(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
þ
) which are n ot mathemati-
cally independent can lead to physically irrelevant quan-
tities such as ð∂Q=∂ΦÞ
T
, 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
2
− Ψ plane, where
Ψ ¼ 1=2r
þ
is the conjugate of Q
2
, and thus the equation of
state is written as Q
2
¼ Q
2
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
s
as
a thermodynamic variable with Ψ ¼ð∂M=∂Q
s
Þ
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
s
¼ Q
s
ðT;ΨÞ and a Smarr relation
based on this new phase space as M ¼ MðS; Q
s
;PÞ.
Clearly, for p ¼ 1, the power Maxwell field reduces to
the standard Maxwell field and Q
s
→ Q
2
. 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 −
1
16π
Z
M
d
nþ1
x
ffiffiffiffiffiffi
−g
p
R −
4
n − 1
ð∇ΦÞ
2
− 2Λ þð−e
−4=ðn−1Þλ
1
Φ
FÞ
p
−
X
3
i¼2
e
−4=ðn−1Þλ
i
Φ
H
i
;
ð5Þ
where R is the Ricci scalar on manifold M, Φ is the dilaton
field, and λ
1
and λ
i
are constants. In Eq. (5) F ¼ F
μν
F
μν
and H
i
are the Maxwell invariants of electromagnetic
fields, where F
μν
¼ ∂
½μ
A
ν
and ðH
i
Þ
μν
¼ ∂
½μ
ðB
i
Þ
ν
, with
A
μ
and ðB
i
Þ
μ
the electromagnetic potentials. Varying the
action (5) with respect to the metric g
μν
, the dilaton field Φ,
with electromagnetic potentials A
μ
and ðB
i
Þ
μ
, leads to the
following field equations [31]:
R
μν
¼
g
μν
n − 1
2Λ þð2p − 1Þð−Fe
−4λ
1
Φ=ðn−1Þ
Þ
p
−
X
3
i¼2
H
i
e
−4λ
i
Φ=ðn−1Þ
þ
4
n − 1
∂
μ
Φ∂
ν
Φ
þ 2pe
−4λ
1
pΦ=ðn−1Þ
ð−FÞ
p−1
F
μλ
F
ν
λ
þ 2
X
3
i¼2
e
−4λ
i
Φ=ðn−1Þ
ðH
i
Þ
μλ
ðH
i
Þ
ν
λ
; ð6Þ
CRITICAL BEHAVIOR OF LIFSHITZ DILATON BLACK HOLES PHYS. REV. D 98, 104026 (2018)
104026-3