Macroscopic Limits and Phase Transition in a System of Self-propelled Particles

Abstract

We investigate systems of self-propelled particles with alignment interaction. Compared to previous work (Degond and Motsch, Math. Models Methods Appl. Sci. 18:1193–1215, 2008a; Frouvelle, Math. Models Methods Appl. Sci., 2012), the force acting on the particles is not normalized, and this modification gives rise to phase transitions from disordered states at low density to aligned states at high densities. This model is the space-inhomogeneous extension of (Frouvelle and Liu, Dynamics in a kinetic model of oriented particles with phase transition, 2012), in which the existence and stability of the equilibrium states were investigated. When the density is lower than a threshold value, the dynamics is described by a nonlinear diffusion equation. By contrast, when the density is larger than this threshold value, the dynamics is described by a similar hydrodynamic model for self-alignment interactions as derived in (Degond and Motsch, Math. Models Methods Appl. Sci. 18:1193–1215, 2008a; Frouvelle, Math. Models Methods Appl. Sci., 2012). However, the modified normalization of the force gives rise to different convection speeds, and the resulting model may lose its hyperbolicity in some regions of the state space.

Appendices

Appendix A: Poincaré Constant

In this appendix, we prove the following.

Proposition A.1

We have the following Poincaré inequality, for  $\psi\in H^{1}(\mathbb{S})$:

(A.1)

The best constant Λ _κ in this inequality is the smallest positive eigenvalue of the operator

(A.2)

We define the linear operator  $L_{\kappa}^{*}$ by

(A.3)

Then one of the following three possibilities is true:

• (i) Λ _κ is the smallest eigenvalue of the Sturm–Liouville problem

  

  (A.4)

  for g∈C ²([0,π]) with Neumann boundary conditions (g′(0)=g′(π)=0) and such that  $\int_{0}^{\pi}(\sin\theta)^{n-2} e^{\kappa\cos\theta }g(\theta)\,\mathrm{d} \theta=0$, and the eigenspace of  $L_{\kappa\varOmega }^{*}$ associated to the eigenvalue Λ _κ is of dimension 1, spanned by  $\omega\mapsto h^{0}_{\kappa}(\omega\cdot\varOmega )$, where the function θ↦h ₀(cosθ) is smooth, positive for 0⩽θ<θ ₀, and negative for θ ₀<θ⩽π.

• (ii) Λ _κ is the smallest eigenvalue of the Sturm–Liouville problem

  

  (A.5)

  for g∈C ²([0,π]) with Dirichlet boundary conditions (g(0)=g(π)=0), and the eigenspace of  $L_{\kappa\varOmega}^{*}$ associated to Λ _κ is of dimension n−1, consisting in the functions ψ _A of the form  $\psi_{A}(\omega)=h_{\kappa}^{1}(\omega\cdot \varOmega)A\cdot\omega$ for any vector A∈ℝⁿ such that Ω⋅A=0, with  $\theta\mapsto h_{\kappa}^{1}(\cos\theta)$ a smooth positive function for 0<θ<π.

• (iii) The two preceding Sturm–Liouville problems have the same smallest eigenvalue Λ _κ , and the eigenspace of  $L_{\kappa \varOmega}^{*}$ associated to Λ _κ is of dimension n, spanned by the two types of function of the above cases.

Proof

First of all, we have

(A.6)

and

(A.7)

The second inequality of (A.6) follows from the Poincaré inequality on the sphere:

$$\int_\mathbb{S} \biggl(\psi-\int_\mathbb{S} \psi \biggr)^2 \leqslant\frac{1}{n-1} \int_\mathbb{S} |\nabla_\omega\psi|^2. $$

The first inequality of (A.7) follows from the fact that

$$\biggl\langle\biggl(\psi-\int_\mathbb{S} \psi\biggr)^2\biggr\rangle_{M_{\kappa\varOmega}} - \bigl\langle\bigl(\psi-\langle\psi\rangle_{M_{\kappa\varOmega}}\bigr)^2\bigr\rangle _{M_{\kappa\varOmega}} = \biggl( \int_\mathbb{S} \psi- \int_\mathbb{S} \psi M_{\kappa\varOmega} \biggr)^2 \geqslant0 . $$

Equations (A.6) and (A.7) lead to the Poincaré inequality (A.1) with

$$\varLambda_\kappa \geqslant (n-1) \frac{\min M_{\kappa\varOmega }}{\max M_{\kappa\varOmega}} = (n-1) e^{2\kappa} . $$

We use the inner product $(\varphi,\psi)\mapsto\langle\varphi\psi\rangle _{M_{\kappa\varOmega}}$, adapted to M _κΩ . We denote by $\dot{L}^{2}_{\kappa}(\mathbb{S})$ (resp. $\dot{H}^{1}_{\kappa}(\mathbb{S})$) the functions $\psi\in L^{2}(\mathbb{S})$ (resp. in $H^{1}(\mathbb{S})$) such that $\langle\psi\rangle_{M_{\kappa \varOmega}}=0$.

The operator $L_{\kappa\varOmega}^{*}$ given by (A.2) is self-adjoint since $\langle\nabla_{\omega}\psi\cdot\nabla_{\omega}\varphi \rangle_{M_{\kappa\varOmega}}=\langle\psi L_{\kappa\varOmega}^{*}\varphi \rangle_{M_{\kappa\varOmega}}$. It is then easy to see, using the Lax-Milgram theorem, that if φ belongs to $\dot{L}^{2}_{\kappa}(\mathbb{S})$, then there is a unique solution $\psi\in\dot{H}^{1}_{\kappa}(\mathbb{S})$ to the equation $L_{\kappa\varOmega}^{*}\psi=\varphi$. The so-obtained inverse operator is then compact and self-adjoint. By the spectral theorem, we get a basis of eigenfunctions, in the Hilbert space $\dot{L}^{2}_{\kappa}(\mathbb{S})$, which are also eigenfunctions of $L_{\kappa\varOmega}^{*}$. If we denote $\varLambda_{\kappa}^{-1}$ the largest eigenvalue of the inverse of $L_{\kappa\varOmega}^{*}$, then it is easy to see that Λ _κ is the best constant for the following Poincaré inequality, in the space $\dot{H}^{1}_{\kappa}(\mathbb{S})$:

Since the constants trivially satisfy this inequality, this shows that Λ _κ is the best constant for the Poincaré inequality (A.1) in $H^{1}(\mathbb{S})$.

The goal is now to reduce the computation of the eigenvalues to simpler problems, using separation of variables. We write ω=cosθΩ+sinθ  v, where v belongs to the unit sphere, orthogonal to Ω. We identify Ω with the last element of an orthogonal basis of ℝⁿ, and we write $v\in \mathbb{S}_{n-2}$.

By spherical harmonic decomposition in an adapted basis (see for example Frouvelle and Liu 2012, Appendix A), we have a unique decomposition of the form

(A.8)

where  is a given orthonormal basis of the spherical harmonics of degree m on $\mathbb {S}_{n-2}$, for m∈ℕ, with $k_{m}=\tbinom{n+m-2}{n-2}- \tbinom {n+m-4}{n-2}$. If ψ is continuous, $g_{m}^{k}$ is given by

(A.9)

We now show that the decomposition (A.8) remains stable under the action of the operator L _κΩ , so that its spectral decomposition can be performed independently for each term of the decomposition.

First, we examine the case of dimension n⩾3. Let ψ(ω)=g(θ)Z(v). We have

where the unit vector e _θ is given by

$$e_\theta= \nabla_\omega\theta=-\frac{1}{\sin\theta}(\mbox{Id}-\omega \otimes\omega)\varOmega. $$

We take functions $\psi(\omega)= g_{m}^{k}(\theta)Z_{m}^{k}(v)$ and $\varphi (\omega)=\sum_{k,m} f_{m}^{k}(\theta)Z_{m}^{k}(v)$. Since the spherical harmonics are orthonormal, and are eigenfunctions of Δ _v for the eigenvalues −m(m+n−3), we get:

(A.10)

Suppose m⩾1. Then, it is easy to see that the function ψ belongs to $\dot{H}^{1}_{\kappa}(\mathbb{S})$ if and only if $(\sin\theta )^{\frac{n}{2}-1} g'\in L^{2}(0,\pi)$ and $(\sin\theta)^{\frac{n}{2}-2} g'\in L^{2}(0,\pi)$. This condition is equivalent to the fact that g∈V, where V is defined by (4.5), and which we denote by $V_{\kappa}^{m}$ for convenience:

Suppose now that m=0. Then $Z^{k}_{m}$ is a constant, and the condition $\psi\in\dot{H}^{1}_{\kappa}(\mathbb{S})$ is equivalent to the first condition only: $(\sin\theta)^{\frac{n}{2}-1} g'\in L^{2}(0,\pi)$, under the constraint that $\int_{0}^{\pi}(\sin\theta)^{n-2} e^{\kappa\cos \theta}g(\theta)\,\mathrm{d} \theta=0$. We will denote this space by $V^{0}_{\kappa}$:

Formula (A.10) then suggests to define the operator $L_{\kappa,m}^{*}:V_{\kappa}^{m}\to(V_{\kappa}^{m})^{*}$ by

(A.11)

From (A.10), it follows that, if we decompose $\psi(\omega )=\sum_{k,m} g_{m}^{k}(\theta)Z_{m}^{k}(v)$, then

showing that $L_{\kappa\varOmega}^{*}$ is block diagonal on each of these spaces $V_{\kappa}^{m}$ (tensorized by the spherical harmonics of degree m on ${\mathbb{S}}_{n-2}$). So we can perform the spectral decomposition of $L_{\kappa\varOmega}^{*}$ by means of the spectral decomposition of each of the $L_{\kappa,m}^{*}$. It is indeed easy to prove, using the Lax-Milgram theorem, that the operators $L_{\kappa,m}^{*}$ have self-adjoint compact inverses for the dot product $(f,g)=\int_{0}^{\pi}fg(\sin\theta)^{n-2} e^{\kappa\cos\theta }\,\mathrm{d} \theta$. Therefore, the eigenfunctions and eigenvalues of $L_{\kappa\varOmega}^{*}$ correspond to those of the operators $L_{\kappa,m}^{*}$, for all m∈ℕ. If we denote by λ _κ,m the smallest eigenvalue of $L_{\kappa,m}^{*}$, we finally get

We notice that

but since all the $V_{\kappa}^{m}$ are the same for m⩾1, and since

we get

Finally, Λ _κ is the minimum between λ _κ,0 and λ _κ,1. The eigenfunctions for the operator $L_{\kappa\varOmega}^{*}$ being smooth, this is also true for the operators $L_{\kappa,m}^{*}$, by formula (A.9). So we can transform the definitions (A.11) by integration by parts.

Indeed, if g ₀ is an eigenfunction (in $V_{\kappa}^{0}$) associated to $L_{\kappa,0}^{*}$ and an eigenvalue λ, then g ₀ is smooth and satisfies the Sturm–Liouville eigenvalue problem

Conversely, a smooth function with the condition $\int_{0}^{\pi}(\sin\theta )^{n-2} e^{\kappa\cos\theta}g(\theta)\,\mathrm{d} \theta=0$ belongs to $V_{\kappa}^{0}$. Actually, in dimension n⩾3, we do not need to impose the Neumann boundary conditions: they appear naturally, since we have

Therefore, by continuity at θ=0 and π, $g_{0}'(0)=g_{0}'(\pi)=0$. Then, using classical Sturm–Liouville oscillation theory (see Weidmann 1987, for example), we find that the first eigenspace of $L_{\kappa}^{*}$ is of dimension 1, spanned by a function g _κ,0(θ), which is positive for 0⩽θ<θ ₀ and negative for θ ₀<θ⩽π.

Similarly, if g ₁ is an eigenfunction (in $V_{\kappa}^{1}$) associated to $L_{\kappa,1}^{*}$ and an eigenvalue λ, then g ₁ is smooth, with g ₁(0)=g ₁(π)=0 and satisfies the Sturm–Liouville eigenvalue problem

And conversely, if a function with Dirichlet boundary conditions is in C ²([0,π]), then it belongs to $V_{\kappa}^{1}$. Once again, if n⩾3, we do not need to impose the Dirichlet boundary conditions in the C ²([0,π]) framework, since we have

So, by continuity at θ=0 and π, g ₁(0)=g ₁(π)=0, and then a first order expansion shows that continuity holds, whatever the values of $g_{0}'(\theta)$ at the endpoints are. Again, using classical Sturm–Liouville theory, we find that the first eigenspace of $L_{\kappa}^{*}$ is of dimension 1, spanned by a function g _κ,1(θ), which keeps the same sign on (0,π).

The case λ _κ,0<λ _κ,1 corresponds to case (i) of the proposition. Since a spherical harmonic of degree 0 on the sphere $\mathbb{S}_{n-2}$ is a constant, introducing the function $h_{\kappa}^{0}$ such that $h_{\kappa}^{0}(\cos\theta)=g_{\kappa ,0}(\theta)$ allows us to state that the eigenspace of $L_{\kappa \varOmega}^{*}$ associated to the lowest eigenvalue is spanned by $\omega \mapsto h_{\kappa}^{0}(\omega\cdot\varOmega)$.

The case λ _κ,0>λ _κ,1 corresponds to case (ii) of the proposition. The spherical harmonics of degree 1 on the sphere $\mathbb{S}_{n-2}$ are the functions of the form v↦A⋅v, with A⋅Ω=0. Introducing $h_{\kappa}^{0}$ such that $h_{\kappa}^{0}(\cos\theta) \sin\theta= g_{\kappa,0}(\theta)$ allows us to state that the eigenspace of $L_{\kappa\varOmega}^{*}$ associated to the lowest eigenvalue is of dimension n−1, consisting of the functions of the form $\omega\mapsto h_{\kappa}^{1}(\omega\cdot\varOmega) A\cdot\omega$, with A any vector in ℝⁿ such that A⋅Ω=0.

Finally, the case λ _κ,0=λ _κ,1 corresponds to case (iii) of the proposition, and this ends the proof in the case of dimension n⩾3.

We now examine the special case of dimension n=2. We identify $H^{1}(\mathbb{S})$ with the 2π-periodic functions in $H^{1}_{loc}(\mathbb{R})$. So, Λ _κ is the smallest eigenvalue of the periodic Sturm–Liouville problem

for functions g such that $\int_{-\pi}^{\pi}e^{\kappa\cos\theta }g(\theta)\,\mathrm{d} \theta=0$. Here the decomposition corresponding to (A.8) is the even-odd decomposition (there are only two spherical harmonics on $\mathbb{S}_{0}$: the constant function of degree 0 and the odd function of degree 1). The odd part g _o of g can be identified with a function of $H^{1}_{0}(0,\pi)$, and it is easy to see that the odd part of $L_{\kappa}^{*}(g)$ is $L_{\kappa}^{*}(g_{o})$, and similarly for the even part g _e . So, we can perform the spectral decomposition of $L^{*}_{\kappa}$ separately on the spaces of even and odd functions.

Actually, if g is a solution of the Sturm–Liouville periodic problem, the function $\widetilde{g}$ defined by $\widetilde{g}(\theta)=e^{-\kappa \cos\theta}\partial_{\theta}g(\pi-\theta)$ is another solution with the same eigenvalue. Furthermore, if g is odd, then $\widetilde{g}$ is even and conversely. So the eigenvalues are the same for the odd and even spaces problems. Therefore, in dimension n=2, Proposition A.1 can be refined, and we can state that case (iii) is the only possibility: the eigenspace of $L_{\kappa\varOmega}^{*}$ associated to Λ _κ is of dimension 2, spanned by an odd function $g_{\kappa}^{o}$, positive on (0,π), and an even function $g_{\kappa}^{e}=\widetilde{g}_{\kappa}^{o}$, positive for 0<θ<θ ₀ and negative for θ ₀<θ<π. The proof of Proposition A.1 is complete. □

We can now state a conjecture, which refines Proposition A.1, if true, and which is based on numerical experiments.

Conjecture A.1

• (i) When κ>0 and n⩾3, only statement (ii) of Proposition A.1 is true.

• (ii) The function κ↦Λ _κ is increasing.

We also observe numerically that λ ₁∼κ when kappa is large.

Some investigations are in progress to prove the monotonicity of the eigenvalue with respect to κ, based on formal expansions similar to those used in Sect. 5 of Frouvelle (2012).

Remark A.1

At the end of the proof of Proposition A.1, we have seen that in dimension n=2 only statement (iii) is true. The proof uses a transformation of the solution of an eigenvalue problem into the solution of another eigenvalue problem. We can try to find a similar transformation in dimensions n⩾3: if f satisfies L _κ,0 f=λf (with Neumann boundary conditions), then $\widetilde{f}=e^{-\kappa\cos\theta} \partial_{\theta}f(\pi-\theta)$ (with Dirichlet boundary conditions) satisfies

so if we can prove that $\int_{0}^{\pi}\cos\theta \widetilde{f}^{\,2} (\sin\theta)^{n-2} e^{\kappa\cos\theta} \,\mathrm{d} \theta>0$, we can deduce that λ ₀>λ ₁. So far we have been unable to prove this estimate.

Appendix B: Numerical Computations of the Coefficients

We adopt a finite difference approach to compute the function g _κ associated to the GCIs and defined by (4.6). We consider the function f _κ such that $f_{\kappa}(\theta)=(\sin \theta)^{\frac{n}{2}-1} g_{\kappa}(\theta)$. In particular, since g _κ ∈V defined by (4.5), f _κ belongs to $H^{1}_{0}(0,\pi)$. Since g _κ satisfies (4.6), f _κ satisfies

We discretize the interval (0,π) with N+1 points $\theta_{i}=\frac{1}{N}i\pi$, and denote by $f^{i}_{\kappa}$ an approximation of f _κ at these points. Since $f_{\kappa}\in H^{1}_{0}(0,\pi)$, we have $f^{0}_{\kappa}=f^{N}_{\kappa}=0$. We define $e_{\kappa}^{i}=e^{\kappa\cos\theta_{i}}$. A second order approximation of $(e^{\kappa\cos\theta}f_{\kappa}')'$ at θ _i is then given by

Introducing

the vector  is the solution of the linear system AF=S, where the vector S is , and the tridiagonal matrix A is defined by

(B.1)

We use the trapezoidal method to perform the integrations in the definitions (2.11) and (4.9) of c and $\widetilde{c}$. The other coefficients ρ, λ, and θ _c are then directly computed from c and $\widetilde{c}$. The numerical results provided in Figs. 3 and 4 have been obtained for N=3000.

We now detail how we obtain an approximation of the Poincaré constant Λ _κ . By Appendix A, Λ _κ is the minimum between λ _κ,1 and λ _κ,0, which are the smallest eigenvalues of two Sturm–Liouville problems. Several algorithms exist to compute eigenvalues of singular Sturm–Liouville problems (which is the case here whenever n⩾3) with a good precision (Bailey et al. 1991). However, we use a simpler method based on finite differences.

Actually, λ _κ,1 is the smallest eigenvalue associated to problem (A.5), with g∈V. So, considering once again the function f such that $f(\theta)=(\sin\theta)^{\frac{n}{2}-1} g(\theta )$, the vector AF, with A defined by (B.1), gives a second order approximation of $(\sin\theta)^{\frac{n}{2}-1}\widetilde{L}_{\kappa}^{*}g(\theta)=\lambda f(\theta)$ at the points θ _i . So we can take the smallest eigenvalue of A as an approximation of λ _κ,1.

We now look for an approximation of λ _κ,0.

Let g be a solution of the Sturm–Liouville problem (A.4) with Neumann boundary conditions. We introduce , the vector of approximations of g at the points $\theta_{i+\frac{1}{2}}=\frac{1}{N}(i+\frac{1}{2})\pi$. Introducing $m_{\kappa}^{i}=(\sin\theta)^{n-2} e^{\kappa\cos\theta_{i}}$, a second order approximation of $L_{\kappa}^{*}g$ at the point $\theta _{i+\frac{1}{2}}$, with i∈〚1,N−2〛, is then given by

With the Neumann boundary conditions, the approximations at the points $\theta_{\frac{1}{2}}$ and $\theta_{N-\frac{1}{2}}$ are given by

Introducing

a second order approximation of $L_{\kappa}^{*}g$ is given by BG, where the tridiagonal matrix B is defined by

(B.2)

So we can take the smallest positive eigenvalue of B as an approximation of λ _κ,0 (excluding the constant functions). The computations of Fig. 1 have been performed with N=300 points.