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We investigate the relations between black hole thermodynamics and holographic transport coefficients in this paper. The formulae for DC conductivity and diffusion coefficient are verified for electrically single-charged black holes. We examine the correctness of the proposed expressions by taking charged dilatonic and single-charged STU black holes as two concrete examples, and compute the flows of conductivity and diffusion coefficient by solving the linear order perturbation equations. We then check the consistence by evaluating the Brown-York tensor at a finite radial position. Finally, we find that the retarded Green functions for the shear modes can be expressed easily in terms of black hole thermodynamic quantities and transport coefficients.

The experiments at the Relativistic Heavy Ion Collider (RHIC) and at the Large Hadron Collider (LHC) show that the quark-gluon plasma (QGP) does not behave as a weakly coupled gas of quarks and gluons, but rather as a strongly coupled fluid. This places limitations on the applicability of perturbative methods. The AdS/CFT correspondence provides a powerful tool for studying the dynamics of strongly coupled quantum field theories [

Aimed to develop a model independent theory of the hydrodynamics, the membrane paradigm and a holographic version of Wilsonian Renormalization Group (hWRG) have been proposed to describe strongly coupled field theories with a finite cutoff [

On the other side, we know that the calculation of the linear response functions (i.e., the retarded Green functions) of a strongly coupled system is very important but obviously a tough task. Even for the transport coefficients of translational-invariant charged black holes, the complete solutions of the shear modes and sound modes are coupled together and hard to be solved [

A natural question is that why the linear response functions of a two parameters (mass and charge) system is so complicated to deal with and do we have a more general and powerful method to deal with the linear type perturbation of charged black holes? We are going to present a positive answer.

In [

Our logic of this paper is as follows: when a system is perturbed slightly, its response will be linear in the perturbation and this regime is called the linear response regime, although the system is in a nonequilibrium state of which all characteristics can be inferred from the properties of its equilibrium state. Because all the scalar, shear, and sound modes are linear response to small perturbations to the black hole thermodynamic equilibrium state, all the transport coefficients can be determined by the black hole thermodynamic variables in its equilibrium state.

The purpose of this paper is to verify the unified form of DC conductivity and diffusion constant for translational-invariant hydrodynamics with a chemical potential, as a first step towards general formulae of transport coefficients of anisotropic and inhomogeneous hydrodynamics. Formula (

As to the diffusion coefficient, we also conjecture the following formula for holographic hydrodynamics with a chemical potential:

After that, we will prove that

In fact, formula (

Before going on, let us summarize the new features and the main result of this paper.

The unified form of the diffusion coefficient and the retarded Green functions are proposed for the first time.

By evaluating the black hole thermodynamic quantities as a function of radial coordinate, we can write down the transport coefficients and the retarded Green functions in terms of black hole metric line-element and black hole thermodynamic variables. This result implies that there exists deep connection between black hole thermodynamics and the linear response functions.

Our work can be regarded as a first step towards easy computation of the transport coefficients with respect to the sound modes and holographic lattice in which partial differential equations are involved.

The contents of this paper are organized as follows. In Section

In this section, we study the transport coefficients and the RG flow of holographic hydrodynamics for the charge dilatonic black holes. We will first review the black hole geometry and thermodynamics. Then, we will compute the transport coefficients.

We start by introducing the following action for charged dilatonic black hole in

In this paper, we will introduce a dimensionless coordinate

The horizon of the black hole locates at

The black hole is extremal if

The equation of motion for the gauge field

The Einstein equation is written as

The equation of motion for the scalar field is

The Hawking temperature yields

The volume density of Bekenstein-Hawking entropy is given by

According to thermodynamic relation

The charge density and the chemical potential are [

Furthermore, there are several relations satisfied by the thermodynamic variables

The temperature and chemical potential can also be obtained as

The susceptibility can be calculated which is given by

According to the definition of the special heat

In particular, when the temperature of the charged dilatonic black hole becomes very low, there is a linear special heat which can be expressed as

Considering the metric perturbations [

We choose the momentum along the

From symmetry analysis, one can find that off-diagonal perturbation

Following the sliding membrane argument [

We define the shear viscosity as

At zero momentum limit, the flow equation is given by

The shear viscosity is requested to be

This result agrees with [

There are some debate on whether the shear viscosity flow depends on the position of the cutoff surface [

Actually, if we do not consider the quantum corrections to the geometry, the radial evolution of the total entropy remains a constant in nature. Therefore, we can see that the entropy density must depend on the radial coordinate. Isentropic evolution equation [

The right hand side of the above equation is exactly a component of Einstein equation. Here

We will follow [

The linear perturbative Einstein equation can be read off from the

One shall notice that in the limit of zero momentum, the equations for the metric, and the gauge perturbations are completely decoupled.

For convenience of calculation, we would like to define “current” and “strength” for the vector field

We also introduce the effective coupling

The vector part off-shell action in

Note that

One the other hand, we must introduce a new current related to

Then we can define the shifted current

In terms of the defined “current” and “strength,” the equations of motion (

The Bianchi identity holds as

For the same reason, we can define

The equation of motion for

In the zero momentum limit, the equation of motion for

The relevant on-shell action for

By defining

So the flow equation (

The regularity condition at the horizon requires

On the other hand, the DC conductivity can be evaluated by using the Kubo formula

The retarded Green’s function is given by [

The equation of motion (

From (

The DC conductivity can be calculated by using the following relation [

Inserting (

The DC conductivity as a function of the radial coordinate

At the horizon

Without the chemical potential, (

We can see from Figure

As a side note, we will check whether the transport coefficients calculated satisfies the Einstein relation

We will see later that the right hand of equation is not the diffusion constant and, thus, the Einstein relation is not satisfied. In the absence of the chemical potential

As a byproduct, we calculate the thermal conductivity by using the relation

The ratio

Now we are going to calculate the diffusion coefficient. Let us evaluate the “conductivity” introduced by the metric perturbation

In the absence of the momentum, the decoupled flow equation for

Again, the regularity condition at the event horizon gives

Actually, the value of this conductivity is equivalent to shear viscosity. Noting that from (

For the fields and equations of motion, we treat the vector modes in a long wave-length expansion. We will find that the diffusion constant depends on

The in-falling boundary condition at the horizon implies

Through the charge conservation equation (

To zeroth order, (

The first equation of (

The zeroth order of the Bianchi identity becomes

Integrating the above equation from

For the gauge perturbation

Following [

Following the sliding membrane paradigm [

By using (

Note that the dimensional

We can define the proper frequency

We define a dimensionless diffusion constant

Thus, in terms of the normalized momentum and the diffusion constant, the conductivity can be expressed as

The dimensionless diffusion coefficient as a function of radial coordinate

Figure

In this section, we will provide a consistent check by using the black hole thermodynamics. We will verify that by using the formula

On an arbitrary cutoff surface

The induced metric

The stress tensor at an arbitrary cutoff surface can be written as

On the other hand, the stress-energy tensor of a relativistic fluid in equilibrium is

Notice that the entropy density is given by

The local Hawking temperature is given by

We can express thermodynamic variables in terms of metric components

By detailed calculation, we can obtain the value

Apparently, one can verify the following identity

Finally, we verify that

In concrete, it can be written as

The above result agrees with (

Moreover, we can verify the diffusion constant by using

For our case, the diffusion coefficient is given by

This is consistent with (

The hydrodynamics of R-charged black holes was studied by several authors [

We know the effective Lagrangian of a single charged black hole can be written as [

The single-charged metric and gauge fields are given by

The charge density and conjugated chemical potential are

The Newton constant is given by

The vector type perturbation takes the form

The linearized equations derived from (

The above four equations are not independent. Combining (

Substituting (

Now we introduce

By further define

The Bianchi identity holds as

Similarly we can define

The equation of motion for

One can see that

By defining

We can immediately write down the regularity condition at the horizon

It is convenient to define the radial momentum as

The equation of motion for

The regularity at the horizon

According to the following relation [

Finally we can obtain the DC conductivity at the cutoff surface (see Figure

The DC charge conductivity as a function of the radial coordinate

At the horizon

Figure

We can write the same equation of motion according to minimally coupled massless scalar as

The flow equation for shear viscosity is the same as in the previous section

On the horizon, the regularity gives

The entropy density is

So the shear viscosity to entropy density ratio

On any cutoff surface, the entropy density can be expressed as

In order to obtain the diffusion coefficient, we need to evaluate the “conductivity” introduced by the metric perturbation

In the zero momentum limit, the decoupled flow equation for

The regularity condition at the event horizon gives

It is worth noting that (

We take the scaling limit for temporal and spatial derivatives as

With regard to the lowest order, we have

Note that

The zeroth order Bianchi identity yields

After imposing the boundary condition

Then substituting it into (

Following the sliding membrane paradigm [

By further using the boundary condition given in (

The dimensionless diffusion constant

The dimensionless diffusion coefficient of single charge runs on the cutoff surface with different charges. The dashed line corresponds to the chargeless case.

When

From Figure

In the following, we present a consistent check by using the Brown-York tensor. The Brown-York tensor is defined as

The induced metric

Notice the entropy density

On the cutoff surface, the local Hawking temperature

So we can obtain the result according to the previous section (

The diffusion coefficient is given by

In summary, we verified the conjectured ansatz for DC conductivity and diffusion coefficient by exploring the RG flows of charged dilatonic and single-charged STU black holes. The DC conductivity as well as the diffusion coefficient shows its nontrivial flow from the IR horizon to the UV boundary. The results indicate that black hole thermodynamics evaluated on arbitrary

It would be interesting to extend our work to the sound modes of charged black holes and conductivity of anisotropic and inhomogeneous holographic background, because the sound modes of hydrodynamics and conductivity of holographic lattice are very complicated and difficult to solve [

In this appendix, we will write down the unified retarded Green function [

For charged dilatonic black holes, the Green function

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors would like to thank Yi Ling, Yu Tian, Jian-Pin Wu, Xiao-Ning Wu, and Hongbao Zhang for helpful discussions. This work was also supported by NSFC, China (no. 11005072 and no. 11375110) and Shanghai Rising-Star Program (no. 10QA1402300).

_{4}black hole and holographic optics