# Long-Time Behaviour and Phase Transitions for the Mckean–Vlasov Equation on the Torus

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## Abstract

## 1 Introduction

Systems of interacting particles arise in a myriad of applications ranging from opinion dynamics [41], granular materials [6, 11, 25] and mathematical biology [8, 47] to statistical mechanics [50], galactic dynamics [18], droplet growth [29], plasma physics [14], and synchronisation [48]. Apart from being of independent interest, these systems find applications in a diverse range of fields such as particle methods in numerical analysis [35], consensus-based methods for global optimisation [20], and nonlinear filtering [23]. They have also been studied in the context of multiscale analysis [40], in the presence of memory-like effects and in a non-Markovian setting [36], and in the discrete setting of graphs [38].

*N*“particles”,

*W*is a periodic interaction potential, and the \(B^i_t, i = 1 \dots N\) represent

*N*independent \({{\mathbb {T}}}^d\)-valued Brownian motions. The constants \(\kappa ,\beta >0\) represent the strength of interaction and inverse temperature respectively. Since one of the two parameters is redundant, we keep \(\beta \) fixed for the rest of the paper. It is clear that what we have described is a set of interacting overdamped Langevin equations. Based on the choice of

*W*(

*x*), one can then obtain models for numerous phenomena from the physical, biological, and social sciences. We refer to [45, 54, 55, 60] and the references therein for a comprehensive list of such models.

*W*is smooth, as \(N \rightarrow \infty \), \({\mathbb {E}}(\varrho ^{(N)})\) converges in the sense of weak convergence of probability measures to some measure \(\varrho \) satisfying the nonlocal parabolic PDE

Our goals are to study some aspects of the asymptotic behaviour and the stationary states of the McKean–Vlasov equation for a wide class of interaction potentials. In terms of the asymptotic behaviour, we analyse the stability conditions for the homogeneous steady state \(1/L^d\) and the rate of convergence to equilibrium. We extend the \({L}^2\)-decay results of [21] to arbitrary dimensions and arbitrary sufficiently nice interactions and also provide sufficient conditions for convergence to equilibrium in relative entropy.

In addition to presenting an existence and uniqueness theory for the evolution problem, we extend considerably the results of both [66] and [27]. We provide explicit criteria based on the Fourier coefficients of the interaction potential *W* for the existence of local bifurcations by studying the implicit symmetry in the problem. In fact, we show that for carefully chosen potentials it is possible to have infinitely many bifurcation points. Additionally, we extend the results of [27] and provide additional criteria for the existence of continuous and discontinuous phase transitions.

### 1.1 Statement of Main Results

We only state simplified versions of our results in one dimension, so as to avoid the use of notation that will be introduced later. We only need to define the cosine transform, Open image in new window for \(k \in {{\mathbb {Z}}}, k>0\). We work with classical solutions of (1.1) which are constructed in Theorem 2.2.

### Theorem 1.1

*W*. Then we have:

- (a)
If \(0< \kappa < \frac{2 \pi }{3 \beta L ||\nabla W||_\infty }\), then \(\left||\varrho (\cdot ,t)- \frac{1}{L}\right||_2 \rightarrow 0\), exponentially, as \(t \rightarrow \infty \),

- (b)
If \({\widetilde{W}}(k) \geqq 0\) for all \(k \in {{\mathbb {Z}}}, k>0\), or \(0< \kappa < \frac{2 \pi ^2}{ \beta L^2 ||\Delta W||_\infty }\), then \( {\mathcal {H}}\left( \varrho (\cdot ,t)|\frac{1}{L}\right) \rightarrow 0\), exponentially, as \(t \rightarrow \infty \)

The previous theorem implies that the uniform state can fail to be the unique stationary solution only if the interaction potential has a negative Fourier mode, that is, the interaction potential is not *H*-stable. Thus, the concept of *H*-stability introduced by Ruelle [62] is relevant for the study of the stationary McKean–Vlasov equation as noticed in [27]. We have the following conditions for the existence of bifurcating branches of steady states:

### Theorem 1.2

*W*be smooth and even and let \((1/L,\kappa )\) represent the trivial branch of solutions. Then every \(k^* \in {{\mathbb {Z}}}, k^*>0\) such that

- (1)
\({\text {card}}\left\{ k \in {{\mathbb {Z}}}, k>0 : {\widetilde{W}}(k)={\widetilde{W}}(k^*)\right\} =1\),

- (2)
\({\widetilde{W}}(k^*) <0\),

We are also able to sharpen sufficient conditions for the existence of continuous or discontinuous bifurcating branches. The following theorem is a simplified version of the exact statements that are presented in Theorem 5.11 and Theorem 5.19:

### Theorem 1.3

*W*be smooth and even and assume the free energy \({\mathscr {F}}_{\kappa ,\beta }\) defined in (1.2) exhibits a transition point, \(\kappa _c<\infty \), in the sense of Definition 5.1. Then we have the following two scenarios:

- (a)
If there exist strictly positive \(k^a,k^b,k^c \in {{\mathbb {Z}}}\) with \({\widetilde{W}}(k^a)\approx {\widetilde{W}}(k^b)\approx {\widetilde{W}}(k^c)\approx \min _k{\widetilde{W}}(k)<0\) such that \(k^a=k^b +k^c\), then \(\kappa _c\) is a discontinuous transition point.

- (b)
Let \(k^\sharp = {{\,\mathrm{\mathrm{arg\,min}}\,}}_k {\widetilde{W}}(k)\) be uniquely defined with \({\widetilde{W}}(k^\sharp )<0\) and \(\kappa _\sharp =\sqrt{2L}/(\beta {\widetilde{W}}(k^\sharp ))\). Let \(W_\alpha \) denote the potential obtained by multiplying all the negative Fourier modes \({\widetilde{W}}(k)\) except \({\widetilde{W}}(k^\sharp )\) by some \(\alpha \in (0,1]\). Then if \(\alpha \) is made small enough, the transition point \(\kappa _c\) is continuous and \(\kappa _c=\kappa _\sharp \).

The proof of the above theorem relies mainly on Proposition 5.8 which states that if \(\varrho _\infty \) is the unique minimiser of the free energy \({\mathscr {F}}_\kappa \) at \(\kappa =\kappa _\sharp \) then \(\kappa _c=\kappa _\sharp \) is a continuous transition point; on the other hand if \(\varrho _\infty \) is not the global minimiser of \({\mathscr {F}}_\kappa \) at \(\kappa =\kappa _\sharp \), then \(\kappa _c<\kappa _\sharp \) and \(\kappa _c\) is a discontinuous transition point.

We conclude the introduction with a figure to provide the reader with some more intuition about the spectral signature of continuous and discontinuous phase transitions. As it can be seen in Figure 1, the results of Theorem 1.3 essentially apply to two perturbative regimes. Figure 1(a) shows the scenario for the existence of a discontinuous transition point in which there are multiple resonating/near-resonating dominant modes \(k^a,k^b,k^c\) which satisfy the algebraic condition \(k^a=k^b+k^c\) from Theorem 1.3(a). This condition allows us to construct a competitor state at \(\kappa =\kappa _\sharp \) which has a lower value of \({\mathscr {F}}_\kappa \) than \(\varrho _\infty \) by controlling the sign of the higher order terms in the Taylor expansion of the free energy. The statement Theorem 1.3(a) is then a direct consequence of Proposition 5.8.

### 1.2 Organisation of the Paper

The paper is organised in the following manner: in Section 2 we introduce the main notation and assumptions on the interaction potential *W*, state a basic existence and uniqueness theorem for classical solutions of the evolutionary problem and present a series of results about the stationary problem and the associated free energy that we use for our later analysis. In Section 3 we present the proof of Theorem 1.1(b), whereas the proof of Theorem 1.1(a) is similar to the argument in [21] and can be found in Version 1 of the arXiv manuscript. Additionally, we perform a linear stability analysis of the Mckean–Vlasov PDE about \(1/L^d\). Section 4 is dedicated mainly to the the proof of Theorem 1.2, including further details about the structure of the bifurcating branches and the structure of the global bifurcation diagram. In Section 5 we give sufficient conditions for the existence of continuous and discontinuous phase transitions and we present the proofs of Theorem 1.3(a) and Theorem 1.3(b), along with some supplementary results. In Section 6, we apply our results to various models from the biological, physical and social sciences.

## 2 Preliminaries

### 2.1 Set Up and Notation

Let \(U={{{\mathbb {R}}}^{d}/ L {{\mathbb {Z}}}^d} \widehat{=} \left( -\frac{L}{2},\frac{L}{2}\right) ^d \subset {{\mathbb {R}}}^d\) be the torus of size \(L>0\). We denote by \({{\mathbb {N}}}= \{0,1,\dots \}\) the nonnegative integers. Furthermore, we will denote by \(\mathcal {P}(U)\) the space of Borel probability measures on *U*, by Open image in new window the subset of \(\mathcal {P}(U)\) absolutely continuous with respect to the Lebesgue measure, and by Open image in new window the subset of Open image in new window having strictly positive densities almost everywhere. Additionally, \(C^k(U)\) will denote the restriction to *U* of all *L*-periodic and *k*-times continuously differentiable functions, \({\mathcal {D}}(U)\) the space of test functions, and \(\left\langle f,g\right\rangle _\mu \) the \({L}^2(U,\mu )\) inner product.

### 2.2 Assumptions on *W*

*W*(

*x*) is at least integrable and coordinate-wise even, that is

*W*(

*x*) for the evolutionary and stationary problems to be the same, it is important to mention that these assumptions are in no way sharp and the aim of this paper is not to study low regularity theory for this class of PDEs.

*W*(

*x*) is coordinate-wise even. \(\mathrm {Sym}(\Lambda )\) represents the symmetry group of the product of two-point spaces \(\Lambda =\{1,-1\}^d\), which acts on \({{\mathbb {Z}}}^d\) by pointwise multiplication, that is, \((\sigma (k))_i=\sigma _i k_i, k \in {{\mathbb {Z}}}^d, \sigma \in \mathrm {Sym}(\Lambda )\). Another expression that we will use extensively in the sequel is the Fourier expansion of the following bilinear form:

*g*(

*x*) and \(k \in {{\mathbb {Z}}}^d\) the sum \(\sum _{\sigma \in \mathrm {Sym}(\Lambda )}|{\widetilde{g}}(\sigma (k))|^2 \) is translation invariant, that is, the value of the sum is the same for

*g*and \(g_\tau (x) =g(x+\tau )\) for \(\tau \in U\). In later sections we will also use the space \({L}^2_s(U) \subset {L}^2(U)\), which we define as the space of coordinate-wise even functions in \({L}^2(U)\) given by

*g*is assumed to be in \({L}^2_s(U)\), then the above expressions reduce toIn addition, the sign of the individual Fourier modes of

*W*is quite important in the subsequent analysis and we introduce the following definition:

### Definition 2.1

*H*-stable, denoted by \(W \in {{\mathbb {H}}}_\text {s}\), if it has non-negative Fourier coefficients, i.e,

*H*-stable by \(W \in {{\mathbb {H}}}_s^c\).

An immediate consequence of the identity (2.3) is that *H*-stable potentials have nonnegative interaction energy. The above definition can be thought of as a continuous analogue of the notion of *H*-stability encountered in the study of discrete systems (cf. [62]). We refer to [26] for an example of the notion of *H*-stability applied to continuous systems. For the rest of the paper we will drop the subscript *U* under the integral sign and all integrals in space will be taken over *U* unless specified otherwise.

### 2.3 Existence and Uniqueness for the Dynamics

*W*satisfying Assumption (

**A1**) in any dimension

*d*, as opposed to [21, Theorem 4.5] which deals with the Hegselmann–Krause potential in one dimension. Additionally, we prove

*strict*positivity of solutions as opposed to the nonnegativity proved in [21]. We prove below the existence of classical solutions \(\varrho (\cdot ,t) \in C^2 (U)\) to the system

### Theorem 2.2

Assume Assumption (**A1**) holds, then for Open image in new window, there exists a unique classical solution \(\varrho \) of (2.5) such that Open image in new window for all \(t>0\). Additionally, \(\varrho (\cdot ,t)\) is strictly positive and has finite entropy, i.e, \(\varrho (\cdot ,t)>0\) and \(S(\varrho (\cdot ,t))< \infty \), for all \(t>0\).

*W*allows us to pass to the limit as \(n \rightarrow \infty \) and recover weak solutions of the McKean–Vlasov equation which are proved to be unique. Their regularity follows from bootstrapping and using the regularity of

*W*and the initial data.

*C*. Since \(\varrho (x,t)\) is nonnegative and \(||\varrho (x,t)||_{1}=1\) for all \(0\leqq t < \infty \), this implies that \(\inf _U\varrho (x,t)\) is positive for any positive time. The fact that the entropy is finite follows from the fact that \(\varrho (x,t)\) is positive and bounded above.

### 2.4 Characterisation of the Stationary Solutions

We start by discussing the existence and and regularity question for the stationary problem. The proof relies on the link between the stationary PDE and the fixed points of a nonlinear map as was discussed in [66] and [37].

### Theorem 2.3

**A2**) holds. Then we have that

- (a)There exists a weak solution, Open image in new window of (2.6) and any weak solution is a fixed point of the nonlinear map Open image in new window(2.7)
- (b)
Any weak solution Open image in new window is smooth and strictly positive, that is, Open image in new window.

### Proof

*E*is a closed, convex subset of \({L}^2(U)\). We can now redefine \({\mathcal {T}}\) to act on

*E*. Additionally, for any \(\varrho \in E\), we have that

*E*, since

*E*is closed. Furthermore, \({\mathcal {T}}\) is Lipschitz continuous, that is, we have for \(\varrho _1,\varrho _2 \in E:\)\(\left||{\mathcal {T}}\!\varrho _1 -{\mathcal {T}}\!\varrho _2\right||_2 \leqq C\left||\varrho _1-\varrho _2\right||_2\), for some positive constant

*C*. By the Schauder fixed point theorem, there exists a fixed point of \(\varrho \in E\) of \({\mathcal {T}}\) which by (2.10) is in \({H}^1(U)\). Plugging this expression into the weak form of the PDE (2.8) we obtain (a). Also note that fixed points of \({\mathcal {T}}\) are bounded from below by \(e^{-\beta \kappa (\left||W_-\right||_\infty + \left||W\right||_1 \left||{\mathcal {T}}\!\varrho \right||_\infty )}\), proving the positivity of them.

We obtain regularity of solutions by observing that if \(f \in {H}^m(U), g \in {H}^n(U)\), then we have that \(f \star g \in {{\mathcal {W}}}^{m+n,\infty }(U)\). Then we use a bootstrap argument, that is, \(W \in {H}^1(U), \varrho \in {H}^1(U)\) implies that \(\varrho ={\mathcal {T}}\!\varrho \in {{\mathcal {W}}}^{2,\infty }(U)\). This implies that \(W \star \varrho \in {{\mathcal {W}}}^{3,\infty }(U)\) and so on and and so forth. Thus we have that \(\varrho \in {H}^m(U) \cup {{\mathcal {W}}}^{m,\infty }(U)\) for any \(m \in {{\mathbb {N}}}\). The strict positivity follows from the lower bound on \({\mathcal {T}}\!\varrho \). \(\quad \square \)

*U*by

### Proposition 2.4

*W*(

*x*) satisfies Assumption (

**A2**) and fix \(\kappa >0\). Let Open image in new window. Then the following statements are equivalent:

- (1)
\(\varrho \) is a classical solution of the stationary McKean–Vlasov equation (2.6).

- (2)
\(\varrho \) is a zero of the map \(F_\kappa (\varrho )\).

- (3)
\(\varrho \) is a critical point of the free energy \({\mathscr {F}}_\kappa (\varrho )\).

- (4)
\(\varrho \) is a global minimiser of the entropy dissipation functional \({\mathcal {J}}_\kappa (\varrho )\).

### Proof

(1)\(\Leftrightarrow \)(2): Observe that \(\varrho \) is a zero of \(F_\kappa (\varrho )\) if and only if it is a fixed point of \({\mathcal {T}}\). Thus by part (a) of Theorem 2.3 we have the desired equivalence.

- (2)\(\Rightarrow \)(3): The main observation for this is that zeroes of \(F_\kappa \) represent solutions of the Euler–Lagrange equations for \({\mathscr {F}}_\kappa \). Let Open image in new window, we define the standard convex interpolant, \(\varrho _s=(1-s)\varrho +s \varrho _1\), \(s \in (0,1)\) such that \({\mathscr {F}}(\varrho ),{\mathscr {F}}(\varrho _1)< \infty \). Then we have the following form of the Euler–Lagrange equations (which are well-defined for Open image in new window):(2.14)
- (3)\(\Rightarrow \)(2): On the other hand assume that \(\varrho \) is a critical point. If the integrand \(\beta ^{-1}\log \varrho + \kappa W \star \varrho \) in (2.14) is not constant almost everywhere, we can find without loss of generality a set \(A \in \mathcal {B}(U)\) of nonzero Lebesgue measure such that We are now free to choose Open image in new window to befor some \(\varepsilon >0\). For this choice of \(\varrho _1\), we have, From our choice of the set$$\begin{aligned} \varrho _1 = \frac{1}{L^d}\left( {(1- \varepsilon ) \chi _A(x) + \varepsilon \chi _A^c(x)}\right) \ \end{aligned}$$
*A*, it is clear that \(a >0\) and \(b \leqq 0\). Since \(\varepsilon \) can be made arbitrarily small, \((1-\varepsilon )a + \varepsilon b\) can be made positive. Thus we have derived a contradiction, since \(\varrho \) is a critical point of \({\mathscr {F}}_\kappa \) and therefore it must satisfy the Euler–Lagrange equations in (2.14). Thus the integrand must be constant almost everywhere from which we obtain (3)\(\Rightarrow \)(2). (2)\(\Rightarrow \)(4): Clearly, \({\mathcal {J}}_\kappa \) is nonnegative. Thus if \({\mathcal {J}}_\kappa (\varrho )=0\) for some Open image in new window then it is necessarily a global minimiser. Plugging in \(\varrho \) for some zero of \(F_\kappa \) finishes this implication.

- (4)\(\Rightarrow \)(2): Now, any global minimiser \(\varrho \) of \({\mathcal {J}}_\kappa (\varrho )\) must satisfy \({\mathcal {J}}_\kappa (\varrho )=0\) since \({\mathcal {J}}_\kappa (\varrho _\infty )=0\). From the expression for \({\mathcal {J}}_\kappa (\varrho )\) and the fact that \(\varrho \) has full support this is possible only ifThus, we have that, \(\varrho - C e^{-\beta \kappa W \star \varrho }=0\), almost everywhere, for some constant, \(C>0\), which is given precisely by \(Z(\varrho ,\kappa ,\beta )\) since Open image in new window. Thus we have that \(\varrho \) is a zero of \(F_\kappa (\varrho )\) and the reverse implication, (4)\(\Rightarrow \)(2).$$\begin{aligned} \nabla \frac{ \log \varrho }{e^{-\beta \kappa W \star \varrho }} =0, \quad {\mathrm {almost everywhere}}. \end{aligned}$$

The following lemma is taken from [27] in which it is shown that for any unbounded Open image in new window there exists a bounded Open image in new window having a lower value of the free energy:

### Lemma 2.5

*W*satisfies Assumption (

**A2**) and fix \(\kappa \in (0, \infty )\). Then there exists a positive constant \(B_0<\infty \) such that for all Open image in new window with \(\left||\varrho \right||_{\infty } > B_0\) there exists some Open image in new window with \(\left||\varrho ^{\dagger }\right||_{\infty }\leqq B_0 \) satisfying

The next lemma shows that minimisers of \({\mathscr {F}}_\kappa (\varrho )\) over Open image in new window are attained in Open image in new window.

### Lemma 2.6

*W*(

*x*) satisfies Assumption (

**A2**) and let Open image in new window. Then, there exists Open image in new window such that

### Proof

*W*and \({{\mathbb {B}}}_0\) and we have chosen \(\epsilon \) sufficiently small. Combining the two expressions together, we obtain

### Theorem 2.7

(Existence of a minimiser [27]) Assume *W*(*x*) satisfies Assumption (**A2**). For \(\kappa \in (0,\infty )\) the free energy \({\mathscr {F}}_{\kappa }(\varrho )\) has a smooth minimiser Open image in new window.

### Proof

**A2**). Since by (2.15), \({\mathscr {F}}_{\kappa }(\varrho )\) is bounded from below over Open image in new window, there exists a minimising sequence Open image in new window. Furthermore, by Lemma 2.5, the minimising sequence can be chosen such that \(\{\varrho _j\}_{j=1}^\infty \subset {L}^2(U)\) with \(\left||\varrho _j\right||_2 \leqq B_0^{\frac{1}{2}}\), where \(B_0\) is the constant from Lemma 2.5. Thus, there exists a subsequence which we continue to denote by \(\{\varrho _j\}_{j=1}^\infty \) such that \(\varrho _j \rightharpoonup \varrho _{\kappa }\) weakly in \({L}^2(U)\). Clearly we have that Open image in new window. It is also easy to see that \(\varrho _\kappa \geqq 0\), almost everywhere. Thus Open image in new window. The lower semicontinuity of \(S(\varrho )\) follows from standard results (cf. [44], Lemma 4.3.1). Consider now the interaction energy term. For \(W \in {L}^1(U)\), the interaction energy is weakly continuous in \({L}^2(U)\) [27, Theorem 2.2, Equation (9)]. This implies that the free energy \({\mathscr {F}}_\kappa (\varrho )\) has a minimiser \(\varrho _\kappa \) over Open image in new window. A direct consequence of this and Lemma 2.6 is that the minimisation problem is well-posed in Open image in new window since the minimiser \(\varrho _\kappa \) must be attained in Open image in new window. We can then use Theorem 2.3 together with Proposition 2.4 to argue that any such minimiser must be smooth. \(\quad \square \)

### Proposition 2.8

*W*satisfies Assumption (

**A2**) such that \(W_\text {u}\) is bounded from below, where \(W_\text {u}\) is the unstable part defined in Definition 2.1. Then, for \(\kappa \in \left( 0,\kappa _{\mathrm {con}}\right) \), where \(\kappa _{\mathrm {con}}:=\beta ^{-1} \left||W_{\text {u}-}\right||_\infty ^{-1}\), the functional \({\mathscr {F}}_\kappa (\varrho )\) is strictly convex on Open image in new window, that is, for all \(s\in (0,1)\), it holds that

### Proof

### Remark 2.9

It follows from the above result that if \(W_\text {u}\equiv 0\), that is, \(W\in {{\mathbb {H}}}_\text {s}\), then \({\mathscr {F}}_\kappa (\varrho )\) is strictly convex for all \(\kappa \in (0,\infty )\).

## 3 Global Asymptotic Stability

### 3.1 Trend to Equilibrium in Relative Entropy

### Proposition 3.1

### Proof of Theorem 1.1(b)

### Remark 3.2

For the case of the noisy Hegselmann–Krausse model studied in [21], we have Open image in new window with \(\phi (|x|)={\mathbb {1}}_{|x|\leqq R}\). We can estimate by the same arguments \(\left||W_\text {u}''(x)\right||_{\infty }\leqq \left||W''(x)\right||_{\infty }={R}\). Thus for \(\kappa <\frac{2 \pi ^2}{\beta L^2}\), we have exponential convergence to equilibrium. See §6.2 for a detailed analysis of this model.

### 3.2 Linear Stability Analysis

*W*is coordinate-wise even, which implies thatThus, we have the following expression for the value of the parameter \(\kappa _\sharp \) at which the first eigenvalue of \({\mathcal {L}}\) crosses the imaginary axis:

**point of critical stability**. We denote by \(k^\sharp \) the critical wave number (if it is unique) and define it as

## 4 Bifurcation Theory

*W*satisfies Assumption (

**A2**). We do not make this assumption for the whole section as we want the bifurcation theory to be valid for more singular potentials, for example, the Keller–Segel model which we treat in a later section. It is also clear that the map \(F(\varrho ,\kappa )\) is translation invariant on the whole space \({L}^2_s(U)\), that is, if \(\varrho \) is a zero of \(F(\varrho ,\kappa )\) then so is any translate \(\varrho (\cdot - y)\) of \(\varrho (\cdot )\) for any \(y\in U\). This is the motivation for the restriction of

*F*to the space \({L}^2_s(U)\). We will further justify our choice of the space \({L}^2_s(U)\) in Lemma 5.18.

The first result is an easy consequence of the characterisation of stationary solutions from §2.4, but could be also derived by standard contraction mapping argument on the map *F* as done in [66, Theorem 4.1] and [53, Theorem 3].

### Proposition 4.1

Assume *W*(*x*) satisfies Assumption (**A2**). Then, for \(\kappa \) sufficiently small, the uniform state \(\varrho _\infty \) is the only solution of \(F(\varrho ,\kappa )=0\).

### Proof

Proposition 2.8 implies that \({\mathscr {F}}_\kappa (\varrho )\) is strictly convex for \(\kappa < \kappa _{\mathrm {con}} = \beta ^{-1} \left||W_\text {u}\right||_\infty ^{-1}\). Hence, using Theorem 2.7, it has a unique minimiser and exactly one critical point. This implies from Proposition 2.4 that \(F(\varrho ,\kappa )\) has a unique solution. \(\quad \square \)

### Theorem 4.2

- (1)
\({\text {card}}\left\{ k : \frac{{\widetilde{W}}(k)}{\Theta (k)}=\frac{{\widetilde{W}}(k^*)}{\Theta (k^*)}\right\} =1\);

- (2)
\({\widetilde{W}}(k^*) <0\).

*V*of \(\kappa _*\) with \(\kappa (0)=\kappa _*\). Moreover, it holds that \(\kappa '(0)=0\), \(\kappa ''(0)=\frac{2\beta \kappa _*}{3 \varrho _\infty }>0\), and \(\varrho _*\) is the only nontrivial solution in a neighbourhood of \((0,\kappa _*)\) in \({L}^2_s(U) \times {{\mathbb {R}}}\).

### Proof of Theorem 1.2

The statment of Theorem 4.2 becomes more transparent in one dimension.

### Corollary 4.3

- (1)
\({\text {card}}\left\{ k : {\widetilde{W}}(k)={\widetilde{W}}(k^*)\right\} =1\);

- (2)
\({\widetilde{W}}(k^*) <0\).

### Remark 4.4

It should also be noted that one can obtain the existence of bifurcations with higher-dimensional kernels as well, i.e, when \(\dim (\ker ({\widehat{T}})) >1\). Since \({\widehat{T}}\) is self adjoint, for any eigenvalue its algebraic and geometric multiplicities are the same. From [33, Theorem 28.1] it follows that any characteristic values (the reciprocals of the eigenvalues of \({\widehat{T}}\)) of odd algebraic multiplicity correspond to a bifurcation point. This implies that we could replace Condition (1) in Theorem 4.2 with \({\text {card}}\left\{ k : \frac{{\widetilde{W}}(k)}{\Theta (k)}=\frac{{\widetilde{W}}(k^*)}{\Theta (k^*)}\right\} =m\), where *m* is odd. However, it is not easy to obtain detailed information about the structure and regularity of the bifurcating branches in this case.

### Remark 4.5

Condition (1) of Theorem 4.2 is in particular satisfied for an interaction potential \(W \in {L}^2_s(U)\) if the map \({\widetilde{W}} : {{\mathbb {N}}}^d \rightarrow {{\mathbb {R}}}\) is injective. In this case, every \(k_\alpha \in {{\mathbb {N}}}^d\) such that \({\widetilde{W}}(k) <0\), corresponds to a unique bifurcation point \( \kappa _\alpha \) of \(F(\varrho ,\kappa )\) through the relation (4.8). For example consider the interaction potential \(W(x)=x^2/2\). In this case \({\widetilde{W}}\) is injective and therefore the system has infinitely many bifurcation points. On the other hand, when \(W(x)=-w_k(x)\) for some \(k \in {{\mathbb {N}}}^d\), the system has only one bifurcation point.

### Remark 4.6

*d*coordinates. In this case it is easy to check that \(\left\langle W,w_k\right\rangle = \left\langle W,w_{\Pi (k)}\right\rangle \) for all \(k \in {{\mathbb {N}}}^d\). We can then define the equivalence relation, \(k \sim k'\) if \(k' = \Pi (k)\) for some permutation \(\Pi \) and write \(\left[ k\right] \) for the corresponding equivalence class. Thus, the consequence of

*W*(

*x*) having this symmetry is that the value \({\widetilde{W}}(k)/\Theta (k)\) is constant on \([k]\). This implies that kernel of \(D_\varrho {\widehat{F}}\) is can never be one-dimensional. We can quotient out this symmetry by defining the space \({L}^2_{{\text {ex}}}(U)={\text {span}}\{w_{[k]}\}\), where \(\{w_{[k]}\}\) is an orthonormal basis defined by

- (1)
\({\text {card}}\left\{ \left[ k\right] : \frac{{\widetilde{W}}([k])}{\Theta ([k])} = \frac{{\widetilde{W}}([k^*])}{\Theta ([k^*])}\right\} = 1 \),

- (2)
\({\widetilde{W}}([k^*]) <0\),

### Remark 4.7

If we now assume that *W* satisfies assumption (**A2**) we can see that the zeros of \(F(\varrho ,\kappa )\) are fixed points of the map \({\mathcal {T}}\) which by Proposition 2.4 are equivalent to smooth solutions of the stationary McKean–Vlasov equation. Theorem 4.2 also provides us information about the structure of the branches, that is, if \(w_k(x)\) is the mode such that \(k\in {{\mathbb {N}}}^d\) satisfies the conditions of Theorem 4.2, then to leading order the nontrivial solution is of the form \(\varrho _\infty + s w_k(x)\). One may think of this as a “proto-cluster”, with the nodes of \(w_k(x)\) corresponding to the positions of the peaks and valleys of the cluster.

So far the analysis in this section has been local. We conclude this section by providing a characterisation of the global structure of the bifurcation diagram for \({\widehat{F}}\) as defined in (4.2).

### Proposition 4.8

*V*be an open neighbourhood of \((0,\kappa _*)\) in \({L}^2_s(U) \times {{\mathbb {R}}}\), where \((0,\kappa _*)\) is a bifurcation point of the map \({\widehat{F}}\) in the sense of Theorem 4.2. We denote by \({\mathcal {C}}_V\) the set of nontrivial solutions of \({\widehat{F}}(\varrho ,\kappa )=0\) in

*V*and by \({\mathcal {C}}_{V,\kappa _*}\) the connected component of \(\overline{{\mathcal {C}}_V}\) containing \((0,\kappa _*)\). Then \({\mathcal {C}}_{V,\kappa _*}\) has at least one of the following two properties:

- (1)
\({\mathcal {C}}_{V,\kappa _*} \cap \partial V \ne \emptyset \);

- (2)
\({\mathcal {C}}_{V,\kappa _*}\) contains an odd number of characteristic values of \({\widehat{T}}\), \((0,\kappa _i) \ne (0,\kappa _*)\), which have odd algebraic multiplicity.

### Proof

*G*is completely continuous and \(o\left( \left||\varrho \right||_2\right) \) uniformly in \(\kappa \) as \(\left||\varrho \right||_2 \rightarrow 0 \). For the first result, it is enough to show that

*G*is compact since \({L}^2_s(U)\) is reflexive. We establish the following estimate:

*G*is a bounded map on \({L}^2(U)\). Together with this and (4.14), and using the fact that the convolution is uniformly continuous, one can check that that

*G*(

*A*) satisfies the conditions of the Kolmogorov–Riesz theorem, where

*A*is any bounded subset of \({L}^2_s(U)\). Thus

*G*is compact. The fact that

*G*is \(o\left( \left||\varrho \right||_2\right) \) follows by Taylor expanding \(e^{-\beta \kappa W \star \varrho }/Z\).

One can now check that if condition (1) of Theorem 4.2 is satisfied for some \(k \in {{\mathbb {N}}}^d\), the associated eigenvalue \(\kappa ^{-1}\)(which could be negative) of \({\widehat{T}}\) is simple, that is, it has algebraic multiplicity one. This implies that all bifurcation points predicted by Theorem 4.2 are associated with simple eigenvalues of \({\widehat{T}}\). Thus, we can apply Theorem A.3 to complete the proof. \(\quad \square \)

## 5 Phase Transitions for the McKean–Vlasov Equation

We know from Proposition 2.8 that \(\varrho _\infty \) is the unique minimiser of the free energy for \(\kappa \) sufficiently small. We are interested in studying under what criteria there is a change in the qualitative structure of the set of minimisers of \({\mathscr {F}}_\kappa \). For the rest of this section we will assume that *W* satisfies Assumption (**A2**), i.e, \(W \in {H}^1(U)\) and bounded below. We build on and extend the notions introduced by [27]. The first definition introduces what we mean by a transition point.

### Definition 5.1

*transition point*of \({\mathscr {F}}_\kappa \) if it satisfies the following conditions:

- (1)
For \(0<\kappa < \kappa _c\), \(\varrho _\infty \) is the unique minimiser of \({\mathscr {F}}_\kappa (\varrho )\).

- (2)
For \(\kappa =\kappa _c\), \(\varrho _\infty \) is a minimiser of \({\mathscr {F}}_\kappa (\varrho )\).

- (3)
For \(\kappa >\kappa _c\), there exists some Open image in new window, not equal to \(\varrho _\infty \), such that \(\varrho _\kappa \) is a minimiser of \({\mathscr {F}}_\kappa (\varrho )\).

In the present work, we are only interested in the first transition point by increasing \(\kappa \) starting from 0, also called the lower transition point. To convince the reader that the above definition makes sense we include the following result from [27]:

### Proposition 5.2

In addition, the following result from [39] shows that *H*-stability of the potential is a necessary and sufficient condition for the nonexistence of a transition point:

### Proposition 5.3

([39]) \({\mathscr {F}}_\kappa \) has a transition point at some \(\kappa =\kappa _c<\infty \) if and only if \(W \in {{\mathbb {H}}}_\text {s}^c\). Additionally for \(\kappa >\kappa _\sharp \), with \(\kappa _\sharp \) the point of critical stability as defined in (3.5) in §3.2, \(\varrho _\infty \) is not the minimiser of \({\mathscr {F}}_\kappa \).

From this result it follows directly that if the system possesses a transition point \(\kappa _c\), \(\varrho _\infty \) can no longer be a minimiser beyond this point. We are also interested in understanding how this transition occurs. In the infinite-dimensional setting it is not always possible to obtain a well-defined order parameter for the system characterizing first and second order phase transitions in the sense of statistical physics. For this reason, it may be better to define such transitions in terms of discontinuity in some norm or metric.

### Definition 5.4

*(Continuous and discontinuous transition point)*A transition point \(\kappa _c >0\) is said to be a

*continuous transition point*of \({\mathscr {F}}_\kappa \) if it satisfies the following conditions:

- (1)
For \(\kappa =\kappa _c\), \(\varrho _\infty \) is the unique minimiser of \({\mathscr {F}}_\kappa (\varrho )\);

- (2)Given any family of minimisers, \(\{\varrho _\kappa |\kappa > \kappa _c \}\), we have that$$\begin{aligned} \limsup _{\kappa \downarrow \kappa _c} \left||\varrho _\kappa -\varrho _\infty \right||_1=0. \end{aligned}$$

*discontinuous*.

We now include a series of results from [27] that we need for our subsequent analysis.

### Proposition 5.5

([27]) Open image in new window is nonincreasing in \(\kappa \).

### Proposition 5.6

([27]) Assume \(W \in {{\mathbb {H}}}_s^c\) and that condition (2) of Definition 5.4 is violated. Then there exists a discontinuous transition point \(\kappa _c < \infty \) and some \(\varrho _{\kappa _c} \ne \varrho _\infty \) such that \({\mathscr {F}}_{\kappa _c}(\varrho _{\kappa _c})={\mathscr {F}}_{\kappa _c}(\varrho _\infty )\).

### Proposition 5.7

([27]) Assume \(W \in {{\mathbb {H}}}_s^c\) and that the free energy \({\mathscr {F}}_\kappa \) exhibits a continuous transition point at some \(\kappa _c < \infty \). Then it follows that \(\kappa _c=\kappa _\sharp \).

By combining certain properties of transition points with the previous analysis on critical stability in §3.2, we obtain more streamlined sufficient conditions for the identification of transition points, which is the basis for the proof of Theorem 1.3, or more precisely Theorem 5.11 and Theorem 5.19.

### Proposition 5.8

- (a)
If \(\varrho _\infty \) is the unique minimiser of \({\mathscr {F}}_{\kappa _\sharp }\), then \(\kappa _c = \kappa _{\sharp }\) is a continuous transition point.

- (b)
If \(\varrho _\infty \) is not a global minimiser of \({\mathscr {F}}_{\kappa _\sharp }\), then \(\kappa _c < \kappa _{\sharp }\) and \(\kappa _c\) is a discontinuous transition point.

### Remark 5.9

The statements of Proposition 5.8(a) and Proposition 5.8(b) are only necessary conditions for the characterisation of transition points. In particular, they are not logical complements of each other, that is, \(\varrho _\infty \) could be a global minimiser of \({\mathscr {F}}_{\kappa _\sharp }\) without being the unique one or vice versa.

### Proof

A consequence of the assumption in the first statement (a) of the proposition is that \(\varrho _\infty \) is the unique minimiser for all \(\kappa \leqq \kappa _\sharp \). Indeed, from Proposition 5.5, we know that Open image in new window for \(\kappa \leqq \kappa _c\). Thus, if \(\varrho _\infty \) is the unique minimiser at some \(\kappa =\kappa _c\), it must be a minimiser for all \(\kappa \leqq \kappa _c\). In fact, using Proposition 5.2 we can assert that \(\varrho _\infty \) is the unique minimiser of \({\mathscr {F}}_\kappa \) for all \(\kappa \leqq \kappa _c\). Indeed, if this were not the case then there exists some Open image in new window not equal to \(\varrho _\infty \) such that \({\mathscr {F}}_{\kappa _T}(\varrho _{\kappa _T}) = {\mathscr {F}}_{\kappa _T}(\varrho _\infty )\) for some \(\kappa _T<\kappa _\sharp \). Proposition 5.2 then tells us that \(\varrho _\infty \) can no longer be a minimiser for any \(\kappa >\kappa _T\), which is a contradiction. It follows that conditions (1) and (2) from Definition 5.1 are satisfied. That condition (3) is satisfied follows directly from Proposition 5.3. This implies that \(\kappa _\sharp \) satisfies the three conditions of being a transition point.

Now, we have to verify condition (2) of Definition 5.4 (condition (1) is already satisfied from the statement of the proposition). Assume condition (2) doesn’t hold, that is, there exists a family of minimisers \(\{\varrho _\kappa |\kappa > \kappa _c \}\) of \({\mathscr {F}}_\kappa (\varrho )\) such that \(\limsup _{\kappa \downarrow \kappa _c} \left||\varrho _\kappa -\varrho _\infty \right||_1\ne 0\). Then we know from Proposition 5.6 that there exists some Open image in new window not equal to \(\varrho _\infty \) such that it is a minimiser of the free energy \({\mathscr {F}}_\kappa (\varrho )\) at \(\kappa =\kappa _c\). Applied in the present setting with \(\kappa _c=\kappa _\sharp \), we would deduce that \(\varrho _\infty \) is no longer the unique minimiser of \({\mathscr {F}}_{\kappa _\sharp }(\varrho )\), in contradiction to statement (a) of the proposition. Thus both conditions (1) and (2) of Definition 5.4 are satisfied from which it follows that \(\kappa _c=\kappa _\sharp \) is a continuous transition point.

To prove the second statement (b) of the proposition, let \(\varrho \) be such that \({\mathscr {F}}_{\kappa _\sharp }(\varrho ) < {\mathscr {F}}_{\kappa _\sharp }(\varrho _\infty )\). Then for any \(\kappa \) close enough to \(\kappa _\sharp \), we also have \({\mathscr {F}}_{\kappa }(\varrho ) < {\mathscr {F}}_{\kappa }(\varrho _\infty )\). Hence by a combination of Proposition 5.2 and Proposition 5.3 there exists a transition point \(\kappa _c < \kappa _\sharp \) and, in particular \(\kappa _\sharp \), cannot be a transition point. From Proposition 5.7, we have the fact that if \(\kappa _c\) is a continuous transition point of \({\mathscr {F}}_{\kappa }\), then necessarily \(\kappa _c =\kappa _\sharp \). This implies that \(\kappa _c< \kappa _\sharp \) cannot be a continuous transition point. \(\quad \square \)

Before proceeding to present the main results of this section, we remind the reader that for the rest of the paper \(\kappa _c\) denotes a transition point, \(\kappa _\sharp \) denotes the point of critical stability, and \(\kappa _*\) denotes a bifurcation point.

### 5.1 Discontinuous Transition Points

We provide below a characterisation of potentials which exhibit discontinuous transition points, which proves Theorem 1.3(a).

### Definition 5.10

Assume \(W\in {{\mathbb {H}}}_\text {s}^c\) and let \(K^{\delta }:=\Big \{k' \in {{\mathbb {N}}}^d\setminus \{{\mathbf {0}}\}: \frac{{\widetilde{W}}(k')}{\Theta (k')}\leqq \min _{k \in {{\mathbb {N}}}^d\setminus \{{\mathbf {0}}\}} \frac{{\widetilde{W}}(k)}{\Theta (k)} +\delta \Big \}\) for some \(\delta \geqq 0\). We define \(\delta _*\) to be the smallest value, if it exists, of \(\delta \) for which the following condition is satisfied:

### Theorem 5.11

Let *W*(*x*) be as in Definition 5.10. Then if \(\delta _*\) exists and is sufficiently small, \({\mathscr {F}}_\kappa \) exhibits a discontinuous transition point at some \(\kappa _c<\kappa _\sharp \).

### Proof

**C1**), it holds thatwhere the constant

*a*is independent of \(\delta _*\). Indeed, the cube of the sum of

*n*numbers \(a_i\), \(i=1, \dots , n\) consists of only three types of terms, namely: \(a_i^3\), \(a_i^2 a_j\) and \(a_i a_j a_k\). Setting the \(a_i=w_{s(i)}\), with \(s(i) \in K^{\delta _*}\), one can check that the first type of term will always integrate to zero. The other two will take nonzero and in fact positive values if and only if condition (

**C1**) is satisfied. This follows from the fact that Thus, for \(\delta _*\) sufficiently small considering the fact that \(|K^{\delta _*}| \geqq 2\) and is nonincreasing as \(\delta _*\) decreases, \(\varrho \) has smaller free energy and \(\varrho _\infty \) is not a minimiser at \(\kappa =\kappa _\sharp \). \(\quad \square \)

### Remark 5.12

The case of the above result for \(\delta _*=0\) can be thought of as the pure resonance case. In this case the set \(K^0\) will denote the set of all resonant modes. Similarly, the above result for \(\delta _*\) small but positive can be thought of as the near resonance case.

The corollary below tells us that if we have a have a sequence of potentials whose Fourier modes grow closer to each other then it will eventually have a discontinuous transition point, as long as the potentials do not lose mass too fast.

### Corollary 5.13

Let \(\{W^n\}_{n \in {{\mathbb {N}}}} \in {{\mathbb {H}}}_\text {s}^c\) be a sequence of interaction potentials such that \(\delta _*(n) \rightarrow 0\) as \(n \rightarrow \infty \), where \(\delta _*\) is as defined in Definition 5.10. Assume further that for all *n* greater than some \(N \in {{\mathbb {N}}}\), there exists a constant \(C>0\) such that \(\left|\min \limits _{k \in {{\mathbb {N}}}^d\setminus \{{\mathbf {0}}\}} \frac{\widetilde{W^n}(k)}{\Theta (k)}\right| \geqq C \delta _*(n)^{ \gamma }\) for some \(\gamma <1/2\). Then for *n* sufficiently large, the associated free energy \({\mathcal {F}}^n_\kappa (\varrho )\) possesses a discontinuous transition point at some \(\kappa _c^n < \kappa _\sharp ^n\).

### Proof

*n*. We also note that the error term is independent of the potential \(W^n\). Using our assumption on the potential (for \(n>N\)), we haveSince \(\gamma <1/2\) and \(\delta _* \rightarrow 0 \) as \(n \rightarrow \infty \), the result follows. \(\quad \square \)

To conclude our discussion of discontinuous transition points, we present the following corollary to provide some more intuition of the types of interaction potentials that exhibit a discontinuous transition point:

### Corollary 5.14

Let \(\{W^n\}_{n \in {{\mathbb {N}}}} \) be a sequence of interaction potentials with \( ||W^n||_1 =C>0\) for all \( n \in {{\mathbb {N}}}\) such that \(W^n\rightarrow -C \delta _0\) in the sense of distributions as \(n \rightarrow \infty \). Then for *n* large enough, the associated free energy \({\mathcal {F}}^n_\kappa (\varrho )\) possesses a discontinuous transition point at some \(\kappa _c^n < \kappa _\sharp ^n\).

### Proof

*N*large enough such that \(\frac{\widetilde{W^n}(k)}{\Theta (k)},\frac{\widetilde{W^n}(2 k)}{\Theta (2 k)} \in {\left( -C L^{-d/2},-C L^{-d/2}+\epsilon \right) }\) for all \(n >N\), for some \(k \in {{\mathbb {N}}}^d \setminus \{{\mathbf {0}}\}\). This and (5.2) tells us that \(\delta _* \leqq \epsilon \) and since \(\epsilon \) is arbitrary \(\delta _* \rightarrow 0\) as \(n \rightarrow \infty \). From similar arguments we assert that for all \(n>N\), \(\left( \min \limits _{k \in {{\mathbb {N}}}^d\setminus \{{\mathbf {0}}\}} \frac{\widetilde{W^n}(k)}{\Theta (k)}\right) < -C L^{-d/2}+\epsilon \). Thus we have that \(\left|\left( \min \limits _{k \in {{\mathbb {N}}}^d\setminus \{{\mathbf {0}}\}} \frac{\widetilde{W^n}(k)}{\Theta (k)}\right) \right|> C \frac{2^{d/2}}{L^{d/2}}-\epsilon \) for \(n >N\). Since the conditions of Corollary 5.13 are satisfied, we have the desired result. \(\quad \square \)

### Remark 5.15

As examples of potentials that satisfy the conditions of Corollary 5.14, we have the negative Dirichlet kernel \(W^n(x)=-1 -2 \sum _{k=1}^{n} w_k(x)\), the negative Féjer kernel \(W^n(x)=-\frac{1}{n}\left( \frac{1- w_n(x)}{1- w_1(x)}\right) \), and any appropriately scaled negative mollifier.

### 5.2 Continuous Transition Points

We now present a couple of technical lemmas starting with a functional inequality that gives a bound on the defect in the Gibbs inequality from below by the size of individual Fourier modes. These will be useful for the characterisation of continuous transition points provided in Theorem 5.19 and, in particular, in the proof of Theorem 1.3(b).

### Lemma 5.16

*f*is nonegative and Open image in new window, then we have, for any \(b\in {{\mathbb {R}}}\) and any \(k\in {{\mathbb {Z}}}\), the following estimate:

### Definition 5.17

### Lemma 5.18

Let \(W_\alpha (x)\) be as in Definition 5.17 and let Open image in new window denote the set of nontrivial solutions of \(F_{\kappa _\sharp }(\varrho ,\alpha )=0\) for \(\alpha \in [0,\alpha ^*) \subset [0,1]\). Then, for \(\alpha ^*\) sufficiently small, we have the uniform lower bound \(\sum \limits _{\sigma \in \mathrm {Sym}(\Lambda )}|{\widetilde{\varrho }}(\sigma (k^\sharp ))|^2 >c\) for all \(\varrho \in {\mathcal {C}}\) and for some \(c>0\) independent of \(\alpha \in [0,\alpha ^*)\).

We are now in the position to give the precise statement of Theorem 1.3(b) and prove it. We present the proofs of Lemma 5.16 and Lemma 5.18 after the proof of Theorem 5.19.

### Theorem 5.19

Let \(W_\alpha (x)\) be as in Definition 5.17 such that \( {\Theta (k^\sharp ) \leqq 2}\) where \(\Theta (k)\) is as defined in (2.2). Assume further that \(W_\text {u}\) and \(W_\text {s}\) are bounded below. Then, for \(\alpha \) sufficiently small, the system exhibits a continuous transition point at \(\kappa _c=\kappa _\sharp \).

### Proof

*c*and the monotonicity of the function \({\mathcal {G}}\) to further estimate

*c*is precisely the constant from Lemma 5.18 for \(\alpha \in [0,\alpha ^*)\). Since \(\varrho \) is a zero of \(F_{\kappa _\sharp }(\varrho ,\alpha )=0\), we have the following estimate:

### Proof of Lemma 5.16

*g*is admissible in (5.5). The estimate (5.3) follows by plugging this specific choice of

*g*into (5.5):

*z*with \({{\widetilde{{\mathcal {G}}}}}(0)=0\). Once we have shown (5.8), the proof concludes by combining this with (5.7) to deduce that

*z*. A sufficient condition for the monotonicity of this quotient is that quotient of the coefficients of the individual power series expansion of numerator and denominator are also increasing (cf. [42, Theorem 4.4], [15]). First of all, we observe that the odd coefficients are zero. We are left to investigate

### Proof of Lemma 5.18

*e*represents the identity element. For fixed \(\alpha \), we consider the map, \({\overline{F}} : {L}^2_{k^\sharp }(U) \times {{\mathbb {R}}}^+ \rightarrow {L}^2(U)\), \((\varrho ,\kappa ) \mapsto F_\kappa (\varrho ,\alpha )\) and note that any \(\varrho \) such that \({\overline{F}}(\varrho ,\kappa )=0\) is obviously in \({L}^2_{k^\sharp }(U)\). Additionally, any zero of \({\overline{F}}\) defined above is also a zero of \(F^* :{L}^2_{k^\sharp }(U) \times {{\mathbb {R}}}^+ \rightarrow {L}^2_{k^\sharp }(U)\), which is defined as

Now, since we need a lower bound which is uniform in \(\alpha \), we redefine \(F^*\) to be a function of \(\alpha \), that is, \(F^*: X \times {{\mathbb {R}}}^+ \rightarrow {L}^2_{k^\sharp }(U)\), where \(X:= {L}^2_{k^\sharp }(U) \times {{\mathbb {R}}}\) is Banach space equipped with the norm \(\left||\cdot \right||_2 + |\cdot |\) and \(f=(\varrho ,\alpha ) \in X\) a typical element of the space. We will now show that due to the particular structure of the problem one can still apply a Crandall–Rabinowitz type argument and obtain existence of local bifurcations. What follows below is a description of the Lyapunov–Schmidt decomposition for the map \(F^*\) and a slightly modified version of the proof of the Crandall–Rabinowitz theorem as presented in [46].

*X*to \({L}^2_{k^\sharp }(U)\), withwhere \(w_1 \in {L}^2_{k^\sharp }(U)\). We will also need \(D^2_{f \kappa }F^*(f,\kappa )= \begin{pmatrix}D_{\varrho \kappa } F^*&D_{\alpha \kappa }F^* \end{pmatrix}\), withThen by using the arguments of Theorem 4.2, we see that \(N:=\ker (D_f F^*((0,0),\kappa _\sharp ))=\mathrm {span}[w_{k^\sharp }] \times {{\mathbb {R}}}\widetilde{=} {{\mathbb {R}}}^2\) and \(Z_0 := R^\perp = ({{\,\mathrm{Im}\,}}(D_f F^*((0,0),\kappa _\sharp )))^\perp = \mathrm {span}[w_{k^\sharp }]\). Thus, \(D_f F^*((0,0),\kappa _\sharp ))\) is Fredholm and we have the following decompositions into complementary subspaces:

*U*of \(((0,0),\kappa _\sharp )\) in \(N \times {{\mathbb {R}}}\) and

*V*of (0, 0) in \(X_0\) along with a \(C^1\) function \(\Psi :U \rightarrow V\) such that every solution of \(G(v,w,\kappa )=0\) in \(U\times V\) is of the form \((v,\kappa ,\Psi (v,\kappa ))\) with \(\Psi ((0,0),\kappa _\sharp )= (0,0)\). Thus in

*U*we are left to solve

*N*giving \(D_\kappa \Psi ((0,0),\kappa _\sharp ),0)=0\). Using an argument similar to the above one, one can show that \(D_{{\widetilde{\varrho }}(k^\sharp )} \Psi ((0,0),\kappa _\sharp )=0 \in L(N,X_0)\).

*N*can be represented by \(({\widetilde{\varrho }}(k^\sharp ),\alpha )=(s,\alpha )\) we proceed by rewriting \(\Phi \) as follows:where we have used the fact that \(\Phi ((0,\alpha ),\kappa )=0\), since \(\varrho _\infty \) is always a trivial solution. Now, \({\widetilde{\Phi }}:{{\mathbb {R}}}^2 \times {{\mathbb {R}}}\rightarrow {{\mathbb {R}}}\) is the map, which we analyse in the neighbourhood

*U*and nontrivial solutions correspond to \(s \ne 0\). Let \({\widehat{v}}= (ts w_{k^\sharp },\alpha ) \in N\), then we compute

*Q*maps the range of \( D_\varrho F^*((0,0),\kappa _\sharp )\) to zero. Noting that \(D_s \Psi ((0,0),\kappa _\sharp ))=D_{{\widetilde{\varrho }}(k^\sharp )} \Psi ((0,0),\kappa _\sharp )=0\), we finally haveThus we can apply the implicit function theorem to obtain a function \(C^1(V_1;V_2)\), \(\varphi (s,\alpha )\) such that \({\widetilde{\Phi }}((s,\alpha ),\varphi (s,\alpha ))=0\), where \(V_1\) and \(V_2\) are neighbourhoods of (0, 0) and \(\kappa _\sharp \) respectively and \(V_1 \times V_2 \subset U\). Additionally, in \(V_1 \times V_2\) every solution of \({\widetilde{\Phi }}\) (and hence \(\Phi \)) is of the form \(((s,\alpha ),\varphi (s,\alpha ))\) and \(\varphi ((0,\alpha ))=\kappa _\sharp \). We know however from Theorem 4.2 that we could apply the same set of arguments for fixed \(\alpha \in [0,1]\) to obtain single locally increasing branches which, at least for some small neighbourhood around 0, must coincide with \(\varphi (s,\alpha )\). Thus, we now know that for each \(\alpha \in [0,1]\), we can find \(\epsilon _\alpha >0\) such that \(\varphi (s,\alpha )>\kappa _\sharp \) for \(0<|s|<\epsilon _\alpha \). Now, let \(\alpha \in [0,\alpha ^*)=A\). If we show that \(\inf _A \epsilon _\alpha =\epsilon '>0\) for \(\alpha ^*\) small enough, we can conclude the proof. To see this, set \(V_1'= V_1 \cap (-\epsilon ',\epsilon ')\times [0,\alpha ^*)\) and observe that \(((s,\alpha ),\varphi (s,\alpha ))\) are the only solutions in \(V_1' \times V_2\) and \(\varphi (s,\alpha )=\kappa _\sharp \) implies \((s,\alpha )=(0,\alpha )\). Thus in \(V_1'\), \((0,\alpha )\) is the only solution of the bifurcation equation which would provide the desired result. Assume now that there exists no \(\alpha ^*\), such that \(\inf _A \epsilon _\alpha >0\). It is straightforward to check that this would violate the continuity of \(\varphi \) since \(\epsilon _0>0\). \(\quad \square \)

As an immediate consequence of Theorem 5.19 we have

### Corollary 5.20

Let \(W_\alpha (x)\) be as in Definition 5.17 such that \(W_\text {u}\) and \(W_\text {s}\) are bounded below. Then, for \(\alpha \) sufficiently small, \(\varrho _\infty \) is the unique minimiser of the free energy \(\mathcal {F}_\kappa (\varrho )\) for \(\kappa \in (0, C(n)\kappa _\sharp ]\), where *C*(*n*) is as defined in Lemma 5.16.

### Proof

The proof follows the same arguments as Theorem 5.19 with \(\kappa _\sharp \) replaced by \(C(n)\kappa _\sharp \). \(\quad \square \)

*W*has only one negative mode, say \(w_{k^\sharp }\), then we have

*W*, Corollary 5.20 provides a sharper estimate on the range of \(\kappa \) for which the uniform state is a unique minimiser of the free energy.

### Remark 5.21

We conclude this section with the following useful proposition which provides us with a comparison principle for interaction potentials to check if they possess continuous transition points.

### Proposition 5.22

Let \(W\in {{\mathbb {H}}}_\text {s}^c\) be an interaction potential such that the associated free energy \({\mathscr {F}}^W_\kappa (\varrho )\) has a continuous transition point. Additionally, assume that \(G\in {{\mathbb {H}}}_\text {s}^c\) is such that \({{\,\mathrm{\mathrm{arg\,min}}\,}}_{k \in {{\mathbb {N}}}^d/ \{{\mathbf {0}}\}}{\widetilde{G}}(k)={{\,\mathrm{\mathrm{arg\,min}}\,}}_{k \in {{\mathbb {N}}}^d/ \{{\mathbf {0}}\}}{\widetilde{W}}(k)= k^\sharp \) and \({\widetilde{G}}(k^\sharp )={\widetilde{W}}(k^\sharp )\) with \({\widetilde{G}}(k)\geqq {\widetilde{W}}(k)\) for all \(k \ne k^\sharp ,k \in {{\mathbb {N}}}^d\). Then \({\mathscr {F}}^G_{\kappa }(\varrho )\) exhibits a continuous transition point.

### Proof

*G*, the value of \(\kappa _\sharp \) is the same for

*G*and

*W*, we have for Open image in new window that

*W*possesses a continuous transition point, we obtain

## 6 Applications

### 6.1 The Generalised Kuramoto Model

Finally, let us mention that there is a larger picture in the Kuramoto model when different frequency oscillators are allowed, see [1] for a nice review of the subject and [19] for recent numerical work on phase transitions for this problem.

Although it is possible to directly apply Theorem 5.19 to prove the existence of a continuous phase transition for this system, we employ an alternative approach that gives us more qualitative information about the structure of the nontrivial solutions.

### Proposition 6.1

The generalised Kuramoto model exhibits a continuous transition point at \(\kappa _c=\kappa _\sharp \). Additionally, for \(\kappa >\kappa _c\), the equation \(F(\varrho ,\kappa )=0\) has only two solutions in \({L}^2(U)\) (up to translations). The nontrivial one, \(\varrho _\kappa \) minimises \({\mathscr {F}}_\kappa \) for \(\kappa >\kappa _c\) and converges in the narrow topology as \(\kappa \rightarrow \infty \) to a normalised linear sum of equally weighted Dirac measures centred at the minima of *W*(*x*).

### Proof

The strategy of proof is similar to that of Theorem 5.19, i.e, we show that at \(\kappa =\kappa _\sharp \), \(\varrho _\infty \) is the unique minimiser of the free energy. We do this by showing that \(F(\varrho ,\kappa )=0\) has a unique solution at \(\kappa =\kappa _\sharp \), which implies, by Proposition 2.4 (since *W* satisfies Assumption (**A2**)), uniqueness of the minimiser.

*n*, \(r_n(a):=\frac{I_{n+1}(a)}{I_n(a)}\), and \(a= \beta \kappa {\widetilde{\varrho }}(k^\sharp )\). This equation is similar to the one derived in Section VI of [5] (cf. [4, 53]). It is also qualitatively similar to the self-consistency equation associated with the two-dimensional Ising model.

For \(\varrho =\varrho _\infty \), we know that \({\widetilde{\varrho }}(\kappa _\sharp )=0\). We argue that any nontrivial solution of \(F(\varrho ,\kappa )=0\) must have \({\widetilde{\varrho }}(k^\sharp ) \ne 0\). Assume this is not the case, that is, there exists \( \varrho _\kappa \ne \varrho _\infty \) which satisfies \(F(\varrho _\kappa ,\kappa )=0\) and \(\widetilde{\varrho _\kappa }(k^\sharp ) =0\), then from (6.1) we have that \(\varrho =\varrho _\infty \). Thus \(F(\varrho ,\kappa )\) has non-trivial solutions if and only if (6.2) has nonzero solutions. One should note that since \(I_1\) is odd and \(I_0\) is even, nonzero solutions to (6.2) come in pairs, i.e, if *a* is a solution so is \(-a\). However, these two solutions are simply translates of each other.

*a*large enough, \(a>M(a,\kappa )\) (since \(r_0(a) \rightarrow 1\), as \(a \rightarrow \infty \), and is strictly increasing). Thus by the intermediate value theorem, there exists at least one positive

*a*such that (6.2) holds for every \(\kappa >\kappa _\sharp \). One can now show that \(\frac{\partial {M}}{\partial {a}}(a,\kappa )\) is strictly decreasing for \(a>0\). This is equivalent to showing that \(r_0''(a)\) is strictly negative. We have

*a*. Using previous arguments, the above inequality tells us that \(0<\frac{\kappa _2}{\kappa _1}a_{\kappa _1}<a_{\kappa _2}\) which implies that \(a_{\kappa } \rightarrow \infty \), as \(\kappa \rightarrow \infty \). Finally, we have the following form for the solution:

*A*be a continuity set of \(\delta _0\), then if \(0 \notin A\) it follows that \(0 \notin \partial A\). By a large deviations argument, Laplace’s principle, we have thatThus, Open image in new window for every Borel set not containing 0 and thus Open image in new window for \(0 \in A\). By the portmanteau theorem (cf. [13, Theorem 2.1]), we have the desired convergence. For arbitrary

*k*, one can apply the same argument on periods of the function \(\cos (2 \pi k x /L)\), and due to the periodicity/symmetry of the solution the masses in each Dirac point are equal. \(\quad \square \)

### 6.2 The Noisy Hegselmann–Krause Model for Opinion Dynamics

*N*interacting agents such that each agent is only influenced by the opinions of its immediate neighbours. In the large

*N*limit, we obtain again the McKean–Vlasov PDE with the interaction potential \(W_{\mathrm {hk}}(x)=-\frac{1}{2}\left( \left( |x|-\frac{R}{2}\right) _-\right) ^2\) for some \(R>0\). The ratio

*R*/

*L*measures the range of influence of an individual agent with \(R/L=1\) representing full influence, that is, any one agent influences all others. In order to analyse this system further, we compute the Fourier transform of \(W_{\mathrm {hk}}(x)\) given by

*R*/

*L*the problem reduces to a computational one, namely checking that the conditions of Theorem 4.2 are satisfied. Also, \(W_{\text {hk}}(x)\) is normalised and decays to 0 uniformly as \(R \rightarrow 0\), that is, as the range of influence of an agent decreases so does its corresponding strength. We could define a rescaled version of the potential, \(W_{\mathrm {hk}}^R(x)=-\frac{1}{2R^3}\left( \left( |x|-\frac{R}{2}\right) _-\right) ^2\) which does not lose mass as \(R \rightarrow 0\). We conclude this subsection with the following result:

### Proposition 6.2

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

### Proof

We define \(C:=||W_{\mathrm {hk}}^R||_1\) and note that it is independent of *R*. The proof follows from the observation that \(W_{\mathrm {hk}}^R \rightarrow - C \delta _0\) as \(R \rightarrow 0\) and applying Corollary 5.14. \(\quad \square \)

### 6.3 The Onsager Model for Liquid Crystals

### Proposition 6.3

- (a)
The trivial branch of the Onsager model, \(W_1(x)\), has infinitely many bifurcation points.

- (b)
The trivial branch of the Maiers–Saupe model, \(W_2(x)\), has exactly one bifurcation point.

- (c)
The trivial branch of the model \(W_\ell (x)\) for \(\ell \) even has at least \(\frac{\ell }{4}\) bifurcation points if \(\frac{\ell }{2}\) is even and \(\frac{\ell }{4} + \frac{1}{2}\) bifurcation points if \(\frac{\ell }{2}\) is odd.

- (d)
The trivial branch of the model \(W_\ell (x)\) for \(\ell \) odd has infinitely many bifurcation points if \(\frac{\ell -1}{2}\) is even and at least \(\frac{\ell +1}{4}\) bifurcation points if \(\frac{\ell -1}{2}\) is odd.

### Proof

*n*and positive otherwise. For the rest of the proof we will always assume that \(k>0\). We will now attempt to show that all nonzero values of \({\widetilde{W}}_\ell (k)\) for \(k>0\) are distinct. Assumeing

*l*is even, we have the following explicit form of \({\widetilde{W}}_\ell (k)\):

*k*is assumed to be even and \(k < \ell +2\)(since it is zero for

*k*odd or \(k\geqq l+2\)). From the above expression one can check that the denominator is strictly increasing as

*k*increases from 2 to \(\ell \), thus \(|{\widetilde{W}}_\ell (k)|\) is strictly decreasing. Thus the nonzero values of \({\widetilde{W}}_\ell (k)\) are distinct for \(\ell \) even. For \(\ell \) odd, we first note that by simple integration by parts we can derive the following recursion relation:

*k*is even(and thus not equal to \(\ell \)). For \(\ell =1\), we have the following alternative formula for \({\widetilde{W}}_\ell (k)\) for even

*k*:

*k*even. From the recursion formula in (6.6) it follows that this holds true for all odd \(\ell \), that is, \(|{\widetilde{W}}_\ell (k)|\) takes distinct values for

*k*even.

Assume now that \(\ell =1\)(that is the Onsager model), then as mentioned earlier we can deduce from (6.7) that \({\widetilde{W}}_1(k)\) is distinct and negative for all *k* even. It follows that \({\widetilde{W}}_1(k)\) satisfies the conditions of Theorem 4.2 for all even *k*, thus completing the proof of (a).

Now let \(\ell >2\) and even. It is clear from the expression in (6.5) that then \({\widetilde{W}}_\ell (k)\) can be negative only if \(\cos (k \pi /2)/\Gamma \left( \frac{1}{2} (-k+\ell +2)\right) \) is negative. This happens if and only if \(\frac{k}{2}\) is odd and \(k<\ell +2\) since if \(k \geqq \ell +2\), \(\frac{1}{\Gamma \left( \frac{1}{2} (-k+\ell +2)\right) }\) is evaluated at a negative integer and thus \({\widetilde{W}}_\ell (k)=0\). Since by the previous arguments each \(\frac{k}{2}\) odd with \(k<\ell +2\) corresponds to a distinct value of \({\widetilde{W}}_\ell (k)\), we can apply Theorem 4.2 to deduce that such *k* correspond to bifurcation points. Given an \(\ell >2\) and even, there are \(\frac{\ell }{4} + \frac{1}{2}\) such *k* if \(\frac{\ell }{2}\) is odd and \(\frac{\ell }{4}\) if \(\frac{\ell }{2}\) is even. This completes the proof of (c).

Now, we let \(\ell >2\) and odd. One can check again that \({\widetilde{W}}_\ell (k)\) is negative if and only if \(\frac{k}{2}\) is odd and \(k < \ell +2\) when \(\frac{\ell -1}{2}\) is odd and if *k* is even, but \(\frac{k}{2}\) is odd if \(k < \ell +2\), when \(\frac{\ell -1}{2}\) is even. For \(\frac{\ell -1}{2}\) odd there are \(\frac{\ell +1}{4}\) such *k*, while for \(\frac{\ell -1}{2}\) even there are infinitely many such *k*. Applying Theorem 4.2 again, gives us (d). \(\quad \square \)

The above result provides us with a finer analysis to that presented in [24], as we are able to count the solutions for general odd and even \(\ell \), instead of just proving the existence of nontrivial solutions. The above result also generalises the work in [49] which studied a truncated version of the Onsager model with only a finite number of modes and proved the existence of nontrivial solutions. It also partially recovers results from [57, Theorem 2] in which the non-truncated Onsager model is analysed. We refer the reader to [70] for an analysis of the Onsager model in 2 dimensions, that is, for liquid crystals that live in 3 dimensions with two degrees of freedom.

### 6.4 The Barré–Degond–Zatorska Model for Interacting Dynamical Networks

### Proposition 6.4

- (a)
If \(2 {\widetilde{W}}(2) - {\widetilde{W}}(1) >0\), then the system exhibits a continuous transition point;

- (b)
If \(2 {\widetilde{W}}(2) - {\widetilde{W}}(1) <0\), then the system exhibits a discontinuous transition point.

The assumptions in the proposition essentially imply a separation of the Fourier modes. It follows immediately under these assumptions that \(k=1\) satisfies the conditions of Theorem 4.2 and thus \(\kappa _*=-\frac{ (2L)^{\frac{1}{2}} }{\beta {\widetilde{W}}(1)} \) corresponds to a bifurcation point of the system. Additionally, looking at Figure 1 one can see that the conditions (a) and (b) from the above proposition are consistent with our analysis for the existence of continuous and discontinuous transition points. If \({\widetilde{W}}(1)\) and \({\widetilde{W}}(2)\) are resonating/near-resonating then it follows that condition (b), that is, \(2 {\widetilde{W}}(2) - {\widetilde{W}}(1) <0\) must hold for \(\delta _*\) small, where \(\delta _*\) is as introduced in Definition 5.10. Indeed, let \(k=1,2\) be elements of the set \(K^{\delta _*}\), then we have \(2 {\widetilde{W}}(2) - {\widetilde{W}}(1) ={\widetilde{W}}(1) + 2({\widetilde{W}}(2)- {\widetilde{W}}(1) ) \leqq {\widetilde{W}}(1) +2\delta _* <0\), for \(\delta _*\) sufficiently small. Similarly, using Lemma 5.22 and comparing with an \(\alpha \)-stabilised potential say \(G_\alpha \), one can argue that if \({\widetilde{W}}(1)\) is the dominant mode then condition (a), that is, \(2 {\widetilde{W}}(2) - {\widetilde{W}}(1) >0\) must hold for \(\alpha \) small, where \(\alpha \) is as defined in Definition 5.17.

### 6.5 The Keller–Segel Model for Bacterial Chemotaxis

*chemotaxis*in the biology literature [47]. For this system, \(\varrho (x,t)\) 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:

*c*. Thus, the stationary Keller–Segel equation is given by

**A2**), Theorem 2.3 does not apply directly. However we can circumvent this issue to obtain the following result:

### Theorem 6.5

Consider the stationary Keller–Segel equation (6.9). For \(d\leqq 2\) and \(s \in (\frac{1}{2},1]\), it has smooth solutions and its trivial branch \((\varrho _\infty ,\kappa )\) has infinitely many bifurcation points.

### Proof

*c*(

*x*) are bounded, one can then check that \(\left||\partial _\alpha \varrho \right||_2 < \infty \) and thus \(\varrho \in {H}^{\ell +1}(U)\). We can then bootstrap to obtain smooth solutions.

*d*coordinates. Our strategy will be to apply Theorem 4.2 after reducing the problem to the symmetrised space \({L}^2_{{\text {ex}}}(U)\) and then use the discussion in Remark 4.6. Then, showing that a particular [

*k*] corresponds to a bifurcation point reduces to the condition

*p*is a prime, and \(n \in {{\mathbb {N}}}\), satisfy the conditions of being a bifurcation point. We need to check that

*p*there is a unique way (up to permutations) of expressing \(p^{2n}\) as the sum of two squares and this is precisely \((p^n)^2 + 0^2\). Jacobi’s two square theorem tells us that number of representations,

*r*(

*z*), of a positive integer

*z*as the sum of two squares is given by the formula

*z*of the form \(4 k +\ell , k\in {{\mathbb {N}}},\ell \geqq 1\). If \(p=2\), then \(d_{1,4}(2^{2n})=1\) and \(d_{3,4}(2^{2n})=0\) and thus \(r(2^{2n})=1\). For any odd prime,

*p*, we know that it is either of the form \(4k+1\) or \(4k+3\). For either case, one can check that we have \(d_{1,4}(p^{2n})=1+n\) and \(d_{3,4}(p^{2n})=n\) and thus \(r(p^{2n})=1\). The expression for the bifurcation points then follows from the discussion in Remark 4.6. \(\quad \square \)

## Notes

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