Analytical Derivation of Correlation Function for Vorticity-Depleted Regions Formed Around Strong Vortex

Yuichi Yatsuyanagi; Tadatsugu Hatori; Hiroshi Ohtsuka; Akio Sanpei; Yukihiro Soga

doi:10.12688/f1000research.176213.1

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Research Article

Analytical Derivation of Correlation Function for Vorticity-Depleted Regions Formed Around Strong Vortex

[version 1; peer review: awaiting peer review]

Yuichi Yatsuyanagi ¹, Tadatsugu Hatori², Hiroshi Ohtsuka³, Akio Sanpei⁴, Yukihiro Soga³

Yuichi Yatsuyanagi ¹, Tadatsugu Hatori², [...] Hiroshi Ohtsuka³, Akio Sanpei⁴, Yukihiro Soga³

PUBLISHED 02 Mar 2026

Author details Author details

¹ Institute of Liberal Arts and Science, Kanazawa University, Kanazawa, 920-1192, Japan
² National Institute for Fusion Science, Toki, Gifu, 509-5292, Japan
³ Faculty of Mathematics and Physics, Institute of Science and Engineering, Kanazawa University, Kanazawa, 920-1192, Japan
⁴ Faculty of Electrical Engineering and Electronics, Kyoto Institute of Technology, Sakyo, Kyoto, 606-8585, Japan

Yuichi Yatsuyanagi
Roles: Conceptualization, Formal Analysis, Methodology, Visualization, Writing – Original Draft Preparation, Writing – Review & Editing

Tadatsugu Hatori
Roles: Conceptualization, Formal Analysis, Methodology, Writing – Review & Editing

Hiroshi Ohtsuka
Roles: Conceptualization, Formal Analysis, Methodology, Validation, Writing – Original Draft Preparation, Writing – Review & Editing

Akio Sanpei
Roles: Data Curation, Investigation, Resources, Visualization

Yukihiro Soga
Roles: Data Curation, Investigation, Resources, Visualization

OPEN PEER REVIEW

REVIEWER STATUS AWAITING PEER REVIEW

This article is included in the Japan Institutional Gateway gateway.

Abstract

Background

Two-dimensional turbulence exhibits distinct behavior from three-dimensional turbulence, characterized by an inverse energy cascade that transports energy from small to large scales, leading to the formation of large-scale vortices and organized structures. Onsager proposed the concept of negative absolute temperature states to explain this large-scale structure formation. Pure electron plasma experiments, where electron motion perpendicular to a strong axial magnetic field obeys the two-dimensional incompressible Euler equation, provide an ideal platform for studying such phenomena. In these experiments, the electron number density corresponds to vorticity, and the electrostatic potential to the stream function. Experiments at Kyoto University revealed the formation of depleted vorticity regions around strong vortices, which remained theoretically unexplained.

Methods

We employ linear response theory combined with mean-field approximation for point vortex systems with negative absolute temperature to investigate this depletion mechanism. Point vortices represent the vorticity field as a collection of delta-function singularities. We consider a delta-function vortex impulsively injected into an equilibrium state of uniform background vorticity and analytically derive the two-body correlation function that characterizes the system’s response to this perturbation. The mean-field approximation treats the continuous particle distribution emerging in the infinite particle limit.

Results

Our analysis demonstrates that the two-body correlation function exhibits negative values in the vicinity of the injected vortex. This negative correlation corresponds to a depleted vorticity region surrounding the strong vortex, consistent with experimental observations.

Conclusions

The analytical framework provides a qualitative explanation for the long-standing question of vorticity depletion formation around strong vortices in two-dimensional flows. However, our linear response treatment does not capture longer-time dynamics, and quantitative agreement with experiments requires further investigation through large-scale numerical simulations.

Keywords

Two-dimensional turbulence, Negative absolute temperature, Point vortex system, Linear response theory, Mean-fileld theory

Corresponding author: Yuichi Yatsuyanagi

Competing interests: No competing interests were disclosed.

Grant information: The author(s) declared that no grants were involved in supporting this work.

Copyright: © 2026 Yatsuyanagi Y et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

How to cite: Yatsuyanagi Y, Hatori T, Ohtsuka H et al. Analytical Derivation of Correlation Function for Vorticity-Depleted Regions Formed Around Strong Vortex [version 1; peer review: awaiting peer review]. F1000Research 2026, 15:335 (https://doi.org/10.12688/f1000research.176213.1) First published: 02 Mar 2026, 15:335 (https://doi.org/10.12688/f1000research.176213.1) Latest published: 02 Mar 2026, 15:335 (https://doi.org/10.12688/f1000research.176213.1)

I. Introduction

Main topic of this paper is a vortex dynamics in a two-dimensional (2D) system.

We analytically investigate the formation mechanism of a depleted vorticity region around a strong vortex injected into a uniform background vorticity field. A delta-function-type strong vortex is impulsively injected as a perturbation into an equilibrium state of uniformly distributed vorticity. In response to the injected impulse vortex, a region of depleted vorticity is formed in the vicinity of the impulse vortex. This phenomenon was discovered in two-dimensional vortex experiments using pure electron plasmas. An overview of these experiments is presented later. In this paper, we provide an analytical explanation for the formation of the depleted vorticity region using linear response theory and mean-field approximation for two-dimensional point vortex systems.

In three-dimensional turbulence, the Richardson cascade is a well-known phenomenon in which system energy is transported from large scales to small scales and is ultimately dissipated through viscous effects.^1,2 In contrast, two-dimensional turbulence is characterized by the inverse cascade, which differs from three-dimensional turbulence.^3,4 In the inverse cascade, energy is transported from small scales to large scales, leading to the formation of large-scale vortices. Furthermore, when numerous small-scale vortices aggregate, they form highly organized structures resembling a crystalline lattice. This phenomenon is sometimes referred to as self-organization.

To observe structure formation in two-dimensional turbulence, it is necessary to track the time evolution over long periods, which requires inviscid or approximately high Reynolds number flow experiments. A pure electron plasma system is well-suited for this purpose.⁵ Let us consider electrons confined in a vacuum chamber with a strong axial magnetic field. The 2D motion of the electrons perpendicular to the axial magnetic field is described by the 2D incompressible, inviscid Euler equation.^6–8 The electron number density and the self-induced electrostatic potential are proportional to the vorticity and the stream function, respectively. Two groups at Kyoto University and at University of California, San Diego have independently reported pure electron plasma experimental results on two-dimensional turbulence and structure formation.^9–14

Onsager proposed the concept of negative absolute temperature $β$ as a key to understanding the inverse cascade phenomenon.^15–17 The state with $β < 0$ corresponds to a phenomenon in which the statistically defined (inverse) temperature $dS / dE$ becomes negative, where $S$ is the entropy and $E$ is the (internal) energy. The entropy and the number of accessible states $W (E)$ of the system are related by $S = k_{B} log W (E)$ where $k_{B}$ is the Boltzmann constant. Therefore, although unusual for conventional systems, negative absolute temperature can occur in systems where the number of accessible states decreases as the energy increases. Point vortex systems are known as representative examples of systems that can exhibit negative absolute temperature states.

The point vortex system serves as a tool for representing two-dimensional flows. In this system, the vorticity field is represented by a collection of point vortices discretized as delta functions (described later). As $N$ same-sign point vortices coalesce into a single location, the degrees of freedom regarding their configuration decrease while the energy increases. That is, since the number of accessible states decreases with increasing energy, this system can exhibit states with $β < 0$ . It was numerically demonstrated by Joyce et al. that point vortices confined in a rectangular domain with periodic boundary conditions form a clustered state.^18,19 Using large-scale simulations with special-purpose supercomputer MDGRAPE-2, the negative temperature properties observed in point vortex systems confined within a circular boundary was examined by Yatsuyanagi et al.²⁰

The point vortex system can serve as a powerful tool not only for numerical simulations but also for theoretical analysis.^21,22 In experiments conducted by the Kyoto University group, the formation of a depleted region around a strong vortex was observed, but the mechanism behind this phenomenon remained unclear for a long time.²³ In the present study, to describe the phenomenon, we derive the two-body correlation function analytically, employing the linear response theory and the mean-field approximation for the point vortex system. The linear response is the response of a system in equilibrium to a perturbation imposed by an external field, where the perturbation is weak enough not to destroy the equilibrium state. In a discrete particle system, the mean field is defined as the continuous particle distribution function that emerges in the infinite particle limit. This approach is referred to as the mean-field approximation. Injecting a delta-function-type strong vortex impulsively as a perturbation into an equilibrium state of uniformly distributed vorticity, we evaluate the response as the two-body correlation function. The obtained function indicates that a region exhibiting a negative response to the impulse input emerges.

Organization of this paper is as follows: In Sec. II, we first introduce the two-dimensional fluid experiments using pure electron plasma and the experimental result that motivated this study (Sec. II A). Subsequently, we define the two-dimensional point vortex system used as an analytical tool (Sec. II B), and present the results calculated via the route of Path (1) (Sec. II C) and the route of Path (2) (Sec. II D) in Figure 1. Then, by imposing the condition that the results obtained from the above two routes should be equal, we derive the equation that the two-body correlation function must satisfy (Sec. II E). In Section III, we obtain the homogeneous solution (Sec. III A 1) and the special solution (Sec. III A 2) of the equation derived in Sec. II E. Finally, in Section IV, we examine the conditions that the two-body correlation function must satisfy in a system with circular boundaries, and present the explicit solution of the two-body correlation function.

Figure 1. The linear response followed by the infinite particle limit, and the linear response to the mean field after taking the infinite particle limit.

II. Methods

A. Two-dimensional pure electron plasma experiment

In this section, we present the experimental results obtained with pure electron plasmas. Electrons are confined in a cylindrical vacuum chamber radially by a strong magnetic field along the axis of the chamber and axially by the negatively biased electrostatic potentials at both ends. When the electrostatic potential at one end of the chamber is turned off, electrons are released along the axis and strike a phosphor screen positioned perpendicular to the magnetic field. The resulting luminosity distribution is recorded by a Charge Coupled Device (CCD) camera. As this measurement is destructive, the experiment requires high reproducibility. Details of the experimental configuration are found in Refs. 9, 24, 25. The electron motion perpendicular to the magnetic field is described by the 2D Euler equation,^6,7 in which the electron density is proportional to the vorticity.

The time evolution of the electron distribution is shown in Figure 2. The brightness in this figure is proportional to the electron density, i.e., the vorticity. In this experiment, we first established an uniform equilibrium electron distribution as the background vortex, and then injected an electron population with a density higher than that of the background. A region where electrons are concentrated at a density higher than the background distribution is referred to as a clump. In this case, the vorticity inside the clump is about 180 times higher than that of the background.

Figure 2. Experimental results are shown.

Shown are snapshots of the electron density at (a) 15 μs, (b) 80 μs, (c) 5 ms, (d) 10 ms, and (e) 200 ms.

The clump migrates toward the center of the background vortex through the interaction of its self-induced swirling flow with the background vortex. As the clump moves, it entrains low-vorticity regions from outside the background vortex, creating a ring hole of reduced vorticity around it. We will explain the mechanism of ring hole formation using the two-point correlation function of vorticity.

The density depression around the clump is depicted in Figure 3. Figure 3 plots the density profiles along lines A-A’, B-B’, and C-C’ shown in Figures 2 (c), (d), and (e), respectively. Points A, A’, B, B’, C, and C’ are located in the background vortex, and regions with density lower than the background are highlighted in green. The green regions indicate areas exhibiting negative correlation.

Figure 3. Radial profile of electron density centered on the clump is shown.

B. Point vortex system

In this section, we define a two-dimensional (2D) system considered and introduce prerequisite for calculating the explicit formula of a two-body correlation function for the system.

1. 2D point vortex system

Let us consider a collection of $N$ singular point vortices, which defines a 2D vorticity field $\hat{ω} (r, t)$ :

(1)

\hat{ω} (r, t) ≔ \sum_{i = 1}^{N} Ω δ (r - r_{i} (t)) .

The position vector of the $i$ -th point vortex is given by $r_{i}$ . Each vortex has a constant circulation $Ω$ . The notation $δ (r)$ is the 2D Dirac delta function. A quantity depending on the positions $r_{i}$ of the singular point vortices is indicated by $\hat{\cdot}$ . We call a variable with $\hat{\cdot}$ as the microscopic one. The total circulation $N Ω$ is assumed to be finite and constant, namely

(2)

Ω \sim \frac{1}{N} .

A stream function is defined by

(3)

\hat{ψ} (r) ≔ \sum_{i} Ω G (r, r_{i})

where

G (r, r^{'})

is the 2D Green function for 2D Laplacian

(4)

G (r, r^{'}) ≔ - \frac{1}{2 π} log | r - r^{'} | .

The stream function (3) and the vorticity (1) are related by the Poisson equation

(5)

- Δ \hat{ψ} (r) = \hat{ω} (r)

2. Hamiltonian

The following integral corresponding to energy of the system diverges at $r = r^{'}$

(6)

\begin{array}{l} \frac{1}{2} \int_{R^{2}} \int_{R^{2}} G (r, r^{'}) \hat{ω} (r) \hat{ω} (r^{'}) d r d r^{'} \\ = \frac{1}{2} \int_{R^{2}} \hat{ψ} (r) \hat{ω} (r) d r ≕ {\hat{H}}_{0} + E_{\inf}, \end{array}

where

E_{\inf}

is a constant including a diverging part. Notation

{\hat{H}}_{0}

is the Hamiltonian of the point vortex system in the usual sense.

(7)

{\hat{H}}_{0} ≔ \frac{Ω^{2}}{2} \sum_{i \neq j} G (r_{i}, r_{j})

A quantity without an external perturbation (see below) is indicated by the suffix $\cdot_{0}$ .

3. External perturbation

A macroscopic stream function $ψ_{e} (r)$ without $\hat{\cdot}$ is introduced as an external arbitrary perturbation, which defines a microsopic stream function ${\hat{ψ}}_{e}$

(8)

Ω {\hat{ψ}}_{e} ≔ \int_{R^{2}} ψ_{e} (r) \hat{ω} (r) d r = Ω \sum_{i} ψ_{e} (r_{i})

(9)

{\hat{H}}_{e} ≔ {\hat{H}}_{0} + Ω {\hat{ψ}}_{e}

The quantity ${\hat{H}}_{e}$ denotes Hamiltonian with an external perturbation ${\hat{ψ}}_{e}$ .

For convenience, we introduce a suffix $\cdot_{*}$ where the asterisk matchs either 0 or $e$ . In an equation which contais two or more asterisks, all the asteriscs are unifiedly replaced by either 0 or $e$ .

4. Partition function

A partition function is defined by

(10)

Z_{*} ≔ \int_{R^{2 N}} exp (- β {\hat{H}}_{*}) d r_{1} \dots d r_{N} .

According to the assumption (2), the inverse temperature $β$ is chosen to be

(11)

β \sim O (N) .

Under the above condition and the definition (7), the dependence of Hamiltonian ${\hat{H}}_{*}$ on $N$ is estimated as

(12)

β {\hat{H}}_{*} \sim β Ω^{2} (N - 1) G \sim N \cdot \frac{1}{N^{2}} \cdot N \sim O (1)

with respect to each

r_{i}

. This ensures that the orders of the partition function

Z_{*}

and the free energy (see Eq. (21)) are estimated as

(13)

\begin{array}{l} Z_{*} \sim O (C^{N}), \\ F_{*} = - \frac{1}{β} ln Z_{*} \sim O (1) . \end{array}

This shows the free energy is well-defined regardless of the magnitude of $N$ .

5. Canonical average

The canonical average converts a microscopic quantity into a macroscopic one. There are two kinds of canonical average, ${⟨ \cdot ⟩}_{0}$ and ${⟨ \cdot ⟩}_{e}$ . The notation ${⟨ \cdot ⟩}_{0}$ represents the average with $H_{0}$ , and the notation ${⟨ \cdot ⟩}_{e}$ represents the average with $H_{e}$ , namely,

(14)

ω_{*} (r) ≔ {⟨ \hat{ω} (r) ⟩}_{*} = \frac{1}{Z_{*}} \int_{R^{2 N}} \hat{ω} (r) exp (- β {\hat{H}}_{*}) d r_{1} \dots d r_{N} .

6. Fluctuation

The fluctuation $δ {\hat{ω}}_{*} (r)$ is defined by the subtraction of the canonical averaged quantity from the microscopic one which depends on the particle position $r_{i}$ :

(15)

δ {\hat{ω}}_{*} (r) ≔ \hat{ω} (r) - ω_{*} (r)

7. One-body distribution function

The one-body distribution function $P_{1, *}^{N} (r)$ is defined by

(16)

P_{1, *}^{N} (r) ≔ {⟨ δ (r - r_{i}) ⟩}_{*} .

Due to the symmetry of ${\hat{H}}_{*}$ under the permutation of the $i$ -th and the $j$ -th particles, it satisfies

(17)

P_{1, *}^{N} (r) = \frac{1}{Z_{*}} \int_{R^{2 (N - 1)}} exp (- β {\hat{H}}_{*} (r, r_{2}, \dots, r_{N})) d r_{2} \dots d r_{N}

for any

i = 1, \dots, N

. Thus, the following relation also holds:

(18)

ω_{*} (r) = {⟨ \hat{ω} (r) ⟩}_{*} = N Ω P_{1, *}^{N} .

8. Two-body distribution function

The two-body distribution function $P_{2, *}^{N} (r, r^{'})$ is defined by

(19)

P_{2, *}^{N} (r, r^{'}) ≔ {⟨ δ (r - r_{i}) δ (r^{'} - r_{j}) ⟩}_{*} .

and satisfies

(20)

P_{2, *}^{N} (r, r^{'}) = \frac{1}{Z_{*}} \int_{R^{2 (N - 2)}} exp (- β {\hat{H}}_{*} (r, r^{'}, r_{3}, \dots, r_{N})) d r_{3} \dots d r_{N} .

Note that $P_{2, *}^{N} (r, r^{'})$ is not defined on $r = r^{'}$ as ${\hat{H}}_{0} (r, r^{'})$ is not defined on $r_{i} = r_{j} (i, j = 1, \dots, N)$ .

C. Linear response theory

The goal of this section is to derive a linear response formula for the $N$ -point vortex system to a fluctuation (path (1) in Figure 1).

1. Linear response formula

The free energy is defined by

(21)

F_{*} ≔ - \frac{1}{β} ln Z_{*} .

As a general procedure, we calculate a functional derivative two times to obtain a correlation of the fluctuation $δ \hat{ω} (r)$ _.²⁶ The functional derivative is carried out by introducing an arbitrary external perturbation $ϕ_{e} (r) ≔ s {δϕ}_{e}$ ( $s \in R$ ) and differentiating with respect to $s$ twice.

(22)

\frac{d}{ds} F_{e} = \frac{1}{Z_{e}} \int_{R^{2 N}} Ω \sum_{i} {δϕ}_{e} (r_{i}) exp (- β {\hat{H}}_{e}) d r_{1} \dots d r_{N} = \int_{R^{2}} (\frac{1}{Z_{e}} \int_{R^{2 N}} \hat{ω} (r) exp (- β {\hat{H}}_{e}) d r_{1} \dots d r_{N}) {δϕ}_{e} (r) d r

We represent the integrand as the functional delivative

(23)

\frac{δ F_{e}}{{δϕ}_{e} (r)} = \frac{1}{Z_{e}} \int_{R^{2 N}} \hat{ω} (r) exp (- β {\hat{H}}_{e}) d r_{1} \dots d r_{N} = {⟨ \hat{ω} (r) ⟩}_{e} = ω_{e} (r) .

Similarly, the second-order delivative is obtained:

(24)

- \frac{1}{β} \cdot \frac{d^{2}}{d s^{2}} F_{e} = - \frac{1}{β} \cdot \frac{d}{ds} (\frac{1}{Z_{e}} \int_{R^{2 N}} Ω \sum_{i} δ ϕ_{e} (r_{i}) exp (- β {\hat{H}}_{e}) d r_{1} \dots d r_{N}) = - \frac{1}{Z_{e}^{2}} {(\int_{R^{2 N}} Ω \sum_{i} δ ϕ_{e} (r_{i}) exp (- β {\hat{H}}_{e}) d r_{1} \dots d r_{N})}^{2} + \frac{1}{Z_{e}} \int_{R^{2 N}} {(Ω \sum_{i} δ ϕ_{e} (r_{i}))}^{2} exp (- β {\hat{H}}_{e}) d r_{1} \dots d r_{N} = - \int_{R^{2}} (\frac{1}{Z_{e}} \int_{R^{2 N}} \hat{ω} (r) exp (- β {\hat{H}}_{e}) d r_{1} \dots d r_{N}) δ ϕ_{e} (r) d r \times \int_{R^{2}} (\frac{1}{Z_{e}} \int_{R^{2 N}} \hat{ω} (r^{'}) exp (- β {\hat{H}}_{e}) d r_{1} \dots d r_{N}) δ ϕ_{e} (r^{'}) d r^{'} + \int_{R^{2}} \int_{R^{2}} (\frac{1}{Z_{e}} \int_{R^{2 N}} \hat{ω} (r) \hat{ω} (r^{'}) exp (- β {\hat{H}}_{e}) d r_{1} \dots d r_{N}) δ ϕ_{e} (r) δ ϕ_{e} (r^{'}) d r d r^{'}

We represent this result as

(25)

\begin{array}{l} - \frac{1}{β} \frac{δ^{2} F_{e}}{{δϕ}_{e} (r^{'}) {δϕ}_{e} (r)} = - \frac{1}{β} \frac{{δω}_{e} (r)}{{δϕ}_{e} (r^{'})} \\ = - {⟨ \hat{ω} (r) ⟩}_{e} {⟨ \hat{ω} (r^{'}) ⟩}_{e} + {⟨ \hat{ω} (r) \hat{ω} (r^{'}) ⟩}_{e} \\ = {⟨ δ {\hat{ω}}_{e} (r) δ {\hat{ω}}_{e} (r^{'}) ⟩}_{e} . \end{array}

In the limit $s \to 0$ , i.e., $ϕ_{e} = 0 \cdot {δϕ}_{e} = 0$ , Eq. (25) is written as

(26)

{- \frac{1}{β} \frac{{δω}_{e} (r)}{{δϕ}_{e} (r^{'})} |}_{ϕ_{e} (r^{'}) = 0} = {⟨ δ {\hat{ω}}_{0} (r) δ {\hat{ω}}_{0} (r^{'}) ⟩}_{0}

As an important result in relation to Eq. (26), a response to an external perturbation defined by

(27)

{δω}_{e} (r) ≔ {⟨ \hat{ω} (r) ⟩}_{e} - {⟨ \hat{ω} (r) ⟩}_{0} = ω_{e} (r) - ω_{0} (r)

is linearly approximated by

(28)

{δω}_{e} (r) \sim - β \int {⟨ δ {\hat{ω}}_{0} (r) δ {\hat{ω}}_{0} (r^{'}) ⟩}_{0} ϕ_{e} (r^{'}) d r^{'} .

It should be noted that Eq. (28) has the same form as the general linear-response formulae.²⁶

2. Two-body correlation function

The two-body correlation funcntion $g_{0} (r, r^{'})$ in a thermal equilibrium state is defined by

(29)

P_{2, 0} (r, r^{'}) ≔ P_{1, 0} (r) P_{1, 0} (r^{'}) (1 + g_{0} (r, r^{'})) .

The aim of the present analysis is to determine $g_{0} (r, r^{'})$ in the limit of vanishing external perturbation. Characteristics of the correlation function for long-range interacting particle system was discussed by Ornstein and Zernike.^27,28 A basic idea introduced by them is to split an effect of the correlation function into a self-correlation and a mutual correlation, which is called Ornstein-Zernike formula. For the point vortex system, the formula is rewritten as

(30)

{⟨ δ {\hat{ω}}_{0} (r) δ {\hat{ω}}_{0} (r^{'}) ⟩}_{0} = Ω ω_{0} (r) δ (r - r^{'}) + ω_{0} (r) ω_{0} (r^{'}) g_{0} (r, r^{'}) .

The first term of Eq. (30) in the right hand side corresponds to the self-correlation and the second term the mutual correlation. Substituting Eq. (30) into (28), we obtain the formula for the linear response in the point vortex system.

(31)

{δω}_{e} (r) = - β Ω ω_{0} (r) ϕ_{e} (r) - {βω}_{0} (r) \int g_{0} (r, r^{'}) ω_{0} (r^{'}) ϕ_{e} (r^{'}) d r^{'} .

This formula is the main conclusion, representing the linear response of the system derived from the two-body correlation function.

D. Mean-field theory

Mean-field theory is an approximation technique in which terms proportional to square of the fluctuation are neglected. The goal of this section is to derive a linear response formula for the mean-field approximated point vortex system.

1. Hamiltonian in the mean-field approximation

In the mean-field theory, physical quantities are represented by the sum of the mean-filed quantity and a fluctuation. There are two possible representations, with (suffix “e”) and without (suffix “0”) the external field:

(32)

\hat{ω} (r) ≔ {\bar{ω}}_{*} (r) + δ {\hat{\bar{ω}}}_{*} (r),

where the notation

\bar{\cdot}

indicates the quantity measured with the mean-field approximation. Namely,

{\bar{ω}}_{*} (r)

is the vorticity in the mean-filed approximation. Mean-field vorticity is related to the stream function by the following Poisson equation.

(33)

- Δ {\bar{ψ}}_{*} (r) = {\bar{ω}}_{*} (r) .

Note that the second term in the right hand side of Eq. (32) depends on the particle coordinate $r_{i}$ . Thus, the term has the hat $\hat{\cdot}$ .

The Hamiltonians ${\hat{\bar{H}}}_{0}$ and ${\hat{\bar{H}}}_{e}$ in the mean-filed approximation are given by

(34)

\begin{matrix} {\hat{H}}_{0} + E_{\inf} = \frac{1}{2} \int_{R^{2}} \int_{R^{2}} G (r, r^{'}) \hat{ω} (r) \hat{ω} (r^{'}) d r d r^{'} \\ \approx \frac{1}{2} \iint G (r, r^{'}) {\bar{ω}}_{0} (r) {\bar{ω}}_{0} (r^{'}) d r d r^{'} \\ + \iint G (r, r^{'}) δ {\hat{\bar{ω}}}_{0} (r) {\bar{ω}}_{0} (r^{'}) d r d r^{'} \\ = - \frac{1}{2} \iint G (r, r^{'}) {\bar{ω}}_{0} (r) {\bar{ω}}_{0} (r^{'}) d r d r^{'} \\ + Ω \sum_{i} {\bar{ψ}}_{0} (r_{i}) \\ ≕ {\hat{\bar{H}}}_{0} (r_{1}, \dots, r_{N}) \end{matrix}

(35)

{\hat{H}}_{e} + E_{\inf} = {\hat{H}}_{0} + E_{\inf} + \int_{R^{2}} ϕ_{e} (r) \hat{ω} (r) d r \approx - \frac{1}{2} \iint G (r, r^{'}) {\bar{ω}}_{e} (r) {\bar{ω}}_{e} (r^{'}) d r d r^{'} + Ω \sum_{i} {{\bar{ψ}}_{e} (r_{i}) + ϕ_{e} (r_{i})} ≕ {\hat{\bar{H}}}_{e} (r_{1}, \dots, r_{N})

2. Partition function

The partition functions ${\bar{Z}}_{e}$ and ${\bar{Z}}_{0}$ are defined by

(36)

{\bar{Z}}_{*} ≔ \int_{R^{2 N}} exp (- β {\hat{\bar{H}}}_{*}) d r_{1} \dots d r_{N}

(37)

\begin{matrix} {\bar{Z}}_{e} = exp (\frac{β}{2} \iint G (r, r^{'}) {\bar{ω}}_{e} (r) {\bar{ω}}_{e} (r^{'}) d r d r^{'}) \\ \times {[\int exp (- β Ω ({\bar{ψ}}_{e} (r^{'}) + ϕ_{e} (r^{'})) d r^{'}]}^{N} \end{matrix}

(38)

\begin{matrix} {\bar{Z}}_{0} = exp (\frac{β}{2} \iint G (r, r^{'}) {\bar{ω}}_{0} (r) {\bar{ω}}_{0} (r^{'}) d r d r^{'}) \\ \times {[\int exp (- β Ω ({\bar{ψ}}_{0} (r^{'})) d r^{'}]}^{N} . \end{matrix}

3. Free energy

The free energy ${\bar{F}}_{e}$ and ${\bar{F}}_{0}$ are defined by

(39)

{\bar{F}}_{*} ≔ - \frac{1}{β} ln {\bar{Z}}_{*}

(40)

\begin{matrix} {\bar{F}}_{e} = - \frac{1}{2} \iint G (r, r^{'}) {\bar{ω}}_{e} (r) {\bar{ω}}_{e} (r^{'}) d r d r^{'} \\ - \frac{N}{β} ln \int exp (- β Ω ({\bar{ψ}}_{e} (r_{1}) + ϕ_{e} (r_{1}))) d r_{1} \end{matrix}

(41)

\begin{matrix} {\bar{F}}_{0} = - \frac{1}{2} \iint G (r, r^{'}) {\bar{ω}}_{0} (r) {\bar{ω}}_{0} (r^{'}) d r d r^{'} \\ - \frac{N}{β} ln \int exp (- β Ω ({\bar{ψ}}_{0} (r_{1}))) d r_{1} \end{matrix}

4. Thermal equilibrium state

A thermal equilibrium state is defined by minimization of the free energy. The condition is given by

(42)

\frac{δ {\bar{F}}_{*}}{δ {\bar{ω}}_{*} (r)} = 0 .

Upon explicit calculation, we obtain:

(43)

\begin{matrix} \frac{δ {\bar{F}}_{e}}{δ {\bar{ω}}_{e} (r)} = - \int G (r, r^{'}) {\bar{ω}}_{e} (r^{'}) d r^{'} \\ - \frac{N}{β} (- β Ω) \frac{\int exp [- β Ω ({\bar{ψ}}_{e} (r^{'}) + ϕ_{e} (r^{'}))] \frac{δ {\bar{ψ}}_{e} (r^{'})}{δ {\bar{ω}}_{e} (r)} d r^{'}}{\int exp [- β Ω ({\bar{ψ}}_{e} (r^{'}) + ϕ_{e} (r^{'}))] d r^{'}} \\ = - {\bar{ψ}}_{e} (r) \\ + N Ω \frac{\int exp [- β Ω ({\bar{ψ}}_{e} (r^{'}) + ϕ_{e} (r^{'}))] G (r, r^{'}) d r^{'}}{\int exp [- β Ω ({\bar{ψ}}_{e} (r^{'}) + ϕ_{e} (r^{'}))] d r^{'}} = 0 . \end{matrix}

By operating $- Δ$ on Eq. (43), the following formula is obtained.

(44)

- Δ {\bar{ψ}}_{e} (r) = {\bar{ω}}_{e} (r) = N Ω \frac{exp (- β Ω ({\bar{ψ}}_{e} (r) + ϕ_{e} (r)))}{\int exp (- β Ω ({\bar{ψ}}_{e} (r^{'}) + ϕ_{e} (r^{'}))) d r^{'}} .

Similarly, the formula without a perturbation is also obtained.

(45)

- Δ {\bar{ψ}}_{0} (r) = {\bar{ω}}_{0} (r) = N Ω \frac{exp (- β Ω {\bar{ψ}}_{0} (r))}{\int exp (- β Ω {\bar{ψ}}_{0} (r^{'})) d r^{'}}

The solution to this Poisson equation is derived here, as it will be required in subsequent sections. Substituting

(46)

u_{0} = - β Ω {\bar{ψ}}_{0} (r)

into Eq. (45), we get

(47)

- Δ u_{0} = λ \frac{e^{u_{0}}}{\int e^{u_{0} (r^{'})} d r^{'}} (= - β Ω {\bar{ω}}_{0}),

where

(48)

λ ≔ - βN Ω^{2} .

The solution to Eq. (47) in $R^{2}$ was discussed by Chen and Li.²⁹ The solution to Eq. (47) exists only when $λ = 8 π$ and the solution is given by

(49)

u_{0} (r) = ln \frac{8 ϵ^{2}}{{(1 + {| ϵ r |}^{2})}^{2}} - ln \frac{8 π}{Z}

and its parallel translation with the parameter

(50)

Z = \int e^{u_{0} (r^{'})} d r^{'} = \int e^{- β Ω {\bar{ψ}}_{0} (r^{'})} d r^{'}, ϵ > 0 .

Thus, the solution of Eq. (45) is given by the following formulae and those of the parallel translation:

(51)

{\bar{ψ}}_{0} (r) = - \frac{1}{β Ω} u_{0} (r) = - \frac{1}{β Ω} ln \frac{8 ϵ^{2}}{{(1 + {| ϵ r |}^{2})}^{2}} + \frac{1}{β Ω} ln \frac{8 π}{Z}

(52)

{\bar{ω}}_{0} (r) = - Δ {\bar{ψ}}_{0} (r) = (- \frac{1}{β Ω}) (- Δ u_{0}) = - \frac{1}{β Ω} \frac{8 ϵ^{2}}{{(1 + {| ϵ r |}^{2})}^{2}}

5. Response in the mean-field theory

The responses in the mean-field theory is defined by

(53)

δ {\bar{ω}}_{e} (r) ≔ {\bar{ω}}_{e} (r) - {\bar{ω}}_{0} (r),

(54)

δ {\bar{ψ}}_{e} (r) ≔ {\bar{ψ}}_{e} (r) - {\bar{ψ}}_{0} (r)

Using Eqs. (53) and (54), Eq. (44) is linearlized.

(55)

\begin{matrix} δ {\bar{ω}}_{e} (r) = N Ω \frac{exp (- β Ω ({\bar{ψ}}_{e} (r) + ϕ_{e} (r)))}{\int exp (- β Ω ({\bar{ψ}}_{e} (r^{'}) + ϕ_{e} (r^{'}))) d r^{'}} \\ - N Ω \frac{exp (- β Ω {\bar{ψ}}_{0} (r))}{\int exp (- β Ω {\bar{ψ}}_{0} (r^{'})) d r^{'}} \\ \approx N Ω \frac{exp (- β Ω {\bar{ψ}}_{0} (r)) (- β Ω) (δ {\bar{ψ}}_{e} (r) + ϕ_{e} (r))}{\int exp (- β Ω {\bar{ψ}}_{0} (r^{'})) d r^{'}} \\ - N Ω \frac{exp (- β Ω {\bar{ψ}}_{0} (r))}{{\int exp (- β Ω {\bar{ψ}}_{0} (r^{'})) d r^{'}}^{2}} \\ \times \int exp (- β Ω {\bar{ψ}}_{0} (r^{'})) (- β Ω) (δ {\bar{ψ}}_{e} (r^{'}) + ϕ_{e} (r^{'})) d r^{'} \\ = - β Ω {\bar{ω}}_{0} (r) (δ {\bar{ψ}}_{e} (r) + ϕ_{e} (r) - \frac{1}{N Ω} \int {\bar{ω}}_{0} (r^{'}) (δ {\bar{ψ}}_{e} (r^{'}) + ϕ_{e} (r^{'})) d r^{'}) \end{matrix}

Equation (55) is the final formula of the response to the external field in the mean-field theory.

E. Comparison between the two results

We have otaintained two formulae for the response, Eq. (31) and Eq. (55). We assume that ${δω}_{e} (r)$ , $ω_{0} (r)$ in Eq. (31) coincde with $δ {\bar{ω}}_{e} (r)$ , ${\bar{ω}}_{0} (r)$ in Eq. (55), respectively:

(56)

\begin{matrix} - β Ω {\bar{ω}}_{0} (r) ϕ_{e} (r) - β {\bar{ω}}_{0} (r) \int g_{0} (r, r^{'}) {\bar{ω}}_{0} (r^{'}) ϕ_{e} (r^{'}) d r^{'} \\ = - β Ω {\bar{ω}}_{0} (r) (δ {\bar{ψ}}_{e} (r) + ϕ_{e} (r) - \frac{1}{N Ω} \int {\bar{ω}}_{0} (r^{'}) {δ {\bar{ψ}}_{e} (r^{'}) + ϕ_{e} (r^{'})} d r^{'}), \\ \leftrightarrow \frac{1}{Ω} \int g_{0} (r, r^{'}) {\bar{ω}}_{0} (r^{'}) ϕ_{e} (r^{'}) d r^{'} \\ = δ {\bar{ψ}}_{e} (r) - \frac{1}{N Ω} \int {\bar{ω}}_{0} (r^{'}) {δ {\bar{ψ}}_{e} (r^{'}) + ϕ_{e} (r^{'})} d r^{'} \end{matrix}

Applying the operator $- Δ_{r} + β Ω {\bar{ω}}_{0} (r)$ to Eq. (56), where $Δ_{r}$ denotes the Laplacian with respect to $r$ , we obtain

(57)

\frac{1}{Ω} \int {- Δ_{r} + β Ω {\bar{ω}}_{0} (r)} g_{0} (r, r^{'}) {\bar{ω}}_{0} (r^{'}) ϕ_{e} (r^{'}) d r^{'}

(58)

= δ {\bar{ω}}_{e} (r) + β Ω {\bar{ω}}_{0} (r) (δ {\bar{ψ}}_{e} (r) - \frac{1}{N Ω} \int {\bar{ω}}_{0} (r^{'}) {δ {\bar{ψ}}_{e} (r^{'}) + ϕ_{e} (r^{'})} d r^{'})

Substituting Eq. (55) into Eq. (58), the following equation is obtained.

(59)

\int {- Δ_{r} + β Ω {\bar{ω}}_{0} (r)} \frac{g_{0} (r, r^{'})}{- β Ω^{2}} {\bar{ω}}_{0} (r^{'}) ϕ_{e} (r^{'}) d r^{'} = {\bar{ω}}_{0} (r) ϕ_{e} (r)

As $ϕ_{e} (r)$ is an arbitrary function, Eq. (59) is reduced to

(60)

- (Δ_{r} - β Ω {\bar{ω}}_{0} (r)) \frac{g_{0} (r, r^{'})}{- β Ω^{2}} = δ (r - r^{'}) .

In the next section, we will solve Eq. (60) and obtain an explicit formula for $g_{0} (r, r^{'})$ .

III. Results

A. Solution for the correlation function $g_{0} (r, r^{'})$

1. Homogeneous solution

Now, we are ready to solve Eq. (60). At first, we will obtain the homogeneous solution to Eq. (60), namely, the solution to

(61)

- (Δ_{r} - β Ω {\bar{ω}}_{0} (r)) \frac{g_{0} (r, r^{'})}{- β Ω^{2}} = 0

where the explicit formula for

{\bar{ω}}_{0} (r)

is given in Eq. (52). The partial derivative of Eq. (47) with respect to

ϵ

at fixed

Z

is given by

(62)

- Δ \frac{\partial u_{0}}{∂ϵ} = \frac{\partial}{∂ϵ} (λ \frac{e^{u_{0}}}{\int e^{u_{0}}}) = λ \frac{e^{u_{0}}}{\int e^{u_{0}}} \frac{\partial u_{0}}{∂ϵ} = - β Ω {\bar{ω}}_{0} \frac{\partial u_{0}}{∂ϵ}

Here, we have used the following relation with fixed $Z$ .

(63)

0 = \frac{\partial Z}{∂ϵ} = \frac{\partial}{∂ϵ} \int e^{u_{0}} = \int e^{u_{0}} \frac{\partial u_{0}}{∂ϵ}

Equation (62) is reduced to

(64)

- (Δ - β Ω {\bar{ω}}_{0} (r)) \frac{\partial u_{0}}{∂ϵ} = 0

and the homogeneous solution

G_{h} (r)

to Eq. (61) is obtained

(65)

\frac{G_{h} (r)}{- β Ω^{2}} = \frac{\partial u_{0}}{∂ϵ} = \frac{2}{ϵ} \frac{1 - {| ϵ r |}^{2}}{1 + {| ϵ r |}^{2}}

where the explicit formula for

u_{0}

is given by Eq. (49).

2. Special solution

In the next we will obtain a special solution to Eq. (60) with $r^{'} = 0$ .

Consider Eq. (47) with a perturbation $cδ (r)$ whose solution is denoted by $u_{c} (r)$ :

(66)

- Δ u_{c} (r) = λ \frac{e^{u_{c} (r)}}{\int e^{u_{c} (r^{'})} d r^{'}} + cδ (r)

where

c

is a parameter. Equation (66) is reduced to (47) in the limit of

c \to 0

. A solution of (66) has been discussed by J. Prajapat and G. Tarantello.³⁰ In the case of sufficiently small

| c |

, the solution of Eq. (66) in

R^{2}

exists only when

(67)

λ = 8 πn

(68)

n ≔ 1 - \frac{c}{4 π}

and is given by

(69)

u_{c} (r) = \frac{c}{2 π} ln {| r |}^{- 1} + ln \frac{8 ϵ^{2 n}}{{(1 + {| ϵ r |}^{2 n})}^{2}} - ln \frac{8 π}{nZ} .

where

(70)

Z = \int e^{u_{c} (r^{'})} d r^{'}, ϵ > 0 .

The limit $c \to 0$ of the partial derivative of Eq. (66) with respect to $c$ at fixed $Z$ is given by

(71)

\begin{matrix} - Δ {\frac{\partial u_{c}}{\partial c} |}_{c = 0} = {\frac{\partial}{\partial c} (λ \frac{e^{u_{c}}}{\int e^{u_{c} (r^{'})} d r^{'}}) |}_{c = 0} + δ (r) \\ = - 2 \frac{e^{u_{0}}}{\int e^{u_{0} (r^{'})} d r^{'}} + 8 π \frac{e^{u_{0}}}{\int e^{u_{0} (r^{'})} d r^{'}} {\frac{\partial u_{c}}{\partial c} |}_{c = 0} + δ (r) \end{matrix}

The above equation is reduced to the following form:

(72)

- (Δ - β Ω {\bar{ω}}_{0}) ({\frac{\partial u_{c}}{\partial c} |}_{c = 0} - \frac{1}{4 π}) = δ

By employing the practical form of

(73)

{\frac{\partial u_{c}}{\partial c} |}_{c = 0} - \frac{1}{4 π} = \frac{1}{2 π} (\frac{1 - {| ϵ r |}^{2}}{1 + {| ϵ r |}^{2}} ln \frac{1}{| ϵ r |} - 1),

we obtain the special solution

G_{s} (r)

to Eq. (60)

(74)

\frac{G_{s} (r)}{- β Ω^{2}} = {\frac{\partial u_{c}}{\partial c} |}_{c = 0} - \frac{1}{4 π} = \frac{1}{2 π} (\frac{1 - {| ϵ r |}^{2}}{1 + {| ϵ r |}^{2}} ln \frac{1}{| ϵ r |} - 1) .

We conclude that the complete solution $G (r)$ to Eq. (60) is given by the combination of the homogeneous solution (65) and the imhomogeneous solution (74), namely,

(75)

G (r) = G_{s} (r) + α G_{h} (r)

where

α

is an arbitrary constant.

IV. Discussion

A. Normalization condition for correlation function

In general, the correlation function satisfies the following two conditions:

(76)

g (x_{0}, 0) = 0 (| x_{0} | ≫ 1)

(77)

\int_{R^{2}} w (x) g (x, 0) d x = 0

where

w (x)

is a one-body distribution function in an equilibrium state. The reason why the second condtion (77) holds is discussed in Appendix A. In this section, we examine if the obtained distribution function (52) and correlation function (75) satisfies the above condition.

To derive the correlation function, we have used an mean field approximation. Mean-field approximation generally fails in systems with strong correlations. That is, we consider that the first condition is already fulfilled.

Let us examine the second condition. One-body distribution function in an equilibirium state is given by Eq. (52)

(78)

{\bar{ω}}_{0} (x) = \frac{ϵ^{2}}{π} \frac{1}{{(1 + {| ϵ x |}^{2})}^{2}}

where the coefficient is determined to hold the normalization condition

(79)

\int_{R^{2}} {\bar{ω}}_{0} (x) d x = 1 .

Integrating Eq. (75) in $R^{2}$ , we obtain

(80)

\begin{array}{l} \int_{R^{2}} {\bar{ω}}_{0} (x) g (x, 0) d x \\ = - β Ω^{2} \int_{R^{2}} \frac{ϵ^{2}}{π} \frac{1}{{(1 + {| ϵ x |}^{2})}^{2}} \\ \times {\frac{1}{2 π} (\frac{{| ϵ x |}^{2} - 1}{{| ϵ x |}^{2} + 1} ln | ϵ x | - 1) + α \frac{{| ϵ x |}^{2} - 1}{{| ϵ x |}^{2} + 1}} d x \end{array}

Perform a change of variables in the integral

(81)

ϵ x = y, d x = \frac{1}{ϵ^{2}} d y

and continue the calculation.

(82)

\begin{array}{l} = - β Ω^{2} \int_{R^{2}} \frac{1}{π} \frac{1}{{(1 + {| y |}^{2})}^{2}} \\ \times {\frac{1}{2 π} (\frac{{| y |}^{2} - 1}{{| y |}^{2} + 1} ln | y | - 1) + α \frac{{| y |}^{2} - 1}{{| y |}^{2} + 1}} d y \end{array}

Axial symmetry is assumed.

(83)

\begin{array}{l} = - β Ω^{2} \int_{0}^{\infty} \frac{1}{π} \frac{1}{{(1 + r^{2})}^{2}} \\ \times {\frac{1}{2 π} (\frac{r^{2} - 1}{r^{2} + 1} ln r - 1) + α \frac{r^{2} - 1}{r^{2} + 1}} 2 πr d r \end{array}

Perform a change of variables in the integral

(84)

r^{2} = s, 2 rdr = ds

and continue the calculation.

(85)

\begin{array}{l} = - β Ω^{2} \int_{0}^{\infty} \frac{1}{{(1 + s)}^{2}} {\frac{1}{2 π} (\frac{s - 1}{s + 1} \frac{1}{2} ln s - 1) + α \frac{s - 1}{s + 1}} ds \\ = - β Ω^{2} {\frac{1}{4 π} \int_{0}^{\infty} \frac{s - 1}{{(s + 1)}^{3}} ln sds - \frac{1}{2 π} \int_{0}^{\infty} \frac{ds}{1 + s} + α \int_{0}^{\infty} \frac{s - 1}{{(s + 1)}^{3}} ds} \\ = \frac{β Ω^{2}}{4 π} \end{array}

We find that the $α$ -dependence does not remain in the integral result. To overcome this difficulty, cutoff radius $R$ is introduced. According to this change, we replace the integral domain $R^{2}$ with $B_{R} (0)$ and search the value of $α$ which satisfies

(86)

G_{R} (x) ≔ G_{s} (x) + α G_{h} (x)

(87)

\int_{B_{R} (0)} {\bar{ω}}_{0} (x) G_{R} (x) d x = 0

Returning to Eq. (85), we re-evaluate the above integral.

(88)

\int_{B_{R} (0)} {\bar{ω}}_{0} (x) G_{R} (x) d x = - β Ω^{2} {\frac{1}{4 π} \int_{0}^{{(ϵR)}^{2}} \frac{s - 1}{{(s + 1)}^{3}} ln sds - \frac{1}{2 π} \int_{0}^{{(ϵR)}^{2}} \frac{ds}{{(s + 1)}^{2}} + α \int_{0}^{{(ϵR)}^{2}} \frac{s - 1}{{(s + 1)}^{3}} ds}

(89)

= - β Ω^{2} {\frac{1}{4 π} (- \frac{{(ϵR)}^{2}}{{({(ϵR)}^{2} + 1)}^{2}} ln | εR | + \frac{{(ϵR)}^{2}}{{(ϵR)}^{2} + 1}) - \frac{1}{2 π} \frac{{(ϵ R)}^{2}}{{(ϵ R)}^{2} + 1} - α \frac{{(ϵ R)}^{2}}{{({(ϵ R)}^{2} + 1)}^{2}}} = 0

Thus, we obtain $α$ satisfying condition (87).

(90)

α = - \frac{1}{4 π} ln {(ϵ R)}^{2} - \frac{1}{4 π} ({(ϵ R)}^{2} + 1)

By introducing a conditional probability that assumes the presence of a particle at the origin as the source of perturbation, the two-body distribution function effectively reduces to a one-body distribution function. Applying the above condition to Eq. (29) yields

(91)

P_{2, 0} (0, r^{'}) = 1 \cdot P_{1, 0} (r^{'}) (1 + g_{0} (0, r^{'})) = P_{1, 0} (r^{'}) (1 + G (r^{'}))

Among the terms in the above expression, $P_{1, 0} (r^{'}) G (r^{'})$ corresponds to the linear response part, where $G (r^{'})$ is defined in Eq. (75). Figure 4 shows a plot of this function. As anticipated, a region with negative correlation forms in the vicinity of the origin in response to the perturbation imposed at the origin. It should be noted that since the term $G (r^{'})$ is proportional to $Ω \sim O (1 / N)$ , the magnitute of the response is small compared to the leading-order term of 1. Thus, although we successfully obtained the two-body correlation function $G (r)$ analytically, its effect is small and only qualitative agreement is achieved. In other words, additional quantitative analysis may be required to refine the explanation for the formation of the depleted region observed in the experiments.

Figure 4. The profiles of the function P₁_,₀( $x$ )G( $x$ ) for parameters $ϵ$ = 4.5 and 6.0 are shown.

The value of R is 1.0. As anticipated, a region with negative correlation forms in the vicinity of the origin in response to the perturbation imposed at the origin.

B. Future work

Further efforts are necessary to quantitatively explain the experimental results. The two-body correlation function that we analytically derived in this work remains within the linear response regime and does not address the longer-time dynamics. Since analytical treatment of the time evolution of the system is expected to be difficult, we are preparing large-scale numerical simulations using a supercomputer equipped with the latest PEZY-SC4s processor, a platform that one of the authors (Y.Y.) has been utilizing. Numerical simulations will enable us to capture the detailed particle transport behavior, allowing us to investigate in detail aspects such as the destination of particles during the formation of the depleted region.

Ethical considerations

Not applicable. This study is purely theoretical and does not involve human participants.

Data availability

Underlying data

Kanazawa University Repository for Academic Resources (KURA): Experimental data on “Analytical Derivation of Correlation Function for Vorticity-Depleted Regions Formed Around Strong Vortex” https://doi.org/ 10.24517/0002003728.³¹

The project contains the following underlying data:

RingHole0001.pdf: (Experimental data shown in Figure 2 (a). Visualization of electron density distribution from FITS format files, obtained by subtracting Charge Coupled Device (CCD) background noise from CCD camera images. The same applies to panels (b)-(e).)

RingHole0007.pdf: (Experimental data shown in Figure 2 (b))

RingHole0022.pdf: (Experimental data shown in Figure 2 (c))

RingHole0024.pdf: (Experimental data shown in Figure 2 (d))

RingHole0030.pdf: (Experimental data shown in Figure 2 (e))

Data are available under the terms of the Creative Commons Attribution 4.0 International license (CC-BY 4.0).

Reporting guidelines

Not applicable. This article presents a theoretical analysis of two-dimensional vortex dynamics and does not involve clinical trials, animal studies, observational studies, or qualitative research. Therefore, the listed reporting guidelines (CONSORT, ARRIVE, STROBE, COREQ, SRQR) do not apply.

Appendices

Appendix A: Condition satisfied by the correlation function

The second condition that should be satisfied by the correlation function (77) is derived as follows.

Coordinates of $N$ particles arranged in 6 dimensional phase space are represented by

(A1)

x_{i} = (q_{i}, p_{i}) (i = 1, \dots, N)

Distribution of $N$ particles in this space is specified by $N$ -body distribution function $P_{N} (x_{1}, x_{2}, \dots, x_{N})$ . The one- and two-body distribution functions $P_{1} (x_{1})$ and $P_{2} (x_{1}, x_{2})$ are defined by $P_{N} (x_{1}, \dots, x_{N})$ :

(A2)

P_{1} (x_{1}) = \int_{R^{6 (N - 1)}} P_{N} (x_{1}, \dots, x_{N}) d x_{2} \dots d x_{N},

(A3)

P_{2} (x_{1}, x_{2}) = \int_{R^{6 (N - 2)}} P_{N} (x_{1}, \dots, x_{N}) d x_{3} \dots d x_{N} .

Let’s introduce conditional probability. The conditional probability $P (x_{i} | x_{j})$ represents the probability of finding the $i$ -th particle at position $x_{i}$ , given that the $j$ -th particle is located at position $x_{j}$ . The two-body distribution function is related to the conditional probability and one-body distribution function:

(A4)

P_{2} (x_{1}, x_{2}) = P (x_{2} | x_{1}) P_{1} (x_{1})

In general, the two-body distribution function does not equal the product of one-body distribution functions. This is because the presence of a certain particle 1 affects another particle 2 through interactions such as Coulomb forces or gravity. The two-body correlation function $g (x_{1}, x_{2})$ is introduced as a quantity that characterizes the deviation from the product of one-body distribution functions. If the particle system is perfectly uniformly distributed, it becomes uncorrelated, yielding $g (x_{1}, x_{2}) = 0$ .

(A5)

P_{2} (x_{1}, x_{2}) = P_{1} (x_{1}) P_{1} (x_{2}) (1 + g (x_{1}, x_{2}))

Substituting (A4) into (A5), we obtain the relation

(A6)

P (x_{2} | x_{1}) = P_{1} (x_{2}) (1 + g (x_{1}, x_{2}))

Normalization condition for the conditional probability is give by

(A7)

\int_{R^{6}} P (x_{2} | x_{1}) d x_{2} = 1

Inserting the relation (A6) into the above integral, we obtain the following relation:

(A8)

\begin{array}{l} \int_{R^{6}} P (x_{2} | x_{1}) d x_{2} \\ = \int_{R^{6}} P_{1} (x_{2}) (1 + g (x_{1}, x_{2})) d x_{2} \\ = \int_{R^{6}} P_{1} (x_{2}) d x_{2} + \int P_{1} (x_{2}) g (x_{1}, x_{2}) d x_{2} \end{array}

The first term in the last line in (A8) equals 1 due to the normalization condition for the one-body distribution function. Thus we obtain finnaly the relation

(A9)

\int_{R^{6}} P_{1} (x_{2}) g (x_{1}, x_{2}) d x_{2} = 0

Thus, the relation (77) holds.

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24. Kiwamoto Y, Mohri A, Ito K, et al.: Yuyama. Non-neutral Plasma Physics. AIP; 1999; III. : pp. 99–105.
25. Kiwamoto Y, Ito K, Sanpei A, et al.: J. Phys. Soc. Jpn. 1999; 85: 3173.
26. Bellac ML, Mortessagne F, Batrouni GG: Equilibrium and non-equilibrium statistical thermodynamics. Cambridge: Cambridge University; 2004. Chap. 9.1 and A.6.
27. Thompson CJ: Classical Equilibrium Statistical Mechanics. Oxfird: Clarendon Press; 1988.
28. Stanley HE: Introduction to Phase Transitions and Critical Phenomena. Oxfird: Oxford University; 1992.
29. Chen W, Li C: Duke Math. J. 1991; 63: 615.
30. Prajapat J, Tarantello G: On a class of elliptic problems in R2: symmetry and uniqueness results. Proc. R. Soc. Edinburgh A. 2001; 131: 967–985. Publisher Full Text
31. Sanpei A: Experimental data on”Analytical Derivation of Correlation Function for Vorticity-Depleted Regions Formed Around Strong Vortex”. [Dataset]. Kanazawa University Repository for Academic Resources (KURA); 2024. Publisher Full Text

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¹ Institute of Liberal Arts and Science, Kanazawa University, Kanazawa, 920-1192, Japan
² National Institute for Fusion Science, Toki, Gifu, 509-5292, Japan
³ Faculty of Mathematics and Physics, Institute of Science and Engineering, Kanazawa University, Kanazawa, 920-1192, Japan
⁴ Faculty of Electrical Engineering and Electronics, Kyoto Institute of Technology, Sakyo, Kyoto, 606-8585, Japan

Yuichi Yatsuyanagi
Roles: Conceptualization, Formal Analysis, Methodology, Visualization, Writing – Original Draft Preparation, Writing – Review & Editing

Tadatsugu Hatori
Roles: Conceptualization, Formal Analysis, Methodology, Writing – Review & Editing

Hiroshi Ohtsuka
Roles: Conceptualization, Formal Analysis, Methodology, Validation, Writing – Original Draft Preparation, Writing – Review & Editing

Akio Sanpei
Roles: Data Curation, Investigation, Resources, Visualization

Yukihiro Soga
Roles: Data Curation, Investigation, Resources, Visualization

Competing interests

No competing interests were disclosed.

Grant information

The author(s) declared that no grants were involved in supporting this work.

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Yatsuyanagi Y, Hatori T, Ohtsuka H et al. Analytical Derivation of Correlation Function for Vorticity-Depleted Regions Formed Around Strong Vortex [version 1; peer review: awaiting peer review]. F1000Research 2026, 15:335 (https://doi.org/10.12688/f1000research.176213.1)

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Analytical Derivation of Correlation Function for Vorticity-Depleted Regions Formed Around Strong Vortex

Abstract

Background

Methods

Results

Conclusions

Keywords

I. Introduction

Figure 1. The linear response followed by the infinite particle limit, and the linear response to the mean field after taking the infinite particle limit.

II. Methods

A. Two-dimensional pure electron plasma experiment

Figure 2. Experimental results are shown.

Figure 3. Radial profile of electron density centered on the clump is shown.

B. Point vortex system

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C. Linear response theory

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D. Mean-field theory

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E. Comparison between the two results

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III. Results

A. Solution for the correlation function g0(r,r′)

(61)

A. Solution for the correlation function $g_{0} (r, r^{'})$

Figure 4. The profiles of the function P₁_,₀( $x$ )G( $x$ ) for parameters $ϵ$ = 4.5 and 6.0 are shown.