Relaxation Theory for Point Vortices Relaxation Theory for Point Vortices

We study relaxation dynamics of the mean field of many point vortices from quasi- equilibrium to equilibrium. Maximum entropy production principle implies four con-sistent equations concerning relaxation-equilibrium states and patch-point vortex models. Point vortex relaxation equation coincides with Brownian point vortex equation in micro-canonical setting. Mathematical analysis to point vortex relaxation equation is done in accordance with the Smoluchowski-Poisson equation.


Introduction
The physical object studied in this chapter is non-viscous, noncompressible fluid with high Reynolds number occupied in bounded, simply-connected domain. Ω ∈ R 2 . Motion of this fluid is described by the Euler-Poisson equation and u, ω and ψ stand for the velocity, vorticity and stream function, respectively.
ωðx, tÞ ¼ X N i¼1 α i δ xiðtÞ ðdxÞ (2) system of Eq. (1) is reduced to associated with the Hamiltonian where G ¼ Gðx, x ′ Þ is the Green's function of -Δ provided with the Dirichlet boundary condition and Onsager [1] proposed to use statistical mechanics of Gibbs to Eq. (3). In the limit N ! ∞ with αN = 1, local mean of vortex distribution is given by where α i ¼α i α,α i ∈ I ¼ ½−1; 1 is the intensity of the i-th vortex, ρα ðxÞ is the existence probability of the vortex at x with relative intensityα, which satisfies and PðdαÞ is the numerical density of the vortices with the relative intensityα. Under H N ¼ E ¼ constant, α 2 Nβ N ¼ β ¼ constant and N ! ∞, mean field equation is derived by several arguments [2][3][4][5][6][7], that is, Since Ref. [8], structure of the set of solutions to Eq. (6) has been clarified in accordance with the Hamiltonian given by Eq. (4) (see [9] and the references therein).
Quasi-equilibria, on the other hand, are observed for several isolated systems with many components [10]. Thus, we have a relatively stationary state, different from the equilibrium, which eventually approaches the latter. Relaxation indicates this time interval, from quasi-equilibrium to equilibrium. To approach relaxation dynamics of many point vortices, patch model is used. It describes detailed vortex distribution, where N p , σ i and Ω i (t) denote the number of patches, the vorticity of the i-th patch and the domain of the i-th patch, respectively. Mean field equations for equilibrium and for relaxation time are derived by the principles of maximum entropy [11,12] and maximum entropy production [13,14], respectively. For the latter case, one obtains a system on p ¼ pðx, σ, tÞ, with the diffusion coefficient D ¼ Dðx, tÞ > 0: In this chapter, we regard Eq. (2) as a limit of Eq. (7). First, point vortex model valid to the relaxation time is derived from Eq. (8), that is, a system on ρα ¼ ρα ðx, tÞ,α ∈ I, in the form of Second, the stationary state of Eq. (9) is given by Eq. (6). Third, Eq. (9) coincides with the Brownian point vortex model of Chavanis [15]. Finally, system of Eq. (9) provided with the boundary condition satisfies the requirements of isolated system in thermodynamics.
In fact, averaging Eq. (9) implies Then, we obtain mass and energy conservations d dt where (,) stands for the L 2 inner product. Assuming ρα > 0, we write the first equation of (9) as Then, it follows that d dt from Eq. (10), where Hence, it follows that d dt from Eq. (13), that is, entropy increasing.

Vorticity patch model
In Eq. (7), the vorticity σ i is uniform in a region with constant area Ω i ðtÞ, called vorticity patch. A patch takes a variety of forms as the time t varies. We collect all the vorticity patches in a small region, called cell. Cell area Δ thus takes the relation jΩ i j ≪ Δ ≪ jΩj. The probability that the average vorticity at x is σ is denoted by pðx, σ, tÞdx which satisfies Z pðx, σ, tÞdσ ¼ 1: equality (18) means conservation of total area of patches of the vorticity σ. Then, the macroscopic vorticity is defined by which is associated with the stream function ψ ¼ ψðx, tÞ and the velocity u ¼ uðx, tÞ through To formulate equilibrium, we apply the principle of maximum entropy [11,12], seeking the maximal state of under the constraint Eqs. (17), (18) and With the Lagrange multipliers β p , cðσÞ, ζðxÞ , it follows that which is reduced to Here, β p and c(σ) may be called inverse temperature and chemical potential, respectively. We put c(0) = 0 because of the degree of freedom of c(σ) admitted by Eq. (19). Then, it follows that and hence, Eq. (17) implies From Eqs. (18) and (26), similarly, it follows that The equilibrium mean field equation of vorticity patch model is thus given by Eqs. (20), (21), (27) and (28), which is reduced to One may use the principle of maximum entropy production to describe near from equilibrium dynamics [13,14]. We apply the transport equation where ν denotes the outer unit normal vector. We obtain the total patch area conservation for each σ, where J ω ¼ Z σJðx, σ, tÞdσ stands for the local mean vorticity flux. Since J ω Á ν ¼ 0 on ∂Ω, Furthermore, J ω is associated with the detailed fluctuation of (ω, u) from ðω, uÞ by Eq. (1).
Here, we ignore the diffusion energy as the total energy of this system. Using maximum entropy production principle, we chose the flux J to maximize entropy production rate S _ under the constraint where Using Lagrange multipliers ðβ p , D, ζÞ ¼ β p ðtÞ, Dðx, tÞ, ζðx, tÞ , we obtain From the constraint of Eq. (34), it follows that which implies Thus, we end up with by Eqs. (30), (37), (40) and (41), where D ¼ Dðx, tÞ > 0.

Point vortex model
Point vortex model is regarded as a special case of vorticity patch model, where the patch size shrinks to zero [16]. Here, we give a quantitative description of this limit process, using localization. First, we derive the equilibrium mean field equation of point vortices from that of vorticity patches. Then, we derive relaxation equation for the point vortex model. Fundamental quantities of point vortex model are circulation αα, probability ρα ðx, tÞ and number density PðdαÞ. Circulation of each vortex is set to be small to preserve total energy and total circulation in the mean field limit. In the vorticity patch model, on the other hand, vorticity σ and probability pðx, σ, tÞ are the fundamental quantities ( Figure 1).
Here, we use the following localization in order to transform vorticity patch to point vortex ( Figure 2): 1. Divide each patch into two patches with half area and the same vorticity.
2. Again, divide each patch into two patches with half area: one has doubled vorticity and the other has 0 vorticity.
Under this procedure, the number of nonzero patches is doubled and their vorticities are also doubled. At the same time, the area of each patch becomes 1/4 and the number of total patches is quadrupled, while the total circulation is preserved. First, we describe the detailed process for the stationary state of Eq. (7).
Let Ω be divided into many cells with uniform size Δ and let each cell be composed of many patches. Let N ðkÞ ðx, σÞdxdσ be the number of patches in the cell after k-times of the above procedure centered at x of which vorticity was originally σ and let σ (k) be the vorticity of these patches after k-times localization. We assume that the number of total vorticity patches in the cell, is independent of x. Then, the number of total patches in Ω, the total area of the patches and the total circulation of the patches after k-times localization procedures, with original vorticity σ, are given by and respectively.
We obtain recalling Eq. (7). Since it holds that We also have Fundamental quantities constituting of the mean field limit of point vortex model thus arise as k ! ∞.
To explore the relationship between the quantities in two models, we take regards to circulation of one patch, total circulation of patches with original vorticity σ and local mean vorticity. Based on σ ðkÞ Á jΩj and Eq. (47), we reach the ansatz σjΩj ¼α, 1 These relations are summarized in the following After k-times localization, the first equation in Eq. (29) takes the form From Table 1, the right-hand side on Eq. (64) is replaced by We reach Therefore, after k-times localization procedure, it holds that (71) Here, we assume lim

Relaxation dynamics
If PðdαÞ ¼ δ 1 ðdαÞ, it holds that ω ¼ ω 2 in Eq. (11). Then, we obtain assuming D = 1. Conservations of total mass and energy ‖ωðÁ, are derived from Eq. (13), while increase in entropy of Eq. (16) is reduced to d dt where ΦðsÞ ¼ sðlogs−1Þ þ 1. In the stationary state, we obtain logω þ βψ ¼ constant by Eq. (77). Hence, it follows that from Eq. (76). Here, the third equation implies the fourth equation as Using In fact, to see the third equality of (81), we note and hence If μ < 0, system of Eq. (81) except for the third equation is equivalent to the Gel'fand problem If Ω is simply connected, there is a non-compact family of solutions as μ ↑ 0, which are uniformly bounded near the boundary [8,9] for any orbit to Eqs. (74), (75) to be global-in-time and compact, for any λ, e > 0 in Eq. (76).
If Ω ¼ B ≡ fx ∈ R 2 j jxj < 1g, it actually holds that Eq. (87). In this case, we have v ¼ vðrÞ, r ¼ jxj, and the result follows from an elementary calculation. More precisely, putting with c ↑ 1 as μ ↑ +∞. It follows that and hence lim c↑1 I ¼ 0: If β is constant in Eq. (9), it is the mean field limit of Brownian vortices [15]. It is nothing but the Smoluchowski-Poisson equation [9,17] and obeys the feature of canonical ensemble, provided with total mass conservation and decrease of free energy: Then, there arises the blowup threshold β ¼ −8π=λ [18]. Here, we show the following theorem, where G ¼ Gðx, x 0 Þ denotes the Green's function for the Poisson part, and where It holds that ρ ϕ ∈ L ∞ ðΩ · ΩÞ. The proof is similar as in Lemma 5.2 of [17] for the case of Neumann boundary condition.

Conclusion
We study the relaxation dynamics of the point vortices in the incompressible Euler fluid, using the vorticity patch which varies with uniform vorticity and constant area. The mean field limit equation is derived, which has the same form as the one derived for the Brownian point vortex model. This equation governs the last stage of self-organization, not only in the point vortices but also in the two-dimensional center guiding plasma and the rotating superfluid helium, from quasi-equilibrium to equilibrium. Mathematical analysis assures this property for radially symmetric case, provided that the inverse temperature is bounded below.