Lifshitz dilaton黑洞的热力学相变与临界行为

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"这篇论文探讨了（n+1）维度Lifshitz dilaton黑洞的临界行为。在扩展的P-v（压力-体积）空间中，由于任意Lifshitz指数无法找到状态解析方程，之前对于Lifshitz黑洞的临界行为研究一直存在困难。作者提出了一种新的热力学相空间方法，这种方法允许在不改变宇宙常数的情况下研究黑洞的临界行为。他们将状态方程表示为Qs = Qs(T, Ψ)，其中Ψ是黑洞质量关于Qs的导数与熵S的比值，P是Qs的共轭变量。然后构建了一个基于新相空间的Smarr关系，即M = M(S, Qs, P)，其中s是幂律Maxwell拉格朗日函数的幂。通过数学分析，证明了这种方法的合理性，并发现系统在固定宇宙常数（等同于压力）时表现出临界行为，类似于范德华流体系统，电荷会随着相变而变化。进一步，作者计算出系统的吉布斯自由能，发现了吉布斯图中的燕尾形状，这标志着一阶相变。最后，他们确定了临界指数，并指出虽然热力学量依赖于度量参数（如z，p和n），但这些临界指数与范德华流体-气体系统的临界指数相同。" 这篇论文的核心知识点包括： 1. **Lifshitz dilaton黑洞**：这是一种具有特殊动力学尺度对称性的黑洞，其动力学由Lifshitz指数z控制，它改变了时空的标度行为。 2. **临界行为**：在热力学中，临界行为是指系统在临界点附近展现出的特殊现象，如相变时的连续性或不连续性。在这里，是指Lifshitz dilaton黑洞在特定条件下可能出现类似范德华流体的相变行为。 3. **扩展的P-v空间**：通常的热力学中，压力和体积是描述系统状态的基本变量，但在扩展的P-v空间中，还考虑了其他可能影响系统行为的变量。 4. **状态解析方程P = P(v, T)**：在Lifshitz黑洞的背景下，找到这样的方程对于理解其热力学性质至关重要，但通常对于任意的Lifshitz指数难以得到解析解。 5. **新方法**：作者提出的新方法是将状态方程改写为Qs = Qs(T, Ψ)，并构造了一个与新相空间关联的Smarr关系，允许在不改变宇宙常数的情况下研究临界行为。 6. **Smarr关系**：在热力学中，Smarr关系是一个能量平衡公式，它连接了黑洞的总质量与其组成成分（如熵、电荷和压力）的关系。 7. **吉布斯自由能**：是系统在恒温恒压下做功的能力，它的变化揭示了相变的可能性。作者通过计算吉布斯自由能找到了相变的证据。 8. **临界指数**：描述临界点附近物理量的幂律行为，如磁化率、比热容等。作者发现，尽管热力学量依赖于度量参数，但临界指数保持不变，与范德华系统一致。 9. **宇宙常数**：在本研究中被视为一个固定的参数，而不是作为一个可变的热力学变量。这一处理方式有助于分析黑洞的稳定性和相变特性。通过这些知识点，我们可以深入理解Lifshitz dilaton黑洞在热力学相变中的复杂行为，并为未来的研究提供了新的视角和工具。

(see Fig. 1). A close look at the temperature expression of

Lifshitz dilaton black holes [see [31] and Eq. (27) of the

present paper] shows that it is nearly impossible to solve

this equation for P (or more precisely for l). Therefore, we

cannot have an analytical equation of state, P ¼ Pðv; TÞ,

to investigate the critical behavior or calculate critical

quantities of Lifshitz black holes. It is important to note

that as one can see from the three-dimensional diagram in

Fig. 1, there is indeed a P − v criticality for the Lifshitz

dilaton black holes, similar to the van der Waals liquid-gas

system. This implies that one may use the numerical

calculations to investigate P − v criticality of this system

and obtain the critical quantities in an extended phase

space, at least approximately. However, in this work, we

would like to consider a more fa scinating and straightfor-

ward way to investigate the critical behavior of this

general type of black hole, which includes dilaton, power

Maxwell field, and Lifshitz effects. In particular, we shall

show that the system admits a critical behavior similar to

the van der Waals fluid, without needing to extend the

phase space.

Another way to investigate the critical behavior of the

black holes is to use the method of Refs. [32,33], but as

shown in [34], such a view of thermodynamic conjugate

variables (Q and Φ ¼ Q=r

) which are n ot mathemati-

cally independent can lead to physically irrelevant quan-

tities such as ð∂Q=∂ΦÞ

, which is supposed to be a

thermodynamic response function, but mathematically ill

defined.

To address this problem, an alternative viewpoint toward

the thermodynamic phase space of black holes was

developed in [34] by treating the cosmological constant

as a fixed parameter and considering the charge of the black

hole as a thermodynamic variable [35,36]. The cosmologi-

cal constant is assumed as a constant related to the

background of AdS geometry in general relativity or the

zero point energy of the background of spacetime in field

theory. So, from a physical standpoint, it is difficult to

consider the cosmological constant as a pressure of the

black hole which can take an arbitrary value. It seems more

physical to consider the variation of charge Q of a black

hole instead of the cosmological constant, because the

charge of the black hole is really a natural external variable

which can vary by absorbing or emitting charged particles.

It was argued that, with a fixed cosmological constant, the

critical behavior indeed occurs in the Q

− Ψ plane, where

Ψ ¼ 1=2r

is the conjugate of Q

, and thus the equation of

state is written as Q

¼ Q

ðT;ΨÞ. We find out that in the

case of Lifshitz dilaton black holes, the system admits a

critical behavior provided we take the electrodynamics in

the form of a power Maxwell field and considering Q

a thermodynamic variable with Ψ ¼ð∂M=∂Q

S;P

as its

conjugate, where s ¼ 2p=ð2p − 1 Þ. In this case we can

define a new response function which naturally leads to a

physically relevant quantity. Thus, the equation of state is

written in the form of Q

¼ Q

ðT;ΨÞ and a Smarr relation

based on this new phase space as M ¼ MðS; Q

;PÞ.

Clearly, for p ¼ 1, the power Maxwell field reduces to

the standard Maxwell field and Q

→ Q

. Following [34],

in this approach we keep the cosmological constant (pres-

sure) as a fixed quantity, while the charge of the system can

vary.

This paper is outlined as follows. In the next section, we

present the action and the basic field equations of Lifshitz

dilaton black holes, and we review thermodynamic proper-

ties of this system. In Sec. III A, we study the phase

structure of the solution and present the modified Smarr

relation. In Sec. III B, we obtain the equation of state and

study the critical behavior of the solutions and compare

them with a van der Waals fluid system. We investigate the

Gibbs free energy and the critical exponents of the system

in Secs. III C and III D, respectively. The last section is

devoted to a summary and conclusion.

II. THERMODYNAMICS OF LIFSHITZ

DILATON BLACK HOLES

In this section we are going to review the solutions of

charged Lifshitz black holes with a power Maxwell field

[31], with an emphasis on their thermodynamic properties.

The (n þ 1)-dimensional action of Einstein-dilaton gravity

in the presence of a power Maxwell electromagnetic and

two linear Maxwell fields can be written as

S ¼ −

16π

nþ1

ﬃﬃﬃﬃﬃﬃ

−g



R −

n − 1

ð∇ΦÞ

− 2Λ þð−e

−4=ðn−1Þλ

FÞ

−

i¼2

−4=ðn−1Þλ



;

ð5Þ

where R is the Ricci scalar on manifold M, Φ is the dilaton

field, and λ

and λ

are constants. In Eq. (5) F ¼ F

μν

and H

are the Maxwell invariants of electromagnetic

fields, where F

μν

¼ ∂

½μ

ν

and ðH

μν

¼ ∂

½μ

ðB

ν

, with

and ðB

the electromagnetic potentials. Varying the

action (5) with respect to the metric g

μν

, the dilaton field Φ,

with electromagnetic potentials A

and ðB

, leads to the

following field equations [31]:

μν

n − 1



2Λ þð2p − 1Þð−Fe

−4λ

Φ=ðn−1Þ

−

i¼2

−4λ

Φ=ðn−1Þ



n − 1

∂

Φ∂

þ 2pe

−4λ

pΦ=ðn−1Þ

ð−FÞ

p−1

μλ

þ 2

i¼2

−4λ

Φ=ðn−1Þ

ðH

μλ

ðH

; ð6Þ

CRITICAL BEHAVIOR OF LIFSHITZ DILATON BLACK HOLES PHYS. REV. D 98, 104026 (2018)

104026-3

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