Long-time behaviour and phase transitions for the McKean--Vlasov equation on the torus

We study the McKean-Vlasov equation \[ \partial_t \varrho= \beta^{-1} \Delta \varrho + \kappa \nabla \cdot (\varrho \nabla (W \star \varrho)) \, , \] with periodic boundary conditions on the torus. We first study the global asymptotic stability of the homogeneous steady state. We then focus our attention on the stationary system, and prove the existence of nontrivial solutions branching from the homogeneous steady state, through possibly infinitely many bifurcations, under appropriate assumptions on the interaction potential. We also provide sufficient conditions for the existence of continuous and discontinuous phase transitions. Finally, we showcase these results by applying them to several examples of interaction potentials such as the noisy Kuramoto model for synchronisation, the Keller--Segel model for bacterial chemotaxis, and the noisy Hegselmann--Krausse model for opinion dynamics.


Introduction
Systems of interacting particles arise in a myriad of applications ranging from opinion dynamics [HK02], granular materials [BCCP98,CMV03,BGG13] and mathematical biology [KS71,BCM07] to statistical mechanics [MA01], galactic dynamics [BT08], droplet growth [CS17], plasma physics [Bit86], and synchronisation [Kur81]. Apart from being of independent interest, these systems find applications in a diverse range of fields such as : particle methods in numerical analysis [DMH10], consensus-based methods for global optimisation [CCTT18], and nonlinear filtering [CL97]. They have also been studied in the context of multiscale analysis [GP18], in the presence of memory-like effects and in a non-Markovian setting [DP18], and in the discrete setting of graphs [EFLS16].
In this paper, we analyse the partial differential equation (PDE) associated to the system of interacting stochastic differential equations (SDEs) on T d (the torus of size L d > 0) of the following form where the X i t ∈ T d , i = 1 . . . N represent the positions of the N "particles", W is a periodic interaction potential, and the B i t , i = 1 . . . N represent N independent T d -valued Brownian motions. The constants κ, β > 0 represent the strength of interaction and inverse temperature respectively (one of the two parameters is redundant; thus we fix β 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 [KCB + 13, PT13, MT14b, MT14a] and the references therein for a comprehensive list of such models.
Systems of interacting diffusions have been studied extensively. They were first analysed by McKean (cf. [McK66,McK67]) who noticed an interesting relation between them and a class of nonlinear parabolic partial differential equations. In particular, it is well known (cf. [Oel84,Szn91]) that for this class of SDEs one can pass to the so-called mean field limit: if we consider the empirical measure defined as follows then as N → ∞, E(̺ (N ) ) converges, in the sense of weak convergence of probability measures, to some measure ̺ which satisfies the following nonlocal parabolic PDE The above equation is commonly referred to as the McKean-Vlasov equation, the latter name stemming from the fact that it also arises as the overdamped limit of the Vlasov-Fokker-Planck equation. Equation (1.1) can also be thought of as a nonlinear Fokker-Planck equation for the following nonlinear SDE, commonly referred to as the McKean SDE, dX t = −κ(∇W ⋆ ̺)(X t , t) dt + 2β −1 dB t .
The PDE (1.1) itself has a very rich structure -associated to it we have the following free energy functional (1.2) where S(̺) and E(̺, ̺) represent the entropy and interaction energy associated with ̺ respectively. It is well known, starting from the seminal work in [JKO98,Ott01], that this equation belongs to a larger class of dissipative PDEs including the heat equation, the porous medium equation, and the aggregation equation, which can be written in the form for some free energy F and are gradient flows for the associated free energy functional with respect to the d 2 transportation distance defined on probability measures having finite second moment, see [CMV03,Vil03]. We refer the reader to [AGS08,San15] for more information on the abstract theory of gradient flows in the space of probability measures. Our goals are to study some aspects of the asymptotic behaviour and the stationary states of 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 equilbrium. We extend the L 2 -decay results of [CJLW17] to arbitrary dimensions and arbitrary sufficiently nice interactions and also provide sufficient conditions for convergence to equilibrium in relative entropy.
The rest of the paper is devoted to the analysis of the properties of non-trivial stationary states of the Mckean-Vlasov system, i.e., nontrivial solutions of Previous results in this direction include those by Tamura [Tam84], who provided some criteria for the existence of local bifurcations on the whole space by using tools from nonlinear functional analysis, in particular, the Crandall-Rabinowitz theorem. Unfortunately, his analysis depends crucially on the unphysical assumption that the interaction potential is an odd function. One of the main results of the present work is complete, quantitative, local bifurcation analysis under physically realistic assumptions. Dawson [Daw83] studied for the first time the existence of nontrivial stationary states for a particular double-well confinement and Curie-Weiss interaction on the line. The existence of nontrivial stationary states or the bifurcation of nontrivial solutions from the homogeneous steady state is usually referred as phase transition in the literature. We also mention that more recently several authors [Tug14,DFL15,BCnCD16] looked at the existence of phase transitions in the whole space with different confinement and interactions. The most related work to us in the literature is due to Chayes and Panferov [CP10], who studied the problem on the torus and provided some criteria for the existence of continuous and discontinuous phase transitions.
In addition to presenting an existence and uniqueness theory for the evolution problem, we extend considerably the results of both [Tam84] and [CP10]. 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 [CP10] 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, W (k) := (2/L) 1/2 W (x) cos 2πk L x dx for k ∈ Z, k > 0. We work with classical solutions of (1.1) which are constructed in Theorem 2.2. Mckean-Vlasov equation (1.1) with smooth initial data and smooth, even, interaction potential W . Then we have:
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, i.e., the interaction potential is not H-stable. Thus, the concept of H-stability introduced by Ruelle [Rue99] is relevant for the study of the stationary McKean-Vlasov equation as noticed in [CP10]. We have the following conditions for the existence of bifurcating branches of steady states.
We are also able to sharpen sufficient conditions for the existence continuous or discontinuous bifurcating branches. The following theorem is a simplified version of the exact statements that are presented in Theorem 5.7 and Theorem 5.15.
such that k a = k b + k c , then κ c is a discontinuous transition point. (b) Let k ♯ = arg min k W (k) be well-defined with W (k ♯ ) < 0. Let W α denote the potential obtained by multiplying all the negative Fourier modes W (k) except W (k ♯ ) by some α ∈ (0, 1]. Then if α is made small enough, the transition point κ c is continuous. As it will become clear in § 5, the condition in Theorem 1.3 (b) is essentially an assumption on the size of the spectral gap of the linearised McKean-Vlasov operator. 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 continuous transition point in which there is one dominant negative mode and all other negative modes are restricted to a small neighbourhood of 0. Figure 1(b) shows the scenario for the existence of a discontinuous transition point in which there are multiple resonating/near-resonating dominant modes which satisfy an algebraic condition corresponding to the conditions of Theorem 5.7. This work provides a complete local and global bifurcation analysis for the Mckean-Vlasov equation on the torus. This enables us to study phase transitions for several important models that have been introduced in the literature. This is  As an example of the typical bifurcation diagram expected for this kind of system, we discuss the noisy Kuramoto model which has the interaction potential W (x) = −(2/L) 1/2 cos(2πx/L). For κ sufficiently small, the uniform state is the unique stationary solution. At some critical κ = κ c a clustered solution branches out from the uniform state and for all κ > κ c this clustered state is preferred solution, i.e., it is the global minimiser of the free energy, F κ,β . The bifurcation diagram and a plot of the clustered solution can be seen in Figure 2. The model is discussed in more detail in § 6.1.

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 relegated to Appendix A. 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.

Preliminaries
2.1. Set up and notation.  . An example of a clustered solution representing phase synchronisation of the oscillators measure, and by P + ac (U ) the subset of P ac (U ) having strictly positive densities a.e. Additionally, C k (U ) will denote the restriction to U of all L-periodic and k-times continuously differentiable functions, D(U ) the space of test functions, and f, g µ the L 2 (U, µ) inner product.

Assumptions on W .
Throughout the subsequent discussion we will assume that W (x) is at least integrable and coordinate-wise even, that is For the evolutionary problem we will assume while for the stationary problem we will assume where the L p (U ) with 1 ≤ p ≤ ∞ represent the periodic Lebesgue spaces and W k,p (U ) represent the periodic Sobolev spaces with H k (U ) = W k,2 (U ). Wherever required, weaker or stronger assumptions will be indicated in the text. As one may expect, the assumptions on 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. For the space L 2 (U ) we define the orthonormal basis, {w k } k∈Z d , k = (k 1 , k 2 , . . . , k d ), as follows: 1) and N k is defined as where δ i,j denotes the Kronecker delta. We then have the following form for the discrete Fourier transform of any f ∈ L 2 (U ) We denote by "⋆" the convolution of any two functions, f (x), g ∈ L 2 (U ) and for f (x) = W (x) we have the following representation in Fourier space: Here, we have used the fact that W (x) is coordinate-wise even. Sym(Λ) represents the symmetry group of the product of two-point spaces, Λ = {1, −1} d , which acts on Z d by pointwise multiplication, i.e., . Another expression that we will use extensively in the sequel is the Fourier expansion of the following bilinear form It will be useful to note that for any function g(x) and k ∈ Z d the sum σ∈Sym(Λ) | g(σ(k))| 2 is translation invariant. In later sections we will also use the space L 2 s (U ) ⊂ L 2 (U ), which we define as the space of coordinate-wise even functions in L 2 (U ) given by It should be noted that any pointwise properties (like being coordinate-wise even) should be understood in a limiting sense, i.e., L 2 s (U ) is the completion of coordinate-wise even C ∞ (U ) functions in the · 2 norm. The space L 2 s (U ) is a closed subspace of L 2 (U ) and thus is a Hilbert space in its own right. It is also easy to check that {w k } k∈N d ⊂ {w k } k∈Z d forms an orthonormal basis for L 2 s (U ). If g is assumed to be in L 2 s (U ), then the above expressions reduce to, In addition, the sign of the individual Fourier modes of W is quite important in the subsequent analysis and we introduce the following definition.
where, W k = W, w k . This is by (2.2) equivalent to the condition that, Thus every potential is decomposed into two parts Hereby, (a) + = max{0, a} (resp. (a) − = min{0, a}) denotes the positive (resp. negative) part for a real number a ∈ R.
An immediate consequence of the identity (2.2) 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. [Rue99]). We refer to [CnCP15] 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. We present an existence and uniqueness result for the McKean-Vlasov equation and provide only a sketch of the proof. The proof is quite standard. Our result is an extension of [CJLW17, Theorem 4.5] since we consider all potentials W satisfying Assumption (A1) in any dimension d, as opposed to [CJLW17,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 [CJLW17]. We prove below the existence of classical solutions ̺(·, t) ∈ C 2 (U ) to the system (2.4) is strictly positive and has finite entropy, i.e, ̺(·, t) > 0 and S(̺(·, t)) < ∞, for all t > 0.
Sketch of proof. Existence. We start by arguing that for smooth initial data, ̺ ′ ∈ P ac (U ) ∩ C ∞ (U ), the following sequence of linear parabolic PDE have unique smooth solutions, The proof of this follows from classical results in the theory of linear parabolic equations (cf. [Eva10,Chapter 7]). Thus, we have a sequence {̺ n } n∈N ∈ C ∞ (U × [0, T ]) ∩ P ac (U ). We define a weak solution of our nonlocal McKean-Vlasov PDE as any function ̺ ∈ L 2 (0, T ; H 1 (U )) ∩ L ∞ (0, T ; L 2 (U )) and ∂ t ̺ ∈ L 2 (0, T ; H −1 (U )) such that The idea then is to show that the sequence {̺ n } n∈N has some compactness properties so that up to a subsequence we obtain limits which we would then expect to satisfy our definition of a weak solution. Using essentially the same ideas as in [CJLW17] and the given regularity of W we obtain the following results where ̺ n k is a subsequence and ̺ is the limiting object which we expect to be a weak solution. Additionally, both the limit and the sequence satisfy the following estimate: Now we multiply (2.5) by some η ∈ L 2 (0, T ; H 1 (U )) for some n = n k and integrate by parts to obtain Consider the following term, For the first term on the right hand side notice that For the second term we have We know that ̺ n k −1 − ̺ 1 ≤ ̺ n k −1 1 + ̺ 1 = 2 therefore we obtain Using (2.6a) and (2.6b) along with the estimates (2.9) and (2.10) we obtain Additionally, we can use (2.6c) to argue that Finally using (2.6b) we obtain Putting it all together we obtain What remains is to check that ̺ satisfies the initial condition, i.e., ̺(·, 0) = ̺ ′ . This condition makes sense as we know from [Eva10, §5.9.2, Theorem 3] that ̺ ∈ C(0, T ; L 2 (U )). Picking η ∈ C 1 (0, T ; H 1 (U )) such that η(x, T ) = 0 the weak form of the PDE reduces to: Similarly, choosing the same η for (2.8), we obtain As k → ∞, we obtain the following relationship: Since η(x, 0) can be any arbitrary H 1 (U ) function it follows that ̺(x, 0) = ̺ ′ . It should also be noted that (2.6a) implies that ̺(·, t) ∈ P ac (U ), t a.e. This completes the proof of existence of a weak solution for smooth initial data. We can now relax the regularity assumption on the initial condition to ̺ ∈ L 2 (U ) ∩ P ac (U ) to obtain existence of a weak solution. We do this by mollifying the initial data ̺ ′ ǫ = ϕ ǫ ⋆ ̺ 0 (here ϕ ǫ is the standard Friedrichs mollifier), applying the above arguments again, and then passing to the limit as ǫ → 0. Uniqueness. Assume ̺ 1 and ̺ 2 are two weak solutions for some initial data ̺ 0 ∈ L 2 (U ) ∩ P ac (U ) and set We can estimate the first terms on the right hand side of the above expression as follows The second term follows in a very similar manner Using the above estimates and setting η = ξ in (2.11), we obtain Applying [Eva10, § 5.9.2, Theorem 3], we know that ∂ t ξ, ξ H 1 , Thus for almost every t ∈ (0, T ], it must hold that Applying Grönwall's inequality and noticing from [Eva10, § 5.9.2, Theorem 3] that ̺ 1 , ̺ 2 ∈ C(0, T ; L 2 (U )), we have the desired uniqueness, i.e., ξ(t) 2 = 0 for all t ∈ (0, T ]. Regularity. We return to the sequence of linear PDEs presented in (2.5). If we mollify the initial data, ̺ ′ǫ = ϕ ǫ ⋆̺ 0 , we obtain sequence of linear parabolic PDE with smooth and bounded coefficients which thus have smooth solutions, {̺ ǫ n } ∞ n=0 . This leaves us free to take derivatives to any order. After establishing the required regularity estimates on this mollified sequence, we can pass to to the limit as n → ∞. We will again omit the standard arguments for passing to the limit as ǫ → 0 and will suppress the dependence on ǫ in our calculations. For regularity in space, we need to derive the following estimate: where k ≥ 0 and C k is a constant that depends on the H ℓ (U ) norms of the initial data for ℓ ≤ k. We prove this by induction noting that the k = 0 case follows from estimate (2.7). We assume the estimate is true for k − 1 ≥ 0 and try to prove it for k. Differentiating (2.5) by ∂ α for some multiindex α with |α| = k, multiplying it by ∆∂ α ̺ n and integrating by parts, we obtain Applying Young's inequality, we deduce All that remains now is to obtain a useful estimate on the second term on the right hand side given the regularity on the interaction potential that we have, i.e., W ∈ W 2,∞ (U ). We have by Poincaré's inequality For the second term(for some multiindex γ with |γ| = k + 1) by the Leibniz rule we obtain If d ≥ k + 1, then it is clear that the second sum vanishes and we have Let us assume that this is not the case, i.e, min(d, k + 1) = d. We now note that for |ℓ| > d we have |γ − ℓ| = |γ| − |ℓ| < k + 1 − d. Thus for |ℓ| > d, ∂ γ−ℓ ̺ n ∈ H d (U ) and by the Sobolev embedding theorem, Using this and retaining only the highest order Sobolev norms we can obtain the following estimate Thus finally we have the following bound Substituting this into (2.12), summing over all |α| = k, and integrating with respect to time we derive where in the last step we have used the induction hypothesis for ̺ n and ̺ n−1 . We are now free to pass to the n → ∞ limit and obtain The method for obtaining regularity in time is identical to [CJLW17,Theorem 4.3] and we have the following estimates for ̺ 0 ∈ H 2j (U ) ∩ P ac (U ) : Finally, we are in a position to apply our assumptions on the initial data. Setting k + 1 = 3 + d in (2.13), we have that ̺ ∈ L ∞ (0, T ; H 3+d (U )). It follows by the Sobolev embedding theorem that ̺(t) ∈ C 2 (U ). Similarly, setting 2j = 3 + d it follows that, ∂ t ̺ ∈ L 2 (0, T ; H 2+d (U )) and ∂ 2 tt ̺ ∈ L 2 (0, T ; H d (U ))(since 2 ≤ (3 + d)/2, for any d ≥ 1). Since the embedding H 2+d ֒→ H 1+d is compact, by the Aubin-Lions lemma it follows that ∂ t ̺ ∈ C(0, T ; H 1+d (U )). Again by the Sobolev embedding theorem we have ∂ t ̺ ∈ C(0, T ; C(U)), which gives us the required pointwise differentiability in time. Positivity for 0 < t 1 < t 2 < ∞ for some positive constant C. Since we know that ̺(x, t) is nonnegative this implies that inf In this subsection we present standard results about the stationary McKean-Vlasov equation that will be useful for our later analysis. The main results in this section are Theorem 2.3 which discusses the existence of solutions and their regularity, Proposition 2.4 which connects stationary solutions to minimisers of the free energy, and Theorem 2.8 which discusses the existence of minimisers for the free energy.
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 [Tam84] and [Dre87].
Proof. The weak formulation of (2.14) is where we look for solutions ̺ ∈ H 1 (U ) ∩ P ac (U ). Consider now the nonlinear map, T : P ac (U ) → P ac (U ), We have the following estimate Thus it makes sense to search for fixed points of this equation in the set E := {̺ ∈ L 2 (U ) ∩ P ac (U ) : } as all fixed points must be in this set. It is easy to check that E is a closed, convex subset of L 2 (U ). We can now redefine T to act on E. Additionally, for any ̺ ∈ E, we have that where we have used the fact that W ∈ H 1 (U ). Thus using (2.17), we have that T (E) ⊂ E is uniformly bounded in H 1 (U ). By Rellich's compactness theorem, this implies that T (E) is relatively compact in L 2 (U ), and therefore in E, since E is closed. Furthermore, T is Lipschitz continuous, i.e., we have for Before proceeding to the proof of (b), we argue that every weak solution in H 1 (U ) ∩ P ac (U ) is a fixed point of the nonlinear map, T . Consider the "frozen" version of the weak form in (2.15), where ̺ ∈ H 1 (U ) ∩ P ac (U ) is a weak solution of (2.14) and ϑ is the unknown function. The above equation is the weak form of a uniformly elliptic PDE whose associated bilinear form is coercive in the weighted space,

after transforming to the corresponding stationary backward Kolmogorov equation). To see this, set ϑ(x) = h(x)T ̺.
We then obtain the following integral formulation of the transformed PDE, Let h 1 and h 2 be two weak solutions of the above equation.
Since T ̺ has full support (it is bounded below), the weak solution to (2.19)(which exists) is unique up to normalisation and therefore if chosen to be a probability measure is unique. We observe that ϑ = T ̺ is such a weak solution, as is ̺ (by its very definition). This implies that any weak solution must be such that ̺ = T ̺.
We obtain regularity of solutions by observing that if f ∈ H m (U ), g ∈ H n (U ), then we have that f ⋆ g ∈ W m+n,∞ (U ). Then we use a bootstrap argument, i.e., W ∈ H 1 (U ), ̺ ∈ H 1 (U ) implies that ̺ = T ̺ ∈ W 2,∞ (U ). This implies that W ⋆ ̺ ∈ W 3,∞ (U ) and so on and and so forth. Thus we have that ̺ ∈ H m (U )(or W m,∞ (U )) for any m ∈ N. The strict positivity follows from the lower bound on T ̺. ■ We already know that associated with this PDE we have a free energy functional F κ,β : P + ac (U ) → R, where P + ac (U ) is the space of strictly positive absolutely continuous probability measures on U . The functional is given by where β, κ > 0, and we take W ∈ W 2,∞ . In the present work β is a fixed parameter and we write F κ = F κ,β and S = S β . Additionally, we also have the functional, J κ,β (J κ for fixed β) which is commonly referred to as the entropy dissipation functional and is given by It is obtained by computing the time rate of change of the free energy along the flow of the PDE, i.e., − d dt F κ,β (̺(x, t)). Finally we have the Gibbs state map F κ,β : P ac (U ) → P ac (U ) (F κ for fixed β). This equation encodes the stationary states as fixed points of the nonlinear mapping T from (2.16) For the subsequent arguments, let β be fixed and positive. The identification of stationary states (2.14), critical points of F κ and J κ , and zeros of F κ is given by the following proposition which we shall use later on several occasions. Proposition 2.4. Assume W (x) satisfies Assumption (A2) and fix κ > 0. Let ̺ ∈ P + ac (U ). Then the following statements are equivalent : Then we have the following form of the Euler-Lagrange equations (which are well-defined for ̺, ̺ 1 ∈ P + ac (U )), is easy to check that the above expression is zero for any ̺ 1 ∈ P + ac (U ).
(3)⇒(2) On the other hand assume that ̺ is a critical point. If the integrand β −1 log ̺ + κW ⋆ ̺ in (2.22) is not constant a.e., we can find without loss of generality a set A ∈ B(U ) of nonzero Lebesgue measure such that We are now free to choose ̺ 1 ∈ P + ac (U ) to be for some ε > 0. For this choice of ̺ 1 , we have, where, From our choice of the set A, it is clear that a > 0 and b ≤ 0. Since ε can be made arbitrarily small, (1 − ε)a + εb can be made positive. Thus we have derived a contradiction since ̺ is a critical point of F κ and therefore it must satisfy the Euler-Lagrange equations in (2.22). Thus the integrand must be constant a.e. from which we obtain (3)⇒(2).
. Then the following statements are equivalent: (1) ̺ is a classical solution of the stationary McKean-Vlasov equation (2.14).
From the expression for J κ (̺) and the fact that ̺ has full support this is possible only if Thus, we have that, ̺ − Ce −βκW ⋆̺ = 0, a.e., for some constant, C > 0, which is given precisely by Z(̺, κ, β) since ̺ ∈ P ac (U ). Thus we have that ̺ is a zero of F κ (̺) and by Proposition 2.4 the reverse implication, (1)⇒ (2). ■ The following lemma is taken from [CP10] in which it is shown that for any unbounded ̺ ∈ P ac (U ) there exists a bounded ̺ † ∈ P ac (U ) having a lower value of the free energy. For the convenience of the reader, we include the entire proof.
Proof. We start by noticing that we can control the entropy and interaction energy from below: where the bound on the entropy follows from Jensen's inequality and the bound on the interaction energy follows from Assumption (A2). Thus, the free energy F κ is bounded below. Now fix some B > 0 and ̺ ∈ P ac . Let ε B be the ̺-measure of the Borel set, Following Chayes and Panferov [CP10], we consider two different cases of (̺, B), namely, ε B ≥ 1 2 and ε B < 1 2 . For the case, ε B ≥ 1 2 , we have by an application of Jensen's inequality Since from (2.24) the interaction energy E(̺, ̺) is bounded below, we choose B large enough such that there exists some positive constant It should be noted that ̺ r is not a probability measure as it is not normalised. For B > 1, ε B > 0 we have Consider the case F κ (̺) ≤ F κ (̺ ∞ )(otherwise the proof is complete). The interaction energy E(̺, ̺) is bounded below and we have the following estimate We also write Using this and the fact that Set the ̺ r = (1 − ε B ) −1 ̺ r which is a probability measure. We are now in a position to compute the difference in free energy, F κ (̺) − F κ (̺ r ): For the interaction energy we know that The above estimate together with ε B < 1 2 gives Thus if we pick B to be larger than some B 2 , we have that F κ (̺ r ) < F κ (̺). However, ̺ r can take values as large as 2B 2 a.e. due to the normalisation. Thus we set, B 0 = max{B 1 , 1, 2B 2 } and pick for The next lemma shows that minimisers of F κ (̺) over P ac (U ) are attained in P + ac (U ).
Lemma 2.7. Assume W (x) satisfies Assumption (A2) and let ̺ ∈ P ac (U ) \ P + ac (U ). Then, there exists ̺ + ∈ P + ac (U ) such that, and show that for ǫ > 0 sufficiently small ̺ ǫ has smaller free energy. We first compute its entropy where we have used the fact that S(̺) > S(̺ ∞ ), ∀̺ ∈ P ac (U ), ̺ = ̺ ∞ . For computing the interaction term, we use the fact that E(̺, ̺) > − W − ∞ to estimate where C 1 , C 2 < ∞ depend on W and B 0 and we have chosen ǫ sufficiently small. Combining the two expressions together we obtain, Thus for ǫ sufficiently small but positive the logarithmic term will dominate and give us the required result. ■ Proof. Since by (2.23) and (2.24), F κ (̺) is bounded from below over P ac (U ), there exists a minimising sequence {̺ j } ∞ j=1 ⊂ P ac (U ). Furthermore, by Lemma 2.6, the minimising sequence can be chosen such Clearly we have that ̺ κ dx = 1. It is also easy to see that ̺ κ ≥ 0, a.e. Thus ̺ κ ∈ L 2 (U ) ∩ P ac (U ). The lower semicontinuity of S(̺) follows from standard results (cf. [JLJ98], Lemma 4.3.1). Consider now the interaction energy term. For k, l ∈ N d , we write k < l if k i < l i for i = {1, . . . , d}. Taking the Fourier transform, we obtain for any where the constant C depends only on L and k 0 . The second term can be made arbitrarily small by choosing k 0 sufficiently large by the Riemann-Lebesgue lemma. The first term can be made arbitrarily small for any k 0 due to the weak convergence result by choosing j sufficiently large. Thus the interaction energy is continuous, which implies that the free energy F κ (̺) has a minimiser ̺ κ over P ac (U ). A direct consequence of this and Lemma 2.7 is that the minimisation problem is well-posed in P + ac (U ) since the minimiser ̺ κ must be attained in P + ac (U ). We can then use Theorem 2.3 together with Proposition 2.4 to argue that any such minimiser must be smooth. ■ Now, we apply Jensen's inequality and use the fact that W s ∈ H s which gives us Finally we bound W u (x) from below to obtain, showing the desired statement. ■

Global asymptotic stability
3.1. The free energy as Lyapunov function. In this section, we will use the free energy as defined in (2.20) to study the global asymptotic stability of the uniform state for the system (2.4). The free energy F κ is a Lyapunov function for the evolution. This follows from rewriting (2.4) as ∂ t ̺ = ∇· ̺ β −1 ∇ log ̺ + ∇W ⋆ ̺ and differentiating the free energy functional along the flow as follows To analyse the trend to equilibrium, it will be convenient to consider the free energy difference using the fact that E(̺ ∞ , ̺) = 0 for any ̺ ∈ P(U ) and W with mean zero. By introducing the relative entropy we observe the following identity between the free energy gap and the relative entropy The identity (3.2) provides us with an intuitive understanding of the two competing mechanisms in the dynamics of this system. On the one hand the relative entropy is minimised for the uniform state ρ ∞ and therefore forces wants the mass to be spread out, where on the other hand the interaction energy with an attractive potential prefers clustered states. A crucial investigation of these two competing mechanisms is the main theme of this work.
3.2. Trend to equilibrium in relative entropy. By either directly differentiating the relative entropy (3.1) or by using the identity (3.2), we obtain the rate of change of the relative entropy We now state two preliminary facts that are useful in the study of convergence to equilibrium. On the torus, the heat semigroup is hypercontractive and satisfies a logarithmic Sobolev inequality [EY87] of the form where the term on the right hand side is the Fisher information of the measure ̺. It will be convenient to compare the L 1 norm with the relative entropy, which is granted by the Csiszár-Kullback-Pinsker These preliminary inequalities allow us to obtain exponential convergence to equilibrium in relative entropy.
Proposition 3.1 (Exponential stability and convergence in relative entropy).
Then the classical solution ̺ of (2.4) is exponentially stable in relative entropy and it holds Especially, in the cases W ∈ H s for any β, κ > 0 and if W / ∈ H s for βκ < 2π 2 L 2 ∆Wu ∞ it holds that we have exponentially fast convergenceto the uniform state in relative entropy for any initial condition ̺ 0 ∈ P ac (U ) ∩ H 3+d (U ).
Proof of Theorem 1.1(b). We know the solution ̺ is classical, thus H(̺(·, t)|̺ ∞ ) ∈ C 1 (0, ∞). Using (3.3) and (3.4), we obtain with another integration by parts Now, we rewrite the interaction term in its Fourier series by (2.2), estimate it in terms of the unstable modes and transform it back to position space Now, we use the fact that ∆W u is mean zero to replace ̺ by ̺ − ̺ ∞ and estimate In combination with (3.6) and (3.5), we obtain the bound Finally, by Gronwall's inequality, we have the desired result. ■

Remark 3.2. For the case of the noisy Hegselmann-Krausse model studied in
βL 2 , we have exponential convergence to equilibrium. See § 6.2 for a detailed analysis of this model. Remark 3.3. By the improved entropy defect estimate of Lemma 5.12, the above statement could be slightly improved under more specific assumptions on the unstable modes of the potential. For the moment, we want to keep the presentation as concise as possible and refer to § 5 for the details.
3.3. Linear stability analysis. We start this subsection by linearising the stationary Mckean-Vlasov equation around some stationary solution, ̺ κ . We obtain the following linear integrodifferential operator: If we pick ̺ κ to be the uniforms state ̺ ∞ the above expression reduces to We are now interested in studying the spectrum of this operator over mean zero L 2 (U ) functions, L 2 0 (U ). From the classical theory for symmetric elliptic operators, it follows that the eigenfunctions of this system form an orthonormal basis in L 2 0 (U ) given by with the eigenvalues given by One can check that we have the following relationship To obtain the above expression we have used the fact that W is coordinate-wise even, which implies that Thus, we have the following expression for the value of the parameter κ ♯ at which the first eigenvalue of L crosses the imaginary axis: We will refer to κ ♯ as the point of critical stability We denote by k ♯ the critical wave number(if it is unique) and define it as:

Bifurcation theory
For the local bifurcation analysis, it is convenient to rewrite the fixed point equation (2.21) of the nonlinear mapping (2.16) by making the parameter κ ∈ (0, ∞) explicit. Hence, in this section we consider the nonlinear map F : where β > 0 is fixed, and W ∈ L 2 s (U ) with L 2 s (U ), the space of coordinate-wise even and square integrable functions as defined in (2.3).
The purpose of this section is to study the bifurcation problem: Any zero of F (̺, κ) is also a coordinate-wise even fixed point of T : P ac → P ac . The converse is true if 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, e.g., the Keller-Segel model which we treat in a later section. It is also clear that the map F (̺, κ) is translation invariant on the whole space L 2 s (U ), i.e., if ̺ is a zero of F (̺, κ) then so is any translate ̺(· − y) of ̺(·) for any y ∈ U . This is the motivation for the restriction of F to the space L 2 s (U ). Equivalently, we could quote out the translation invariance. We will further justify our choice of the space L 2 s (U ) in Lemma 5.14. 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 [Tam84,  Proof. Proposition 2.9 implies that F κ (̺) is strictly convex for κ < κ con = β −1 W u −1 ∞ . Hence, it has a unique minimiser and exactly one critical point. This implies from Proposition 2.4 that F (̺, κ) has a unique solution. ■ We use the trivial branch of solutions F (̺ ∞ , κ) = 0, κ ∈ (0, ∞) with ̺ ∞ ≡ 1/L d to centre the map and define for any u ∈ L 2 s (U ) In this way, we have F (0, κ) = 0. We compute the Frechét derivatives of this map for variations We have the following characterisation of the local bifurcations of F : . Assume there exists k * ∈ N d , such that: (2) W (k * ) < 0 .
Before proceeding, it is useful to characterize Im(I − κ * T ). By using (4.13), we can see that we have the following orthogonal decomposition of L 2 s (U ) into Using the identity [Kie12, (I.6.3)] it follows that κ ′ (0) = 0 provided that D 2 where we have used (4.6) and the fact that w 3 k * dx = 0. Thus we conclude that κ ′ (0) = 0. Likewise, from [Kie12, (I.6.11)], we also have that where we have used (4.5) and (4.7). The first two properties of (4.10) follow from Theorem B.2. To prove the third property in (4.10), we observe that lim |s|+|κ(s)−κ * |→0 Since κ ′ (0) = 0, we also have lim |s|→0 where we have used the fact from Theorem B.2 that κ is continuously differentiable. This completes the proof. ■ The statment of Theorem 4.2 becomes more transparent in one dimension: . Assume that there exists k * ∈ N, such that: that is, there exists a branch of solutions having the following form: = 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
In this case, every k α ∈ N d such that W (k) < 0, corresponds to a unique bifurcation point κ α of F (̺, κ) through the relation (4.8). For example consider the interaction potential W (x) = x 2 /2. In this case 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 ∈ N d , the system has only one bifurcation point.
Remark 4.6. In dimensions higher than one, the space L 2 s (U ) may not be small enough for our purposes, i.e., it is possible that the potential may have additional symmetries. For instance, the potential could be exchangeable, that is W (x) = W (Π(x)) for all possible permutations Π of the d coordinates. In this case it is easy to check that W, w k = W, w Π(k) for all k ∈ N d . We can then define the equivalence relation, k ∼ k ′ if k ′ = Π(k) for some permutation Π and write [ The bifurcation point is given by .

Remark 4.7. Consider the following interaction potential
It is straightforward to check that W s (x) belongs to H s (U ) and thus to C(U ). Additionally, W s (x) → −w 1 (x) uniformly as s → ∞. One can check now that, for any s > 1, W s (x) satisfies the conditions of Theorem 4.2 for all k ∈ N, k = 0 and thus the trivial branch of the system has infinitely many bifurcation points. However, as mentioned in Remark 4.5, the system W (x) = −w 1 (x) has only one bifurcation point. This can be explained by the fact that as s → ∞ all bifurcation points of W s (x) except one are pushed to infinity. This example illustrates however that two potentials may "look" similar but their associated bifurcation structure may be entirely different. Therefore, approximating potentials, even uniformly, by some dense subset, may not reveal all the information about the bifurcation structure of the limiting system.
If we now assume that W satisfies assumption (A2) we can see that the zeros of F (̺, κ) are fixed points of the map 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, i.e., if w k (x) is the mode such that k ∈ N d satisfies the conditions of Theorem 4.2, then to leading order the nontrivial solution is of the form ̺ ∞ + sw 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 F as defined in (4.2).

Proposition 4.8. Let V be an open neighbourhood of
is a bifurcation point of the map F in the sense of Theorem 4.2. We denote by C V the set of nontrivial solutions of F (̺, κ) = 0 in V and by C V,κ * the connected component of C V containing (0, κ * ). Then C V,κ * has at least one of the following two properties: (2) C V,κ * contains an odd number of characteristic values of T , (0, κ i ) = (0, κ * ), which have odd algebraic multiplicity .
Proof. The proof follows from the direct application of the so-called Rabinowitz alternative [Dei85, Theorem 29.1] which we have included as Theorem B.3 for the convenience of the reader. It is easy to check that the map F can be written in the following form, with T as defined in (4.12), and We now need to show that G is completely continuous and o( ̺ 2 ) uniformly in κ as ̺ 2 → 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: (4.14) Similarly we can also deduce the following estimate by bounding W ⋆ (̺ 2 − ̺ 1 ) from above: In the above two expressions, C κ is a constant which tends to 0 as κ → 0. Setting ̺ 2 = 0 in (4.15), it follows that 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( ̺ 2 ) follows by Taylor expanding e −βκW ⋆̺ /Z.
One can now check that if condition (1) of Theorem 4.2 is satisfied for some k ∈ N d , the associated eigenvalue κ −1 (which could be negative) of T is simple, i.e., it has algebraic multiplicity one. This implies that all bifurcation points predicted by Theorem 4.2 are associated with simple eigenvalues of T . Thus, we can apply Theorem B.3 to complete the proof. ■

Phase transitions for the McKean-Vlasov equation
We know from Proposition 2.9 that ̺ ∞ is the unique minimiser of the free energy for κ sufficiently small. We are interested in studying under what criteria there is a change in the qualitative structure of the set of minimisers of F κ . For the rest of this section we will assume that W satisfies Assumption (A2), i.e, W ∈ H 1 (U ) and bounded below. We build on and extend the notions introduced by [CP10]. The first definition introduces what we mean by a transition point. (1) For 0 < κ < κ c , ̺ ∞ is the unique minimiser of F κ (̺) .
In the present work, we are only interested in the first transition point by increasing κ 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 [CP10]. Proposition 5.2 ([CP10, Proposition 2.8]). Assume W ∈ H c s and suppose that for some κ T < ∞ there exists ̺ κT ∈ P + ac (U ) not equal to ̺ ∞ such that: Then, for all κ > κ T , ̺ ∞ no longer minimises the free energy.
In addition, the following result from [GP70] shows that H-stability of the potential is a necessary and sufficient condition for the nonexistence of a transition point. From this result it follows directly that if the system possesses a transition point κ c , ̺ ∞ can no longer be a minimiser beyond this point. We are also interested in understanding how this transition occurs. We will distinguish between continuous and discontinuous transition points, in analog to the concepts of first and second order phase transitions arising from statistical physics 2 .

Definition 5.4 (Continuous and discontinuous transition point). A transition point κ c > 0 is said to be a continuous transition point of F κ if it satisfies the following conditions:
(1) For κ = κ c , ̺ ∞ is the unique minimiser of F κ (̺) .
(2) Given any family of minimisers, {̺ κ |κ > κ c }, we have that A transition point κ c which is not continuous is said to be discontinuous.
By combining certain properties of transition points with the previous analysis on critical stability in § 3.3, we obtain a more streamlined characterisation of transition points, which is the basis for the proof of Theorem 1.3, or rather Theorem 5.7 and Theorem 5.15.
Proposition 5.5. Let F κ have a transition point at some κ c < ∞ and let κ ♯ denote the point of critical stability defined in § 3.3.
Proof. A consequence of the assumption in the first statement (a) of the proposition is that ̺ ∞ is the unique minimiser for all κ ≤ κ ♯ . Indeed, since by assumption W α has zero mean, the free energy of the uniform state ̺ ∞ is independent of κ. From [CP10, Proposition 2.4], we know that the minimum value of F κ is nonincreasing in κ. Thus, if ̺ ∞ is the unique minimiser at some κ = κ c , it must be a minimiser for all κ ≤ κ c . In fact, using Proposition 5.2 we can assert that ̺ ∞ is the unique minimiser of F κ for all κ ≤ κ c . Indeed, if this were not the case then there exists some ̺ κT ∈ P + ac (U ) not equal to ̺ ∞ such that F κT (̺ κT ) = F κT (̺ ∞ ) for some κ T < κ ♯ . Proposition 5.2 then tells us that ̺ ∞ can no longer be a minimiser for any κ > κ 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 κ ♯ 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, i.e., there exists a family of minimisers {̺ κ |κ > κ c } of F κ (̺) such that lim sup κ↓κc ̺ κ − ̺ ∞ 1 = 0. Then we know from [CP10, Proposition 2.13] that there exists some ̺ κc ∈ P + ac (U ) not equal to ̺ ∞ such that it is a minimiser of the free energy F κ (̺) at κ = κ c . Applied in the present setting with κ c = κ ♯ , we would deduce that ̺ ∞ is no longer the unique minimiser of F κ ♯ (̺), 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 κ c = κ ♯ is a continuous transition point.
To prove the second statement (b) of the proposition, let ̺ be such that F κ ♯ (̺) < F κ ♯ (̺ ∞ ). Then for any κ close enough to κ ♯ , we also have F κ (̺) < F κ (̺ ∞ ). Hence by a combination of Proposition 5.2 and Proposition 5.3 there exists a transition point κ c < κ ♯ and, in particular κ ♯ , cannot be a transition point. From [CP10, poposition 2.12], we have the fact that if κ c is a continous transition point of F κ , then necessarily κ c = κ ♯ . This implies that κ c < κ ♯ cannot be a continuous transition point. ■ Before proceeding to present the main results of this section, we remind the reader that for the rest of the paper κ c denotes a transition point, κ ♯ denotes the point of critical stability, and κ * denotes a bifurcation point.

Discontinuous transition points.
We provide below a characterisation of potentials which exhibit discontinuous transition points, which proves Theorem 1.3(a).
Definition 5.6. Assume W ∈ H c s and let Θ(k) + δ for some δ ≥ 0. We define δ * to be the smallest value of δ for which the following condition is satisfied Theorem 5.7. Let W (x) be as in Definition 5.6. Then if δ * is sufficiently small, F κ exhibits a discontinuous transition point at some κ c < κ ♯ .
Proof. We know already from Proposition 5.3 that the system possesses a transition point κ c . We are going to use Proposition 5.5 (b) and construct a competitor ̺ ∈ P + ac (U ) which has a lower value of the free energy than ̺ ∞ . Let for some ǫ > 0, sufficiently small. We denote by |K δ * | the cardinality of K δ * , which is necessarily finite as W ∈ L 2 (U ). Expanding about ̺ ∞ we obtain Using the fact that κ ♯ min One can now check 3 that under at least one of condition of the theorem, it holds that where the constant a is independent of δ * . Thus, for δ * sufficiently small considering the fact that |K δ * | ≥ 2 and is nonincreasing as δ * decreases, ̺ has smaller free energy and ̺ ∞ is no longer a minimiser at κ = κ ♯ . ■ Remark 5.8. The case of the above result for δ * = 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 δ * 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 5.9. Let {W n } n∈N ∈ H c s be a sequence of interaction potentials such that δ * (n) → 0 as n → ∞, where δ * is as defined in Definition 5.6. Assume further that for all n greater than some N ∈ N , there exists a constant C > 0 such that min ≥ Cδ γ * for some γ < 1/2. Then for n sufficiently large, the associated free energy F n κ (̺) possesses a discontinuous transition point at some κ n c < κ n ♯ .
Proof. We return to estimate (5.1) from the proof of Theorem 5.7 where we have suppressed the dependence of δ * on 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 have Since γ < 1/2 and δ * → 0 as n → ∞, the result follows. ■ 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 5.10. Let {W n } n∈N be a sequence of interaction potentials with W n 1 = C > 0 for all n ∈ N such that W n → −Cδ 0 in the sense of distributions as n → ∞. Then for n large enough, the associated free energy F n κ (̺) possesses a discontinuous transition point at some κ n c < κ n ♯ .
Proof. Note first that we have not included the assumption W n ∈ H c s as eventually this must be the case if the potentials converge to a negative Dirac measure. Now we just need to check that the other conditions of Corollary 5.9 hold true. We have the following estimate From the convergence to the Dirac measure it follows that for any ǫ > 0 we can . This and (5.2) tells us that δ * ≤ ǫ and since ǫ is arbitrary δ * → 0 as n → ∞. From similar arguments we assert that for all n > N , min 1−w1(x) , and any appropriately scaled negative mollifier.

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 5.15 and, in particular, in the proof of Theorem 1.3(b).
Lemma 5.12. Let (Ω, Σ, µ) be a probability space and {w k } k∈N be any orthonormal basis for L 2 (Ω, µ). Assume that f ∈ L 2 (Ω, µ) is a probability density with respect µ, that is f is nonegative and f dµ = 1, then we have, for any b ∈ R and any k ∈ Z, the following estimate, In particular, let Ω = U , µ = ̺ ∞ and w k is as defined in (2.1). Moreover, for any k ∈ Z d \ {0} let n = n(k) = |{i : k i = 0}| denote the number of nonzero entries. Then, there exists a strictly increasing function G : R + → R + with G(0) = 0 such that it holds where the constant C(n) > 0 for is given by C(1) = C(2) = 1 and for n > 2 by Definition 5.13. Assume that W ∈ H c s has one dominant negative mode, i.e., there exists a unique as defined in (3.8)). We define the α-stabilised potential W α (x) as follows where α ∈ [0, 1], W s (x), W u (x) are as defined in Definition 2.1, and W 1 (x) = W (x).
The above definition puts into context the discussion around Figure 1(a) in § 1, i.e., the α-stabilised potential W α pushes all negative modes except the dominant one to some small neighbourhood of 0. We define the fixed point equation associated with the interaction potential W α to be Lemma 5.14. Let W α (x) be as in Definition 5.13 and let C ⊂ P + ac (U ) denote the set of nontrivial solutions of F κ ♯ (̺, α) = 0 for α ∈ [0, α * ) ⊂ [0, 1]. Then, for α * sufficiently small, we have the uniform lower bound σ∈Sym(Λ) | ̺(σ(k ♯ ))| 2 > c for all ̺ ∈ C and for some c > 0 independent of α ∈ [0, α * ).
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 5.12 and Lemma 5.14 after the proof of Theorem 5.15.
Theorem 5.15. Let W α (x) be as in Definition 5.13 such that log 2 Θ(k ♯ ) ≤ 1. Assume further that W u and W s are bounded below. Then, for α sufficiently small, the system exhibits a continuous transition point at κ c = κ ♯ .
Proof. By Proposition 5.5 ((a)), it is sufficient to show that at the point of critical stability κ ♯ , i.e., the uniform state ̺ ∞ is the unique minimiser, for α small enough. Let ̺ be any solution of F κ ♯ (̺, α) = 0, i.e., a critical point of F κ ♯ (cf. Proposition 2.4). Then we have We can translate ̺ w.l.o.g so that ̺(σ(k ♯ )) = 0, ∀σ ∈ (Sym(Λ) − e) and throw away all positive W α (k). A consequence of this is that | ̺(k ♯ )| 2 = σ∈Sym(Λ) | ̺(σ(k ♯ ))| 2 . Thus we obtain we can obtain the estimate We apply Lemma 5.12 to the first term on the right hand side Here, we use that the fact that the assumption that log 2 Θ(k ♯ ) ≤ 1 is equivalent to n(k ♯ ) ≤ 2, where n(k ♯ ) is the number of nonzero components in k ♯ as defined in the statement of Lemma 5.14. Now, we use the result of Lemma 5.12 with the constant c and the monotonicity of the function G to further estimate where c is precisely the constant from Lemma 5.14 for α ∈ [0, α * ). Since ̺ is a zero of F κ ♯ (̺, α) = 0 we have the following estimate If we restrict α to [0, α * ) as in Lemma 5.14, we can obtain the following estimates on the norms of W α : Thus for α ∈ [0, α * ) we have ̺ It is easy to check that e g dµ = 1 and hence g is admissible in (5.5), from which we deduce the lower bound In the special case Ω = U and µ = ̺ ∞ , setting f = ̺ ̺∞ , we obtain from (5.6) the lower bound We can pick b = αL d for some α > 0 and set y = L d/2 2 n/2 ̺(k). We thus obtain, where the w ki (x i ) are as defined previously and n ≥ 1 represents the number of k i = 0. Setting x i = L 2πki θ i for all k i = 0, we arrive at We introduce the function We will show that is strictly increasing in z with G(0) = 0. Once we have shown (5.8), the proof concludes by combining this with (5.7) to deduce that from where the result (5.4) follows by setting G(y) = G(y/λ).
It is left now to show (5.8). For its validity, it is sufficient to note that I n (0) = 1 and to show that exp λz 2 /(λ2 n+1 ) /I n (z) is strictly increasing in 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. [HVV09,Theorem 4.4], [BK55]). First of all, we observe that the odd coefficients are zero. We are left to investigate In the case n = 1, the monotonicity follows by the above representation. For n > 1, we consider a l+1 a l = λ 1/(n−1) (l + 1) 1+ n n−1 2 (2l + 2)(2l + 1) = λ 1/(n−1) (l + 1) n n−1 2l + 1 .
We need to find a λ such that the above expression is greater than or equal to 1. Hence, we obtain λ 1/(n−1) = sup l≥1 2l + 1 (l + 1) where we note that the sup is attained for l = n−2 2 , hence proving (5.8). ■ Proof of Lemma 5.14. For the first part of the proof, we fix α ∈ [0, α * ). Then, we know that κ = κ ♯ independent of α is a bifurcation point, i.e., it satisfies the conditions of Theorem 4.2. Then one can check that the same set of arguments can be applied in the larger space where e represents the identity element. For fixed α, we consider the map, F : α) and note that any ̺ such that F (̺, κ) = 0 is obviously in L 2 k ♯ (U ). Additionally, any zero of F defined above is also a zero of , which is defined as One can also notice that F * (̺) does not change any of the local properties of F (̺) near ̺ ∞ , i.e, . The advantage of defining F * in this way is that the Frechét derivative of the map is then Fredholm with index zero, which is not the case with F . We also know from Theorem 4.2 that F has at least one nontrivial solution ̺ κ ∈ L 2 s (U ) in a neighbourhood of (̺ ∞ , κ ♯ ). We can now apply the same bifurcation argument to F * to obtain that F * has exactly one nontrivial solution in some neighbourhood of (̺ ∞ , κ ♯ ). Since every zero of F is a zero of F * it follows that ̺ κ is this nontrivial zero in some neighbourhood of (̺ ∞ , κ ♯ ) and that F has only one nontrivial solution in this neighbourhood. Thus the problem of studying bifurcations of F is reduced to that of studying bifurcations of F * 4 . Now, since we need a lower bound which is uniform in α, we redefine F * to be a function of α, i.e., is Banach space equipped with the norm · 2 + | · | and f = (̺, α) ∈ 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 [Kie12].
Thus we can apply the implicit function theorem to obtain a function C 1 (V 1 ; V 2 ), ϕ(s, α) such that Φ((s, α), ϕ(s, α)) = 0, where V 1 and V 2 are neighbourhoods of (0, 0) and κ ♯ respectively and V 1 × V 2 ⊂ U . Additionally in V 1 ×V 2 every solution of Φ (and hence Φ) is of the form ((s, α), ϕ(s, α)) and ϕ((0, α)) = κ ♯ . We know however from Theorem 4.2, that we could apply the same set of arguments for fixed α ∈ [0, 1] to obtain single locally increasing branches which, at least for some small neighbourhood around 0, must coincide with ϕ(s, α). Thus, we now know that for each α ∈ [0, 1], we can find ǫ α > 0 such that ϕ(s, α) > κ ♯ for 0 < |s| < ǫ α . Now, let α ∈ [0, α * ) = A. If we show that inf A ǫ α = ǫ ′ > 0 for α * small enough, we can conclude the proof. To see this, set Proof. The proof follows the same arguments as Theorem 5.15 with κ ♯ replaced by C(n)κ ♯ . ■ A natural question to ask now is how the estimate from Corollary 5.16 compares to the one obtained in Proposition 2.9 by the convexity argument, i.e., how does C(n)κ ♯ compare to κ con . It is easier to make this comparison whenever we can explicitly compute W u− ∞ . Assume, W = W 0 , i.e, W has only one negative mode, say w k ♯ , then we have with n = n(k ♯ ) as defined in Lemma 5.12. Thus, for all n ≥ 1, we have that C(n)κ ♯ > κ con .. From this we conclude that, for this choice of W , Corollary 5.16 provides a sharper estimate on the range of κ for which the uniform state is a unique minimiser of the free energy.
Let us assume that |λ| > C 1 for some constant C 1 > 0. This implies that for all k ∈ N d such that W (k) < 0 it must hold It is easy to see then that λ being sufficiently large is equivalent to α being sufficiently small.
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 5.18. Let W ∈ H c s be an interaction potential such that the associated free energy F W κ (̺) has a continuous transition point. Additionally, assume that G ∈ H c s is such that arg min Proof. As in the proof of Theorem 5.15, it is sufficient to show that at κ = κ ♯ , the free energy F G κ ♯ (̺) has ̺ ∞ as its unique minimiser. Noting that given the assumptions on G, the value of κ ♯ is the same for G and W , we have for ̺ = ̺ ∞ , ̺ ∈ L 2 (U ) ∩ P ac (U ) Using the fact that the term in the brackets must be strictly positive, since the free energy F W κ ♯ (̺) associated to W possesses a continuous transition point, we obtain In the above estimate we have used the fact that G(k ♯ ) = W (k ♯ ) and that G(k) ≥ W (k) for all other k ∈ N d . Thus, we have the desired result. ■ 6. Applications 6.1. The generalised Kuramoto model. Let W (x) = −w k (x), for some k ∈ N, k = 0. Then we refer to the corresponding McKean-Vlasov system as the generalised Kuramoto model. For k = 1, it corresponds to the so-called noisy Kuramoto system (also referred to as the Kuramoto-Shinomoto-Sakaguchi model (cf. [Kur81,SSK88])) which models the synchronisation of noisy oscillators interacting through their phases. For infinitely many oscillators we obtain a mean field approximation of the underlying particle dynamics given precisely by the McKean-Vlasov equation with W (x) = −w 1 (x). It is well known that this system exhibits a phase transition for some critical, κ c (cf. [BGP10]). For k = 2, it corresponds to the Maiers-Saupe system which is a model for the synchronization of liquid crystals (cf. [CKT04,CV05]). Again, in the mean field limit we obtain the McKean-Vlasov equation with the effective interaction potential, W (x) = −w 2 (x). The system exhibits a continuous transition point which represents the nematic-isotropic phase transition as the temperature is lowered, i.e., as κ is increased. Finally, let us mention that there is a larger picture in the Kuramoto model when different frequency oscillators are allowed, see [ABPV + 05] for a nice review of the subject and [CCP18] for recent numerical work on phase transitions for this problem.
Although it is possible to directly apply Theorem 5.15 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. Proof. The strategy of proof is similar to that of Theorem 5.15, i.e, we show that at κ = κ ♯ , ̺ ∞ is the unique minimiser of the free energy. We do this by showing that F (̺, κ) = 0 has a unique solution at κ = κ ♯ , which implies, by Proposition 2.4(since W satisfies Assumption (A2)) , uniqueness of the minimiser.
For W (x) = −w k ♯ (x), we can explicitly compute, Since F (̺, κ) is translation invariant, one can always translate ̺ so that ̺(−k ♯ ) = 0. Thus we obtain the following simplified equation, Taking the inner product with w k ♯ (x) we obtain, After a change of variables we obtain, We can express the above equation in the following form, where the I n represent modified Bessel functions of the first kind having order n, r n (a) := In+1(a) In(a) , and a = βκ ̺(k ♯ ). This equation is similar to the one derived in Section VI of [Bav91](cf. [MS82,Bat77]). It is also qualitatively similar to the self-consistency equation associated with the two-dimensional Ising model.
We now show that if κ ≤ κ ♯ = √ 2L/β, (6.2) has no nonzero solutions. As mentioned earlier it is sufficient to study the problem on the half line. Note first, that for a > 0, r 0 (a) is increasing, i.e, r ′ 0 (a) > 0 (cf. [Amo74, (15)]). Additionally, we have that, and so r ′ 0 (0) = 1 2 . We can now use the so-called Turan-type inequalities (cf. [TN51,BP13]) to assert that I 0 (a)I 2 (a) − I 1 (a) 2 < 0 for a > 0. This tells us that, with r 0 (a) > 0 for a > 0. Using the fact that κ ≤ κ ♯ , we obtain, We know now that M (a, κ) is increasing for a > 0, M (0, κ) = 0, ∂M ∂a (0, κ) = 1, and ∂M ∂a (a, κ) is bounded above by 1 for a > 0. Thus the curve y = M (a, κ) cannot intersect y = a for any a > 0. Thus ̺ ∞ is Let us denote by ̺(dx, a κ ) the measure associated to the density ̺(x, a κ ). We will now show that for k = 1, ̺(dx, a κ ) converges to δ 0 as a κ → ∞ in the narrow topology, i.e., tested against bounded, continuous functions. The argument for other k ∈ N is then simply an extension of the k = 1 case. Let A be a continuity set of δ 0 , then if 0 / ∈ A it follows that 0 / ∈ ∂A. By a large deviations argument, Laplace's principle, we have that for some R > 0. The ratio R/L measures the range of influence of an individual agent with R/L = 1 representing full influence, i.e., any one agent influences all others. In order to analyse this system further, we compute the Fourier transform of W hk (x) given by A simple consequence of the above expression is that the model has infinitely many bifurcation points for R/L = 1. For the other values of R/L the problem reduces to a computational one, namely checking that the conditions of Theorem 4.2 are satisfied. Also, W hk (x) is normalised and decays to 0 uniformly as R → 0, i.e., as the range of influence of an agent decreases so does its corresponding strength. We could define a rescaled version of the potential, which does not lose mass as R → 0. We conclude this subsection with the following result. Proof. We define C := W R hk 1 and note that it is independent of R. The proof follows from the observation that W R hk → −Cδ 0 as R → 0 and applying Corollary 5.10. ■ 6.3. The Onsager model for liquid crystals. In § 6.1, we discussed the Maiers-Saupe model as a special case of the generalised Kuramoto model. In this subsection we discuss another model for the alignment of liquid crystals, i.e., the Onsager model which has as its interaction potential, W (x) = sin 2π L x . As discussed in [CLW10], one can also study the potential W ℓ (x) = sin 2π L x ℓ ∈ L 2 s (U ) ∩ C ∞ (U ) with ℓ ∈ N, ℓ ≥ 1, so that the Onsager and Maiers-Saupe potential correspond to the cases ℓ = 1 and ℓ = 2, respectively. We have the following representation of W ℓ (x) in Fourier space . (6.5) Any nontrivial solutions to the stationary dynamics correspond to the so-called nematic phases of the liquid crystals. We can obtain the following characterisation of bifurcations associated to the W ℓ (x) and thus of the Onsager model.
The above result provides us with a finer analysis to that presented in [CLW10], as we are able to count the solutions for general odd and even ℓ, instead of just proving the existence of nontrivial solutions. The above result also generalises the work in [LV10] 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 [NY15, Theorem 2] in which the non-truncated Onsager model is analysed. We refer the reader to [Vol17] for an analysis of the Onsager model in 2 dimensions, i.e., for liquid crystals that live in 3 dimensions with two degrees of freedom.
6.4. The Barré-Degond-Zatorska model for interacting dynamical networks. The Barré-Degond-Zatorska system [BDZ17] models particles that interact through a dynamical network of links. Each particle interacts with its closest neighbours through cross-links modelled by springs which are randomly created and destroyed. Taking the combined mean field and overdamped limits one obtains the McKean-Vlasov equation with the interaction potential given by for two positive constants 0 < ℓ ≤ R ≤ L/2. In [BDZ17, Theorem 6.1], using formal asymptotic analysis, it was shown (and later numerically verified in [BCD + 18]) that one can provide conditions for continuous and discontinuous transitions for the above potential based on the values of the Fourier modes. We restate their result using our notation for the convenience of the reader. 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 κ * = − (2L) 1 2 β 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 W (1) and W (2) are resonating/nearresonating then it follows that condition (b), i.e., 2 W (2) − W (1) < 0 must hold for δ * small, where δ * is as introduced in Definition 5.6. Indeed, let k = 1, 2 be elements of the set K δ * , then we have 2 W (2) − W (1) = W (1) + 2( W (2) − W (1)) ≤ W (1) + 2δ * < 0, for δ * sufficiently small. Similarly, using Proposition 5.18 and comparing with an α-stabilised potential say G α , one can argue that if W (1) is the dominant mode then condition (a), i.e., 2 W (2) − W (1) > 0 must hold for α small, where α is as defined in Definition 5.13. 6.5. The Keller-Segel model for bacterial chemotaxis. The (elliptic-parabolic) Keller-Segel model is used to describe the motion of a group of bacteria under the effect of the concentration gradient of a chemical stimulus, whose distribution is determined by the density of the bacteria. This phenomenon is referred to as chemotaxis in the biology literature [KS71]. For this system, ̺(x, t) represents the particle density of the bacteria and c(x, t) represents the availability of the chemical resource. The dynamics of (a) The associated wave numbers which correspond to bifurcation points of the stationary system the system are then described by the following system of coupled PDEs: with ̺ ∈ C 2 (U ) and where Φ s is the fundamental solution of −(−∆) s . Since Φ s does not, in general, satisfy assumption (A2), Theorem 2.3 does not apply directly. However we can circumvent this issue to obtain the following result.
Indeed, let ̺ be such a fixed point and 0 < ǫ < s − 1 2 , then Thus c ∈ H 1+ǫ (U ) which by the Sobolev embedding theorem for d ≤ 2 implies that c ∈ C 0 (U ). This tells us that ̺ ∈ H 1 (U ) ∩ P ac (U ) with ∇̺ = −βκZ −1 e −βκc ∇c. Plugging ̺ into (6.10), we see immediately that it is a solution. The reverse implication follows by arguments identical to those in Theorem 2.3.
Thus ∂ α c enters the expression for ∂ α ̺ linearly. Since all lower derivatives of c(x) are bounded, one can then check that ∂ α ̺ 2 < ∞ and thus ̺ ∈ H ℓ+1 (U ). We can then bootstrap to obtain smooth solutions. Observe now that for d ≤ 2 and s ∈ ( 1 2 , 1], Φ s ∈ L 2 s (U ). For d = 1, Theorem 4.2 applies directly and the bifurcation points are given by: For d = 2 one can notice that Φ 1 (x) = Φ 1 (Π(x)) for any permutation Π of the d coordinates. Our strategy will be to apply Theorem 4.2 after reducing the problem to the symmetrised space L 2 ex (U ) and then use the discussion in which is equivalent to checking that given a prime 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 r(z) = (d 1,4 (z) − d 3,4 (z)) , where d ℓ,4 (z) is the number of divisors of z of the form 4k + ℓ, k ∈ N, ℓ ≥ 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. ■

Remark A.2. A sharper estimate based on the diameter of the support can be obtained if it is assumed
that the gradient of the interaction potential has compact support strictly contained in U . Let ε = R/L denote the ratio of the diameter of the support to the size of the domain. Then we can obtain exponential convergence to equilibrium in L 2 , for κ < 2π L ∇W ∞ (2+ε) .
Appendix B. Results from bifurcation theory Let X be a separable Hilbert space and denote by L(X) the set of bounded, linear, operators on X. For F : X × R + → X a twice Frechét-differentiable mapping, we define N = ker D x F (x 0 , κ * ) and R = Im D x F (x 0 , κ * ). Furthermore, we assume that, F (x 0 , κ * ) = 0 for some (x 0 , κ * ) ∈ X × R + . We also assume that D x F (x 0 , κ * ) is a Fredholm operator with index zero and that dim N = 1 from which also follows that codim R = 1. Then, we have the following decompositions into complementary subspaces of X: where N = span[v 0 ] and Z 0 = span[z 0 ] for some v 0 , z 0 ∈ X. We can also pick X 0 to be orthogonal to N and closed, i.e., X 0 = {x ∈ X : x, v 0 X = 0}, where ·, · X denotes the inner product on X. This allows us to define the following canonical projection operators: which, by the closed graph theorem, are continuous.

r1(s v0,ψ(s))
|s|+|ψ(s)−κ * | = 0. We finally present the following result from [Dei85, Theorem 29.1], often referred to as the Rabinowitz alternative (cf. [Rab71]). Theorem B.3. Let X be a real Banach space, V ⊂ X × R a neighbourhood of (0, κ * ), G : V → X completely continuous, and G(x, κ) = o(|x|) as x → 0 uniformly in κ on compact subsets of R + . Let K be a compact, linear operator on X and κ * be a characteristic value of K having odd algebraic multiplicity with F (̺, κ) = x − κKx + G(x, κ). If C V ⊂ V is the set of nontrivial solutions of F (x, κ) = 0 in V and C V,κ * is the connected component of C V containing (0, κ * ), then C V,κ * has at least one of the following two properties: