Relation between vorticity patch model and point vortex model for
Abstract
We study relaxation dynamics of the mean field of many point vortices from quasiequilibrium to equilibrium. Maximum entropy production principle implies four consistent equations concerning relaxationequilibrium states and patchpoint vortex models. Point vortex relaxation equation coincides with Brownian point vortex equation in microcanonical setting. Mathematical analysis to point vortex relaxation equation is done in accordance with the SmoluchowskiPoisson equation.
Keywords
 point vortex
 quasiequilibrium
 relaxation dynamics
 maximum entropy production
 globalintime solution
1. Introduction
The physical object studied in this chapter is nonviscous, noncompressible fluid with high Reynolds number occupied in bounded, simplyconnected domain.
where
and
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
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
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
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
for
Then, we obtain mass and energy conservations
where (,) stands for the
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
Let
be independent of
equality (18) means conservation of total area of patches of the vorticity
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,
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
where
Using Lagrange multipliers
Since
Eq. (35) is reduced to
From the constraint of Eq. (34), it follows that
and
which implies
and
Thus, we end up with
by Eqs. (30), (37), (40) and (41), where
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
is independent of
and
respectively.
We obtain
recalling Eq. (7). Since
it holds that
From Eq. (48), the related probability
satisfies
and hence,
We also have
which implies
by
It holds also that
and
Fundamental quantities constituting of the mean field limit of point vortex model thus arise as
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
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
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
and hence
Flux is thus given by
We reach
with
Therefore, after
Putting
from
Here, we assume
4. Relaxation dynamics
If
assuming
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
with
for this family. For
for any orbit to Eqs. (74), (75) to be globalintime and compact, for any
If
where
with
with
and hence
If
Then, there arises the blowup threshold
and
where
Theorem 1: Let
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
from
Then,
Now, we assume
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
Then,
If the conclusion in Eq. (97) is false, we have the
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.
Acknowledgments
This work was supported by GrantinAid for Scientific Research (A) 26247013 and GrantinAid for Challenging Exploratory Research 15K13448.
References
 1.
Onsager L. Statistical hydrodynamics. Il Nuovo Cimento. 1949; 6 (2):279–287. doi:10.1007/BF02780991  2.
Caglioti E, Lions PL, Marchioro C, Pulvirenti M. A special class of stationary flows for twodimensional Euler equations: A statistical mechanics description. Communications in Mathematical Physics. 1992; 143 (3):501–525. doi:10.1007/BF02099262  3.
Eyink GL, Spohn H. Negativetemperature states and largescale, longlived vortices in twodimensional turbulence. Journal of Statistical Physics. 1993; 70 (3):833–886. doi:10.1007/BF01053597  4.
Joyce G, Montgomery D. Negative temperature states for the twodimensional guidingcentre plasma. Journal of Plasma Physics. 1973; 10 (01):107–121. doi:10.1017/S0022377800007686  5.
Kiessling MKH. Statistical mechanics of classical particles with logarithmic interactions. Communications on Pure and Applied Mathematics. 1993; 46 (1):27–56. doi:10.1002/cpa.3160460103  6.
Pointin YB, Lundgren TS. Statistical mechanics of two‐dimensional vortices in a bounded container. Physics of Fluids. 1976; 19 (10):1459–1470. doi:10.1063/1.861347  7.
Sawada K, Suzuki T. Derivation of the equilibrium mean field equations of point vortex system and vortex filament system. Theoretical and Applied Mechanics Japan. 2008; 56 :285–290. doi:10.11345/nctam.56.285  8.
Nagasaki K, Suzuki T. Asymptotic analysis for twodimensional elliptic eigenvalue problems with exponentially dominated nonlinearities. Asymptotic Analysis. 1990; 3 (2):173–188. doi:10.3233/ASY19903205  9.
Suzuki T. Mean Field Theories and Dual Variation—Mathematical Structures of the Mesoscopic Model. 2nd ed. Paris: Atlantis Press; 2015. 444 p. doi:10.2991/9789462391543  10.
LyndenBell D. Statistical mechanics of violent relaxation in stellar systems. Monthly Notices of the Royal Astronomical Society. 1967; 136 (1):101–121. doi:10.1093/mnras/136.1.101  11.
Robert R. A maximumentropy principle for twodimensional perfect fluid dynamics. Journal of Statistical Physics. 1991; 65 (3):531–553. doi:10.1007/BF01053743  12.
Robert R, Sommeria J. Statistical equilibrium states for twodimensional flows. Journal of Fluid Mechanics. 1991; 229 :291–310. doi:10.1017/S0022112091003038  13.
Robert R, Sommeria J. Relaxation towards a statistical equilibrium state in twodimensional perfect fluid dynamics. Physical Review Letters. 1992; 69 (19):2776–2779. doi:10.1103/PhysRevLett.69.2776  14.
Robert R, Rosier C. The modeling of small scales in twodimensional turbulent flows: A statistical mechanics approach. Journal of Statistical Physics. 1997; 86 (3):481–515. doi:10.1007/BF02199111  15.
Chavanis PH. Twodimensional Brownian vortices. Physica A. 2008; 387 (28):6917–6942. doi:10.1016/j.physa.2008.09.019  16.
Chavanis PH, Sommeria J, Robert R. Statistical mechanics of twodimensional vortices and collisionless stellar systems. The Astrophysical Journal. 1996; 1 (471):385–399. doi:10.1086/177977  17.
Suzuki T. Free Energy and SelfInteracting Particles. 1st ed. Boston: Birkhäuser; 2005. 370 p. doi:10.1007/0817644369  18.
Suzuki T. Brownian point vortices and DDmodel. Discrete and Continuous Dynamical Systems—Series S. 2014; 7 (1):161–176. doi:10.3934/dcdss.2014.7.161