rule: P[ (1 + ε) αA , N] ∼ exp[-N1/6 Φ(ε N1/3) ]. Exponents are related to the critical behaviour of random graphs, and the scaling function Φ is exactly calculated through a mapping onto a diffusion-and-death problem.
A quantum field theoretic formulation of the dynamics of the Contact Process on a regular graph of degree z is introduced. A perturbative calculation in powers of 1/z of the effective potential for the density of particles φ(t) and an instantonic field ψ(t) emerging from the quantum formalism is performed. Corrections to the mean-field distribution of densities of particles in the out-of-equilibrium stationary state are derived in powers of 1/z. Results for typical (e.g. average density) and rare fluctuation (e.g. lifetime of the metastable state) properties are in very good agreement with numerical simulations carried out on D-dimensional hypercubic (z=2D) and Cayley lattices.
We study the dynamics of the q-state random bond Potts ferromagnet on the square lattice at its critical point by Monte Carlo simulations with single spin-flip dynamics. We concentrate on q=3 and q=24 and find, in both cases, conventional, rather than activated, dynamics. We also look at the distribution of relaxation times among different samples, finding different results for the two q values. For q=3 the relative variance of the relaxation time tau at the critical point is finite. However, for q=24 this appears to diverge in the thermodynamic limit and it is ln(tau) which has a finite relative variance. We speculate that this difference occurs because the transition of the corresponding pure system is second order for q=3 but first order for q=24.
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