Exponential corrected thermodynamics of black holes

Thermodynamics and related topics are important ways to study black hole physics. Holographic principles help us to relate the black hole surface quantities to the black hole thermodynamics [1]. Black hole entropy and temperature holographically are related to the horizon area and surface gravity, respectively [2, 3]. Black hole parameters like mass, charges, or angular momenta are thermodynamics variables of the first law which yields to the Smarr formula [4]. In that case, generalization of the Smarr formula for both static and rotating black holes with nonlinear electromagnetic fields has been proven and analyzed by reference [5]. According to quantum mechanics, a black hole can emit radiation which is known as Hawking radiation [6, 7]. Due to the Hawking radiation, a black hole decreases its size, which may yield to evaporation [8]. Also, a black hole may stop evaporation and become stable at the quantum ground state. From the string theory point of view, it is described by a space-time metric whose gauge fields are denoted by mass, charges, or angular momenta [9]. One of the best ways to investigate what happened to the black hole at the quantum scale is the study of thermal fluctuations [10–12]. It increases knowledge about the microscopic origin of entropy [13]. Indeed, it is the statistical fluctuations that may be interpreted as the quantum correction [14]. In order to study such quantum corrections in a strong gravitational system, we need a theory of quantum gravity. It can be done via some quantum theories of gravity like string theory or loop quantum gravity.

Already, it is found that leading order correction to the black hole entropy may be logarithmic [15–20]. It is indeed an important term when the black hole size is small. Hence, it can be considered to test the quantum gravity [21–25]. Such thermal fluctuations are considered as small perturbations around the equilibrium temperature. In that case, the effects of logarithmic correction on a BTZ black hole [26] in a massive gravity investigated to find that thermal fluctuations modify the black hole stability [27–29]. Logarithmic corrected (leading order) thermodynamics of Horava–Lifshitz black hole also investigated [30], while the extension to the higher order correction is including inverse of the entropy [31]. The higher order corrections are indeed a way to calculate the microcanonical black hole entropy [32]. It can be applied to the Schwarzschild and BTZ black holes [33]. It has been argued that corrections to the black hole entropy have a universal form, but different quantum gravity theories yield to the different correction coefficients.

The first and the second order corrections are indeed perturbative corrections. These are important when the black hole size is reduced due to the Hawking radiation. Decreasing more, we need non-perturbative analysis, which is the main issue of this paper. Recently, an exponential term is proposed to correct the black hole entropy [34]. It has been claimed that the exponential corrections in the black hole entropy may arise in any quantum theory of gravity [34]. This term is negligible for a large horizon radius, while it is crucial when the black hole seize becomes small. Until now, there is no thermodynamics analysis of black holes with exponential entropy corrected except [35]. Hence it has been done, in this paper, for some kinds of famous black holes solutions.

As we know, any thermodynamics systems consist of some thermal fluctuations, which yield to the ordinary entropy plus a logarithmic term as a perturbative correction. However, there is a contribution of non-perturbative correction. Since any given thermodynamics system like black holes can lead including both corrections, there is an additional term in the entropy. In this paper, we show that the non-perturbative correction yields an exponential term as discussed by [34]. Then, we try to obtain the effect of this correction on the partition function, hence study the black hole exponential corrected thermodynamics.

This paper is organized as follows. In the next section, we introduce exponential correction on the entropy. We use it in section 3 to obtain partition function and hence study thermodynamics quantities. In section 4, we consider Schwarzschild black hole to apply the obtained formula of section 3. In section 5, we consider Reissner–Nordström black hole and find the effect of exponential term on the black hole stability. In section 6, we extend our calculations to the case of the Schwarzschild-AdS black hole. The more general case of charged AdS black hole will discuss in section 7. Finally, in section 8, we give a conclusion and summary of results.

The entropy of large black holes (comparing with the Planck scale) is proportional to the event horizon area. Also, we know that the black hole size is reduced due to the Hawking radiation. Hence, the small black hole entropy needs some corrections. These corrections may be interpreted as a quantum effect, coming from thermal fluctuations, which yields to the modification of the holographic principle [36, 37]. It has been found that the leading order correction to the Bekenstein–Hawking entropy is logarithmic [38]. However, for the space-time dimension other than four (D ≠ 4), the black hole entropy corrections may be power law [39] which is like higher order corrections [40]. We should note that the entropy corrected black holes may be investigated using the non-perturbative quantum theory of general relativity. In that case, the leading order entropy corrected AdS black hole in the large area limit of a four-dimensional Einstein gravity with negative cosmological constant has been studied by [41]. Moreover, it is interesting to consider a matter field near the extremal Reissner–Nordström and dilatonic black holes and study the effect of quantum corrections on the black hole thermodynamics [42].

Logarithmic corrected thermodynamics is an interesting subject in various contexts like that discussed by references [43, 44]. Several black objects are considered to investigate universality in the form of correction terms [45, 46]. It is also possible to use the partition function to study the corrected thermodynamics of black hole [47], which yields to the fact that the quantum correction of space-time structure would produce thermal fluctuations [48].

Assuming a black hole consists of total N particles, we can write total microstates to obtain the entropy. From the statistical mechanics, we know that, the total number of microstates of a given system expressed as

$$\begin{equation}{\Omega}=\frac{\left({\Sigma}{n}_{i}\right)!}{{\Pi}{n}_{i}!},\end{equation} \tag{ 1 }$$

Assuming each number n_i is shared by s_i pieces, therefore,

$$\begin{equation}{\Omega}=\frac{\left({\Sigma}{s}_{i}\right)!}{{\Pi}{s}_{i}!}.\end{equation} \tag{ 2 }$$

Also, ɛ_i is the energy of the ith microstate with number n_i . Hence

$$\begin{equation}N={\Sigma}{s}_{i}{n}_{i},\end{equation} \tag{ 3 }$$

is total number and

$$\begin{equation}E={\Sigma}{s}_{i}{n}_{i}{\varepsilon }_{i},\end{equation} \tag{ 4 }$$

is total energy. Then, using the Stirling formula for the large N limit,

$$\begin{equation}\mathrm{ln}\enspace N!=N\enspace \mathrm{ln}\enspace N-N,\end{equation} \tag{ 5 }$$

and varying ln Ω under the following conditions,

$$\begin{align}\delta {\Sigma}{s}_{i}{n}_{i}& =0,\\ {\Sigma}{\varepsilon }_{i}\delta ({s}_{i}{n}_{i})& =0,\end{align} \tag{ 6 }$$

one can obtain the most likely configuration as

$$\begin{equation}{s}_{i}=\left(\sum {n}_{i}\right){\text{e}}^{-\lambda {n}_{i}},\end{equation} \tag{ 7 }$$

where λ is called the variation parameter (plays the role of Lagrange multipliers), which satisfied the following condition [34],

$$\begin{equation}\sum {\text{e}}^{-\lambda {n}_{i}}=1,\end{equation} \tag{ 8 }$$

which yields,

$$\begin{equation}\lambda \approx \mathrm{ln}\enspace 2-{2}^{-N},\end{equation} \tag{ 9 }$$

where $\mathcal{O}({2}^{-2N})$ neglected. So, the entropy given by S = λN. Hence, eliminating N using the equation (9) tells that the quantum correction to the black hole entropy is exponential and given by,

$$\begin{equation}S={S}_{0}+{\text{e}}^{-{S}_{0}},\end{equation} \tag{ 10 }$$

with

$$\begin{equation}{S}_{0}=\frac{A}{4{l}_{\text{p}}^{2}},\end{equation} \tag{ 11 }$$

where l_p is Planck length and A is the black hole horizon area.

All above mentioned corrections can be expressed as following,

$$\begin{equation}S={S}_{0}+\alpha \enspace \mathrm{ln}\enspace {f}_{1}({S}_{0})+\frac{\gamma }{{S}_{0}}+\eta \enspace {\text{e}}^{{f}_{2}({S}_{0})},\end{equation} \tag{ 12 }$$

where α, γ, and η are some infinitesimal constants which are called correction coefficients. Also, f₁(S₀) and f₂(S₀) are suitable functions of the uncorrected black hole entropy. Several forms of function f₁(S₀) like S₀ T², CT² or S₀ already discussed in literatures [49–53]. The logarithmic correction with coefficient α [54, 55] and higher order correction with coefficient γ [56, 57] may be negligible. Hence, it is possible to consider the case α = γ = 0 which is interested in this paper. In [34], it is argued that one can set f₂(S₀) = −S₀ and η = 1, independent of the theory of quantum gravity. The exponential term also is negligible for the large black hole areas (entropy), but its effect is important when the black hole area is small. Therefore, it is considered as a quantum effect for the small black hole as well as logarithmic correction [58].

It should be noted that the exponential correction on the statistical entropy of supersymmetric string theories using the quantum entropy function formalism [59]. It is compatible with the non-perturbative features of the string theory [17] and is valid in the Planckian regime of the black hole event horizon area. It is clear that for the large black hole we have A ≫ 1 so S₀ ≫ 1, hence S = S₀. On the other hand, for the case of S₀ = 0 (A = 0), we have S = 1, so we may use it for the two-dimensional black holes as well as entropy function formalism [60, 61]. Although, the quantum entropy function may also yield to the logarithmic correction [62]. In general, according to the relation (18) we can find S ⩾ 1.

A general spherically symmetric black hole metric in D dimensional space-time given by,

$$\begin{equation}\mathrm{d}{s}^{2}=-f(r)\mathrm{d}{t}^{2}+\frac{\mathrm{d}{r}^{2}}{f(r)}+{r}^{2}\enspace \mathrm{d}{{\Omega}}_{D-2}^{2}.\end{equation} \tag{ 13 }$$

Black hole area, and hence, black hole entropy depends on $\mathrm{d}{{\Omega}}_{D-2}^{2}$ elements, while black hole temperature (at outer horizon radius r₊) is given by,

$$\begin{equation}T=\frac{1}{4\pi }{\left(\frac{\mathrm{d}f(r)}{\mathrm{d}r}\right)}_{r={r}_{+}},\end{equation} \tag{ 14 }$$

which is independent of $\mathrm{d}{{\Omega}}_{D-2}^{2}$ elements. Therefore, the black hole entropy may correct, but the black hole temperature remains unchanged. However, there are some possible ways to obtain corrected temperature [63]. In this work, we assume that the Hawking temperature of the black hole, given by the equation (14), do not affect by quantum corrections. Now, we justify this assumption by proposing the following line element,

$$\begin{equation}\mathrm{d}{s}^{\prime 2}=-f(r)\mathrm{d}{t}^{2}+\frac{\mathrm{d}{r}^{2}}{f(r)}+{r}^{2}K(r)\mathrm{d}{{\Omega}}_{D-2}^{2},\end{equation} \tag{ 15 }$$

where

$$\begin{align}K(r)& =1+\alpha \frac{4{l}_{\text{p}}^{2}}{{r}_{\text{H}}^{D-2}{{\Omega}}_{D-2}}\enspace \mathrm{ln}{f}_{1}\left(\frac{{r}_{\text{H}}^{D-2}{{\Omega}}_{D-2}}{4{l}_{\text{p}}^{2}}\right)\\ & \quad +\gamma \frac{{(4{l}_{\text{p}}^{2})}^{2}}{{({r}_{\text{H}}^{D-2}{{\Omega}}_{D-2})}^{2}}+\eta \frac{4{l}_{\text{p}}^{2}}{{r}_{\text{H}}^{D-2}{{\Omega}}_{D-2}}\enspace \mathrm{exp}{f}_{2}\left(\frac{{r}_{\text{H}}^{D-2}{{\Omega}}_{D-2}}{4{l}_{\text{p}}^{2}}\right)+\cdots \enspace ,\end{align} \tag{ 16 }$$

where dots represent higher order terms of correction coefficients which are neglected. So, the modified black hole horizon area is given by,

$$\begin{equation}\mathcal{A}=K({r}_{+}){r}_{+}^{D-2}{{\Omega}}_{D-2}.\end{equation} \tag{ 17 }$$

Hence the corrected entropy (12) reproduced by,

$$\begin{equation}S=\frac{\mathcal{A}}{4{l}_{\text{p}}^{2}}.\end{equation} \tag{ 18 }$$

Using the metric (15), it is clear that the event horizon radius remains unchanged and the black hole temperature, as before, given by the equation (14). Because the additional terms of (16) are small perturbations around the equilibrium, hence have not any effective impact on the field equations and hence the geometry of dΩ². It is like the situation that happen for the Hořava–Lifshitz black hole [31]. This is also true for the corrections obtained using AdS/CFT correspondence [64].

The extra exponential term of the black hole entropy (18) affects some thermodynamics quantities, which will be discussed in the next section. Also, it modifies the partition function of statistical physics, which will be obtained in the next section.

In this section, we would like to use corrected entropy (18) to see a modification of some thermodynamics and statistics quantities in a general form. To find a general formalism, we need to specify the temperature dependence of the entropy. It is a power law for several systems like Schwarzschild black holes. Hence, we use the following ansatz,

$$\begin{equation}T={c}_{1}{S}_{0}^{n},\end{equation} \tag{ 19 }$$

where c₁ is a constant, and n is a number. Also, it is easy to extend this relation as,

$$\begin{equation}T=\sum\limits _{i}\;{c}_{i}{S}_{0}^{{n}_{i}}.\end{equation} \tag{ 20 }$$

Hence, the equation (19) is a special case of the equation (20) where there is only one coefficient c₁.

To begin, we use a well-known relation between entropy and partition function (Z) of a canonical ensemble which is,

$$\begin{equation}S=\mathrm{ln}\enspace Z+T{\left(\frac{\partial \enspace \mathrm{ln}\enspace Z}{\partial T}\right)}_{V}.\end{equation} \tag{ 21 }$$

By using the exponential corrected entropy (18) and temperature (19) in the relation (21) we obtain the following relation,

$$\begin{equation}\mathrm{ln}\enspace Z=\frac{n}{n+1}{S}_{0}+\frac{n}{{S}_{0}^{n}}\int {S}_{0}^{n-1}\enspace {\text{e}}^{-{S}_{0}}\mathrm{d}{S}_{0}+g({S}_{0}),\end{equation} \tag{ 22 }$$

where g(S₀) is an unknown function depend on quantum correction, which will be obtained using other thermodynamics quantities. Indeed, quantum effects are encoded in both the last terms of expression (22). Hence, at the first step, we try to solve the above integral numerically. By using the following relation,

$$\begin{equation}{\int }_{0}^{\infty }{x}^{n}\enspace {\text{e}}^{-x}\enspace \mathrm{d}x=n!,\end{equation} \tag{ 23 }$$

we can rewrite the equation (22) as following,

$$\begin{equation}\mathrm{ln}\enspace Z=\frac{n}{n+1}{S}_{0}+\frac{n(n-1)!}{{S}_{0}^{n}}+g({S}_{0}).\end{equation} \tag{ 24 }$$

The function g(S₀) includes two parts; the first one is corresponding to the ordinary entropy (classical), and the other is corresponding to the quantum correction. Hence we can write,

$$\begin{equation}g({S}_{0})\equiv \frac{{g}_{\text{c}}}{{S}_{0}^{n}}+{g}_{\text{c}}({S}_{0}),\end{equation} \tag{ 25 }$$

where g_c is a constant related to the classical part, while g_q(S₀) is an unknown function due to the quantum correction. This part was added to recover the missed entropy-dependence part due to the numerical calculations of (23). Therefore,

$$\begin{equation}\mathrm{ln}\enspace Z=\mathrm{ln}\enspace {Z}_{0}+\mathrm{ln}\enspace {Z}_{\text{c}}=\left[\frac{n}{n+1}{S}_{0}+\frac{{g}_{\text{c}}}{{S}_{0}^{n}}\right]+\left[\frac{n(n-1)!}{{S}_{0}^{n}}+{g}_{\text{q}}({S}_{0})\right],\end{equation} \tag{ 26 }$$

where

$$\begin{equation}\mathrm{ln}\enspace {Z}_{\text{c}}=\frac{n(n-1)!}{{S}_{0}^{n}}+{g}_{\text{q}}({S}_{0}),\end{equation} \tag{ 27 }$$

referred to the correction of partition function due to the thermal fluctuations, while Z₀ denotes uncorrected partition function. Therefore, the corrected partition function can express as,

$$\begin{equation}Z=\mathrm{exp}\left(\frac{n}{n+1}{S}_{0}+\frac{{g}_{\text{c}}+n(n-1)!}{{S}_{0}^{n}}+{g}_{\text{q}}({S}_{0})\right).\end{equation} \tag{ 28 }$$

Having the partition function, helps us to obtain all thermodynamics quantities. For example, internal energy given by,

$$\begin{align}U& ={T}^{2}\frac{\mathrm{d}\enspace \mathrm{ln}\enspace Z}{\mathrm{d}T}\\ & =\frac{{c}_{1}}{n}{S}_{0}^{n+1}\left[\frac{n}{n+1}-\frac{n}{{S}_{0}^{n+1}}({g}_{\text{c}}+n(n-1)!)+{g}_{\text{q}}^{\prime }({S}_{0})\right],\end{align} \tag{ 29 }$$

where the prime denotes derivatives with respect to S₀.

On the other hand, having corrected entropy and temperature also gives us the thermodynamics of a given system. Both ways should coincide, and this helps us to determine the g_q(S₀) function. We can do that by using the Helmholtz free energy F. According to the statistical mechanical we have,

$$\begin{equation}F=-T\enspace \mathrm{ln}\enspace Z=-{c}_{1}{S}_{0}^{n}\left[\frac{n}{n+1}{S}_{0}+\frac{{g}_{\text{c}}+n(n-1)!}{{S}_{0}^{n}}+{g}_{\text{q}}({S}_{0})\right],\end{equation} \tag{ 30 }$$

while according to the thermodynamics we have,

$$\begin{equation}F=U-TS=U-T({S}_{0}+{\text{e}}^{-{S}_{0}}).\end{equation} \tag{ 31 }$$

Both equations (30) and (31) should make the same result. It yields to the following differential equation,

$$\begin{equation}{S}_{0}{g}_{\text{q}}^{\prime }({S}_{0})+n{g}_{\text{q}}({S}_{0})=n\enspace {\text{e}}^{-{S}_{0}}.\end{equation} \tag{ 32 }$$

Solution of the above equation obtained as,

$$\begin{equation}{g}_{\text{q}}({S}_{0})=\frac{{g}_{\text{q}}}{{S}_{0}^{n}}+\left[\frac{WM\left(\frac{n}{2},\frac{n+1}{2},{S}_{0}\right){S}_{0}^{-\frac{n}{2}}}{n+1}+WM\left(\frac{n+2}{2},\frac{n+1}{2},{S}_{0}\right){S}_{0}^{-\frac{n+2}{2}}\right]{\text{e}}^{-\frac{{S}_{0}}{2}},\end{equation} \tag{ 33 }$$

where WM(μ, ν, z) is the Whittaker function is constructed from the hypergeometric function. Also, g_q is an integration constant. In the case of n = 0 we have g_q(S₀) ≈ 1 + g_q; in absence of exponential corrections, the functions g_q takes the form of the first term, proportional to g_s, as is evident from the solution to equation (33). In other words, it only changes the value of the constant g_s. However, due to the exponential correction, g_q may be interpreted as renormalizing the semi-classical constant g_s. It is in this spirit that the term g_q has a meaning, and its plots against S₀ may be considered as a flow of semi-classical constants as length scales approach the Plank scale. In figure 1 we can see behavior of g_q(S₀) in terms of S₀ for some values of n. Hence, using the solution (33) in equation (26) one can write,

$$\begin{align}\mathrm{ln}\enspace Z& =\frac{n}{n+1}{S}_{0}+\frac{g+n(n-1)!}{{S}_{0}^{n}}\\ & \quad +\left[\frac{WM\left(\frac{n}{2},\frac{n+1}{2},{S}_{0}\right){S}_{0}^{-\frac{n}{2}}}{n+1}+WM\left(\frac{n+2}{2},\frac{n+1}{2},{S}_{0}\right){S}_{0}^{-\frac{n+2}{2}}\right]{\text{e}}^{-\frac{{S}_{0}}{2}},\end{align} \tag{ 34 }$$

where g ≡ g_c + g_q is used.

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To investigate further about thermodynamics quantities like pressure, enthalpy, and Gibbs free energy, we need black hole volume, which can be expressed as following,

$$\begin{equation}V={v}_{d}{S}_{0}^{\frac{d}{d-1}},\end{equation} \tag{ 35 }$$

where v_d is a constant and d is a spatial dimension. For example, in five-dimensional space-time (d = 4) we find ${v}_{d}={(\frac{2}{{\pi }^{2}})}^{1/3}$.

In that case, pressure given by,

$$\begin{equation}p=T\frac{\mathrm{d}\enspace \mathrm{ln}\enspace Z}{\mathrm{d}V}={p}_{0}+{p}_{\text{c}},\end{equation} \tag{ 36 }$$

where p₀ is uncorrected pressure given by,

$$\begin{equation}{p}_{0}=\frac{(d-1){c}_{1}n({S}_{0}^{n+1}-{g}_{\text{c}}(n+1))}{(n+1)\mathrm{d}{v}_{d}{S}_{0}^{\frac{d}{d-1}}},\end{equation} \tag{ 37 }$$

while p_c is corrected pressure which is obtained as,

$$\begin{equation}{p}_{\text{c}}=\frac{(d-1){c}_{1}n}{(n+1)\mathrm{d}{v}_{d}}\left[{S}_{0}^{\frac{n(d-1)-2d}{d-1}}\enspace {\text{e}}^{-\frac{{S}_{0}}{2}}WM\left(\frac{n}{2},\frac{n+1}{2},{S}_{0}\right)+{S}_{0}^{-\frac{d}{d-1}}\left((n+1)(n!+{g}_{\text{q}})\right)\right].\end{equation} \tag{ 38 }$$

In plots of figure 2 we can see the effects of quantum correction on the black hole pressure. Increasing pressure is one of the important quantum effects.

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Enthalpy is given by the following general formula,

$$\begin{equation}H=T\left[T\frac{\mathrm{d}\enspace \mathrm{ln}\enspace Z}{\mathrm{d}T}+V\frac{\mathrm{d}\enspace \mathrm{ln}\enspace Z}{\mathrm{d}V}\right]\equiv {H}_{0}+{H}_{\text{c}},\end{equation} \tag{ 39 }$$

where H₀ is uncorrected entropy, while H_c denotes the effect of quantum correction, which are respectively obtained as,

$$\begin{equation}{H}_{0}=\frac{{c}_{1}((d-1)n+d)({S}_{0}^{n+1}-{g}_{\text{c}}(n+1))}{(n+1)d},\end{equation} \tag{ 40 }$$

and

$$\begin{equation}{H}_{\text{c}}=-\frac{{c}_{1}((d-1)n+d)}{(n+1)d}\left[{S}_{0}^{\frac{n}{2}{\text{e}}^{-\frac{{S}_{0}}{2}}}WM\left(\frac{n}{2},\frac{n+1}{2},{S}_{0}\right)+(n+1)!+{g}_{\text{q}}(n+1)\right].\end{equation} \tag{ 41 }$$

In figure 3 we can see the behavior of the entropy and find the effect of quantum correction which is decreasing enthalpy.

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Then, we can obtain Gibbs free energy via,

$$\begin{equation}G=T\left[-\mathrm{ln}\enspace Z+V\frac{\mathrm{d}\enspace \mathrm{ln}\enspace Z}{\mathrm{d}V}\right]\equiv {G}_{0}+{G}_{\text{c}}.\end{equation} \tag{ 42 }$$

In figure 4 we can see the effect of quantum correction on the Gibbs free energy.

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Finally, we can discuss the specific heat at constant volume,

$$\begin{equation}C=T\left[2\frac{\mathrm{d}\enspace \mathrm{ln}\enspace Z}{\mathrm{d}T}+T\frac{{\mathrm{d}}^{2}\enspace \mathrm{ln}\enspace Z}{\mathrm{d}{T}^{2}}\right]\equiv {C}_{0}+{C}_{\text{c}},\end{equation} \tag{ 43 }$$

where

$$\begin{equation}{C}_{0}=\frac{2n{S}_{0}^{n+1}-{g}_{\text{c}}(n-1)(n+1)}{n(n+1){S}_{0}^{n}},\end{equation} \tag{ 44 }$$

and

$$\begin{equation}{C}_{\text{c}}=-\frac{{S}_{0}^{-\frac{n}{2}}\enspace {\text{e}}^{-\frac{{S}_{0}}{2}}WM\left(\frac{n}{2},\frac{n+1}{2},{S}_{0}\right)+(n+1)\left((n-1)(n{\Gamma}(n)+{g}_{\text{q}}){S}_{0}^{-n}+{S}_{0}\enspace {\text{e}}^{-{S}_{0}}\right)}{n(n+1)},\end{equation} \tag{ 45 }$$

In plots of figure 5 we draw specific heat in terms of S₀ for some values of n. We can see that quantum correction is different for a given system with different n. For example, the system with n = −1.5 is stable (C₀ < 0) while in the presence of a quantum effect, corrected specific heat is a negative, and the black hole is unstable. In the case of n = −0.5, also the effect of quantum correction is the stability of the black hole (see the solid red line of figure 5). It means that some black holes, reduced their size due to the Hawking radiation and goes to a stable phase, hence do not evaporate.

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Then, in the figure 6, we draw specific heat in terms of n, and see some phase transitions for specific values of n like n = −1, −2.

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Now, we can consider some famous black hole solutions to exam the above general formulation. We begin with the Schwarzschild black hole.

The simplest spherically symmetric solution is a Schwarzschild black hole. We consider it in both four and five dimensions separately, to study exponential corrected thermodynamics. The black hole mass (M) is the only important parameter of this kind, hence the first law of thermodynamics for the Schwarzschild black hole given by,

$$\begin{equation}\mathrm{d}M=T\enspace \mathrm{d}S=T\enspace \mathrm{d}{S}_{0}(1-{\text{e}}^{-{S}_{0}}),\end{equation} \tag{ 46 }$$

where in the last equality we used (18). Using the equation (19) in the thermodynamics first law (46), then applying numerical integration (23), we find,

$$\begin{equation}M={c}_{1}\left(\frac{{S}_{0}^{n+1}}{n+1}-n!\right)={M}_{0}-{c}_{1}n!,\end{equation} \tag{ 47 }$$

which means quantum corrections reduced the value of the black hole mass. For obtaining expected results for usual Schwarzschild black hole thermodynamics, we should set g = 0.

The Smarr–Gibbs–Duhem relation for the Schwarzschild black hole is given by [65],

$$\begin{equation}\frac{D-3}{D-2}M=TS.\end{equation} \tag{ 48 }$$

Following, we study two separate cases of Schwarzschild black hole in four and five dimensions.

4.1. 4D

Four-dimensional Schwarzschild black hole given by the metric (13) with D = 4 and (see for example [65]),

$$\begin{equation}f(r)=1-\frac{2{M}_{0}}{r},\end{equation} \tag{ 49 }$$

where M₀ = TdS₀ denotes the uncorrected black hole mass satisfying ordinary first law of thermodynamics, so r₊ = 2M₀ is the black hole event horizon radius. The black hole entropy and thermodynamic volume given by,

$$\begin{equation}{S}_{0}=\pi {r}_{+}^{2},\end{equation} \tag{ 50 }$$

and

$$\begin{equation}V=\frac{4}{3}\pi {r}_{+}^{3},\end{equation} \tag{ 51 }$$

respectively. Using the relations (14) and (49) one can obtain,

$$\begin{equation}T=\frac{1}{4\pi {r}_{+}}.\end{equation} \tag{ 52 }$$

Combination of (50) and (52) yields $T=\frac{1}{4\sqrt{\pi {S}_{0}}}$, hence we find $n=-\frac{1}{2}$ and ${c}_{1}=\frac{1}{4\sqrt{\pi }}$ in the equation (19). Also, combination of (35), (50) and (51) gives, ${v}_{d}=\frac{4}{3\pi }$, so $V=\frac{4}{3\pi }{S}_{0}^{\frac{3}{2}}$.

The first law of black hole thermodynamics reads as,

$$\begin{equation}\mathrm{d}M=\mathrm{d}{M}_{0}(1-{\text{e}}^{-{S}_{0}}),\end{equation} \tag{ 53 }$$

which yield,

$$\begin{equation}M={M}_{0}-\frac{1}{4}.\end{equation} \tag{ 54 }$$

In that case, the Smarr–Gibbs–Duhem formula (48) gives

$$\begin{equation}{M}_{0}-\frac{1}{4}=2T({S}_{0}+{\text{e}}^{-{S}_{0}}),\end{equation} \tag{ 55 }$$

which hold only in the special radius r₊ ≈ 0.867, where we used M₀ = 2TS₀ as ordinary (uncorrected) Smarr–Gibbs–Duhem relation.

In figure 7 we can see the important effect of quantum correction on the partition function of 4D Schwarzschild black hole. Both solid and dashed lines of figure 7 are coincide for large area (large S₀).

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The right plot of figure 2 shows the effect of quantum correction on the Schwarzschild black hole pressure. In the case of a large area, the pressure is negative. However, the pressure becomes positive at a small area due to the thermal fluctuations. In order to see a variation of specific heat with the horizon radius, we draw figure 8. As we found earlier, the effect of quantum fluctuations may be black hole stability.

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4.2. 5D

Five-dimensional Schwarzschild black hole given by the metric (13) with D = 5 and (see for example [66]),

$$\begin{equation}f(r)=1-\frac{{r}_{+}^{2}}{{r}^{2}},\end{equation} \tag{ 56 }$$

where ${r}_{+}^{2}=\frac{8}{3\pi }{M}_{0}$ is event horizon radius. The black hole entropy and thermodynamic volume is given by,

$$\begin{equation}{S}_{0}=\frac{{\pi }^{2}}{2}{r}_{+}^{3},\end{equation} \tag{ 57 }$$

and

$$\begin{equation}V=\frac{{\pi }^{2}}{2}{r}_{+}^{4},\end{equation} \tag{ 58 }$$

respectively. Using relations (14) and (56) one can obtain,

$$\begin{equation}T=\frac{1}{2\pi {r}_{+}}.\end{equation} \tag{ 59 }$$

Combination of (57) and (59) yields $T=\frac{1}{2{(2\pi )}^{1/3}{S}_{0}^{-1/3}}$, hence we find $n=-\frac{1}{3}$ and ${c}_{1}=\frac{1}{2{(2\pi )}^{1/3}}$ in the equation (19). Also, combination of (35), (57) and (58) gives, ${v}_{d}={(\frac{2}{{\pi }^{2}})}^{1/3}$, so $V={(\frac{2}{{\pi }^{2}})}^{1/3}{S}_{0}^{\frac{4}{3}}$.

Using the first law of thermodynamics we find

$$\begin{equation}M={M}_{0}-0.24.\end{equation} \tag{ 60 }$$

In that case, relation (48) hold if the following condition satisfied,

$$\begin{equation}{r}_{+}^{4}-\frac{2}{{\pi }^{2}}{r}_{+}-\frac{3}{2{\pi }^{8/3}}=0.\end{equation} \tag{ 61 }$$

We can analyze black hole stability by using a sign of specific heat. We can find that specific heat is negative for all area ranges. Therefore, opposite to the previous case, the 5D Schwarzschild black hole is completely unstable even under quantum correction.

We can express exponential corrected entropy in terms of corrected mass to see that the second law of thermodynamics is also satisfied, which is illustrated by figure 9.

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Now, we consider a charged black hole. Four-dimensional Reissner–Nordström black hole given by the metric (13) with D = 4 and (see for example [65]),

$$\begin{equation}f(r)=1-\frac{2{M}_{0}}{r}+\frac{{Q}^{2}}{{r}^{2}},\end{equation} \tag{ 62 }$$

where Q is the black hole charge. The first law of black hole thermodynamics reads as,

$$\begin{equation}\mathrm{d}M=T\enspace \mathrm{d}S+{\Phi}\enspace \mathrm{d}Q,\end{equation} \tag{ 63 }$$

where electrostatic potential given by,

$$\begin{equation}{\Phi}=\frac{Q}{{r}_{+}}.\end{equation} \tag{ 64 }$$

It is clear that the first law of black hole thermodynamics is satisfied as 4D Schwarzschild black hole. Hence, the corrected mass is the same as (54) and the Smarr–Gibbs–Duhem formula (48) should be extended to [65],

$$\begin{equation}\frac{D-3}{D-2}M=TS+\frac{D-3}{D-2}{\Phi}Q,\end{equation} \tag{ 65 }$$

which will be hold if the following equation satisfied approximately,

$$\begin{equation}2\pi {r}_{+}^{4}-\pi {r}_{+}^{3}-2(6\pi {Q}^{2}+1){r}_{+}^{2}+8{Q}^{2}=0.\end{equation} \tag{ 66 }$$

This case is described by equation (20) where we have only two coefficients. Hence, equation (19) extends to

$$\begin{equation}T={c}_{1}{S}_{0}^{{n}_{1}}+{c}_{2}{S}_{0}^{{n}_{2}},\end{equation} \tag{ 67 }$$

where ${c}_{1}=\frac{1}{4\sqrt{\pi }}$ and ${n}_{1}=-\frac{1}{2}$ are as 4D Schwarzschild black hole, while ${c}_{2}=-\sqrt{\pi }{Q}^{2}$ and ${n}_{2}=-\frac{3}{2}$ are corresponding to Reissner–Nordström black hole.

Therefore, similar to the previous section, we can obtain the corrected partition function as,

$$\begin{equation}\mathrm{ln}\enspace Z=\mathrm{ln}\enspace {Z}_{0}+\mathrm{ln}\enspace {Z}_{\text{c}},\end{equation} \tag{ 68 }$$

where we defined,

$$\begin{equation}\mathrm{ln}\enspace {Z}_{0}=\frac{\left(g+\frac{12\pi {Q}^{2}+{S}_{0}}{\sqrt{{S}_{0}}}\right){S}_{0}^{\frac{3}{2}}}{4\pi {Q}^{2}-{S}_{0}},\end{equation} \tag{ 69 }$$

and

$$\begin{equation}\mathrm{ln}\enspace {Z}_{\text{c}}=-\left(\frac{12\pi {Q}^{2}\left(\frac{2\sqrt{\pi }}{3}+\frac{1}{3\sqrt{{S}_{0}}\enspace {\text{e}}^{{S}_{0}}}\left(2-\frac{1}{{S}_{0}}\right)\right)+\frac{1}{\sqrt{{S}_{0}}\enspace {\text{e}}^{{S}_{0}}}+\sqrt{\pi }}{4\pi {Q}^{2}-{S}_{0}}\right){S}_{0}^{\frac{3}{2}},\end{equation} \tag{ 70 }$$

In order to obtain the above solution, we used the following numerical integral,

$$\begin{equation}{\int }_{0}^{\infty }{\text{e}}^{-x}\enspace \mathrm{d}x=\frac{\sqrt{\pi }}{2}.\end{equation} \tag{ 71 }$$

By using (29) and (30) we can obtain the behavior of internal energy and Helmholtz free energy as

$$\begin{align}U={U}_{0}+{U}_{\text{c}},\\ F={F}_{0}+{F}_{\text{c}},\end{align} \tag{ 72 }$$

which are illustrated by plots of figure 10. Here, U₀ and F₀ denote uncorrected energies, while U_c and F_c are correction terms. We can see that the corrected internal energy is completely negative while the ordinary one is completely positive. Also, we can see that Helmholtz free energy decreased due to quantum corrections. All of these may be a sign of instability in the presence of quantum correction, which will be discussed soon by analyzing sign of specific heat.

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A combination of relations (43) and (68) gives us the specific heat. The presence of Q makes Reissner–Nordström black hole stable at a small radius while unstable at a large area. So, one can see an unstable/stable phase transition. However, the presence of quantum correction is cause by instability in a small area. It is illustrated in figure 11.

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However, for the charged black holes, we should confirm our stability analysis by using the Hessian matrix of the Helmholtz free energy, which is given by,

$$\begin{equation}\left(\begin{matrix}\hfill \frac{{\partial }^{2}F}{\partial {T}^{2}}\hfill & \hfill \frac{{\partial }^{2}F}{\partial T\partial {\Phi}}\hfill \\ \hfill \frac{{\partial }^{2}F}{\partial {\Phi}\partial T}\hfill & \hfill \frac{{\partial }^{2}F}{\partial {{\Phi}}^{2}}\hfill \end{matrix}\right)=\mathcal{H}.\end{equation} \tag{ 73 }$$

It is easy to see that the determinant of the $\mathcal{H}$-matrix vanishes, which means that, one of the eigenvalues is zero. Hence, we should consider the other one, which is the trace of the matrix (73) given by,

$$\begin{equation}\mathrm{Tr}(\mathcal{H})=\left(\frac{{\partial }^{2}F}{\partial {T}^{2}}\right)+\left(\frac{{\partial }^{2}F}{\partial {{\Phi}}^{2}}\right).\end{equation} \tag{ 74 }$$

Black hole is in stable phase when $\mathrm{Tr}(\mathcal{H}){\geqslant}0$. Our numerical analysis indicated that $\mathrm{Tr}(\mathcal{H}){< }0$ at a small radius, which confirms black hole instability at quantum scales.

In this section, we consider the Schwarzschild-AdS black hole in four-dimensional space-time. It is given by the metric (13) with D = 4 and (see for example [65]),

$$\begin{equation}f(r)=1-\frac{2{M}_{0}}{r}+\frac{{r}^{2}}{{l}^{2}},\end{equation} \tag{ 75 }$$

where l is the AdS radius. The first law of black hole thermodynamics read as,

$$\begin{equation}\mathrm{d}M=T\enspace \mathrm{d}S+V\enspace \mathrm{d}P,\end{equation} \tag{ 76 }$$

where V is given by equation (51), and thermodynamic pressure is given by,

$$\begin{equation}P=\frac{3}{8\pi {l}^{2}}.\end{equation} \tag{ 77 }$$

It is clear that the first law of black hole thermodynamics is satisfied. Hence, corrected mass is the same as (54) and the Smarr–Gibbs–Duhem formula (48) should be extended to [65],

$$\begin{equation}\frac{D-3}{D-2}M=TS-\frac{2}{D-2}VP,\end{equation} \tag{ 78 }$$

which will be held if the following equation satisfied approximately (for D = 4),

$$\begin{equation}10\pi {r}_{+}^{4}+2(\pi {l}^{2}-3){r}_{+}^{2}-\pi {l}^{2}{r}_{+}-2{l}^{2}=0.\end{equation} \tag{ 79 }$$

Here we have similar relation with equation (67) with ${c}_{1}=\frac{1}{4\sqrt{\pi }}$ and ${n}_{1}=-\frac{1}{2}$ are as 4D Schwarzschild black hole, while ${c}_{2}=\frac{3}{4{\pi }^{3/2}{l}^{2}}$ and ${n}_{2}=\frac{1}{2}$ are corresponding to Schwarzschild-AdS black hole.

The corrected partition function is obtained as follows,

$$\begin{equation}Z=\mathrm{exp}\left(\frac{{S}_{0}^{2}-\pi {l}^{2}{S}_{0}+g\sqrt{{S}_{0}}}{\pi {l}^{2}+3{S}_{0}}\right)\mathrm{exp}\left(\frac{\sqrt{{S}_{0}}}{\pi {l}^{2}+3{S}_{0}}\left[\pi {l}^{2}\left(\sqrt{\pi }+\frac{{\text{e}}^{-{S}_{0}}}{\sqrt{{S}_{0}}}\right)+\frac{3\sqrt{\pi }}{2}\right]\right),\end{equation} \tag{ 80 }$$

where the last exponential is due to the quantum correction. We find that increasing partition function is a consequence of quantum corrections.

Using equation (43) we can obtain specific heat and find that quantum corrections make Schwarzschild-AdS black hole stable at the small areas (see figure 12). In the absence of the thermal fluctuations, Schwarzschild-AdS black hole is unstable at a small horizon radius. At the same time, in the presence of quantum correction, it will be stable by reducing size.

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It is the most general case which considers in this paper. The charged AdS black hole is described by mass, charge, and AdS radius, which all of them introduced already. It is given by the metric (13) with D = 4 and (see for example [65]),

$$\begin{equation}f(r)=1-\frac{2{M}_{0}}{r}+\frac{{Q}^{2}}{{r}^{2}}+\frac{{r}^{2}}{{l}^{2}}.\end{equation} \tag{ 81 }$$

The first law of black hole thermodynamics read as,

$$\begin{equation}\mathrm{d}M=T\enspace \mathrm{d}S+{\Phi}\enspace \mathrm{d}Q+V\enspace \mathrm{d}P,\end{equation} \tag{ 82 }$$

where S is given by (50), Φ is given by (64), V is given by equation (51), and pressure P is given by (77). Since the exponential corrected entropy is independent of Q and P, it is clear that the first law of black hole thermodynamics is satisfied with the corrected mass (54). Hence, the Smarr–Gibbs–Duhem formula given by [65],

$$\begin{equation}\frac{D-3}{D-2}M=TS+\frac{D-3}{D-2}{\Phi}Q-\frac{2}{D-2}VP.\end{equation} \tag{ 83 }$$

Using the corrected mass and entropy for D = 4 we find that the Smarr–Gibbs–Duhem formula satisfied if,

$$\begin{equation}4\pi {r}_{+}^{6}+2\pi ({l}^{2}+3){r}_{+}^{4}-\pi {l}^{2}{r}_{+}^{3}-2(3\pi {Q}^{2}{l}^{2}+{l}^{2}+3){r}_{+}^{2}+2{Q}^{2}{l}^{2}=0.\end{equation} \tag{ 84 }$$

Here, we have equation (19) with three terms as,

$$\begin{equation}T=\frac{1}{4\sqrt{\pi }}{S}_{0}^{-\frac{1}{2}}-\sqrt{\pi }{Q}^{2}{S}_{0}^{-\frac{3}{2}}+\frac{3}{4{\pi }^{3/2}{l}^{2}}{S}_{0}^{\frac{1}{2}}.\end{equation} \tag{ 85 }$$

In figure 13 we can see the typical behavior of the black hole temperature. In the case of charged AdS black hole, we can see a minimum horizon radius where temperature vanishes. Hence, it is interesting to find what happens for the black hole in the presence of quantum corrections where its temperature becomes zero.

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In that case, corrected partition function obtained similar to (68) where,

$$\begin{equation}\mathrm{ln}\enspace {Z}_{0}=\frac{(g-3\sqrt{\pi })\sqrt{{S}_{0}}+24{\pi }^{2}{Q}^{2}{l}^{2}+2\pi {l}^{2}{S}_{0}-2{S}_{0}^{2}}{8{\pi }^{2}{Q}^{2}{l}^{2}-2\pi {l}^{2}{S}_{0}-6{S}_{0}^{2}}{S}_{0},\end{equation} \tag{ 86 }$$

is ordinary partition function and

$$\begin{equation}\mathrm{ln}\enspace {Z}_{\text{c}}=-\frac{\pi {l}^{2}\left(\sqrt{\pi }{S}_{0}^{3/2}\enspace {\text{e}}^{{S}_{0}}(8\pi {Q}^{2}+1)+4\pi {Q}^{2}(2{S}_{0}-1)+{S}_{0}\right)}{(8{\pi }^{2}{Q}^{2}{l}^{2}-2\pi {l}^{2}{S}_{0}-6{S}_{0}^{2}){\text{e}}^{{S}_{0}}},\end{equation} \tag{ 87 }$$

is corrected terms, hence corrected partition function is ln Z = ln Z₀ + ln Z_c.

Now, we can study specific heat graphically. By using the above relations in equation (43) we can find that the specific heat of charged AdS black hole is negative at a small radius, while in the presence of quantum correction, specific heat is positive. These are illustrated in figure 14. The dashed green line of figure 14 shows that charged AdS black hole is stable when r₁ < r₊ < r₂ in agreement with the claim of [65] (r₁ and r₂ denoted by circles in figure 14). We can confirm such stability by analyzing the Hessian matrix of the Helmholtz free energy similar to the section 5.

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By using the result of figure 13 we can see that the black hole temperature is zero at about r₊ = 0.2 (for Q = 0.1). However, at this point, we have non-zero mass, entropy, and specific heat. We can interpret it as a black remnant.

Then we can see the typical behavior of Gibbs free energy in figure 15. We can see some local extremum in Gibbs free energy; anyway the effect of quantum correction is decreasing it. We find that opposite to the ordinary case [65] there is no holographically dual Van der Waals fluid in the presence of quantum corrections (in a small area). Hence, there are no any critical points and phase transition at quantum scales.

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In this paper, we considered the recent idea that the black hole entropy should be corrected by an exponential term [34] at quantum scales, and show that, such corrected entropy obtained by modified black hole metric. The main goal of this paper was to study such an effect on the black hole thermodynamics. Hence, the effect of the exponential correction term of the black hole entropy in the canonical partition function is calculated. Therefore, the corrected partition function due to the quantum fluctuations obtained, which is used to study exponential corrected thermodynamics of some black holes. We found that such a correction term affects the black hole stability when the black hole size becomes small. We found that a 4D Schwarzschild black hole, which decreased its size due to the Hawking radiation, becomes stable at a small area limit hence does not evaporate. We obtained a similar result for the Schwarzschild-AdS black hole in four-dimensional space-time. On the other hand, we found that the quantum corrections do not affect on the 5D Schwarzschild black hole. Opposite to the 4D Schwarzschild black hole, we found that Reissner–Nordström black hole (charged black hole) is unstable due to the thermal fluctuations. It means that the black hole charge is cause by a stable/unstable phase transition which happens when the black hole size is large. However, the charged AdS black hole will be stable at a small area due to the quantum correction, yielding to the black remnant. It means that the stable black hole evaporates and creates the black remnants at zero temperature while non-zero mass. It may be used to solve the information loss paradox of black holes.

For future work, it is interesting to find the exponential corrected partition function of Kerr or Kerr-AdS black holes (rotating black holes [65]) and discuss the quantum effect on thermodynamics stability. Also, it would be interesting to apply the method used in this paper for BTZ [67] or Hořava–Lifshitz black holes [31].

Already, we used logarithmic corrected entropy [68] to study shear viscosity to entropy ratio and found the universal lower bound violated. It is interesting to consider exponential corrected entropy to study the holographic description of a quark–gluon plasma [69].

B P would like to thank Iran Science Elites Federation, Tehran, Iran, and Canadian Quantum Research Center 204-3002 32 Ave Vernon, BC V1T 2L7 Canada.