# Holographic Brownian motion in \(2+1\) dimensional hairy black holes

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## Abstract

In this paper, we investigate the dynamics of a heavy quark for plasmas corresponding to three dimensional hairy black holes. We utilize the AdS/CFT correspondence to study the holographic Brownian motion of this particle with different kinds of hairy black holes. For an uncharged black hole in the low frequency limit we derive analytic expressions for the correlation functions and the response functions and verify that the fluctuation–dissipation theorem holds in the presence of a scalar field against a metric background. In the case of a charged black hole, we think that the results are similar to that derived for an uncharged black hole.

### Keywords

Black Hole Brownian Motion Scalar Field Open String Brownian Particle## 1 Introduction

Heavy-ion collision experiments at RHIC are believed to create a strongly coupled quark gluon plasma (sQGP) [1, 2, 3]. The QGP is a phase of QCD that is thought to be very similar to the plasma of N \(=\) 4 super Yang Mills theory at finite temperature. One of the current challenges in theoretical particle physics is to compute the properties of this strongly coupled plasma. The AdS/CFT correspondence [4, 5, 6, 7] has led to many profound insights into the nature of strongly coupled gauge theories. This gauge/gravity duality provides the possibility of computing some properties of QGP [8, 9, 10, 11]. QGPs contain quarks and gluons, like hadrons, but unlike hadrons, the mesons and baryons lose their identities and dissolve into a fluid of quarks and gluons. A heavy quark immersed in this fluid undergoes the Brownian motion [12, 13, 14, 15] at finite temperature. The AdS/CFT correspondence can be utilized to investigate the Brownian motion of this particle. In the context of this duality, the dual statement of the quark in QGP corresponds to the end point of an open string that extends from the boundary to the black-hole horizon. The black-hole environment excites the modes of the string by Hawking radiation. It was found that, once these modes are quantized, the end point of the string at the boundary shows a Brownian motion which is described by the Langevin equation [13, 14, 15].

In the formulation of the AdS/CFT correspondence, the fields of gravitational theory would be related to the corresponding boundary theory operators [6, 7], such as that their boundary value should couple to the operators. In this way, instead of using the boundary field theory to obtain the correlation function of the quantum operators, one can determine these correlators by the thermal physics of black holes and use them to compute the correlation functions. For different theories of gravity one can make an association with various plasmas in the boundary. In this paper we follow different works [16, 17, 18, 19, 20, 21, 22] to investigate the Brownian motion of a particle in a two dimensional plasma of which the gravity dual is described by a three dimensional hairy metric background [23, 24, 25, 26, 27, 28, 29, 30, 31]. We obtain the solutions to the equation of motion of uncharged hairy black holes by using a matching technique in the low frequency limit. We utilize these solutions in investigating the Brownian motion of the particle. We obtain an expression for the response functions and correlation functions and show that the fluctuation–dissipation theorem holds in the presence of a scalar field against a metric background. For the case of a charged black hole we make some comments.

This paper is structured as follows: in Sect. 2, we review different kinds of three dimensional hairy black holes. Section 3 is assigned to looking for a holographic realization of Brownian motion on the boundary and bulk side of theory. We study the holographic Brownian motion in hairy black holes and the fluctuation–dissipation theorem in the presence of a scalar field in a metric background in Sect. 4. In Sect. 5, we summarize our work in this paper and make some comments on our results and close with our conclusions.

## 2 Hairy black holes in \(2+1\) dimensions

## 3 Holographic Brownian motion

### 3.1 Dictionary of Brownian motion in the boundary

### 3.2 Brownian motion in the bulk

## 4 Brownian motion in Hairy black holes

### 4.1 String dynamics in Hairy black holes and the response function

- (I)
the near horizon solution \((\rho \sim 1)\) for arbitrary \(\upsilon \) (where \(\upsilon =\frac{l^{2}\omega }{r_{h}}\)),

- (II)
the solution for arbitrary \(\rho \) in the limit \(\upsilon \ll 1\), and

- (III)
the asymptotic \(\rho \rightarrow \infty \) solution for arbitrary \(\upsilon \),

#### 4.1.1 Uncharged black hole with the special mass (conformal black hole)

#### 4.1.2 Uncharged black hole with the general mass

#### 4.1.3 Strings in the charged hairy black hole

In the case of a charged hairy black hole, the method of solving the equation of motion is the same as in the last section. It means that we must work in the same three regimes as introduced before. The first step is to derive the tortoise coordinate from the relation (28). The behavior of this parameter near the horizon helps to derive the solutions in the regime (I). In the next step we obtain solutions in the regime (II), through the relation (44). By expanding these solutions near the horizon and matching them with the solutions in regime (I), one can obtain the exact solutions in regime (II). The final step is to derive the asymptotic solutions from expanding the solutions in the regime (II) for \(\rho \rightarrow \infty \) and comparing them with Eq. (46) in regime (III). In working through the above processes for a charged hairy black hole, we encountered some problems. In fact, solving the integrals (28) and (44) for \(f(r)\) defined through (3) is rather difficult. It seems that for this kind of black hole, we will achieve a similar relation for the admittance, i.e., \(\mu (0)=\frac{2\pi \alpha ^{\prime }m}{r_{+}^{2}}\). So, the admittance parameter will be related to the charge, mass, and temperature of the black hole (e.g. Eq. (6) for the charged BTZ black hole). This statement is an opinion and we would like to confirm it in future work.

### 4.2 Displacement square and the fluctuation–dissipation theorem

## 5 Conclusion

In this paper, we obtain the normalized asymptotic solutions (including outgoing and ingoing modes) to the equation of motion of uncharged hairy black hole at low frequencies. By using those solutions, we derive the response function and correlation function for an uncharged black hole with general mass and special mass \(M=\frac{3B^{2}}{l^{2}}\) separately. We found that the admittance and diffusion constant are dependent on the scalar parameter \(B\) and the mass of the black hole through the radius of the horizon. We prove that the fluctuation–dissipation theorem holds in the plasma where the corresponding gravity is for a three dimensional uncharged hairy black hole. In the case of a charged hairy black hole, we cannot get an explicit solution to the equation of motion, but we think that its behavior of as regards the asymptotic solution is similar to the uncharged case in the low frequency limit. It means that the dependence of the admittance and the diffusion constant on the radius of the horizon is as before and in the case of a charged black hole is related to the scalar parameter, charge, and mass of the black hole. This statement should be confirmed in future work.

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