Long-Time Behaviour and Phase Transitions for the Mckean-Vlasov Equation

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 functional 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 distribution 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 parameter of the system, are also obtained by use of the fluctuation-dissipation theorem for stochastic systems. 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.. This is because the corre- cal anomaly such as critical slowing down is generally ex- sponding linear Fokker-Planck equation pected to occur at its transition points. Recently the con- cept of phase transition has been extended to include 2 2 nonthermodynamic or nonequilibrium phase transi- Q Q —p (t,x) = — (yx —x )p (t,' x)+ (t,x) (1.2) tions. A problem arises of how the dynamical behav- Bx 2 p ior of nonequilibrium phase transitions compares with the one for phase transitions in thermodynamic systems. To discuss this sort of problem, stochastic approaches or is con6rmed to have the property of global stability with models have been extensively employed, ' ' since such respect to its uniquely determined stationary solution, as stochastic methods as using Langevin equation models are will be noted later. Thus, one cannot expect any critical often considered to be capable of simulating the dynami- divergence at y=O for such physical quantities as the cal behavior of phase transitions both in thermodynamic variance or relaxation time for the variable x in Eq. and in nonequilibrium systems. (1.1). In fact, the finiteness of the variance is easily In particular, the Langevin equation of the form " checked with use of the stationary solution of Eq. (1.2) — and the absence of critical divergence of the relaxation x =yx x'+f(t), time was shown through the investigation of the eigen- ' ' ( (t)f(t') ) =ct'5(t —t') values of Eq. (1.2). In view of the fact that changes in f the shape of the stationary distribution for Eq. (1.2) sure- has been one of the most popular models used to discuss ly occur at y=O, however, the term "phase transition" the dynamical behavior of systems undergoing phase tran- seems to have been extensively used in a somewhat wider sitions involving symmetry-breaking instabilities with sense to describe a qualitative change in most probable change in certain control parameters as expressed by y in values as well as shapes of a distribution function accom- Eq. (1.1). We must however be cautious of the use of this panied by changes in control parameters, in the study of equation. Although the ordinary differential equation ob- symmetry-breaking instabilities observed in far-from-

36 2393 Q~ 1987 The American Physical Society The ﬁnite dimensional dynamics (5) is reversible with respect to the Gibbs measure

1 N Z 1 N 1 −βWN (x ,...x ) 1 N −βWN (x ,...x ) 1 N µN (dx) = e dx . . . dx ,ZN = e dx . . . dx ZN RN (10)

where WN (·) is given by (6).

the McKean dynamics (7) can have more than one invariant measures, for nonconvex conﬁning potentials and at suﬃciently low temperatures (Dawson1983, Tamura 1984, Shiino 1987, Tugaut 2014).

The density of the invariant measure(s) for the McKean dynamics (7) satisﬁes the stationary nonlinear Fokker-Planck equation Z ∂ 0 −1 ∂p∞ V (x)p∞ + θ x − xp∞(x) dx p∞ + β = 0. (11) ∂x R ∂x

June 25, 2018 Phase transitions for the McKean-Vlasov eqn 11 For the quadratic interaction potential a one-parameter family of solutions to the stationary McKean-Vlasov equation (11) can be obtained:

1 −β V (x)+θ 1 x2−xm p (x; θ, β, m) = e ( ( 2 )), (12a) ∞ Z(θ, β; m) Z −β V (x)+θ 1 x2−xm Z(θ, β; m) = e ( ( 2 )) dx. (12b) R

These solutions are subject, to the constraint that they provide us with the correct formula for the ﬁrst moment: Z m = xp∞(x; θ, β, m) dx =: R(m; θ, β). (13) R

June 25, 2018 Phase transitions for the McKean-Vlasov eqn 12 This is the selfconsistency equation:

m = R(m; θ, β).

The critical temperature can be calculated from 1 Varp∞ (x) = . (14) m=0 βθ

June 25, 2018 Phase transitions for the McKean-Vlasov eqn 13 2 1

1 0.5

0 0

-0.5 -1

-1 -2 -2 -1 0 1 2 0 10 20 30 40 50

Figure: Plot of R(m; θ, β) and of the straight line y = x for θ = 0.5, β = 10, and bifurcation diagram of m as a function of β for θ = 0.5 for the bistable potential x4 x2 x2 V (x) = 4 − 2 and interaction potential F (x) = 2 .

June 25, 2018 Phase transitions for the McKean-Vlasov eqn 14 -0.6

-0.7

-0.8 F -0.9

-1 5 0 5 0 10 15 -5 β m

Figure: Free energy surface corresponding to the bistable potential, as a function of the inverse temperature β and the ﬁrst moment m with θ = 0.5.

June 25, 2018 Phase transitions for the McKean-Vlasov eqn 15 We can study the eﬀect of breaking the symmetry of polynomial potentials by adding a tilt to a bistable potential.

We consider potentials of the form

1 x4 x2 V (x) = − + κx. (15) a0 4 2

June 25, 2018 Phase transitions for the McKean-Vlasov eqn 16 1

0.5

0 m V (x) 2 V (x)+0.01x 2 -0.5 V (x)+0.1x 2 V (x)+x 2

-1 0 0.5 1 1.5 2 2.5 3 3.5 β

Figure: Bifurcation diagrams of m as a function of β for tilted bistable potentials with a0 = 0.25, and κ = 0, 0.01, 0.1, 1 (see legend). Here, we used θ = 2.5 The symmetric pitchfork bifurcation is broken at any κ > 0. Note that the locus of the critical points forms a distinct critical line.

June 25, 2018 Phase transitions for the McKean-Vlasov eqn 17 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 (16)

where V (x) = V0(x) + V1(x, x/).

The full N-particle potential is

N N N N X 1 X X X U(x , . . . , x , y , . . . , y ) = V (x )+ F (x −x )+ V (x , y ). 1 N 1 N 0 i 2N i j 1 i i i=1 i=1 j=1 i=1

The homogenization theorem applies to the N-particle system.

June 25, 2018 Phase transitions for the McKean-Vlasov eqn 18 The homogenized equation is ! 1 X dXi = − M(Xi) ∇V (Xi) + ∇F Xj − Xi + ∇ψ(Xi) dt t t 0 t N t t t i6=j (17)

−1 i p −1 i i + β ∇ · M(Xt )dt + 2β M(Xt )dWt ,

for i = 1,...,N, where Z ψ(x) = −β−1∇ ln Z(x), with Z(x) = e−βV1(x,y) dy. Td

The stochastic integral in (17) can be interpreted in the Klimontovich sense:

p −1 Klim dXt = −M(Xt)∇U(Xt) dt + 2β M(Xt) ◦ dWt.

The dynamics is ergodic with respect to the Gibbs measure 1 −βU(x) µβ (dx) = e dx. Zb June 25, 2018 Phase transitions for the McKean-Vlasov eqn 19 d d×d The diﬀusion tensor M : R → Rsym is deﬁned by Z Z 1 −βV1(x,y) d M(x) = (I + ∇yθ(x, y))e dy, x ∈ R , Z(x) Td

d and where, for ﬁxed x ∈ R , θ is the unique mean zero solution to

−βV1(x,y) d ∇ · e (I + ∇yθ(x, y) = 0, y ∈ T , (18)

June 25, 2018 Phase transitions for the McKean-Vlasov eqn 20 We can pass to the mean ﬁeld limit N → +∞ using the results from e.g. Dawson (1983), Oelschlager (1984) to obtain the McKean-Vlasov-Fokker-Planck equation:

∂p = ∇· M(∇V p+∇Ψp+(∇F ∗p)p)+β−1∇·Mp+β−1∇·(Mp) . (19) ∂t 0

The mean ﬁeld N → +∞ and the homogenization → 0 limits commute over ﬁnite time intervals.

This is a nonlinear equation and more than one invariant measures can exist, depending on the temperature. Eqn (19) can exhibit phase transitions.

The number of invariant measures depends on the number of solutions of the self-consistency equation.

June 25, 2018 Phase transitions for the McKean-Vlasov eqn 21 The phase/bifurcation diagrams can be diﬀerent depending on the order with which we take the limits. For example:

x2 V ε(x) = + cos(x/). 2

The homogenization process tends to "convexify"the potential.

June 25, 2018 Phase transitions for the McKean-Vlasov eqn 22 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 additive (left) and multiplicative (right) ﬂuctuations.

June 25, 2018 Phase transitions for the McKean-Vlasov eqn 23 x2 Consider the case F (x) = θ 2 , take N → +∞ and keep ﬁxed. The invariant distribution(s) are: Z ε 1 −β(V ε(x)+θ( 1 x2−x m)) ε −β(V ε(x)+θ( 1 x2−x m)) p (x; m, θ, β) = e 2 ,Z = e 2 dx, Zε

where Z m = xpε(x; m, θ, β) dx. (20)

Take ﬁrst ε → 0 and then N → +∞. The invariant distribution(s) are Z 1 −β(V (x)+ψ(x)+θ( 1 x2−x m)) −β(V (x)+ψ(x)+θ( 1 x2−x m)) p(x; m, θ, β) = e 0 2 ,Z = e 0 2 dy, Z

where Z m = xp(x; m, θ, β) dx. (21)

June 25, 2018 Phase transitions for the McKean-Vlasov eqn 24 The number of invariant measures is given by the number of solutions to the self-consistency equations (20) and (21).

Separable ﬂuctuations V0(x) + V1(x/ε) do not change the structure of the phase diagram, since they lead to additive noise. Nonseparable ﬂuctuations

V0(x) + V1(x, x/ε) lead to multiplicative noise and change the bifurcation diagram.

Rigorous results for the ε → 0, N → +∞ limits, formal asymptotics for the opposite limit.

June 25, 2018 Phase transitions for the McKean-Vlasov eqn 25 The structure of the bifurcation diagram for the homogenized dynamics is similar to the one for the dynamics in the absence of ﬂuctuations.

The critical temperature is diﬀerent, but there are no additional branches and their stability is the same as in the case V1 = 0.

This is the case both for additive and multiplicative oscillations.

We can study the stability of the diﬀerent branches using the formula for the free energy

θ F[ρ ] = −β−1 ln Z + m2. ∞ β,θ,m 2

June 25, 2018 Phase transitions for the McKean-Vlasov eqn 26 Finite ε: separable ﬂuctuations

2 3 2 1 1 1 0 -1 -2 0 2 0 0

-1 -1

-2 -2 -1 0 1 2 0 10 20 30 40 50

June 25, 2018 Phase transitions for the McKean-Vlasov eqn 27 Finite ε: nonseparable ﬂuctuations

4 2

1 2 1 0 -2 0 2 0 0

-1 -1 -2 -2 -2 -1 0 1 2 0 10 20 30

June 25, 2018 Phase transitions for the McKean-Vlasov eqn 28 Noncommutativity: particle simulations

JuneFigure: 25, 2018Histogram of NPhase= transitions 1000 particles for the McKean-Vlasov for MC eqn simulations of a convex potential with 29 separable ﬂuctuations. Parameters used were θ = 2, β = 8, δ = 1. Left: = 0.1. Right: homogenized system. Noncommutativity: McKean-Vlasov evolution

5 2

4 1.5 3

2 1

1 0.5 0 -1 -0.5 0 0.5 1 0 -1 -0.5 0 0.5 1

x2 x Figure: Time evolution of p(x, t) for V0(x) = 2 + δ cos ε . Parameters used were θ = 2, β = 8, δ = 1. Left: = 0.1. Right: homogenized system.

June 25, 2018 Phase transitions for the McKean-Vlasov eqn 30 non-Periodic multiwell potentials

0.6

0.4

F 0.2

-0.2 0 5 10 10 -10 -5 0 5 β M 1

10 1

0.8 5

0.6 1 0 M V(x) 0.4

-5 0.2

-10 0 0 2 4 6 8 10 -10 -5 0 5 10 β x

Figure: Free energy surface as a function of β and the ﬁrst moment m for potential V (q) = 1 , but the energy barriers are uniformly randomly PN −2 |q−q`| `=−N distributed. June 25, 2018 Phase transitions for the McKean-Vlasov eqn 31 The McKean-Vlasov equation on the torus

June 25, 2018 Phase transitions for the McKean-Vlasov eqn 32 The McKean–Vlasov equation – Setup

Nonlocal parabolic PDE

∂% = β−1∆% + κ∇ · (%∇W?%) in Td × (0,T ] ∂t L

Td Td L L d with periodic boundary conditions, %(·, 0) = %0 ∈ P( L), L ˆ= − 2 , 2 Td %(·, t) ∈ P( L) probability density of particles

W coordinate-wise even interaction potential

β > 0 inverse temperature (ﬁxed)

κ > 0 interaction strength (parameter)

June 25, 2018 Phase transitions for the McKean-Vlasov eqn 33 Example: The noisy Kuramoto model p 2 x The Kuramoto model: W (x) = − L cos 2πk L , k ∈ Z

Figure: * Figure: *

κ < κc, no phase locking κ > κc, phase locking

June 25, 2018 Phase transitions for the McKean-Vlasov eqn 34 Goals and Motivation:

Classiﬁcation of continuous and discontinuous transitions.

Better understanding of the free energy landscape.

Study dynamical properties related to nucleation/coarsening of clustered states.

June 25, 2018 Phase transitions for the McKean-Vlasov eqn 35 H-stability

d Fourier representation f(k) = hf, wki 2 with k ∈ Z e L (TL) 2 Td A function W ∈ L ( L) is H-stable, W ∈ Hs, if

d Wf(k) = hW, wki ≥ 0, ∀k ∈ Z ,

Decomposition of potential W into H-stable and H-unstable part X Ws(x) = (hW, wki)+wk(x) and Wu(x) = W (x) − Ws(x) . k∈Nd

June 25, 2018 Phase transitions for the McKean-Vlasov eqn 36 Functionals for stationary states

Free energy functional Fκ: Driving the W2-gradient ﬂow Z −1 κ Fκ(%) = β % log % dx + x W (x − y)%(x)%(y) dx dy . Td 2 L Td ×Td L L

Dissipation: Fκ is Lyapunov-function Z 2 d % Jκ(%) = − Fκ(%) = ∇ log −βκW ?% % dx , dt Td e L

Kirkwood-Monroe ﬁxed point mapping Z 1 −βκW ?% −βκW ?% Fκ(%) = %−T % = %− e , with Z(%, κ) = e dx . Z(%, κ) Td L

June 25, 2018 Phase transitions for the McKean-Vlasov eqn 37 Characterization of stationary states: The following are equivalent

−1 % is a stationary state: β ∆% + κ∇ · (%∇W?%) = 0.

% is a root of Fκ(%).

% is a global minimizer of Jκ(%).

% is a critical point of Fκ(%).

−d ⇒ %∞ ≡ L is a stationary state for all κ > 0.

June 25, 2018 Phase transitions for the McKean-Vlasov eqn 38 Existence/Uniqueness of Solutions

Theorem 3+d Under appropriate assumptions on the potential, for %0 ∈ H (U) ∩ Pac(U), there exists a unique classical solution % of the McKean-Vlasov equation such that 2 %(·, t) ∈ Pac(U) ∩ C (U) for all t > 0. Additionally, %(·, t) is strictly positive and has ﬁnite entropy, i.e, %(·, t) > 0 and S(%(·, t)) < ∞, for all t > 0.

June 25, 2018 Phase transitions for the McKean-Vlasov eqn 39 Exponential stability/convergence in relative entropy

Theorem (Convergence to equilibrium) Let %(x, t) be a classical solution of the Mckean–Vlasov equation with smooth initial data and smooth, even, interaction potential W . Then we have: 1. If 0 < κ < 2π , then % − 1 → 0, exponentially, as t → ∞, 3βLk∇W k∞ L 2 ˆ 2π2 1 2. If W (k) ≥ 0 for all k ∈ Z or 0 < κ < 2 , then H %| → 0, βL k∆W k∞ L exponentially, as t → ∞, ˆ 1 where W (k) represents the Fourier transform and H %| L represents the relative entropy.

June 25, 2018 Phase transitions for the McKean-Vlasov eqn 40 June 25, 2018 Phase transitions for the McKean-Vlasov eqn 41 Nontrivial solutions to the stationary McKean–Vlasov equation?

W/∈ Hs is a necessary condition for the existence of nontrivial steady states.

Numerical experiments indicate one, multiple, or possibly inﬁnite solutions

What determines the number of nontrivial solutions?

Birfurcation analysis of % 7→ Fκ(%). p 2 Example: Kuramoto model: W (x) = − L cos(2πx/L)

0.03 0.25

0.025 0.2 0.02

0.015 0.15

0.01 0.1 0.005

0 0.05 -0.005

-0.01 0 0 2 4 6 8 10 -5 0 5

⇒ 1-cluster solution and uniform state %∞.

June 25, 2018 Phase transitions for the McKean-Vlasov eqn 42 Local bifurcation result Theorem (Local bifurcations) Let W be smooth and even and let (1/L, κ) represent the trivial branch of solutions. Then every k∗ ∈ Z, k > 0 such that 1. cardk ∈ Z, k > 0 : Wˆ (k) = Wˆ (k∗) = 1 , 2. Wˆ (k) < 0, corresponds to a bifurcation point of the stationary McKean–Vlasov equation through the formula √ L κ∗ = − , (22) βWˆ (k∗)

with (1/L, κ∗) the bifurcation point.

June 25, 2018 Phase transitions for the McKean-Vlasov eqn 43 June 25, 2018 Phase transitions for the McKean-Vlasov eqn 44 Examples of birfucations results

Kuramoto-type of models: W (x) = −wk(x) in d = 1 with W (k) = −1, √ f 2L satisfying both conditions. Thus we have that κ∗ = β . 2 L5/2 cos(πk) For W (x) = x holds W (k) = √ satisfying both conditions for odd 2 f 2 2πk2 4k2 values of k. Hence, every odd k is bifurcation point κ∗ = βL2 .

0.5 ∞ s P 1 W (x) = − k2s+2 wk(x) k=1 For s ≥ 1 : W s(x) ∈ Hs(Td ) L 0 ∀k > 0 : conditions (1) and (2) ok Inﬁnitely many bifurcation points

-0.5 -5 0 5

June 25, 2018 Phase transitions for the McKean-Vlasov eqn 45 Transition points: Qualitative change of minimizers

Deﬁnition (Transition point [Chayes & Panferov ’10])

A parameter value κc > 0 is said to be a transition point of Fκ if it satisﬁes the following conditions,

1. For 0 < κ < κc: %∞ is the unique minimiser of Fκ(%)

2. For κ = κc: %∞ is a minimiser of Fκ(%).

3. For κ > κc: ∃%κ 6= %∞, such that %κ is a minimiser of Fκ(%).

June 25, 2018 Phase transitions for the McKean-Vlasov eqn 46 Deﬁnition (Continuous and discontinuous transition point)

A transition point κc > 0 is a continuous transition point of Fκ if

1. For κ = κc: %∞ is the unique minimiser of Fκ(%).

2. For any family of minimizers {%κ 6= %∞}κ>κc it holds

lim supk%κ − %∞k1 = 0. κ↓κc

A transition point κc > 0 which is not continuous is discontinuous.

June 25, 2018 Phase transitions for the McKean-Vlasov eqn 47 Thermodynamics

Figure: (M1) Figure: (M2) Figure: (M1)+(M2)

Figure: Ways in which a discontinuous transition can occur. The horizontal line represents the uniform solution

June 25, 2018 Phase transitions for the McKean-Vlasov eqn 48 Basic properties of transition points

Summary of critical points:

κc transition point.

κ∗ bifurcation point. d L 2 κ] point of linear stability, i.e., κ] = − with β mink We (k)/Θ(k) k] = arg min Wf(k).

If there is exactly one k], then κ] = κ∗ is a bifurcation point.

June 25, 2018 Phase transitions for the McKean-Vlasov eqn 49 Results from [Gates & Penrose 1970] and [Chayes & Panferov ’10]

Fκ has a transition point κc iﬀ W 6∈ Hs.

min Fκ is non-increasing as a function of κ. 0 If for some κ : %∞ is no longer the unique minimiser, 0 then ∀κ > κ : %∞ is no longer a minimizer.

If κc is continuous, then κc = κ].

(M2) =⇒ (M1)+(M2). c For L suﬃciently large, d ≥ 2, and W (x) ∈ Hs, radially symmetric, the transition point is discontinuous.

June 25, 2018 Phase transitions for the McKean-Vlasov eqn 50 Conclusion:

To prove a discontinuous transition: Show %∞ at κ] is no longer global minimizer.

To prove a continuous transition: If κ∗ = κ], suﬃcient to show that %∞ at κ] is the only global minimizer and investigate a resonance condition.

June 25, 2018 Phase transitions for the McKean-Vlasov eqn 51 Conditions for continuous and discontinous phase transition Theorem (Discontinuous and continuous phase transitions) Let W be smooth and even and

assume the free energy Fκ,β exhibits a transition point, κc < ∞. Then we have the following two scenarios: 1. If there exist strictly positive ka, kb, kc ∈ Z with a b c a b c Wˆ (k ) = Wˆ (k ) = Wˆ (k ) = mink Wˆ (k) < 0 such that k = k + k or a b k = 2k , then κc is a discontinuous transition point. ] ˆ ˆ ] 2. Let k = arg mink W (k) be well-deﬁned with W (k ) < 0. Let Wα denote the potential obtained by multiplying all the negative Wˆ (k) except Wˆ (k]) by

some α ∈ (0, 1]. Then if α is made small enough, the transition point κc is continuous.

June 25, 2018 Phase transitions for the McKean-Vlasov eqn 52 The generalized Kuramoto model Proposition

The generalised Kuramoto model W (x) = −wk(x), for some k ∈ N, k 6= 0

exhibits a continuous transition point at κc = κ]. Additionally, for κ > κc, the equation F (%, κ) = 0 has only two solutions in L2(U) (up to translations). The

nontrivial one, %κ minimises Fκ for κ > κc and converges in the narrow topology as κ → ∞ to a normalised linear sum of equally weighted Dirac measures centred at the minima of W (x).

June 25, 2018 Phase transitions for the McKean-Vlasov eqn 53 The noisy Hegselmann–Krause model for opinion dynamics

The noisy Hegselmann–Krause system models the opinions of N interacting agents such that each agent is only inﬂuenced by the opinions of its immediate neighbours. The interaction potential is

2 1 R Whk(x) = − |x| − 2 2 −

for some R > 0. The ratio R/L measures the range of inﬂuence of an individual agent with R/L = 1 representing full inﬂuence.

The Fourier transform of Whk(x) is

2 2 2 2 πkR πkR −π k R + 2L sin L − 2πkLR cos L Whk(k) = √ , k ∈ N, k 6= 0 . f 3 3p 1 4 2π k L (23)

the model has inﬁnitely many bifurcation points for R/L = 1. June 25, 2018 Phase transitions for the McKean-Vlasov eqn 54 We deﬁne a rescaled version of the potential 2 R 1 R Whk(x) = − 3 |x| − , 2R 2 −

which does not lose mass as R → 0.

Proposition For R small enough, the rescaled noisy Hegselmann–Krause model possesses a discontinuous transition point.

June 25, 2018 Phase transitions for the McKean-Vlasov eqn 55 The Onsager model for liquid crystals

The Onsager/Maiers–Saupe model is described by the interaction potential

2π ` W (x) = sin x ∈ L2(U) ∩ C∞(U¯) ` L s N with ` ∈ , ` ≥ 1, so that the Onsager and Maiers–Saupe potential correspond to the cases ` = 1 and ` = 2, respectively.

The Fourier transform of W`(x) is

√ 1 −` πk π2 2 cos Γ(` + 1) W (k) = 2 . (24) f` 1 1 Γ 2 (−k + ` + 2) Γ 2 (k + ` + 2)

Any nontrivial solutions to the stationary dynamics correspond to the so-called nematic phases of the liquid crystals.

June 25, 2018 Phase transitions for the McKean-Vlasov eqn 56 Proposition

1. The trivial branch of the Onsager model, W1(x), has inﬁnitely many bifurcation points.

2. The trivial branch of the Maiers–Saupe model, W2(x), has exactly one bifurcation point. ` 3. The trivial branch of the model W`(x) for ` even has at least 4 bifurcation ` ` 1 ` points if 2 is even and 4 + 2 bifurcation points if 2 is odd.

4. The trivial branch of the model W`(x) for ` odd has inﬁnitely many bifurcation `−1 `+1 `−1 points if 2 is even and at least 4 bifurcation points if 2 is odd.

June 25, 2018 Phase transitions for the McKean-Vlasov eqn 57 The Keller–Segel model for bacterial chemotaxis

The Keller–Segel model is used to describe the motion of a group of bacteria under the eﬀect of the concentration gradient of a chemical stimulus, whose distribution is determined by the density of the bacteria.

For this system, %(x, t) represents the particle density of the bacteria and c(x, t) represents the availability of the chemical resource.

The dynamics of the system are then described by the following system of coupled PDEs:

−1 ∂t% = ∇· β ∇% + κ%∇c (x, t) ∈ U × (0, ∞) , −(−∆)sc = % (x, t) ∈ U × [0, ∞) , (25) %(x, 0) = %0 x ∈ U × {0} , %(·, t) ∈ C2(U) t ∈ [0, ∞),

1 for s ∈ ( 2 , 1].

June 25, 2018 Phase transitions for the McKean-Vlasov eqn 58 The stationary Keller–Segel equation is given by,

∇·β−1∇% + κ%∇Φs ?% = 0 x ∈ U, (26)

2 ¯ s s with % ∈ C (U) and where Φ is the fundamental solution of −(−∆) .

Theorem 1 Consider the stationary Keller–Segel equation (26). For d ≤ 2 and s ∈ ( 2 , 1], it has smooth solutions and its trivial branch (%∞, κ) has inﬁnitely many bifurcation points.

June 25, 2018 Phase transitions for the McKean-Vlasov eqn 59 19

17 16

9 8 7

5 4 3 2 1

1 2 3 4 5 7 8 9 11 13 16 17 19

Figure: * Figure: *

(a) (b)

Figure: (a). Contour plot of the Keller–Segel interaction potential Φs for d = 2 and s = 0.51. The orange lines indicate the positions at which the potential is singular (b). The associated wave numbers which correspond to bifurcation points of the stationary system.

June 25, 2018 Phase transitions for the McKean-Vlasov eqn 60 Conclusions

Studied the combined homogenization mean-ﬁeld limits; the limits do not necessarily commute.

Complete analysis of local and global bifurcations for the McKean-Vlasov equation on the torus.

Study the eﬀect of memory, colored noise/non-gradient structure, hypoellipticity etc.

Study dynamical metastability phenomena.

Predicting phase transitions, linear response theory, optimal control.

June 25, 2018 Phase transitions for the McKean-Vlasov eqn 61