Relation between vorticity patch model and point vortex model for

## Abstract

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.

### Keywords

- point vortex
- quasi-equilibrium
- relaxation dynamics
- maximum entropy production
- global-in-time solution

## 1. Introduction

The physical object studied in this chapter is non-viscous, noncompressible fluid with high Reynolds number occupied in bounded, simply-connected 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 *i*-th vortex, *x* with relative intensity

and *N* → ∞, mean field equation is derived by several arguments [2–7], that is,

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).

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

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 *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 *t* varies. We collect all the vorticity patches in a small region, called cell. Cell area *Δ* thus takes the relation *x* is *σ* is denoted by

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 *ν* denotes the outer unit normal vector. We obtain the total patch area conservation for each *σ*,

because

where *J* _{ω }is associated with the detailed fluctuation of (*ω*, *u*) from

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

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 *σ* and probability

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 *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

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 *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 *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 *k* → ∞, we obtain the first equation of (6) with

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 *k*-times localization procedure, it holds that

Putting

from

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 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

for any orbit to Eqs. (74), (75) to be global-in-time 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 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

and

where

Theorem 1: Let *Ω* = *B* and *ω* _{0} be a smooth function in the form of *T* ∈ (0, + ∞] be the maximal existence time of the classical solution to Eqs. (74), (75) and *λ* be the total mass defined by Eq. (76). Then, it follows that

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 *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 Smoluchowski-Poisson equation, which guarantees the continuation of *t* = *T* as a measure on *t* < *T*. By Eq. (100), therefore, it holds that

with *c* ≥ 0 and

Then, *k* = 1,2,. Hence, it holds that *c* = 0 in Eq. (111).

If the conclusion in Eq. (97) is false, we have the *ε* regularity in Eqs. (74), (75) [9, 17]. Thus, there is

for *c* = 0, which contradicts to *T* < + ∞.

## 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 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.

## Acknowledgments

This work was supported by Grant-in-Aid for Scientific Research (A) 26247013 and Grant-in-Aid for Challenging Exploratory Research 15K13448.