Long-time behaviour and phase transitions for the McKean-Vlasov equation Grigorios A. Pavliotis Department of Mathematics Imperial College London Interactions Between PDEs and Probability, UIMP August 13, 2018 Joint work with José Carrillo, Susana Gomes, Rishabh Gvalani, Hong Duong, Petr Yatsyshin, Serafim Kalliadasis and André Schlichting (Aachen/Bonn). Phase Transitions for the Desai-Zwanzig model in multiwell and random energy landscapes (with S.N. Gomes, S. Kalliadasis, G.A. Pavliotis, P. Yatsyshin). Preprint (2018). Long time behaviour and phase transitions for the McKean-Vlasov equation on the torus (with R. Gvalani, J.A. Carrillo, A. Schlichting) Preprint (2018) https://arxiv.org/abs/1806.01719. Mean field limits for interacting diffusions in a two-scale potential (S.N. Gomes and G.A. Pavliotis), J. Nonlinear Sci. 28(3), pp. 905-941, (2018). Mean field limits for non-Markovian interacting particles: convergence to equilibrium, GENERIC formalism, asymptotic limits and phase transitions (with M. H. Duong). Preprint (2018). https://arxiv.org/abs/1805.04959 Research Funded by the EPSRC, Grants EP/P031587/1, EP/J009636/1, EP/L024926/1, EP/L020564/1 and EP/L025159/1 June 25, 2018 Phase transitions for the McKean-Vlasov eqn2 Study the mean field limits of weakly interacting diffusions: The Dasai-Swanzing model in a 2-scale potential: N ! 0 i 1 X j p −1 i dXi = −V (Xi,Xi/ε) dt − θ X − X dt + 2β dW . t N t t j=1 Noisy Kuramoto oscillators: N 1 X p −1 x˙ i = − cos(xi − xj ) + 2β W˙ i. N j=1 Models for opinion formation: N 1 X p −1 x˙ i = aij (|xi − xj |)(xi − xj ) + 2β W˙ i. N j=1 June 25, 2018 Phase transitions for the McKean-Vlasov eqn3 Study the mean field limits of weakly interacting diffusions: Interacting non-Markovian Langevin dynamics N N ∂V 1 X 0 X q¨i = − − U (qi − qj ) − γij (t − s)q ˙j (s) ds + Fi(t), i = 1, . N, ∂qi N j=1 j=1 (1) where F (t) = (F1(t),...FN (t)) is a mean zero, Gaussian, stationary process −1 with autocorrelation function E(Fi(t) Fj (s)) = β γij (t − s). Langevin dynamics driven by colored noise N 0 1 X 0 x˙ i = −V (xi) − W (xi − xj ) + ηi, (2a) N j=1 p −1 η˙i = −ηi + 2β B˙ i (2b) applications: Models for systemic risk (Garnier, Papanicolaou...), clustering in the Hegselmann-Krause model (Chazelle, E, .....) June 25, 2018 Phase transitions for the McKean-Vlasov eqn4 We consider a system of weakly interacting diffusions moving in a 2-scale locally periodic potential: N i i 1 X i j p i dX = −∇V (X )dt − ∇F (X − X )dt + 2β−1dB , i = 1, .., N, t t N t t t j=1 (3) where V (x) = V0(x) + V1(x, x/). (4) Our goal is to study the combined mean-field/homogenization limits. In particular, we want to study bifurcations/phase transitions for the McKean-Vlasov equation in a confining potential with many local minima. June 25, 2018 Phase transitions for the McKean-Vlasov eqn5 5 5 4 4 3 3 2 2 1 1 0 0 -1 -1 -2 -1 0 1 2 -2 -1 0 1 2 Figure: Bistable potential with (left) separable and (right) nonseparable fluctuations, ε x4 x2 x ε x4 x x2 V (x) = 4 − 2 + δ cos ε and V (x) = 4 − 1 − δ cos ε 2 . June 25, 2018 Phase transitions for the McKean-Vlasov eqn6 McKean-Vlasov Dynamics in a Bistable Potential Consider a system of interacting diffusions in a bistable potential: N !! i 0 i i 1 X j p i dX = −V (X ) − θ X − X dt + 2β−1 dB . (5) t t t N t t j=1 The total energy (Hamiltonian) is N N N X θ X X W (X) = V (X`) + (Xn − X`)2. (6) N 4N `=1 n=1 `=1 We can pass rigorously to the mean field limit as N → ∞ using, for example, martingale techniques, (Dawson 1983, Gartner 1988, Oelschlager 1984). Formally, using the law of large numbers we obtain the McKean SDE 0 p −1 dXt = −V (Xt) dt − θ(Xt − EXt) dt + 2β dBt. (7) June 25, 2018 Phase transitions for the McKean-Vlasov eqn7 The Fokker-Planck equation corresponding to this SDE is the McKean-Vlasov equation ∂p ∂ Z ∂p = V 0(x)p + θ x − xp(x, t) dx p + β−1 . (8) ∂t ∂x R ∂x The McKean-Vlasov equation is a gradient flow, with respect to the Wasserstein metric, for the free energy functional Z Z θ ZZ F[ρ] = β−1 ρ ln ρ dx + V ρ dx + F (x − y)ρ(x)ρ(y) dxdy, (9) 2 1 2 with F (x) = 2 x . June 25, 2018 Phase transitions for the McKean-Vlasov eqn8 Journal of Statistical Physics, Vol. 31, No. 1, 1983 Critical Dynamics and Fluctuations for a Mean-Field Model of Cooperative Behavior Donald A. Dawson 1'2 Received September 20, 1982 The main objective of this paper is to examine in some detail the dynamics and fluctuations in the critical situation for a simple model exhibiting bistable macroscopic behavior. The model under consideration is a dynamic model of a collection of anharmonic oscillators in a two-well potential together with an attractive mean-field interaction. The system is studied in the limit as the number of oscillators goes to infinity. The limit is described by a nonlinear partial differential equation and the existence of a phase transition for this limiting system is established. The main result deals with the fluctuations at the critical point in the limit as the number of oscillators goes to infinity. It is established that these fluctuations are non-Gaussian and occur at a time scale slower than the noncritical fluctuations. The method used is based on the June 25, 2018 Phaseperturbation transitions theory for the for McKean-Vlasov Markov processes eqn 9developed by Papanicolaou, Stroock, and Varadhan adapted to the context of probability-measure-valued processes. KEY WORDS: Mean field model; cooperative behavior; phase transition; critical fluctuations; universality; probability-measure-valued processes; perturbation theory. 1. INTRODUCTION AND DESCRIPTION OF THE RESULTS One of the principal problems of stochastic system theory is to describe the behavior of a system which is comprised of a large number of interacting subsystems. In addition an important feature of most systems of this type is a degree of randomness inherent in the microscopic subsystems. The i Department of Mathematics and Statistics, Carleton University, Ottawa, Canada K1S 5B6. 2 Research supported by the Natural Sciences and Engineering Research Council of Canada. In addition, a part of this research was supported by SFB 123, University of Heidelberg and the University of Wisconsin-Madison. 29 0022-4715/83/0400-0029503.00/0 9 1983 Plenum Publishing Corporation PHYSICAL REVIE%' A VOLUME 36, NUMBER 5 SEPTEMBER 1, 1987 Dynamical behavior of stochastic systems of infinitely many coupled nonlinear oscillators exhibiting phase transitions of mean-field type: H theorem on asymptotic approach to equilibrium and critical slowing down of order-parameter fiuctuations Masatoshi Shiino Department of App/ied Physics, Faculty of Science, Tokyo Institute of Technology, Oh okay-ama, Meguro ku-, Tokyo 152, Japan (Received 29 September 1986) It is shown that statistical-mechanical properties as well as irreversible phenomena of stochastic systems, which consist of infinitely many coupled nonlinear oscillators and are capable of exhibiting phase transitions of mean-field type, can be successfully explored on the basis of nonlinear Fokker- Planck equations, which are essentially nonlinear in unknown distribution functions. Results of two kinds of approaches to the study of their dynamical behavior are presented. Firstly, a problem of asymptotic approaches to stationary states of the infinite systems is treated. A method of Lyapunov functional is employed to conduct a global as well as a local stability analysis of the systems. By constructing an H functional for the nonlinear Fokker-Planck equation, an H theorem is proved, ensuring that the Helmholtz free energy for a nonequilibrium state of the system decreases monotoni- cally until a stationary state is approached. Calculations of the second-order variation of the H func- tional around a stationary state yield a stability criterion for bifurcating solutions of the nonlinear Fokker-Planck equation, in terms of an inequality involving the second moment of the stationary dis- tribution function. Secondly, the behavior of critical dynamics is studied within the framework of linear-response theory. Generalized dynamical susceptibilities are calculated rigorously from linear responses of the order parameter to externally driven fields by linearizing the nonlinear Fokker- Planck equation. Correlation functions, together with spectra of the fluctuations of the order param- eter of the system, are also obtained by use of the fluctuation-dissipation theorem for stochastic sys- tems. A critical slowing down is shown to occur in the form of the divergence of relaxation time for the fluctuations, in accordance with the divergence of the static susceptibility, as a phase transition point is approached. I. INTRODUCTION tained by omitting the random force f(t) in Eq. (1.1) can exhibit a bifurcation at y=O, the stochastic differential The study of dynamical behavior of systems exhibiting equation (1.1) has nothing to do with bifurcations nor thermodynamic phase transitions has been of considerable phase transitions in that a stationary distribution for the ' June 25, 2018 interest for many years.Phase transitionsIt is well forknown the McKean-Vlasovthat in a random eqnvariable x is always uniquely determined irrespec- 10 thermodynamic system undergoing phase transitions criti- tive of the values of y and o.