# Mean Field Limits for Interacting Diffusions in a Two-Scale Potential

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## Abstract

In this paper, we study the combined mean field and homogenization limits for a system of weakly interacting diffusions moving in a two-scale, locally periodic confining potential, of the form considered in Duncan et al. (Brownian motion in an N-scale periodic potential, arXiv:1605.05854, 2016b). We show that, although the mean field and homogenization limits commute for finite times, they do not, in general, commute in the long time limit. In particular, the bifurcation diagrams for the stationary states can be different depending on the order with which we take the two limits. Furthermore, we construct the bifurcation diagram for the stationary McKean–Vlasov equation in a two-scale potential, before passing to the homogenization limit, and we analyze the effect of the multiple local minima in the confining potential on the number and the stability of stationary solutions.

## Keywords

McKean–Vlasov equation Interacting particles Multiscale diffusions Bifurcation diagram Phase transitions Desai–Zwanzig model Curie–Weiss model## Mathematics Subject Classification

35Q70 35Q83 35Q84 82B26 82B80## 1 Introduction

Systems of interacting particles, possibly subject to thermal noise, arise in several applications, ranging from standard ones such as plasma physics and galactic dynamics (Binney and Tremaine 2008) to dynamical density functional theory (Goddard et al. 2012a, b), mathematical biology (Farkhooi and Stannat 2017; Lućon and Stannat 2016) and even in mathematical models in the social sciences (Garnier et al. 2017; Motsch and Tadmor 2014). As examples of models of interacting “agents” in a noisy environment that appear in the social sciences—which has been the main motivation for this work—we mention the modeling of cooperative behavior (Dawson 1983), risk management (Garnier et al. 2013) and opinion formation (Garnier et al. 2017). Another recent application that has motivated this work is that of global optimization (Pinnau et al. 2017).

*p*(

*x*,

*t*).

^{1}Passing, formally, to the limit as \(N\rightarrow \infty \) in the stochastic differential equation (1.1), we obtain the McKean SDE

*V*being a confining potential is always ergodic, and in fact reversible, with respect to the Gibbs measure (Pavliotis 2014, Ch. 4),

On the other hand, the McKean dynamics (1.3) and the corresponding McKean–Vlasov–Fokker–Planck equation (1.4) can have more than one invariant measures, for nonconvex confining potentials and at sufficiently low temperatures (Dawson 1983; Tamura 1984). This is not surprising, since the McKean–Vlasov equation is a nonlinear, nonlocal PDE and the standard uniqueness of solutions for the linear (stationary) Fokker–Planck equation does not apply (Bogachev et al. 2015).

**self-consistency**equation and it will be the main object of study of this paper. Once a solution to (1.9) has been obtained, substitution back into (1.8) yields a formula for the invariant density \(p_{\infty }(x ; \theta , \beta , m)\).

*m*at \(m=0\) (see Shiino 1987; Frank 2005, Sec 5.1.3 for more details):

*m*and \(R(m;\theta ,\beta )\) cross, i.e., the number of stationary measures, depends on the slope of \(R(m;\theta ,\beta )\) at the origin. This is given precisely by Eq. (1.10).

^{2}In these works, the homogenized SDE for a Brownian particle moving in a two-scale potential in \({\mathbb {R}}^d\) valid in the limit of infinite scale separation \(\epsilon \rightarrow 0\) was obtained and the effect of the multiscale structure on noise-induced transitions was investigated. It was shown, in particular, that the homogenized SDE is characterized by multiplicative noise. For a single Brownian particle in \({\mathbb {R}}^d\) moving in a two-scale potential (1.11) (or, equivalently, for a system of

*d*noninteracting Brownian particles in a two-scale potential), the homogenized equation reads

- 1.
What is the effect of the presence of (infinitely) many local minima in the locally periodic confining potential on the bifurcation diagram? In other words, how do the bifurcation diagrams for \(\epsilon \ll 1\) but finite and \(\epsilon \rightarrow 0\) differ?

- 2.
Do the homogenization and mean field limits commute, in particular when also passing to the long time limit \(T \rightarrow +\infty \)? In other words: are the bifurcations diagrams corresponding to the \(N\rightarrow \infty , \, T\rightarrow \infty , \, \epsilon \rightarrow 0\) and \(\epsilon \rightarrow 0, \, N\rightarrow \infty , \, T\rightarrow \infty \) limits the same?

We will study these problems using a combination of formal multiscale calculations, (some) rigorous analysis and extensive numerical simulations. There are many technical issues that we do not address, such as the rigorous homogenization study of the McKean–Vlasov equation and the rigorous study of bifurcations in the presence of infinitely many local minima. We will address these in future work.

The rest of the paper is organized as follows. In Sect. 2, we study the mean field limit for a system of homogenized interacting diffusions, i.e., the first \(\epsilon \rightarrow 0\), then \(N\rightarrow \infty \) limit. In Sect. 3, we study the homogenization problem for the McKean–Vlasov equation in a two-scale potential. In Sect. 4, we present extensive numerical simulations. Section 5 is reserved for conclusions.

## 2 Mean Field Limit of the Homogenized Interacting Diffusions: First \(\epsilon \rightarrow 0\), then \(N\rightarrow \infty \)

In this section, we consider the one-dimensional version of the system of SDEs (1.14). We first take the homogenization limit (\(\epsilon \rightarrow 0\)) and then the mean field limit (\(N\rightarrow \infty \)). The homogenization theorem for a system of finite-dimensional interacting diffusions moving in a two-scale confining potential is presented in Duncan et al. (2016b). The mean field limit of the homogenized SDE system can be obtained by using the results of Gärtner (1988), Oelschläger (1984).

### 2.1 Homogenization for Finite System of Interacting Diffusions in a Two-Scale Potential

*F*is a smooth even function with \(F(0) = 0\) and \(F'(0)= 0\) and \(V^\epsilon \) is a smooth locally periodic potential of the form (2.7). We introduce the notation \(\mathbf{x}_t = (X_t^1, \dots , X_t^N)\), so that we have

*F*does not depend on the fast scale, the results of Duncan et al. (2016b) apply directly to (2.2) and we deduce that the sequence \(\left\{ \mathbf{x}_t^\epsilon \right\} \) converges, as \(\epsilon \rightarrow 0\), to the solution of the homogenized equation

*x*and \(\Psi \) is given by Equations (2.8)-(2.9).

^{3}Furthermore, the diffusion coefficient of the

*i*th particle depends only on the position of the particle itself, and not of the other particles. The dynamics (2.17) is reversible with respect to the Gibbs measure

### 2.2 Mean Field Limit for the Homogenized SDE

^{4}It is straightforward to check that the homogenized equation satisfies the conditions in the aforementioned papers.

^{5}Taking the mean field limit of (2.17), we obtain the following nonlinear Fokker–Planck equation:

*x*,

*x*, then \(\psi (x)\) becomes a constant. This, in turn, means that the stationary solutions to the homogenized McKean–Vlasov equation are the same to the ones for the system without fluctuations (\(V_1(x,y)=0\))—see Corollary 3.2 in Sect. 3. For example, when the large-scale part of the potential \(V_0(x)\) is convex, there are no phase transitions for the homogenized dynamics. We will show in Sections 3 and 4 that this is not the case if we take the limits in different order.

## 3 Multiscale Analysis for the McKean–Vlasov Equation in a Two-scale Potential

In this section, we consider the homogenization problem for the McKean–Vlasov equation in a locally periodic potential for the case of a quadratic (Curie–Weiss) interaction. In particular, we first pass to the mean field limit (i.e., send \(N\rightarrow \infty \)) in Eq. (1.14) with \(F(x) = \frac{x^2}{2}\) and study the effects of finite (but small) \(\epsilon \) on the bifurcation diagram, before sending \(\epsilon \rightarrow 0\).

### 3.1 Mean Field Limit for Interacting Diffusions in a Two-Scale Potential: \(N\rightarrow \infty \), \(\epsilon >0\) Finite

*L*-periodic in its second argument, \(\theta >0\) is the interaction strength, \(\beta \) the inverse temperature and \(\{ B^i_t, \; i=1, \dots , N \}\) are standard independent one-dimensional Brownian motions.

### Proposition 3.1

Consider equations (2.24), (2.26) and (3.9), (3.10). Assume the potential \(V^\epsilon \) is smooth and has fluctuations which are truncated in an interval \([-a,a]\). Then the limits \(\epsilon \rightarrow 0, \, N\rightarrow \infty , \, T\rightarrow \infty \) (Eqs. (2.24) and (2.26)) and \(N\rightarrow \infty , \, T\rightarrow \infty , \, \epsilon \rightarrow 0\) ((3.9) and (3.10)) do not commute. In particular, the \(\epsilon \rightarrow 0\) limits of the self-consistency equation (3.4) and of the equation for the critical temperature (3.5) are **different** from (2.24) and (2.26).

### Proof

*m*. We use the convergence of \(m^{\epsilon }\) to

*m*and (3.6) to deduce:

**different**, from (2.24) and (2.26). \(\square \)

The two limits \(\epsilon \rightarrow 0, \, N\rightarrow \infty , \, T\rightarrow \infty \) and \(N\rightarrow \infty , \, T\rightarrow \infty , \, \epsilon \rightarrow 0\) commute in the case where the fluctuations in the potential are independent of the macroscale *x*, \(V_1 =V_1(y)\) in (2.7). An immediate corollary of the above proposition is the following.

### Corollary 3.2

Separable fluctuations do not affect the bifurcation diagram in the mean field limit.

### Proof

When the fluctuations are separable (i.e., \(V_1(x,y)\) does not depend on *x*), \(\psi (x,\beta )\) in (2.24), (2.26) becomes a constant that we can ignore since it also appears in the partition function and they cancel out. Similarly, the terms of the form \(\int _{\mathbb {R}}e^{-\beta V_1(x,y)} \ \hbox {d}y\) in Eqs. (3.9) and (3.10) become constants independent of *x* and cancel with the corresponding terms in the partition function (3.7). \(\square \)

Let us consider now the case of nonseparable fluctuations. As we have already discussed, see Fig. 2b and also the inside panels of Figs. 7a and 9a, the resulting two-scale potential does not only contain many additional local minima, but it is also flattened around \(x=0\). In Fig. 4, we present curves \(R(m^\epsilon ;\theta ,\beta )\) for nonseparable fluctuations, compared with the line \(R(m;\theta , \beta ) = m\) (or \(y=x\)). We observe that in the limit \(\epsilon \rightarrow 0\) the curves \(R(m^\epsilon ;\theta ,\beta )\) (various dashed lines) **do not** converge to \(R(m;\theta ,\beta )\) corresponding to the homogenized problem (full black line), in accordance with Prop. 3.1. Notice also the flatness of \(R(m^\epsilon ;\theta ,\beta )\) around \(m=0\) for smaller values of \(\epsilon \), which follows from the flatness of the corresponding potentials \(V^\epsilon \) around \(x=0\).

### 3.2 Multiscale Analysis for the McKean–Vlasov Equation in a Two-Scale Confining Potential

In this section, we study the problem of periodic homogenization for the McKean–Vlasov equation in a locally periodic confining potential, for the Curie–Weiss quadratic interaction and in one dimension. We only present formal arguments. The rigorous analysis of this problem will be presented elsewhere.

*x*. The PDE (3.11) is coupled to the self-consistency equation

*L*-periodic in their second argument. Substituting (3.13) into (3.11) and (3.12) and using the standard tools from the theory of periodic homogenization, e.g., Fredholm’s alternative, we obtain the homogenized equation (2.19), satisfied by the marginal of the first term in the two-scale expansion \(p(x,t) = \int _0^L p(x,y,t) \, \hbox {d}y\) and with the partial free energy \(\psi (x)\) given by (2.20) and with

*m*(

*t*) can be justified using the a priori estimates on moments of the solution to the McKean–Vlasov equation that were derived in Arnold et al. (1996), in particular (Arnold et al. 1996, Eqs. (3.1), (3.2)).

## 4 Numerical Simulations

^{6}We note that this is necessary for the proof of the homogenization theorem in Duncan et al. (2016b) and that, furthermore, it ensures that the a priori estimates on the moments from Arnold et al. (1996) hold.

^{7}

Potentials used for the numerical simulations

Confining potential \(V_0(x)\) | Fluctuating potential \(V_1(x)\) | Case |
---|---|---|

\(V_0^c(x) = \frac{x^2}{2}\) | \(V_1^+(x) = \delta \cos \left( \frac{x}{\epsilon }\right) \) | 1 |

\(V_1^\times (x) = \delta \chi _{[-a,a]}(x)\frac{x^2}{2}\cos \left( \frac{x}{\epsilon }\right) \) | 2 | |

\(V_0^b(x) = \frac{x^4}{4} - \frac{x^2}{2}\) | \(V_1^+(x) = \delta \cos \left( \frac{x}{\epsilon }\right) \) | 3 |

\(V_1^\times (x) = \delta \chi _{[-a,a]}(x)\frac{x^2}{2}\cos \left( \frac{x}{\epsilon }\right) \) | 4 |

Throughout this section, we consider fluctuations which have period \(L=2\pi \). In all cases, we will consider the Curie–Weiss interaction potential \(F(x) = \frac{x^2}{2}\), and throughout Sections 4.1 and 4.2, we will fix the interaction strength to be \(\theta = 5\). We choose this value because larger values of \(\theta \) allow for bifurcations to occur at higher temperatures, i.e., lower \(\beta \), which is easier to handle numerically. In fact, the relevant bifurcation parameter for our problem is given by the combination \(\beta \theta \); see Eq. (1.10). Fixing \(\theta \) allows us to construct the bifurcation diagram by varying only the temperature. It is also clear from Eq. (1.10) that this equation has no solutions for negative values of \(\theta \), i.e., that no (pitchfork) bifurcations can occur for \(\theta < 0\).

*x*and is given by

*m*as a function of the inverse temperature \(\beta \) for a fixed value of the interaction strength \(\theta \). We do this using the Moore–Penrose quasi-arclength continuation algorithm.

^{8}The stability of each branch was determined in two different ways: First, we checked whether it corresponded to a local minimum or maximum of the confining potential. Second, we solved the time-dependent McKean–Vlasov equation—see details in Sect. 4.5—using a perturbation of the steady state belonging to each branch (for a particular value of \(\beta \) and \(\theta \)) as initial condition. Finally, we have confirmed the stability of each branch by computing the free energy (1.5) of a steady state from that branch at a particular value of \(\beta \), chosen so that all the branches plotted were present. Stable branches, plotted in blue in all the figures presented in this section, correspond to local minimizers of the free energy functional; unstable branches, plotted in red, correspond to local maxima of the free energy.

### 4.1 Mean Field Limit of the Homogenized System of SDEs: The \(\epsilon \rightarrow 0, \, N\rightarrow \infty \) Limit

As discussed before (see discussion of Corollary 3.2), when the fluctuations are separable the partial free energy \(\psi (x)\) defined in Eq. (2.20) drops out from the homogenized stationary Fokker–Planck equation. This implies, in particular, that the invariant measure(s) of the homogenized dynamics is(are) independent of the fluctuating part of the potential. In particular, there are still no phase transitions when the large-scale part of the potential is convex and still only one pitchfork bifurcation for the bistable potential case—see Fig. 5—where two new, stable, branches emerge from the zero mean solution. We note that in this case the homogenized confining potential in the homogenized equation depends on the inverse temperature \(\beta \); see the inside panels in Fig. 5. In particular, the values of the local minima of the effective potential are shifted, although their location remains the same, and there are no changes in the topology of the bifurcation diagrams.

For nonseparable fluctuations, the mean field and homogenization limits do not commute (see Prop. 3.1). In fact, the homogenization procedure can convexify the effective potential, and we still see no bifurcations when the large-scale part of the potential is convex, while for the bistable potential there is still only one phase transition. The effect of fluctuations on the bifurcation diagram is visible by a shift of the critical temperature at which the phase transition occurs.

### 4.2 Mean Field Limit of the Multiscale System of SDEs: Effects of Finite \(\epsilon \)

In this section, we present numerical results on the bifurcation diagram when we first pass to the mean field limit, while keeping \(\epsilon \) small but finite. We are particularly interested in the finite \(\epsilon \) effects on the bifurcation diagrams for the two-scale potentials presented in Table 1.

#### 4.2.1 Convex Confining Potential with Separable and Nonseparable Fluctuations

Figure | 6 | 7 | 8 | 9 |
---|---|---|---|---|

\(\beta \) | 45 | 29 | 20 | 8 |

Free Energy | \(\ 0.3080\) | 0.1441 | \(-\,0.5827\) | \(-\,1.7409\) |

0.3066 | 0.3684 | \(-\,0.5674\) | \(-\,0.9933\) | |

\(-\,0.4600\) | 0.1433 | \(-\,1.0918\) | \(-\,0.8241\) | |

\(-\,0.3908\) | 0.3184 | \(-\,0.7727\) | 0.0856 | |

\(-\,0.8593\) | 0.0976 | \(-\,0.8868\) | ||

\(-\,0.6514\) | 0.2425 | \(-\,0.6903\) | ||

0.0625 | ||||

0.0630 | ||||

0.0586 |

We observe in Fig. 7b that no pitchfork bifurcations appear; all new branches that appear do not emerge continuously from the mean-zero solution. This is due to the flatness observed in the potential around \(m=0\) (see Fig. 7a). Furthermore, the mean-zero solution remains the global minimizer of the free energy for all values of \(\beta \). This is tabulated in Table 7c, where they are listed in the same way as in Table 2, i.e., in decreasing order of nonnegative *m*. The free energies of the different branches are presented in Fig. 7d. These new branches correspond to metastable states.

We have checked the stability of each branch by computing the free energy (1.5) of a steady state from that branch at a particular value of \(\beta \), chosen so that all the branches plotted were present. We summarize the results in Table 2. Since we only consider symmetric potentials, it is sufficient to calculate the free energy for the branches with, say, nonnegative values of *m*. In each column of Table 2, the values of the free energy are presented from the branch with largest value of *m* to the lowest; the last value presented in each column corresponds to the branch with \(m=0\). We summarize the results in Table 2.

We observe that the branch corresponding to a pitchfork bifurcation (i.e., second-order phase transition), when present, has the lowest value of the free energy, i.e., it is the globally stable one. Furthermore, when a pitchfork bifurcation does not occur—see Fig. 7—the branch corresponding to \(m=0\) is the one with the lowest value of the free energy. Finally, we observe that the stability of the branches in Fig. 9b does not alternate in the same manner as in the previous figures. This is due to the flatteness of the potential around \(x=0\) for nonseparable oscillations.

The results on the stability of the different branches that are reported in this section are preliminary. A more thorough study of the local (linear) and global stability of the stationary states of the McKean–Vlasov dynamics in multiwell potentials will be presented elsewhere. We mention in passing the early rigorous work on the global stability of the steady states for the McKean–Vlasov equation in Tamura (1987) as well as the careful study of the connection between the loss of linear stability of the uniform state and phase transitions for the McKean–Vlasov equation on the torus (without a confining potential) and with finite-range interactions in Chayes and Panferov (2010).

#### 4.2.2 Bistable Confining Potential with Separable and Nonseparable Fluctuations

Here we consider cases 3 and 4 in Table 1, the bistable potential \(V_0^b(x)\). In this case, the large-scale potential exhibits a second-order phase transition even in the absence of small-scale fluctuations (see the pitchfork bifurcation in Fig. 1b) due to the existence of two local minima for \(V_0^b(x)\). We are interested in analyzing the topological changes that rapid oscillations in the potential induce to the bifurcation diagram.

### 4.3 Numerical Study of the Critical Temperature as a Function of \(\epsilon \)

We present in Fig. 10 plots of the critical temperature, \(\beta _C\) as a function of \(\epsilon \) for a fixed \(\theta = 5\). The results are presented for cases 1 (Fig. 10a), 3 (Fig. 10b) and 4 (Fig. 10c) from Table 1. We do not present the remaining case because, as shown in Fig. 7b, there is no pitchfork bifurcation from the mean-zero solution for case 2. The dependence of the critical temperature on \(\epsilon \) is different for separable and nonseparable potentials. It appears that the critical temperature can change considerable by varying \(\epsilon \), which implies that a different number of branches might be present in the bifurcation diagram at a fixed temperature, for different values of \(\epsilon \). This issue will be studied in detail in future work.

### 4.4 Simulations of the Interacting Particles System

For the full dynamics (2.1), we used \(\delta = 1\) and \(\epsilon = 0.1\). We solved the SDEs using the Euler–Maruyama scheme. For the homogenized dynamics (2.17), since the noise is multiplicative (for nonseparable potentials), we used the Milstein scheme. In both cases, the time step used was \(\hbox {d}t = 0.01\), which is of \(O(\epsilon ^2)\). Finally, in both cases we initialized the *N* particles as being normally distributed, with mean zero and variance 4, which was large enough so that all the local minima were contained within two standard deviations of the Gaussian distribution.

*t*. We observe that in both cases, the average converges to 0 as expected, but that the convergence for the homogenized SDE (2.17) is slower. The position of the

*N*particles follows approximately the same qualitative behavior (with the particles clustering close to 0), but as we can see from the corresponding histogram there exist additional wells (nonconvexity) for the finite \(\epsilon \) case.

We performed similar experiments for case 4 in Table 1 (i.e., \(V^\epsilon (x) = \frac{x^4}{4}-\frac{x^2}{2}\left( 1-\delta \chi _{[-a,a]}(x)\cos \left( \frac{x}{\epsilon }\right) \right) \)). Here we used \(N=500\) particles, and smaller values of \(\theta \) and \(\beta \). The parameters used were \(\theta = 0.5\), \(\beta \approx 5.6\), \(\delta = 1\) and \(\epsilon = 0.1\), and the results are plotted in Figs. 14, 15 and 16.

### 4.5 Time-Dependent McKean–Vlasov Evolution

We performed time-dependent simulations of the evolution of the nonlinear McKean–Vlasov equation both for the full and for the homogenized dynamics. We present below the results corresponding to the cases presented for the Monte Carlo simulations.

To solve the McKean–Vlasov evolution PDE, we used the positivity preserving, entropy decreasing finite volume scheme from Carrillo et al. (2015). We point out that this scheme solves the equations using no-flux boundary conditions. We use these boundary conditions and a sufficiently large domain. We used the same initial conditions for the time-dependent Fokker–Planck simulations as the ones used for the Monte Carlo simulations, i.e., the initial condition was the PDF for a normal distribution with mean zero and variance 4. However, for the bistable large-scale potential with nonseparable fluctuations in the finite but positive \(\epsilon \) case—see left panel in Fig. 18—we needed to use a different initial condition: Here we used a normal distribution with mean \(-\,0.1\) and variance 4. This is likely because the value of \(\beta \) we chose here was close to the bifurcation point and the mean-zero solution was still being picked up on the time evolution.

As expected, the results obtained by solving the time-dependent McKean–Vlasov equation are in agreement with the results obtained from the Monte Carlo simulations and from solving the stationary McKean–Vlasov equation—i.e., the self-consistency equation. We note that, similarly to what we observed in the solution of the system of interacting particles, the solution to the McKean–Vlasov equation converges to its steady state faster for the full dynamics than for the homogenized equation. This observation can be quantified by comparing the convergence rates in the weighted \(L^2\) or relative entropy exponential estimates, in particular by comparing the constants in the Poincaré and logarithmic Sobolev inequalities for the full and for the homogenized dynamics. A preliminary study of this—for the Fokker–Planck operator of the finite-dimensional dynamics—was presented in Duncan et al. (2016b).

As expected, the solutions converge to those computed by solving the stationary McKean–Vlasov equation and are qualitatively similar to those obtained from the particle system simulations; see Fig. 15. In this case, the solution of the time-dependent McKean–Vlasov PDE converges to a steady state slower for the full dynamics, in comparison with the homogenized dynamics. We believe that this is related to the phenomenon of critical slowing down (Shiino 1987) when the dynamics is close to a bifurcation, since the inverse temperature \(\beta ^{-1}\) that we use for the simulations is close to the critical temperature \(\beta _C^{-1}\) for the full dynamics.

## 5 Conclusions and Further Work

The combined mean field and homogenization limit for a system of interacting diffusions in a two-scale confining potential was studied in this paper. In particular, the homogenized McKean–Vlasov equation was obtained and studied and the bifurcation diagram for the stationary states was considered. It was shown, by means of analysis and extensive numerical simulations, that the homogenization and mean field limits, at the level of the bifurcation diagram (i.e., when combined with the long time limit), do not commute for nonseparable two-scale potentials. Furthermore, it was shown that the bifurcation diagrams can be completely different for small but finite \(\epsilon \) and for the homogenized McKean–Vlasov equation.

It should be emphasized, as is clearly explained in Chayes and Panferov (2010), see in particular the remarks at the end of Sec. 2 of this paper, that the connection between bifurcations and phase transitions for the McKean–Vlasov dynamics is not entirely straightforward. In particular, in order for a bifurcation point to correspond to a genuine phase transition, it is not sufficient to have the emergence of a new branch of solutions, but these emergent solutions should have a lower free energy. More precisely, it was shown in Chayes and Panferov (2010) for the McKean–Vlasov dynamics on the torus and with a finite-range interaction potential that the loss of linear stability of the uniform state—which corresponds to the mean-zero Gibbs state in our setting—does not imply a second-order phase transition. Furthermore, the critical temperature (or, equivalently, critical interaction strength) at which first-order phase transitions occur is lower than the temperature at which the pitchfork bifurcation happens. For the problem that we studied, supercritical pitchfork bifurcations occur which correspond to second-order (continuous) phase transitions. On the other hand, when only saddle node bifurcations are present, e.g., in Fig. 7b, then the mean-zero solution is still the global minimizer of the free energy; see Fig. 7d. In particular, no first-order phase transitions seem to appear in the McKean–Vlasov model that we studied in this work.

There are many open questions that are not addressed in this work. First, the rigorous multiscale analysis for the McKean–Vlasov equation in locally periodic potentials needs to be carried out. Perhaps more importantly, the rigorous construction of the bifurcation diagram in the presence of infinitely many local minima in the confining potential, thus extending the results presented in, e.g., Dawson (1983), Tamura (1984), Tugaut (2014), appears to be completely open. Furthermore, the study of the stability of stationary solutions to the McKean–Vlasov equation in the presence of a multiscale structure, as well as the analysis of the problem of convergence to equilibrium in this setting is an intriguing question. Finally, the extension of the work presented in this paper to higher dimensions presents additional challenges. We mention, for example, that the corresponding nonlinear diffusion process does not have to be reversible (Lelievre et al. 2013; Duncan et al. 2016). We believe that the results reported in this work open up a new exciting avenue of research in the study of mean field limits for interacting diffusions in the presence of many local minima, with potentially interesting applications to the study of McKean–Vlasov-based mathematical models in the social sciences.

## Footnotes

- 1.
- 2.
In fact, in these papers a potential with

*N*microscales and one macroscopic scale of the form \(V^{\epsilon }(\mathbf{x}) = V\left( \mathbf{x}, \frac{\mathbf{x}}{\epsilon }, \frac{\mathbf{x}}{\epsilon ^2}, \dots \frac{\mathbf{x}}{\epsilon ^N} \right) \), where*V*is periodic in all the microvariables is studied. For the purposes of this work, it is sufficient to consider a potential with two characteristic, widely separated, length scales. - 3.In fact, the noise in this SDE can be interpreted in the Klimontovich sense:where \(\circ ^{K}\) denotes the Klimontovich stochastic integral; see Duncan et al. (2016b). In particular, the correction to the Itô integral is \(\beta ^{-1}{\mathcal {M}}(X_t^i)\) instead of \(\frac{1}{2}{\mathcal {M}}(X_t^i)\) that corresponds to the Stratonovich stochastic integral. See Pavliotis (2014, Sec. 3.2) for details.$$\begin{aligned} \hbox {d}X_t^i = - {\mathcal {M}}(X_t^i) \partial _{x^i}\Psi '(X_t^1, \dots X_t^N) \, \hbox {d}t + \sqrt{2\beta ^{-1}{\mathcal {M}}(X_t^i)} \circ ^{K} \hbox {d}B_t^i, \end{aligned}$$
- 4.
In fact, these papers consider the more general case, where the diffusion coefficient, \(\sigma \), also depends on the empirical measure, \(\sigma \left( X_t^i, \frac{1}{N}\sum _{j=1}^N X_t^j\right) \).

- 5.
These are variants of boundedness and Lipschitz continuity assumptions for the drift and diffusion coefficients. The estimates on the homogenized coefficients that are obtained in Abdulle et al. (2017), are sufficient in order to invoke the results of Gärtner (1988), Oelschläger (1984). For the purposes of this paper it is sufficient to pass formally to the mean field limit. The rigorous analysis will be presented elsewhere.

- 6.
In Table 1, we denote by \(\chi _A\) the characteristic function of the set

*A*. - 7.
The moment bounds in Arnold et al. (1996) were obtained for confining potentials with no oscillatory terms. However, it can be checked that they are also valid for the class of fluctuating potentials that we consider in this work, and that they provide us with bounds on the moments that uniform in \(\epsilon \).

- 8.Rigorous mathematical construction of the arclength continuation methodology can be found, e.g., in Krauskopf (2007) and Allgower and Georg (1990). Some useful practical aspects of implementing arclength continuation are also given in Dhooge et al. (2006). We use Matlab’s toolboxes to compute the integrals in (2.24) and (2.25) and thus need to solvewhere \(p_\infty (x;\theta ,\beta ,m)\) is a stationary solution of the McKean–Vlasov equation. We start the algorithm at a sufficiently large \(\beta _0\), i.e., at a sufficiently low temperature for which we have a good initial guess for the value of the order parameter.$$\begin{aligned} \mathbf {F}([p,m]) = \left[ \begin{array}{c} p - p_\infty (x;\theta ,\beta ,m)\\ m-R(m;\theta ,\beta ) \end{array} \right] = 0, \quad \text { and } \quad \mathbf {G}(\beta ) = \beta - \frac{1}{\theta \int x^2p_\infty (x;\theta ,\beta ,0) \ \hbox {d}x} = 0, \end{aligned}$$

## Notes

### Acknowledgements

The authors are grateful to S. Kalliadasis, J.A. Carrillo, A.O. Parry and Ch. Kuehn for useful discussions. They are particularly grateful to J.A. Carrillo for making available the code used for the solution of the McKean–Vlasov–Fokker–Planck dynamics presented in Sect. 4.5 and to P. Yatsyshin for help with the arclength continuation methodology. SG is currently supported by the EPSRC under Grant No. EP/K034154/1. Part of the work was done while she was supported by the EPSRC under Grant No. EP/L020564/1. GP is partially supported by the EPSRC under Grants No. EP/P031587/1, EP/L024926/1, EP/L020564/1 and EP/L025159/1.

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