R. Lemoy, S. Grauwin, G. Chevereau, E. Bertin, P. Jensen
Linking the microscopic and macroscopic behavior is at the heart of many natural and social sciences. This apparent similarity conceals essential differences across disciplines: while physical particles are assumed to optimize the global energy, economic agents maximize their own utility. Here, we solve exactly a Schelling-like segregation model, which interpolates continuously between cooperative and individual dynamics. We show that increasing the degree of cooperativity induces a qualitative transition from a segregated phase of low utility towards a mixed phase of high utility. By introducing a simple function which links the individual and global levels, we pave the way to a rigorous approach of a wide class of systems, where dynamics is governed by individual strategies. As a byproduct of this approach, we also find a simple relationship between the utility (a socio-economic concept) and a chemical potential (a physics concept) that can be defined in this model.
We have also considered a simple decision model in which a set of agents randomly choose one of two competing shops selling the same perishable products (typically food). Agents select with a higher probability the store selling the fresher products. Studying the model from a statistical physics perspective, both through numerical simulations and mean-field analytical methods, we find a rich behaviour with continuous and discontinuous phase transitions between a symmetric phase where both stores maintain the same level of activity, and a phase with broken symmetry where one of the two shops attracts more customers than the other.
Competition between collective and individual dynamics
S. Grauwin, E. Bertin, R. Lemoy, P. Jensen, Proc. Natl. Acad. Sci. USA 106, 20622 (2009)
Socio-economic utility and chemical potential
R. Lemoy, E. Bertin, P. Jensen, EPL 93, 38002 (2011)
Symmetry-breaking phase transition in a dynamical decision model
G. Lambert, G. Chevereau, E. Bertin, J. Stat. Mech. P06005 (2011)
K. Martens (LPMCN, Univ. Lyon 1), M. Droz (Univ. Geneve)
On general grounds, a nonequilibrium temperature can be consistently defined from generalized fluctuation-dissipation relations only if it is independent of the observable considered. We argue that the dependence on the choice of observable generically occurs when the phase-space probability distribution is non-uniform on constant energy shells. We relate quantitatively this observable dependence to a fundamental characteristics of nonequilibrium systems, namely the Shannon entropy difference with respect to the equilibrium state with the same energy. This relation is illustrated on a mean-field model in contact with two heat baths at different temperatures.
Another possibility is to define intensive thermodynamic parameters conjugated to conserved quantities, in analogy to equilibrium definition. For instance, if the number of particle is conserved in a steady-state non-equilibrium system, a chemical potential can be defined. Some difficulties however arise in some cases, for instance in the presence of mass fluxes on a branching geometry. In this case, even if a global chemical potential can be defined, local measurements do not always provide equal values of the chemical potential throughout the system.
Dependence of the fluctuation-dissipation temperature on the choice of observable
K. Martens, E. Bertin, M. Droz, Phys. Rev. Lett. 103, 260602 (2009)
Entropy-based characterization of the observable-dependence of the fluctuation-dissipation temperature
K. Martens, E. Bertin, M. Droz, Phys. Rev. E 81, 061107 (2010)
Influence of flux balance on the generalized chemical potential in mass transport models K. Martens, E. Bertin, J. Stat. Mech. P09012 (2011)
O. Dauchot (SPEC, CEA Saclay)
We address, on the example of a simple solvable model, the issue of whether the stationary state of dissipative systems converges to an equilibrium state in the low dissipation limit. We study a driven dissipative Zero Range Process on a tree, in which particles are interpreted as finite amounts of energy exchanged between degrees of freedom. The tree structure mimicks the hierarchy of length scales; energy is injected at the top of the tree ('large scales'), transferred through the tree and dissipated mostly in the deepest branches of the tree ('small scales'). Varying a parameter characterizing the transfer dynamics, a transition is observed, in the low dissipation limit, between a quasi-equilibrated regime and a far-from-equilibrium one, where the dissipated flux does not vanish.
Far-from-equilibrium state in a weakly dissipative model
E. Bertin, O. Dauchot, Phys. Rev. Lett. 102, 160601 (2009)
G. Gregoire (MSC, Univ. Paris 7), M. Droz (Univ. Geneve)
Considering a gas of self-propelled particles with binary interactions, we derive the hydrodynamic equations governing the density and velocity fields from the microscopic dynamics, in the framework of the associated Boltzmann equation. Explicit expressions for the transport coefficients are given, as a function of the microscopic parameters of the model. We show that the homogeneous state with zero hydrodynamic velocity is unstable above a critical density (which depends on the microscopic parameters), signaling the onset of a collective motion. Comparison with numerical simulations on a standard model of self-propelled particles shows that the phase diagram we obtain is robust, in the sense that it depends only slightly on the precise definition of the model. While the homogeneous flow is found to be stable far from the transition line, it becomes unstable with respect to finite-wavelength perturbations close to the transition, implying a non trivial spatio-temporal structure for the resulting flow. We find solitary wave solutions of the hydrodynamic equations, quite similar to the stripes reported in direct numerical simulations of self-propelled particles.
Hydrodynamic equations for self-propelled particles: microscopic derivation and stability analysis
E. Bertin, M. Droz, G. Gregoire, J. Phys. A: Math. Theor. 42, 445001 (2009)
M. Mézard (LPTMS, Univ. Orsay, Paris)
The analysis of the linearization effect in multifractal analysis, and hence of the estimation of moments for multifractal processes, is revisited borrowing concepts from the statistical physics of disordered systems, notably from the analysis of the so-called Random Energy Model, one of the simplest models exhibiting a phase transition. Considering a standard multifractal process (compound Poisson motion), chosen as a simple representative example, we show the existence of a critical order q* beyond which moments, though finite, cannot be estimated through empirical averages, irrespective of the sample size of the observation. In addition, empirical multifractal exponents are found to behave linearly in q for q > q*.
Linearization effect in multifractal analysis: Insights from the Random Energy Model F. Angeletti, M. Mézard, E. Bertin, P. Abry, Physica D 240, 1245 (2011)
G. Gyorgyi (Eotvos Univ., Budapest)
Extreme value statistics is a classical field of probability theory
that found applications many different fields, from physics to hydrology
or finance. Reformulating the evolution of extreme distributions
with the size of the set considered as a partial differential equation
describing a renormalization flow, we recoverin a simple way the well-known
asymptotic distribution (as fixed points of the renormalization flow),
and give a straightforward method to derive finite size correction
by linearizing the flow around the fixed points.
Independently of this formal approach, an interesting connection was
also proposed between extreme value statistics and a simple model
exhibiting the aging phenomenon, the so-called Barrat-Mezard model.
Renormalization flow in extreme value statistics
E. Bertin, G. Gyorgyi, J. Stat. Mech. P08022 (2010)
Entropic aging and extreme value statistics
E. Bertin, J. Phys. A: Math. Theor. 43, 345002 (2010)