1. Introduction
The physical object studied in this chapter is nonviscous, noncompressible fluid with high Reynolds number occupied in bounded, simplyconnected domain.
where
and u, ω and ψ stand for the velocity, vorticity and stream function, respectively.
In the point vortex model
system of Eq. (1) is reduced to
associated with the Hamiltonian
where
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
and
with
where
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).
Quasiequilibria, 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 quasiequilibrium 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 ith patch and the domain of the ith 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
(8) 
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
(9) 
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
(11) 
for
Then, we obtain mass and energy conservations
where (,) stands for the L
^{2} inner product. Assuming
Then, it follows that
from Eq. (10), where
Hence, it follows that
from Eq. (13), that is, entropy increasing.
2. Vorticity patch model
In Eq. (7), the vorticity σ
_{
i
} is uniform in a region with constant area
Let
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
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
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
with the diffusion flux
because
where
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
where
Using Lagrange multipliers
Since
Eq. (35) is reduced to
From the constraint of Eq. (34), it follows that
and
(39) 
which implies
and
Thus, we end up with
3. 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
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 Ω be divided into many cells with uniform size Δ and let each cell be composed of many patches. Let
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 ktimes localization procedures, with original vorticity σ, are given by
and
respectively.
We obtain
recalling Eq. (7). Since
it holds that
From Eq. (48), the related probability
satisfies
(50) 
and hence,
We also have
which implies
(53) 
by
It holds also that
and
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
and Eq. (47), we reach the ansatz
to put
by
Finally, we use the identity on local mean vorticity
to assign
regarding
These relations are summarized in the following Table 1 :
Vorticity patch model  Point vortex model 











After ktimes localization, the first equation in Eq. (29) takes the form
(64) 
From Table 1 , the righthand side on Eq. (64) is replaced by
for
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 J for σ = 0 is
and hence
Flux is thus given by
We reach
with
Therefore, after ktimes localization procedure, it holds that
Putting
from
(73) 
Here, we assume
4. Relaxation dynamics
If
assuming D = 1. Conservations of total mass and energy
are derived from Eq. (13), while increase in entropy of Eq. (16) is reduced to
where
In the stationary state, we obtain
from Eq. (76). Here, the third equation implies the fourth equation as
Using
therefore, Eq. (78) is reduced to
In fact, to see the third equality of (81), we note
which implies
and hence
If μ < 0, system of Eq. (81) except for the third equation is equivalent to the Gel’fand problem
with σ = –μ. If Ω is simply connected, there is a noncompact 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
for any orbit to Eqs. (74), (75) to be globalintime and compact, for any
If
where
with c ↑ 1 as μ ↑ +∞. It follows that
with
and hence
If β is constant in Eq. (9), it is the mean field limit of Brownian vortices [15]. It is nothing but the SmoluchowskiPoisson 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
and
where
Theorem 1: Let Ω = B and ω
_{0} be a smooth function in the form of
and
In particular, we have
Proof: From the assumption, it follows that
Then, we obtain
and hence
It holds also that
which implies
with
The comparison theorem now guarantees
For Eq. (96) to prove, we use the second moment. First, the Poisson part of Eq. (75) is reduced to
Second, it follows that
(106) 
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
by Eq. (100). Second, we have
and hence
Inequality (110) takes place of the monotonicity formula used for the SmoluchowskiPoisson equation, which guarantees the continuation of
with c ≥ 0 and
Then,
If the conclusion in Eq. (97) is false, we have the ε regularity in Eqs. (74), (75) [9, 17]. Thus, there is
for
5. 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 selforganization, not only in the point vortices but also in the twodimensional center guiding plasma and the rotating superfluid helium, from quasiequilibrium to equilibrium. Mathematical analysis assures this property for radially symmetric case, provided that the inverse temperature is bounded below.