Equilibrium state for two-dimensional point vortex system withmultiple temperatures
Yuichi Yatsuyanagi1 , Megumi Ikeda1 , and Tadatsugu Hatori2
1

1

Faculty of Education, Shizuoka University, Japan
2
National Institute for Fusion Science, JapanIntroduction

A concept of a negative temperature state for twodimensional (2D) point vortex system was ﬁrst introduced
by Onsager to explain large-scale structure formation, forexample, Jupiter’s Great Red Spot, Naruto eddy, and soon [1]. If the system temperature is negative, a probability
proportional to exp(−βE) increases as the system energy
increases, so that higher energy conﬁguration is possible,
where β is inverse temperature and E is energy.
Motivated in part by his conjecture, we have tackled thisproblem both numerically and theoretically. In this paper,
we will present recent results revealing that an equilibrium
state at negative absolute temperature consists of multiple
temperature subsystems.
24Point vortex system

We consider a system consisting of N/2 positive andN/2 negative point vortices conﬁned in a circular area withradius R. The word ”positive vortex” means that the circulation of the vortex is positive. The value of the i-thpoint vortex Ωi is either Ω0 or −Ω0 where Ω0 is a positive
constant. System energy H is given byH

=1
Ωi ψi
2=1 −
Ωj log |r i − r j |2πSimulation results

To reveal the characteristics of the point vortex system
in negative temperature state, we have carried out simulation research.

4.1

Equilibrium distribution

(1)

i

ψi

where S is entropy, W is density of state and Boltzmann’s
relation is assumed. The density of state increases as energy increases in normal systems, so that temperature isalways positive. On the other hand, if the total phasespace volume (total number of states available) is limited,the density of state has a peak at some energy E0 anddecreases E > E0 . Thus, T becomes negative at E > E0 .
The equations of motion of the point vortices are given by
Eq. (3). The conﬁguration space coincides with the phasespace for the point vortex system. As the total phase spacevolume is (πR2 )N < ∞, Onsager anticipated the existenceof the negative temperature state for the point vortex system.

Equilibrium distributions in both negative and positive
temperatures are obtained time asymptotically. They are
shown in Fig. 1. For the positive temperature case, bothj=i1 Ωj log |r i − r̄ j |+2π
j1 R−Ωj log2π|r j |(2)jwhere r i is the position vector of the i-th point vortexand ψi is the stream function at r i which corresponds theenergy possessed by the i-th point vortex. The eﬀect of
the circular wall is introduced by the image vortex locatedat r̄ i = R2 r i /|r i |2 . Motions of the vortices are traced by
the following equations of motion:
Ωi3dxi∂H,=dt∂yi

Ωidyi∂H=−dt∂xi(3)
Fig. 1. Typical equilibrium distributions of point vortices
at (a) positive and (b) negative temperatures.

Negative absolute temperature

Here we brieﬂy summarize the concept of the negative
absolute temperature introduced by Onsager [1]. Statistical deﬁnition of temperature T is given by∂ log W∂S1== kBT∂E∂E(4)sign vortices spread uniformly over the circular area. On
the other hand, for the negative temperature case, same
sign vortices tend to cluster and form clumps exclusively
consisting of single sign.

4.2

Population of point vortices categorized byenergy

The energy possessed by each point vortex is deﬁned by
ψi as shown in Eq. (2). Populations (histograms) as afunction of ψi are plotted in Fig. 2. For the positive tem-Fig. 3. Energy population is plotted separately for thevortices outside the clumps and inside the clumps

Fig. 2. Population is plotted as a function of energy possessed by each point vortex ψi .
perature case, the peak is located at ψi ≈ 0. This is dueto the uniform distribution of the point vortices. For thenegative temperature case, there are three peaks. The leftmost and the rightmost peaks correspond the clump distributions. The center peak corresponds the backgrounduniform distribution outside the clumps.
To conﬁrm the origin of the peaks located at the bothends, population is recalculated separately for the point
vortices inside the clump and the others. The resultis shown in Fig. 3. Red line indicates the populationof the vortices inside the clumps, and blue line outside
the clumps. We categorize the energy regions in 8 partsin Fig. 3. In region (1), the population decreases linearly, which corresponds the distribution is proportionalto exp(−βΩi ψi ). The slope is almost kept constant among
the various simulations with N Ω0 = constant. The center peak resembles the positive temperature case shown inFig. 2 (a). Indeed, the blue line originates the backgroundvortices located outside the clumps. These observations
indicate that the equilibrium distribution consists of thesubsystems of diﬀerent temperature including positive andnegative ones.
Another evidence is given by Fig. 4. Equilibrium vortex
distribution is rearranged in conjunction with the 8 energy
ranges shown in Fig. 3. This ﬁgure clearly indicates thatthe vortices in the cores of the clumps yield the leftmost
and rightmost populations. It is surprising that the boundaries of the diﬀerent temperature vortices are parallel toFig. 4. Equilibrium distribution categorized by 8 energy
ranges shown in Fig. 3 is shown. Black line indicates thestream function.
the stream function. As there is no ﬂow (particle motion)across the boundaries, the diﬀerence of the temperature ispreserved in the equilibrium distribution.
5Conclusion

In this paper, we have shown that the equilibrium stateof 2D point vortex system in negative temperature consists of multiple subsystems of diﬀerent temperature. The
diﬀerence of the temperature is maintained due to the noparticle transport across the boundary.
Acknowledgement
This work was supported by KAKENHI (22740254).
References[1] L. Onsager, ”Statistical Hydrodynamics”, Nuovo Cimento
Suppl., 6, 279, 1949.