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We study relaxation dynamics of the mean field of many point vortices from quasi-equilibrium to equilibrium. Maximum entropy production principle implies four consistent 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.
maximum entropy production
chapter and author info
Meteorological College, Kashiwa, Japan
Graduate School of Engineering Science Osaka University, Toyonaka, Japan
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The physical object studied in this chapter is non-viscous, noncompressible fluid with high Reynolds number occupied in bounded, simply-connected domain. . Motion of this fluid is described by the Euler-Poisson equation
and u, ωand ψstand for the velocity, vorticity and stream function, respectively.
where is the Green’s function of –Δ provided with the Dirichlet boundary condition and
Onsager  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 is the intensity of the i-th vortex, is the existence probability of the vortex at xwith relative intensity , which satisfies
and is the numerical density of the vortices with the relative intensity . Under and N→ ∞, mean field equation is derived by several arguments [2–7], that is,
Since Ref. , structure of the set of solutions to Eq. (6) has been clarified in accordance with the Hamiltonian given by Eq. (4) (see  and the references therein).
Quasi-equilibria, on the other hand, are observed for several isolated systems with many components . 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 Np, σiand Ωi(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 ,
with the diffusion coefficient
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 , 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 . Finally, system of Eq. (9) provided with the boundary condition
satisfies the requirements of isolated system in thermodynamics.
In Eq. (7), the vorticity σiis uniform in a region with constant area , called vorticity patch. A patch takes a variety of forms as the time tvaries. We collect all the vorticity patches in a small region, called cell. Cell area Δthus takes the relation . The probability that the average vorticity at xis σis denoted by ， which satisfies
be independent of t. Since
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 and the velocity through
To formulate equilibrium, we apply the principle of maximum entropy [11, 12], seeking the maximal state of
Point vortex model is regarded as a special case of vorticity patch model, where the patch size shrinks to zero . 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 , probability and number density . 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 σand probability are the fundamental quantities ( Figure 1 ).
Here, we use the following localization in order to transform vorticity patch to point vortex ( Figure 2 ):
Divide each patch into two patches with half area and the same vorticity.
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 Ωbe divided into many cells with uniform size Δ and let each cell be composed of many patches. Let be the number of patches in the cell after k-times of the above procedure centered at xof which vorticity was originally σand 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
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 σand local mean vorticity. Based on
and Eq. (47), we reach the ansatz . Similarly, we use
Finally, we use the identity on local mean vorticity
These relations are summarized in the following Table 1 :
Vorticity patch model
Point vortex model
Relation between vorticity patch model and point vortex model for .
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
for . Sending k→ ∞, we obtain the first equation of (6) with by Eq. (51). This means that the vorticity patch model is transformed to the point vortex model applied to the mean field limit by taking the localization procedure.
We can derive also relaxation equation of point vortex model from that of vorticity patch model. By Eq. (37), the value of the diffusion flux Jfor σ= 0 is
Flux is thus given by
Therefore, after k-times localization procedure, it holds that
Putting , similarly, we obtain
Here, we assume , because and the 0-vorticity patch becomes dominant in the system. Then, we obtain Eq. (9) by Eq. (72).
In fact, to see the third equality of (81), we note
If μ< 0, system of Eq. (81) except for the third equation is equivalent to the Gel’fand problem
with σ= –μ. If Ωis simply connected, there is a non-compact family of solutions as μ↑ 0, which are uniformly bounded near the boundary [8, 9]. Hence, there arises
for this family. For μ≥ 0, on the contrary, system of Eq. (81) except for the third equation admits a unique solution . Regarding Eq. (76), therefore, it is necessary that
for any orbit to Eqs. (74), (75) to be global-in-time and compact, for any in Eq. (76).
If , it actually holds that Eq. (87). In this case, we have and the result follows from an elementary calculation. More precisely, putting , , we obtain
where . Using , , we have
with c↑ 1 as μ↑ +∞. It follows that
If βis constant in Eq. (9), it is the mean field limit of Brownian vortices . 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 . Here, we show the following theorem, where denotes the Green’s function for the Poisson part,
where . It holds that . The proof is similar as in Lemma 5.2 of  for the case of Neumann boundary condition.
Theorem 1: Let Ω= Band ω0 be a smooth function in the form of with , . Let T∈ (0, + ∞] be the maximal existence time of the classical solution to Eqs. (74), (75) and λbe the total mass defined by Eq. (76). Then, it follows that
In particular, we have
Proof: From the assumption, it follows that and
Then, we obtain
It holds also that
The comparison theorem now guarantees and hence
For Eq. (96) to prove, we use the second moment. First, the Poisson part of Eq. (75) is reduced to
Second, it follows that
from A(1) = M. Under the hypothesis of Eq. (96), we have δ> 0 such that
Then, T= +∞ gives a contradiction.
Now, we assume T< +∞. First, equality in (106) implies
Inequality (110) takes place of the monotonicity formula used for the Smoluchowski-Poisson equation, which guarantees the continuation of up to t= Tas a measure on [9, 17]. Thus, there is such that for 0 ≤ t< T. By Eq. (100), therefore, it holds that
with c≥ 0 and . From the elliptic regularity, we obtain
Then, implies for k= 1,2,. Hence, it holds that c= 0 in Eq. (111).
If the conclusion in Eq. (97) is false, we have the εregularity in Eqs. (74), (75) [9, 17]. Thus, there is , such that
for . The hypothesis in Eq. (113) is valid for by Eq. (111), c= 0, which contradicts to T< + ∞.
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.
Ken Sawada and Takashi Suzuki (March 1st 2017). Relaxation Theory for Point Vortices, Vortex Structures in Fluid Dynamic Problems, Hector Perez-de-Tejada, IntechOpen, DOI: 10.5772/67075. Available from:
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