11institutetext: School of Mathematical Sciences, Peking University, Beijing,
China 22institutetext: Damo Academy, Alibaba Group, Hangzhou, China
22email: {yuhangguo, xiaoluo, clandzyy<EMAIL_ADDRESS>
# DNA-GCN: Graph convolutional networks for predicting DNA-protein binding

Yuhang Guo Equal Contribution.11 Xiao Luo 1122** Liang Chen 11 Minghua Deng 11

###### Abstract

Predicting DNA-protein binding is an important and classic problem in
bioinformatics. Convolutional neural networks have outperformed conventional
methods in modeling the sequence specificity of DNA-protein binding. However,
none of the studies has utilized graph convolutional networks for motif
inference. In this work, we propose to use graph convolutional networks for
motif inference. We build a sequence k-mer graph for the whole dataset based
on k-mer co-occurrence and k-mer sequence relationship and then learn DNA
Graph Convolutional Network(DNA-GCN) for the whole dataset. Our DNA-GCN is
initialized with a one-hot representation for all nodes, and it then jointly
learns the embeddings for both k-mers and sequences, as supervised by the
known labels of sequences. We evaluate our model on 50 datasets from ENCODE.
DNA-GCN shows its competitive performance compared with the baseline model.
Besides, we analyze our model and design several different architectures to
help fit different datasets.

###### Keywords:

bioinformatics DNA-protein binding Graph convolutional network motif
inference.

##  1 Introduction

DNA-binding proteins play an important role in gene regulation. It’s well-
known that the transcription of each gene is controlled by a regulatory region
of DNA relatively near the transcription start site. There are two fundamental
components in transcription, the short DNA regulatory element, and its
corresponding gene regulatory proteins. DNA binding sites are small but highly
variable, which makes them difficult to detect. Several experimental methods
were developed(e.g. ChIP-seq[31]) to solve this problem, but they are usually
costly, and each has its artifacts, biases, and limitation. Based on sequence-
based data, the problem of predicting DNA-protein binding is to model the
sequence specificity of protein binding (i.e. connect a relationship between
sequence-based data and binary labels of data). Specifically, the task is a
classification problem given training DNA sequences and their binary labels to
predict labels of given testing sequences in the dataset. Recent work on motif
inference includes conventional machine learning-based methods(e.g. SVM,
Random Forest)[16, 8, 11] and deep learning-based methods(e.g. CNN, RNN)[2,
30, 24, 25]. CNN’s and RNNs have shown their superiority compared with
conventional machine learning-based methods. However, when it comes to small
datasets, the performance of the models is often limited. Besides, the models
of CNN’s usually learn truncated motifs that aren’t desired[4].

On the other hand, the binary labels of the sequences are up to whether they
have some specific regions called a motif. If we regard the ”A”, ”C”, ”G”
and”T” as special kinds of characters, k-mer can be treated as words and DNA
sequences can be viewed as sentences. The k-mer related to given motifs can be
viewed as keywords and predicting DNA-protein binding is transformed into the
problem of text classification.

In this paper, we propose a novel method based on Graph Convolutional
Networks[15] – DNA-GCN for predicting DNA-protein binding. In DNA-GCN, firstly
we construct a single large graph from the whole dataset, and then GCN is
utilized to obtain neighborhood information. By this, predicting DNA-protein
binding is turned into a semi-supervised node classification problem. We
choose a lot of different datasets with limited samples to evaluate our model.
The model shows competitive performance on the task of predicting TF binding
sites. All code is public in https://github.com/Tinard/dnagcn. In summary, our
contributions in this paper are twofold. (1) We propose a novel graph
convolutional network for predicting DNA-protein binding. To the best of our
knowledge, we are the first to model the sequence specificity with a graphical
model and utilize GCN to learn the sequences and k-mer embeddings. (2) The
empirical results show that our proposed model has a competitive performance
compared with the baseline models on many datasets with limited sequences. We
suppose that our method could contribute to the study of DNA sequence modeling
and other biological models.

##  2 Related Work

###  2.1 Deep learning for motif inference

The deep learning method for motif inference can be categorized into two
groups – CNN-based and RNN-based methods. As for the CNN-based model(may
contain RNN), DeepBind[2] is the first CNN-based model to predict DNA-protein
binding and since then deep learning has been widely utilized in this field
for its great performance. [30] shows that deploying more convolutional
kernels is always beneficial. iDeepA[23] applies an attention mechanism to
automatically search for important positions. DeeperBind[13] and iDeepS[21]
add an LSTM layer on DeepBind to learn long dependency within sequences to
further improve the prediction performance. As for the RNN-based model,
KEGRU[25] identifies TF binding sites by combining Bidirectional Gated
Recurrent Unit (GRU) network with k-mer embedding. Besides model selection,
CONCISE[3] and iDeep[22] integrate other information(e.g. structured
information) into predicting RBP-binding sites and preference. Other work
includes data augmentation[5], circular filters[4] and convolutional kernel
networks[6, 18, 19].

Figure 1: Architecture of DNA-GCN.

###  2.2 Graph Convolutional Networks

In the past few years, graph convolutional network[15] has attracted wide
attention for its learning hierarchical information on graphs[17, 7, 12]. It
has been shown that the GCN model is a special form of Laplacian smoothing
which makes the features of nodes similar to their neighbors and makes the
subsequent classification task much easier. The layer-wise propagation rule is
formulated as:

$H^{(k+1)}=\phi\left(\tilde{D}^{-\frac{1}{2}}\tilde{A}\tilde{D}^{-\frac{1}{2}}H^{(k)}W^{(k)}\right)$

where $H^{(}k)$ is the node representation matrix, $W^{(}k)$ is the trainable
parameter matrix of the $k$-th layer, $H^{(}0)=X$ is the origin feature
matrix, $\tilde{A}=A+I$ is the adjacency matrix with increased self-
connection, $\ tilde{D}$ is the degree matrix of $\tilde{A}$ (i.e.
$\tilde{D}_{ii}=\sum_{j}\tilde{A}_{ij}$), and $\ phi$ is the activation
function, such as $RELU(\ cdot)$. In addition, there are many variants of GCN,
focusing on improving the performance of GCN and coping with the storage
bottleneck of GCN[7, 12, 26, 28, 32]. In recent years, GCN has been used to
handle many tasks, such as text classification, drug recommendation and
laboratory test classification [29, 20, li2018classifying], and has shown
better performance than baseline on different tasks.

###  2.3 Heterogeneous graph

The heterogeneity is an intrinsic property of a heterogeneous graph, i.e.,
various types of nodes and edges. Apparently, different types of nodes have
different features which fall in different feature space. However, if we
utilize Laplacian smoothing directly, different feature space is mixed which
seems unreasonable. HAN[27] utilizes an attention mechanism to generate node
embedding by aggregating features from meta-path-based neighbors in a
hierarchical manner. MedGCN[20] assumes that there is no edge between nodes in
the same type and the propagation rule is rewritten as

$H_{i}^{(k+1)}=\phi\left(\sum_{j=1}^{n}A_{ij}\cdot H_{j}^{(k)}\cdot
W_{j}^{(k)}\right)$

We assume that the number of types is $n$, $A_{ij}$ is the adjacency matrix
between nodes type $i$ and $j$. $W^{(k)}_{j}$ is the learnable weight matrix
for type $i$ nodes in layer $k$.

##  3 DNA-GCN

###  3.1 Sequence k-mer graph

First of all, we construct a large and heterogenous graph
$G=(\mathcal{V},\mathcal{E})$ where $\mathcal{V},\mathcal{E}$ are sets of
nodes and edges respectively to describe the relationship between sequences
and k-mers. As shown in Figure 1, nodes in the graph have two types: sequences
nodes and k-mers nodes. The number of nodes in sequence k-mer graph
$|\mathcal{V}|$ is the number of sequences including training set, validation
set, and testing set plus the number of possible k-mers. The weight of the
edge between sequences and k-mers is the number of occurrences of k-mer
multiplied by inverse sequence frequency(ISF) in the sequence. ISF is the
logarithmically scaled inverse fraction of the number of sequences that
contain the k-mer. We test two models with or without ISF and found that the
former is better. We suppose that some common k-mers which isn’t related to
the motif in real-world data may disrupt the performance and ISF can ease the
effect of irrelevant k-mers. Point-wise mutual information(PMI) is utilized to
calculate weights between two k-mers. Above all, the adjacent matrix of
sequence k-mer graph is formulated as:

$A_{ij}=\left\\{\begin{array}[]{ l l }{\operatorname{PMI}(i,j)}&{i,j\text{ are
k-mers },\operatorname{PMI}(i,j)>0}\\\
{\mathrm{O}*\operatorname{ISF}_{ij}}&{i\text{ is sequence, }j\text{ is k-mer
}}\\\ {0}&{\text{ otherwise }}\end{array}\right.$

We can computed $PMI(i,j)$ as

$\displaystyle\operatorname{PMI}(i,j)$
$\displaystyle=\log\frac{p(i,j)}{p(i)p(j)}$ $\displaystyle p(i,j)$
$\displaystyle=\frac{\\#W(i,j)}{\\#W}$ $\displaystyle p(i)$
$\displaystyle=\frac{\\#W(i)}{\\#W}$

where $\\#W(i)$ is the number of sequences that contain k-mer $i$, $\\#W(i,j)$
is the number of sequences that contain both k-mer $i$ and $j$, and $\\#W$ is
the total number of sequences in the dataset. We set $k$ to be $4$, and
because a motif length is between 6 and 20, the information of a motif may be
spitted into several k-mers. We believe that if two k-mers co-occur in a
sequence frequently, they are probably to co-decide whether a sequence
contains a motif. From the formulation above, a positive PMI value indicates a
high correlation of k-mers while a negative PMI value indicates little or no
correlation. From the analysis, we set positive PMI values to be the weights
of edges between k-mers.

###  3.2 DNA-GCN

After constructing the sequence k-mer graph, we feed the graph into GCN.
Because the information of sequences is embedded into the graph, we set the
feature matrix $X=I$ for simplicity. At first, we feed the graph into a two-
layer GCN, and the second layer node embeddings only have one dimension and
then are fed into a softmax classifier for classification.

$Z=\operatorname{softmax}\left(\hat{A}\operatorname{ReLU}\left(\hat{A}XW_{0}\right)W_{1}\right)$

where $\hat{A}=\tilde{D}^{-\frac{1}{2}}\tilde{A}\tilde{D}^{-\frac{1}{2}}$ and
$softmax(x_{i})=\frac{1}{\mathcal{Z}}\exp\left(x_{i}\right)$ with partition
function $\mathcal{Z}$(i.e. $\mathcal{Z}=\sum_{i}\exp\left(x_{i}\right)$). The
model has transformed into a semi-supervised model this time and the cross-
entropy error over all labeled sequences determine loss function.

$\mathcal{L}=-\sum_{d\in\mathcal{Y}_{D}}Y_{d}\ln Z_{d}$

where $\mathcal{Y}_{D}$ is the set of sequence indices that have labels and
$Y$ is the label indicator vector.

We give an ideal example. For the dataset with a specific protein, its
positive samples contain the motif ”AACGTC” while negative samples don’t
contain it. AACG, ACGT, CGTC are the key 4-mers which guide classification,
the three k-mers is connected to all the positive sample. By training guided
by labeled sample, the three k-mers can be trained to have the features that
point to positive label, and then the model can predict the label of a testing
sequence by whether the key k-mers is connected to target sequences (i.e.
Features of key k-mers can be transferred to testing sequences or not).
Overall, the information is transferred from labeled samples to k-mers, and
then from k-mers to unlabeled sequences. From the analysis above, we need at
least two times of Laplacian smoothing(i.e. two layers GCN) to construct our
DNA-GCN. By experiments and experience that too many layers lead to over-
smoothing features, we set the number of layers to be two.

We also notice that the sequence k-mer graph is a heterogeneous graph. As a
result, feeding our graph into GCN directly seems to be unreasonable. The
layer-wise propagation rule can be rewritten as

$H^{(k+1)}=\phi\left(\hat{A}(H^{(k)}W^{(k))}\right).$

The formulation above means we feed every node into an identical single-layer
perceptron, and from this point of view, we can feed each type of node into a
specific perception. We assume that $n_{1}$ nodes represent sequences, $n_{2}$
nodes represent k-mers and $n=n_{1}+n_{2}$. $\tilde{A}$ can be partitioned
into two blocks, $(\tilde{A_{1}})_{n_{1}\times n}$ and
$(\tilde{A_{2}})_{n_{2}\times n}$ according to node type. $W_{1}^{k}$ and
$W_{2}^{k}$ are weight matrices with the same shape. The layer-wise
propagation rule for a heterogeneous graph is formulated as

$H^{(k+1)}=\phi\left(\tilde{A}_{1}(H^{(k)}W_{1}^{(k)})+\tilde{A}_{2}(H^{(k)}W_{2}^{(k)})\right).$

We call the second method DNA-HGCN. It’s evident that in DNA-HGCN, the number
of parameters double. As a result, DNA-HGCN is easier to overfit. We also
utilize Simple Graph Convolutional Network(SGC)[28] to build our model. SGC
removes nonlinearities and collapses weight matrices between consecutive
layers to reduce excess complexity while keeping the model’s performance.

###  3.3 Implementation of DNA-GCN

Our model is trained using Adam[14] optimizer with a learning rate of 0.001
for 10000 epochs. $20\%$ sequences are chosen from the labeled set to
construct the validation set. We chose the best model according to the
performance in the validation set. We set the embedding size of the first
convolution layer as 100.

We utilized the area under the ROC(AUC)[9, 10] to assess the prediction
performance of prediction. Our model is implemented using Tensorflow[1] for
Python.

##  4 Result

###  4.1 Datasets

To evaluate the performance of our model, we chose the 50 ChIP-seq ENCODE
datasets. Each of these datasets corresponds to a specific DNA-binding protein
(e.g., transcription factor); its positive samples are 101bp DNA sequences
which were experimentally confirmed to bind to this protein, and its negative
samples were created by shuffling these positive samples. All these datasets
were downloaded from http://cnn.csail.mit.edu/. We didn’t test our performance
in all available datasets for two reasons. On one hand, there is a tendency
that when training sets contain more samples, better performance can be
obtained by DeepBind. From this view, raising performance on these large
datasets with high performance (i.e. AUC is about $99\%$) is useless. On the
other hand, our model needs to be learned with the presence of both training
and test data. Moreover, the recursive neighborhood expansion across layers
poses time and memory challenges for training with large graphs(i.e. large
datasets). As a result, We selected 50 datasets with limited samples to test
the performance of our model.

Figure 2: Overall performance of DNA-GCN.

###  4.2 Baselines

#### Gkm-SVM

[11] introduces alternative feature sets using gapped k-mers and develops an
efficient tree data structure for computing the kernel matrix. Compared to
original kmer-SVM and alternative approaches, gkm-SVM predicts functional
genomic regulatory elements and tissue-specific enhancers with significantly
improved accuracy.

#### CNN-based model

[30] is similar to the architecture of DeepBind. We utilize their best model
with 128 convolutional kernels as our baseline. The overall performance of the
CNN-based model is better than DeepBind in ENCODE datasets. Model 1 and model
2 refer to CNN-based model with 1 and 128 convolutional kernels, respectively.

###  4.3 Our model outperforms on many datasets

Figure 3: Performance of DNA-GCN on specific datasets.

From Figure 2, the performance of DNA-HGCN among the datasets is sightly
better than the best baseline. DNA-GCN-based three different models have close
performance. DNA-GCN based on SGC sacrifices a little performance for its
least calculation while DNA-HGCN has better performance with doubled
parameters.

As for specific datasets[Figure 3], we found that the performance among the
CNN-based model and GCN-based model is inconsistent, which shows that GCN and
CNN predict DNA-protein binding from different views. The performance of the
same kind of models is consistent, regardless of the GCN-based model or CNN-
based model. We believe if two methods can be combined, the best performance
can be arrived at.

##  5 Conclusion

In this paper, a novel method named DNA-GCN is proposed to predict DNA-protein
binding. We build a heterogeneous sequence k-mer graph for the whole dataset
and turn predicting the DNA-protein binding problem into a node classification
problem. Our DNA-GCN can transit information from labeled sequences to key
k-mers and predict labels of unlabeled sequences. By experiments, we show that
a simple two-layer GCN brings up promising results by comparing numerous
models on many datasets with a limited number of sequences.

One the other hand, we believe that we haven’t made the best use of GCN to
predict DNA-protein binding. Much improvement may be achieved by adjusting the
architectures and hyper-parameters on a given dataset. Although DNA-GCN can’t
arrive at motif logos like CNN-based model, it can provide us with information
about which k-mers are important in the classification. Above all, our model
gives a brand new perspective to study the motif inference.

## Acknowledgements

This work was supported by The National Key Research and Development Program
of China (No.2016YFA0502303) and the National Natural Science Foundation of
China (No.31871342).

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# Phase transitions and entropy force of charged de Sitter black holes with
cloud of string and quintessence

Yubo Ma1, Yang Zhang1, Ren Zhao1, Shuo Cao2∗, Tonghua Liu2, Shubiao Geng 2,
Yuting Liu2, Yumei Huang3 1 Institute of Theoretical Physics, Shanxi Datong
University, Datong, 037009, China
2 Department of Astronomy, Beijing Normal University, Beijing 100875, China
3 School of Mathematics and Physics, Mianyang Teachers’ College, Mianyang
621000, China

###### Abstract

In this paper, we investigate the combined effects of the cloud of strings and
quintessence on the thermodynamics of a Reissner-Nordström-de Sitter black
hole. Based on the equivalent thermodynamic quantities considering the
correlation between the black hole horizon and the cosmological horizon, we
extensively discuss the phase transitions of the space-time. Our analysis
prove that similar to the case in AdS space-time, second-order phase
transitions could take place under certain conditions, with the absence of
first-order phase transition in the charged de Sitter black holes with cloud
of string and quintessence. The effects of different thermodynamic quantities
on the phase transitions are also quantitatively discussed, which provides a
new approach to study the thermodynamic qualities of unstable dS space-time.
Focusing on the entropy force generated by the interaction between the black
hole horizon and the cosmological horizon, as well as the Lennard-Jones force
between two particles, our results demonstrate the strong degeneracy between
the entropy force of the two horizons and the ratio of the horizon positions,
which follows the surprisingly similar law given the relation between the
Lennard-Jones force and the ratio of two particle positions. Therefore, the
study of the entropy force between two horizons, is not only beneficial to the
deep exploration of the three modes of cosmic evolution, but also helpful to
understand the correlation between the microstates of particles in black holes
and those in ordinary thermodynamic systems.

##  I Introduction

In the paradigm of de Sitter space-time, the black hole horizon and the
cosmological horizon are always considered to be different thermodynamic
systems with different thermodynamic quantities and radiation temperature.
Therefore, de Sitter space-time does not satisfy the requirements of the
stability of thermodynamic equilibrium, which makes it very difficult to
investigate the thermodynamics of de Sitter (dS) black holes. Moreover, the
discovery of cosmic acceleration is arguably one of the most important
developments in modern cosmology, which has also triggered wide spread
attention to deeply understand the thermodynamic properties of de Sitter
space-time Mbarek et al. (2019); Simovic et al. (2018, 2019); Dolan et al.
(2013); Hendi et al. (2017); Sekiwa et al. (2006); Kubiznak et al. (2016);
McInerney et al. (2016); Urano et al. (2009); Bhattacharya et al. (2013); Cai
et al. (2002, 2002); Koberlein et al. (1994); Azreg-Aiou et al. (2015); Zhang
et al. (2016, 2014); Zhao et al. (2014); Li et al. (2017); Zhang et al.
(2016); Liu et al. (2019); Ma et al. (2018). In the early 1980’s the
inflationary universe scenario (Guth 1981), with its predictions of a
spatially flat universe and almost scale-invariant density perturbations,
changed the cosmological landscape and helped set the stage for the discovery
of cosmic acceleration. In the early inflation period, our universe behaved as
a quasi-de Sitter space-time. On the other hand, with the inclusion of a
mysterious component with negative pressure as a new cosmological component, a
large number of dark energy models have been proposed to explain the cosmic
acceleration. The simplest candidate for dark energy is the cosmological
constant (or vacuum energy density), in which our universe will naturally
evolve into a new de Sitter phase with the equation of state (EoS) of dark
energy equal to $-1$. This model, the so-called LCDM, has provided excellent
agreement with a wide range of astronomical data so far. Finally, there has
also been flourishing interest in the duality relation of de Sitter space-
time, promoted by the recent success of AdS/CFT correspondence in theoretical
physics. Therefore, from observational and theoretical point of view, it is
rewarding to obtain a better understanding of the classical and quantum
properties of de Sitter space-time.

Now it is widely recognized that in de Sitter space-time, the thermodynamic
quantities of the black hole horizon and the cosmological horizon are of a
function of the mass $M$, the charge $Q$, and cosmological constant $\Lambda$.
Therefore, it is necessary to investigate the thermodynamic properties of
thermodynamic quantities, taking into account the correlation between the
black hole s horizon and the cosmological horizon. In the recent analysis of
Reissner-Nordström-de Sitte (RN-dS) space-time Zhang et al. (2016), the mass
$M$, the charge $Q$, and cosmological constant $\Lambda$, which satisfy the
first law of thermodynamics, were taken as the function of the state
parameters of thermodynamic systems. Based on the above assumptions, one could
obtain the equivalent thermodynamic quantities of RN-dS space-time and the
corresponding total entropy as

$S=\pi r_{c}^{2}[1+{x^{2}}+f(x)]$ (1)

where $x={r_{+}}/{r_{c}}$. ${r_{+}}$ and ${r_{c}}$ respectively denote the
radius of black hole horizon and cosmological horizon

$f(x)=\frac{8}{5}{\left({1-{x^{3}}}\right)^{2/3}}-\frac{{2\left({4-5{x^{3}}-{x^{5}}}\right)}}{{5\left({1-{x^{3}}}\right)}}$
(2)

On the other hand, the recent observations showed that the dark energy is
prevailing in our universe Komatsu et al. (2011); Riess et al. (2004). The
state of equation of dark energy approaches to cosmological constant or vacuum
energy Peebles et al. (2003), however, the dynamical dark energy is also
possible Copeland et al. (2006), and the quintessence could affect the black
hole space-time Stuchlk et al. (2005). Kiselev obtained the modified black
hole metric for the Schwarzschild black hole in in quintessence Kiselev et al.
(2003). Toshmatov and Xu obtained the solution of the rotational quintessence
black hole and Kerr-Newman black hole Toshmatov et al. (2017); Xu et al.
(2017). The thermodynamics and phase transition of the Schwarzschild black
hole surrounded by quintessence matter have been discussed by Tharanath and
Ghaderi Tharanath et al. (2014); Ghaderi et al. (2016); Penner et al. (2016).
Penner has studied the Rerssner-Nordström black hole situation Penner et al.
(2016); Thomas et al. (2012); Shao et al. (2018). Li investigate the effects
of dark energy on $P-V$ criticality of charged AdS black holes by considering
the case of the RN-AdS black holes surrounded by quintessence Li et al.
(2014). Employing the effective thermodynamic quantities, the thermodynamic
stability of black holes surrounded by quintessence is studied in Ma et al.
(2017). Toledo and Chatterjee Toledo et al. (2019); Chatterjee et al. (2019)
analyzed the thermodynamics of a Rerssner-Nordström AdS black hole surrounded
cloud of string and quintessence, by calculating different thermodynamic
variables, determining their critical values as well as the efficiency of a
heat engine. Recent progress has been made in Ref. Liu et al. (2019), which
derived the equivalent thermodynamic quantities of Rerssner-Nordström dS black
hole surrounded by quintessence, and furthermore discussed the thermodynamic
properties of this space-time possessing both black hole s horizon and
cosmological horizon.

In this paper, we will focus on the Rerssner-Nordström dS black hole
surrounded cloud of string and quintessence (RN-dSSQ), and obtain its
equivalent thermodynamic quantities. The analysis of the heat capacity,
constant pressure expansion coefficient, isothermal compression system and
Gibbs function show that, second-order phase transitions could take place
under certain conditions (similar to the case in AdS space-time), while first-
order phase transitions do not exist in RN-dSSQ. More importantly, different
from the previous result that the isothermal capacity in spherically symmetric
AdS black holes is equal to zero, our findings demonstrates the nonzero heat
capacity in the case of RN-dSSQ space-time. Such conclusion is well consistent
with the results in the case of general thermodynamic systems. Meanwhile,
concentrated on the entropy force generated by the interaction between the
black hole horizon and the cosmological horizon, we find it is strongly
dependent on the ratio of the two horizons positions, following the exact law
given the relation between the Lennard-Jones force and the ratio of two
particle positions Johnston et al. (2014). This paper is arranged as follows:
in Section 2, we briefly introduce the thermodynamic properties of RN-dSSQ
space-time, and the condition satisfied by the charge Q and the cosmological
constant when the temperature of the two horizons are equal to each other. In
Section 3, the equivalent thermodynamic quantities of the black hole horizon
and the cosmological horizon in RN-dSSQ space-time are derived, based on which
the phase transitions of such pace-time is studied in detail. In Section 4, we
discuss the entropy force generated by the interaction between the horizons,
as well as its comparison with the Lennard-Jones force between two particles.
Finally, we present the main conclusions and discussion in Section 5. The
units of $G=\hbar={k_{B}}=c=1$ are used throughout this work.

##  II The space-time structure

Let s start from the well-known Einstein-Maxwell field equation

${G_{\mu\nu}}+\Lambda{g_{\mu\nu}}=8\pi G{T_{\mu\nu}}$ (3)

with the solution proposed by Kiselev et al. (2003) in the presence of a
cosmological constant $\Lambda$. The additivity and linearity conditions on
the quintessence dark energy stress tensor imply

$T_{t}^{t}=T_{r}^{r}={\rho_{q}}+\frac{a}{{{r^{2}}}}$ (4)

$T_{\theta}^{\theta}=T_{\varphi}^{\varphi}=-\frac{1}{2}{\rho_{q}}(3\omega+1)$
(5)

where

${\rho_{q}}=-\frac{{3\alpha\omega}}{{2{r^{3(\omega+1)}}}}$ (6)

and ${\rho_{q}}$ is the density of quintessence field, $\omega$ is the state
parameter and $\alpha$ is the normalization parameter related to the density
of quintessence field, $a$ is an integration constant which is related to the
presence of the cloud of string. The static spherically symmetric solution of
Einstein equation for a Reissner-Nordström-dS black hole in quintessence
matters have been obtained by Kiselev as Kiselev et al. (2003); Toledo et al.
(2019); Chatterjee et al. (2019)

$d{s^{2}}=-f(r)d{t^{2}}+{f^{-1}}d{r^{2}}+{r^{2}}d\Omega_{2}^{2}$ (7)

With the horizon function

$\displaystyle
f(r)=1-a-\frac{{2M}}{r}+\frac{{{Q^{2}}}}{{{r^{2}}}}-\frac{{{r^{2}}}}{{{l^{2}}}}-\alpha{r^{-3\omega-1}}$
(8)

$M$ and $Q$ are the black hole mass and charge. $l$ is the curvature radius of
dS pace. The parameter $a$ and $\alpha$ codify the presence of the cloud of
string and the quintessencer, respectively.

If we set $a=\Lambda=Q=0$ and put $\omega=-{\textstyle{2\over 3}}$ the space-
time reduces to the Schwarzchild solution surrounded by quintessence whose
geodesics were studied in Uniyal et al. (2015). Null trajectories of charged
black holes surrounded by quintessence were carried out in Fernando et al.
(2015), thermodynamic properties and the Joule-Thomson effect was investigated
in Pradhan et al. (2017). The Shadow of rotating charged black holes with
quintessence was considered in Fernando et al. (2015).

Now, let us analyze the existence of horizons which are determined by imposing
that

$\displaystyle
f(r)=1-a-\frac{{2M}}{r}+\frac{{{Q^{2}}}}{{{r^{2}}}}-\frac{{{r^{2}}}}{{{l^{2}}}}-\alpha{r^{-3\omega-1}}=0$
(9)

The black hole has three positive horizons: the black hole event horizon
$r_{+}$, the internal (Cauchy) horizon $r_{-}$, and a quintessential
cosmological horizon $r_{c}$ Toledo et al. (2019). From Eq.(9), one may
straightforwardly obtain

$\displaystyle M$
$\displaystyle=(1-a)\frac{{{r_{+}}+{r_{c}}}}{4}+\frac{{{Q^{2}}}}{4}\left({\frac{1}{{{r_{+}}}}+\frac{1}{{{r_{c}}}}}\right)-\frac{1}{{4{l^{2}}}}\left({r_{+}^{3}+r_{c}^{3}}\right)-\frac{\alpha}{4}\left({r_{+}^{-3\omega}+r_{c}^{-3\omega}}\right)$
(10)
$\displaystyle=(1-a)\frac{{{r_{c}}(1+x)}}{4}+\frac{{{Q^{2}}(1+x)}}{{4{r_{c}}x}}-\frac{{r_{c}^{3}}}{{4{l^{2}}}}\left({1+{x^{3}}}\right)-\frac{\alpha}{4}r_{c}^{-3\omega}\left({1+{x^{-3\omega}}}\right)$
$\displaystyle=(1-a)\frac{{{r_{c}}x(1+x)}}{{2(1+x+{x^{2}})}}+\frac{{{Q^{2}}(1+x+{x^{2}}+{x^{3}})}}{{2{r_{c}}x(1+x+{x^{2}})}}+\frac{{\alpha
r_{c}^{-3\omega}{x^{3}}}}{{2(1-{x^{3}})}}(1-{x^{-3-3\omega}})$

$\displaystyle\frac{1}{{{l^{2}}}}$
$\displaystyle=\frac{{(1-a)({r_{c}}-{r_{+}})}}{{\left({r_{c}^{3}-r_{+}^{3}}\right)}}-\frac{{{Q^{2}}({r_{c}}-{r_{+}})}}{{\left({r_{c}^{3}-r_{+}^{3}}\right){r_{c}}{r_{+}}}}-\frac{{\alpha\left({r_{c}^{-3\omega}-r_{+}^{-3\omega}}\right)}}{{\left({r_{c}^{3}-r_{+}^{3}}\right)}}$
(11)
$\displaystyle=\frac{{(1-a)}}{{r_{c}^{2}(1+x+{x^{2}})}}-\frac{{{Q^{2}}}}{{r_{c}^{4}x(1+x+{x^{2}})}}-\frac{{\alpha(1-{x^{-3\omega}})}}{{r_{c}^{3+3\omega}(1-{x^{3}})}}$

Some thermodynamic quantities associated with the cosmological horizon are

$\displaystyle S_{c}=\pi r_{c^{2}},\quad\Phi_{c}=\frac{Q}{r_{c}},\quad
T_{c}=-\frac{f^{{}^{\prime}}r_{c}}{4\pi}$ (12)

where $T_{c}$, $S_{c}$ and $\Phi_{c}$ denote the Hawking temperature, the
entropy and the electric potential, respectively.

For the black hole horizon, associated thermodynamic quantities are

$\displaystyle{S_{+}}=\pi
r_{+}^{2},\quad{\Phi_{+}}=\frac{Q}{{{r_{+}}}},\quad{T_{+}}=\frac{{f^{\prime}({r_{+}})}}{{4\pi}}=\frac{1}{{4\pi{r_{+}}}}\left({1-a-\frac{{{Q^{2}}}}{{r_{+}^{2}}}-\frac{{3r_{+}^{2}}}{{{l^{2}}}}+3\omega\alpha
r_{+}^{-3\omega-1}}\right)$ (13)

When the temperature of the black hole horizon is equal to that of the
cosmological horizon, from Eqs. (12)-(13) the cosmological constant ${l^{2}}$
satisfies the following expression

$\displaystyle\frac{3}{{{l^{2}}}}=\frac{{1-a}}{{r_{c}^{2}x}}-\frac{{{Q^{2}}}}{{r_{c}^{4}{x^{3}}}}\frac{{(1+{x^{3}})}}{{(1+x)}}+3\omega\alpha\frac{{(1+{x^{-3\omega-2}})}}{{r_{c}^{3\omega+3}(1+x)}}$
(14)

while from Eqs.(11) and (14), the electric potential ${\Phi_{c}}$ reads Mellor
et al. (1989)

$\displaystyle\frac{{{Q^{2}}}}{{r_{c}^{2}}}=\frac{{{x^{2}}(1-a)}}{{{{(1+x)}^{2}}}}+\frac{{3\alpha{x^{3}}(1-{x^{-3\omega}})}}{{r_{c}^{1+3\omega}{{(1-x)}^{3}}{{(1+x)}^{2}}}}+3\omega\alpha\frac{{{x^{3}}(1+{x^{-3\omega-2}})(1+x+{x^{2}})}}{{r_{c}^{3\omega+1}(1+x){{(1-{x^{2}})}^{2}}}}$
(15)

Now the combination of Eqs.(12)-(15) will generate the radiation temperature
of the two horizons

$\displaystyle T=$ $\displaystyle
T_{+}=T_{c}=\frac{1}{{4\pi{r_{c}}}}\left\\{{2(1-a)\frac{{1-x}}{{{{(1+x)}^{2}}}}}\right.-3\alpha
r_{c}^{-3\omega-1}\frac{{(1-{x^{-3\omega}})(1+{x^{2}})}}{{{{(1-x)}^{2}}{{(1+x)}^{2}}}}$
(16) $\displaystyle-3\omega\alpha
r_{c}^{-3\omega-1}\frac{{x(1-{x^{-3\omega-3}})}}{{(1+x)}}-\left.{3\omega\alpha\frac{{(1+{x^{-3\omega-2}})(1-{x^{3}})(1+{x^{2}})}}{{r_{c}^{3\omega+1}(1+x){{(1-{x^{2}})}^{2}}}}}\right\\}$

##  III Equivalent thermodynamic quantities

Considering the fact that interaction between the black hole horizon and the
cosmological horizon, the equivalent thermodynamic quantities in the two
horizons respectively satisfy the corresponding first law of thermodynamics.
More specifically, following the recent study of the thermodynamic properties
of dS space-time, the thermodynamic volume can be defined as Zhang et al.
(2014); Zhao et al. (2014); Li et al. (2017); Zhang et al. (2016)

$\displaystyle
V=\frac{{4\pi}}{3}\left({r_{c}^{3}-r_{+}^{3}}\right)=\frac{{4\pi}}{3}r_{c}^{3}\left({1-{x^{3}}}\right)$
(17)

Now, taking the mass $M$, the charge $Q$, and the cosmological constant as the
state parameters in the space-time, the first law of thermodynamics can be
written as

$dM={T_{eff}}dS-{P_{eff}}dV+{\phi_{eff}}dQ$ (18)

Note that the total entropy $S$ is an explicit function of the horizon
position, with the definition of

$S=\pi r_{c}^{2}[1+{x^{2}}+f(x)]$ (19)

Here the undefined function $f(x)$ represents the extra contribution from the
correlations of the two horizons. Inserting Eqs. (10), (17) and (19) into Eq.
(18), the effective temperature of space-time is obtained as

$\displaystyle{T_{eff}}={\left({\frac{{\partial M}}{{\partial
S}}}\right)_{Q,V}}=$
$\displaystyle\frac{1}{{2\pi{r_{c}}(1+x+{x^{2}})A(x)}}\left\\{{(1-a)\left[{(1+x)(1+{x^{3}})-2{x^{2}}}\right]}\right.$
(20)
$\displaystyle-\frac{{{Q^{2}}}}{{r_{c}^{2}{x^{2}}}}\left[{(1+x+{x^{2}})(1+{x^{4}})-2{x^{3}}}\right]+\frac{{3\alpha
r_{c}^{-3\omega-1}{x^{2}}}}{{(1-x)}}\left.{\left[{1-{x^{-3\omega}}-\omega({x^{3}}-{x^{-3-3\omega}})}\right]}\right\\}$

where

$\displaystyle
A(x)=\left[{(1-{x^{3}})f^{\prime}(x)+2{x^{2}}f(x)+2x(1+x)}\right]$ (21)

In the lukewarm case, when the electric charge $Q$ and cosmological constant
${l^{2}}$ are characterized by Eqs.(14)-(15), the effective temperature of the
space-time is given by

${T_{eff}}=T$ (22)

Combining Eqs. (15), (20) and (22), one may obtain the following expression

$\displaystyle\frac{{1-x}}{{{{(1+x)}^{2}}}}\left[{1-a-3\alpha
r_{c}^{-3\omega-1}\frac{{(1-{x^{-3\omega}})(1+{x^{2}})}}{{2{{(1-x)}^{3}}}}-3\omega\alpha
r_{c}^{-3\omega-1}\frac{{(1+2x+3{x^{2}})}}{{2(1+x){{(1-x)}^{2}}}}-3\omega\alpha\frac{{r_{c}^{-3\omega-1}{x^{-3\omega-2}}{x^{2}}(3+2x+{x^{2}})}}{{2(1+x){{(1-x)}^{2}}}}}\right]$
(23)
$\displaystyle=\frac{{2x(1+{x^{4}})}}{{(1+x+{x^{2}}){{(1+x)}^{2}}A(x)}}{{1-a-\frac{{3\alpha(1-{x^{-3\omega}})(1+{x^{2}})}}{{r_{c}^{1+3\omega}2{{(1-x)}^{3}}}}}-\frac{{3\alpha\omega(1+2x+3{x^{2}})}}{{r_{c}^{1+3\omega}2(1+x){{(1-x)}^{2}}}}-{\frac{{3\alpha\omega{x^{2}}{x^{-3\omega-2}}(3+2x+{x^{2}})}}{{r_{c}^{1+3\omega}2(1+x){{(1-x)}^{2}}}}}}$

Then we can calculate the expression of $A(x)$ from Eq. (23) and find it to be

$\displaystyle A(x)=\frac{{2x(1+{x^{4}})}}{{1-{x^{3}}}}$ (24)

Now the combination of Eqs. (21) and (24) will generate

$f^{\prime}(x)+\frac{{2{x^{2}}}}{{1-{x^{3}}}}f(x)=\frac{{2{x^{2}}(2{x^{3}}+{x^{2}}-1)}}{{{{(1-{x^{3}})}^{2}}}}$
(25)

with the corresponding solution as (when taking the boundary condition
$f(0)=0$)

$f(x)=\frac{8}{5}{\left({1-{x^{3}}}\right)^{2/3}}-\frac{{2\left({4-5{x^{3}}-{x^{5}}}\right)}}{{5\left({1-{x^{3}}}\right)}}$
(26)

One should note that such solution is well consistent with that in the RN-dS
space-time Zhang et al. (2016).

Now we will focus on the derivation of the equivalent thermodynamic quantities
in RN-dSSQ space-time. First of all, we get the effective temperature of RN-
dSSQ space-time from Eqs. (20) and (26)

$\displaystyle{T_{eff}}=$
$\displaystyle\frac{{1-x}}{{4\pi{r_{c}}x(1+{x^{4}})}}\left\\{{(1-a)\left[{(1+x)(1+{x^{3}})-2{x^{2}}}\right]}-\frac{{{Q^{2}}}}{{r_{c}^{2}{x^{2}}}}\left[{(1+x+{x^{2}})(1+{x^{4}})-2{x^{3}}}\right]\right.$
(27) $\displaystyle+\frac{{3\alpha
r_{c}^{-3\omega-1}{x^{2}}}}{{(1-x)}}\left.{\left[{1-{x^{-3\omega}}-\omega({x^{3}}-{x^{-3-3\omega}})}\right]}\right\\}$
$\displaystyle=$
$\displaystyle\frac{1}{{4\pi{r_{c}}{x^{3}}(1+{x^{4}})}}\left\\{{(1-a){x^{2}}(1-3{x^{2}}+3{x^{3}}}\right.-{x^{5}})-{\phi^{2}}(1-3{x^{3}}+3{x^{4}}-{x^{7}})$
$\displaystyle-\frac{{{\beta_{c}}{x^{4}}}}{{2\pi\omega}}\left.{\left[{1-{x^{-3\omega}}-\omega({x^{3}}-{x^{-3-3\omega}})}\right]}\right\\}$
$\displaystyle=$
$\displaystyle\frac{{B(x,\omega)}}{{4\pi{r_{c}}{x^{3}}(1+{x^{4}})}}$

where

$\displaystyle\frac{{3\alpha}}{{r_{c}^{1+3\omega}}}=\frac{{3\alpha 4\pi
r_{c}^{2}}}{{4\pi
r_{c}^{3+3\omega}}}=-\frac{{{A_{c}}{\rho_{c}}}}{{2\pi\omega}}=-\frac{{{\beta_{c}}}}{{2\pi\omega}},\quad\phi=\frac{Q}{{{r_{c}}}},$
(28) $\displaystyle B(x,\omega)={x^{2}}(1-a)(1-3{x^{2}}$
$\displaystyle+3{x^{3}}-{x^{5}})-{\phi^{2}}(1-3{x^{3}}+3{x^{4}}-{x^{7}})-\frac{{{\beta_{c}}{x^{4}}}}{{2\pi\omega}}\left[{1-{x^{-3\omega}}-\omega({x^{3}}-{x^{-3-3\omega}})}\right]$

In order to clearly see the effect of relevant parameters on the effective
temperature, we illustrate an example of the $T_{eff}-x$ diagram with
different value of $a$, $\omega$, $\beta_{c}$ and $\phi$, which are explicitly
shown in Fig. 1 (by fixing $r_{c}=1$). It is obvious that, compared with the
other two parameters ($\beta_{c}$ and $\omega$), different values of a and
$\phi$ (the electric potential at the cosmological horizon) play a more
important role in the determination of the maximum effective temperature and
its corresponding position $x$. More specifically, when $\phi$ is fixed, the
maximum value of the effective temperature of the system will decrease with
$a$. Such tendency can also be seen from the behavior of the effective
temperature as a function of $\phi$, in term of different $a$.

(a) $\phi=0.01,\beta_{c}=0.1,\omega=-1/2$ (b) $a=0.1,\phi=0.01,\beta_{c}=0.1$

(c) $a=0.1,\phi=0.01,\omega=-1/2$ (d) $a=0.1,\beta_{c}=0.1,\omega=-1/2$

Figure 1: The behavior of the effective temperature as a function of $x$, (a)
For different $a$, (b) for different $\Omega$, (c) For different $\beta_{c}$ ,
and (d) for different $\phi$, when the parameter $r_{c}$ is fixed at 1.

Secondly, the effective pressure of RN-dSSQ space-time can be derived from Eq.
(18), which is given by

$\displaystyle{P_{eff}}=-{\left({\frac{{\partial M}}{{\partial
V}}}\right)_{Q,S}}=\frac{{(1-x)D(x,\omega)}}{{8\pi r_{c}^{2}x(1+{x^{4}})}}$
(29)

where

$\displaystyle D(x,\omega)=$
$\displaystyle-\left[{(1-a)x(1+x)-\frac{{{\phi^{2}}(1+x+{x^{2}}+{x^{3}})}}{x}+\frac{{{\beta_{c}}{x^{3}}(1-{x^{3-3\omega}})}}{{2\pi(1-x)}}}\right](x+f^{\prime}(x)/2)$
(30)
$\displaystyle+\left[{\frac{{(1-a)(1+2x)}}{{(1+x+{x^{2}})}}-\frac{{{\phi^{2}}(1+2x+3{x^{2}})}}{{{x^{2}}(1+x+{x^{2}})}}-\frac{{{\beta_{c}}{x^{2}}(1-(2-\omega){x^{5-3\omega}}+(1-\omega){x^{8-3\omega}})}}{{2\pi\omega(1-{x^{3}})(1-x)}}}\right](1+{x^{2}}+f(x))$

Similarly, we also analyze the behavior of the effective pressure $P_{eff}$.
The results are shown in Fig. 2, which quantify the effects of these
parameters on the effective pressure of RN-dSSQ space-time. One can clearly
see that the maximum value of the effective pressure will significantly will
decrease with the value of a and $\phi$, which is quite similar to the
behavior of effective temperature illustrated in Fig. 1.

(a) $\phi=0.01,\beta_{c}=0.1,\omega=-1/2$ (b) $a=0.1,\phi=0.01,\beta_{c}=0.1$

(c) $a=0.1,\phi=0.01,\omega=-1/2$ (d) $a=0.1,\beta_{c}=0.1,\omega=-1/2$

Figure 2:  The same as Fig. 1, but for the behavior of the effective pressure
$P_{eff}$ as a function of $x$.

Thirdly, we turn to the behavior of the heat capacity of RN-dSSQ space-time.
On the one hand, the heat capacity of isovolumetric systems can be written as

${C_{V}}={T_{eff}}{\left({\frac{{\partial
S}}{{\partial{T_{eff}}}}}\right)_{V}}=\frac{{2\pi
r_{c}^{2}B(x,\omega)}}{{1-{x^{3}}}}\frac{{{x^{2}}{{(1+{x^{4}})}^{2}}}}{{B^{\prime}(x,\omega)x(1-{x^{3}})(1+{x^{4}})-B(x,\omega)(3-2{x^{3}}+7{x^{4}}-6{x^{7}})}}$
(31)

with the differential of $B$ given by $B^{\prime}(x,\omega)=\frac{{\partial
B(x,\omega)}}{{\partial x}}$. When taking $\pi r_{c}^{2}=1$, one can get the
${C_{V}}-x$ graphs with different combinations of relevant parameters ($a$,
$\phi$, $\beta_{c}$, $\omega$) from the above equations. Comparing the two
cases illustrated in Fig. 3, one could easily find that the effective
isovolumetric heat capacity of RN-dSSQ space-time is not equal to zero, which
is quite different from with the case of ${C_{V}}=0$ in AdS space-time.
However, such finding in RN-dSSQ space-time agrees very well with the previous
result obtained in the framework of ordinary thermodynamic systems, which
strongly implies the strong consistency between the two types of thermodynamic
systems. On the other hand, another important thermodynamical quantity is the
heat capacity at constant pressure, which could be defined as

$\displaystyle{C_{{P_{eff}}}}={T_{eff}}{\left({\frac{{\partial
S}}{{\partial{T_{eff}}}}}\right)_{{P_{eff}}}}=2\pi
r_{c}^{2}\frac{{B(x,\omega)E(x,\omega)}}{{F(x,\omega)}}$ (32)

where

$\displaystyle
E(x,\omega)=\left[{(1-x)D^{\prime}(x,\omega)-2\frac{{(1+5{x^{4}}-4{x^{5}})D(x,\omega)}}{{x(1+{x^{4}})}}}\right][1+{x^{2}}+f(x)]+(1-x)D(x,\omega)(2x+f^{\prime}(x))$
(33) $\displaystyle
F(x,\omega)=2(1-x)D(x,\omega)\left[{B^{\prime}(x,\omega)-\frac{{B(x,\omega)(3+7{x^{4}})}}{{x(1+{x^{4}})}}}\right]-B(x,\omega)\left[{(1-x)D^{\prime}(x,\omega)-\frac{{(1+5{x^{4}}-4{x^{5}})D(x,\omega)}}{{x(1+{x^{4}})}}}\right]$

The $C_{V}-x$ graphs with different combinations of relevant parameters ($a$,
$\phi$, $\beta_{c}$, $\omega$) are presented in Fig. 4.

(a) $\phi=0.01,\beta_{c}=0.1,\omega=-1/2$ (b) $a=0.1,\phi=0.01,\beta_{c}=0.1$

(c) $a=0.1,\phi=0.01,\omega=-1/2$ (d) $a=0.1,\beta_{c}=0.1,\omega=-1/2$

Figure 3: The behavior of the heat capacity $C_{v}$ as a function of $x$ (with
$\pi r_{c}^{2}=1)$

(a) $\phi=0.01,\beta_{c}=0.1,\omega=-1/2$ (b) $a=0.1,\phi=0.01,\beta_{c}=0.1$

(c) $a=0.1,\phi=0.01,\omega=-1/2$ (d) $a=0.1,\beta_{c}=0.1,\omega=-1/2$

Figure 4: The same as Fig. 3, but for the behavior of the heat capacity
$C_{P_{eff}}$ as a function of $x$.

Now we are at a crossroads in the thermodynamical study of critical phenomena
at the critical point, as well as the phase transitions with different space-
time parameters. More specifically, in the framework of RN-dSSQ space-time
considered in this analysis, the analog of volume expansion coefficient given
by

$\displaystyle\alpha$ $\displaystyle=\frac{1}{V}{\left({\frac{{\partial
V}}{{\partial{T_{eff}}}}}\right)_{{P_{eff}}}}=\frac{1}{V}\frac{{{{\left({\frac{{\partial
V}}{{\partial{r_{c}}}}}\right)}_{x}}{{\left({\frac{{\partial{P_{eff}}}}{{\partial
x}}}\right)}_{{r_{c}}}}-{{\left({\frac{{\partial V}}{{\partial
x}}}\right)}_{{r_{c}}}}{{\left({\frac{{\partial{P_{eff}}}}{{\partial{r_{c}}}}}\right)}_{x}}}}{{{{\left({\frac{{\partial{P_{eff}}}}{{\partial
x}}}\right)}_{{r_{c}}}}{{\left({\frac{{\partial{T_{eff}}}}{{\partial{r_{c}}}}}\right)}_{x}}-{{\left({\frac{{\partial{P_{eff}}}}{{\partial{r_{c}}}}}\right)}_{x}}{{\left({\frac{{\partial{T_{eff}}}}{{\partial
x}}}\right)}_{{r_{c}}}}}}$ (34)
$\displaystyle=\frac{{12\pi{r_{c}}{x^{3}}(1+{x^{4}})}}{{F(x,\omega)}}\left({(1-x)D^{\prime}(x,\omega)-\frac{{(1+5{x^{4}}-4{x^{5}})D(x,\omega)}}{{x(1+{x^{4}})}}-2\frac{{(1-x)D(x,\omega)}}{{1-{x^{3}}}}{x^{2}}}\right)$

while the analog of isothermal compressibility can be derived as

$\displaystyle{\kappa_{{T_{eff}}}}$
$\displaystyle=-\frac{1}{V}{\left({\frac{{\partial
V}}{{\partial{P_{eff}}}}}\right)_{{T_{eff}}}}=\frac{1}{V}\frac{{{{\left({\frac{{\partial
V}}{{\partial{r_{c}}}}}\right)}_{x}}{{\left({\frac{{\partial{T_{eff}}}}{{\partial
x}}}\right)}_{{r_{c}}}}-{{\left({\frac{{\partial V}}{{\partial
x}}}\right)}_{{r_{c}}}}{{\left({\frac{{\partial{T_{eff}}}}{{\partial{r_{c}}}}}\right)}_{x}}}}{{{{\left({\frac{{\partial{P_{eff}}}}{{\partial
x}}}\right)}_{{r_{c}}}}{{\left({\frac{{\partial{T_{eff}}}}{{\partial{r_{c}}}}}\right)}_{x}}-{{\left({\frac{{\partial{P_{eff}}}}{{\partial{r_{c}}}}}\right)}_{x}}{{\left({\frac{{\partial{T_{eff}}}}{{\partial
x}}}\right)}_{{r_{c}}}}}}$ (35) $\displaystyle=\frac{{48\pi
r_{c}^{2}x(1+{x^{4}})}}{{F(x,\omega)}}\left({B^{\prime}(x,\omega)-\frac{{B(x,\omega)(3+7{x^{4}})}}{{x(1+{x^{4}})}}-\frac{{B(x,\omega){x^{2}}}}{{(1-{x^{3}})}}}\right)$

where the differential of $D$ is given by
$D^{\prime}(x,\omega)=\frac{{\partial D(x,\omega)}}{{\partial x}}$. It is
important to further investigate the thermodynamic properties of a black hole
at the critical point, besides the phase transition and critical behaviors.
The critical point $x={x_{c}}$ is determined at the position where the curves
of $\alpha-x$ and ${\kappa_{{T_{eff}}}}-x$ become infinitely divergent, which
are explicitly illustrated in Fig. 5-6.

(a) $\phi=0.01,\beta_{c}=0.1,\omega=-1/2$ (b) $a=0.1,\phi=0.01,\beta_{c}=0.1$

(c) $a=0.1,\phi=0.01,\omega=-1/2$ (d) $a=0.1,\beta_{c}=0.1,\omega=-1/2$

Figure 5: The analog of volume expansion coefficient varying with $x$ ($\pi
r_{c}^{2}=1$ ), based on different combinations of relevant parameters ($a$,
$\phi$, $\beta_{c}$, $\omega$).

(a) $\phi=0.01,\beta_{c}=0.1,\omega=-1/2$ (b) $a=0.1,\phi=0.01,\beta_{c}=0.1$

(c) $a=0.1,\phi=0.01,\omega=-1/2$ (d) $a=0.1,\beta_{c}=0.1,\omega=-1/2$

Figure 6: The same as Fig. 5, but for the behavior of the analog of isothermal
compressibility.

In term of the thermodynamic quantities discussed above, the extended Gibbs
free energy for our system is given by

$\displaystyle G$ $\displaystyle=M-{T_{eff}}S$ (36)
$\displaystyle={r_{c}}\left[{\frac{{(1-a)x(1+x)}}{{2(1+x+{x^{2}})}}+\frac{{{\phi^{2}}(1+x+{x^{2}}+{x^{3}})}}{{2x(1+x+{x^{2}})}}-\frac{{{\beta_{c}}{x^{3}}}}{{6\pi\omega(1-{x^{3}})}}(1-{x^{-3-3\omega}})-\frac{{B(x,\omega)(1+{x^{2}}+f(x))}}{{4{x^{3}}(1+{x^{4}})}}}\right]$

Now, it is worthwhile commenting on the correlation between the corresponding
Gibbs function and the effective temperature. Following the same procedure
performed on the effective temperature, we obtain the $G-{T_{eff}}$ graphs
presented in Fig. 7. Considering the fact that the entropy $S$ and the volume
$V$ are both a continuous function of temperature $T_{eff}$, the critical
point $x={x_{c}}$ should also be a turning point in the $G-{T_{eff}}$ curve.
In the following analysis of the thermodynamical study of critical phenomena,
we will follow the detailed classification of general types of transition
between phases of matter, firstly introduced by Paul Ehrenfest in 1933 and
extensively applied in the previous literature Ma et al. (2018). More
interestingly, in the framework of Ehrenfest s classification method, such
point satisfies the requirements of a second-order phase transition, while the
existence of first-order phase transitions can be ruled out in RN-dSSQ space-
time. A quantitative analysis shown in Figs. 4-6 indicates that the
criticality does not vary with the two parameters of $\beta_{c}$ and $\omega$,
while the position of the critical point may dramatically change with
different values of $a$ and $\phi$. Such behavior also resembles the behavior
of effective temperature and effective pressure in RN-dSSQ space-time (Fig.
1-2).

(a) $\phi=0.01,\beta_{c}=0.1,\omega=-1/2$ (b) $a=0.1,\phi=0.01,\beta_{c}=0.1$

(c) $a=0.1,\phi=0.01,\omega=-1/2$ (d) $a=0.1,\beta_{c}=0.1,\omega=-1/2$

Figure 7: The Gibbs function $G$ varying with effective temperature $T_{eff}$,
concerning different combinations of relevant parameters ($a$, $\phi$,
$\beta_{c}$, $\omega$).

##  IV Entropy force generated by the horizon interaction

Motivated by the recent work by Ref.Verlinde et al. (2011), it was proposed
that gravity can be explained as an entropic force caused by changes in the
information associated with the positions of material bodies. In this
scenario, the Newton s law of gravitation, the Einstein equations, and the law
of inertia could be successfully derived from the entropic point of view.
Concerning two independent ordinary thermodynamic systems, the entropy of the
two systems are respectively $S_{A}$ and $S_{B}$. Therefore, in initial works
it was always assumed that the entropy of the space-time is the sum of the
entropy of the black hole horizon and the cosmological horizon. However, in
this analysis, considering the interaction between the two space-time
horizons, the entropy of RN-dSSQ space-time can be written as

$S={S_{A}}+{S_{B}}+{S_{AB}}$ (37)

where $S_{AB}$ denotes the external entropy generated by such interaction. In
combination with the entropy definition in Eq. (19), the total entropy of RN-
dSSQ space-time can be divided into two parts, one part from the entropy of
the black hole horizon and the cosmological horizon, and the other part from
the contribution of the overall thermodynamic system, consist of the two
thermodynamic systems corresponding to the two horizons. Moreover, when taking
the mass $M$, the electric charge $Q$, and the space-time volume $V$ as the
state parameters, the additional entropy generated by interactions between the
two thermodynamic systems can be expressed as

${S_{f}}=\pi r_{c}^{2}f(x)=\pi
r_{c}^{2}\left[{\frac{8}{5}{{(1-{x^{3}})}^{2/3}}-\frac{{2(4-5{x^{3}}-{x^{5}})}}{{5(1-{x^{3}})}}}\right]$
(38)

From the entropic point of view, the effective entropic force due to the
change in entropy obeys Verlinde et al. (2011); Plastino et al. (2018, 2018);
Panos et al. (2019); Kharzeev et al. (2014); Cai et al. (2010); Zhang et al.
(2017, 2018); Tahery et al. (2018)

$F=-T\frac{{\partial S}}{{\partial r}}$ (39)

where $T$ is the temperature of the system, and $r$ is the radius location of
the boundary surface. In our analysis, the entropic force due to interactions
between the black hole horizon and the cosmological horizon of RN-dSSQ space-
time is obtained as

$F={T_{eff}}{\left({\frac{{\partial{S_{f}}}}{{\partial
r}}}\right)_{{T_{eff}}}}$ (40)

with the effective temperature ${T_{eff}}$ and the sapec-time radius defined
as $r={r_{c}}-{r_{+}}={r_{c}}(1-x)$. Combined with Eq. (38), the above
expression may be extended to

$\displaystyle F(x)$
$\displaystyle={T_{eff}}\frac{{{{\left({\frac{{\partial{S_{f}}}}{{\partial{r_{c}}}}}\right)}_{x}}{{\left({\frac{{\partial{T_{eff}}}}{{\partial
x}}}\right)}_{{r_{c}}}}-{{\left({\frac{{\partial{S_{f}}}}{{\partial
x}}}\right)}_{{r_{c}}}}{{\left({\frac{{\partial{T_{eff}}}}{{\partial{r_{c}}}}}\right)}_{x}}}}{{(1-x){{\left({\frac{{\partial{T_{eff}}}}{{\partial
x}}}\right)}_{{r_{c}}}}+{r_{c}}{{\left({\frac{{\partial{T_{eff}}}}{{\partial{r_{c}}}}}\right)}_{x}}}}$
(41)
$\displaystyle=\frac{{B(x)}}{{{x^{3}}(1+{x^{4}})}}\times\frac{{2A(x)f(x)+x(1+{x^{4}})B(x)f^{\prime}(x)}}{{(1-x)A(x)-(1+{x^{4}})B(x)}}$

where

$\displaystyle A(x)=$
$\displaystyle{x^{2}}(1-a)(-1-3{x^{2}}+6{x^{3}}-5{x^{4}}-4{x^{5}}+9{x^{6}}-6{x^{7}})+{\phi^{2}}(3+4{x^{4}}-8{x^{7}}+9{x^{8}})$
(42)
$\displaystyle-\frac{{{\beta_{c}}{x^{4}}}}{{2\pi\omega}}\left[{1-3{x^{4}}-4\omega{x^{3}}-(1-3\omega){x^{-3\omega}}-\omega(2+3\omega){x^{-3-3\omega}}+3(1+\omega){x^{4-3\omega}}-3\omega(2+\omega){x^{1-3\omega}}}\right]$
$\displaystyle B(x)=$
$\displaystyle{x^{2}}(1-a)(1-3{x^{2}}+3{x^{3}}-{x^{5}})-{\phi^{2}}(1-3{x^{3}}+3{x^{4}}-{x^{7}})-\frac{{{\beta_{c}}{x^{4}}}}{{2\pi\omega}}\left[{1-{x^{-3\omega}}-\omega({x^{3}}-{x^{-3-3\omega}})}\right]$

In order to have a better illustration, the $F(x)-x$ diagram of the RN-dSSQ
space-time is displayed in Fig. 8 for the specific cases with different
parameter combinations, which highlights the importance of the $a$ parameter.
Therefore, compared with other effects quantified by three relevant parameters
($\phi$, $\beta_{c}$, $\omega$), the presence of the cloud of string plays
significant role in determining the entropy force of RN-dSSQ space-time.

(a) $\phi=0.01,\beta_{c}=0.1,\omega=-1/2$ (b) $a=0.1,\phi=0.01,\beta_{c}=0.1$

(c) $a=0.1,\phi=0.01,\omega=-1/2$ (d) $a=0.1,\beta_{c}=0.1,\omega=-1/2$

Figure 8: The behavior of the entropy force as a function of $x$, (a) For
different $a$, (b) for different $\omega$, (c) For different $\beta_{c}$, and
(d) for different $\phi$.

For a better comparison, in what follows we will concentrate on the
interactions between neutral molecules or atoms with a center of mass
separation $r$, which are often approximated by the so-called Lennard-Jones
potential energy ${\phi_{L,J}}$ Plastino et al. (2018, 2018); Miao et al.
(2018)

${\phi_{L,J}}=4{\phi_{\min}}\left[{{{\left({\frac{{{r_{0}}}}{r}}\right)}^{12}}-{{\left({\frac{{{r_{0}}}}{r}}\right)}^{6}}}\right]$
(43)

Note that the first term is a short-range repulsive interaction and the second
term is a longer-range attractive interaction. A plot of
${\phi_{L,J}}/{\phi_{\min}}$ versus $r/{r_{0}}$ is shown in Fig. 10. The value
$r={r_{0}}$ corresponds to ${\phi_{L,J}}=0$, and the minimum value of
${\phi_{L,J}}$ is ${\phi_{L,J}}/{\phi_{\min}}=-1$ at

${r_{\min}}/{r_{0}}={2^{1/6}}\approx 1.122$ (44)

Based on the definition of potential energy, the force between a molecule and
a neighbor in the radial direction from the first molecule is
${F_{r}}=-d{\phi_{L,J}}/dr$, which is positive (repulsive) for $r<{r_{\min}}$
and negative (attractive) for $r>{r_{\min}}$. When the center of the first
atom (with the radius of ${r_{0}}/2$) is well coincided with the coordinate
center, the corresponding center of the second atom (with the boundary of
${r_{2}}$) is $r={r_{2}}+{r_{0}}/2$. When taking $y={r_{0}}/(2{r_{2}})$, and
$0<y\leq 1$, one may transform Eq.(43) to obtain the expression of Lennard-
Jones potential energy

${\phi_{L,J}}(y)=4{\phi_{\min}}\left[{{{\left({\frac{{{r_{0}}}}{r}}\right)}^{12}}-{{\left({\frac{{{r_{0}}}}{r}}\right)}^{6}}}\right]=4{\phi_{\min}}{2^{6}}\left[{{2^{6}}{{\left({\frac{y}{{1+y}}}\right)}^{12}}-{{\left({\frac{y}{{1+y}}}\right)}^{6}}}\right]$
(45)

the evolution of which is displayed in Fig. 10. Furthermore, it is
straightforward to obtain the interaction between the two atoms as

$F(y)=-\frac{{d{\phi_{L,J}}}}{{dr}}=4{\phi_{\min}}\frac{6}{r}\left[{2{{\left({\frac{{{r_{0}}}}{r}}\right)}^{12}}-{{\left({\frac{{{r_{0}}}}{r}}\right)}^{6}}}\right]=\frac{{3{\phi_{\min}}{2^{10}}}}{{{r_{0}}}}\left[{{2^{7}}{{\left({\frac{y}{{1+y}}}\right)}^{13}}-{{\left({\frac{y}{{1+y}}}\right)}^{7}}}\right]$
(46)

Figure 9: The evolution of $\Phi_{L}-y$ with
$\frac{{3\times{2^{10}}{\phi_{\min}}}}{{{r_{0}}}}=1$.

Figure 10: The evolution of $F(y)-y$ with
$\frac{{3\times{2^{10}}{\phi_{\min}}}}{{{r_{0}}}}=1$.

Now, it is worthwhile commenting on the correlation between the entropy force
generated by the interaction between the two horizons and the Lennard-Jones
force between two particles. Our results demonstrate the strong degeneracy
between the entropy force of the two horizons and the ratio of the horizon
positions, which follows the surprisingly similar law given the relation
between the Lennard-Jones force and the ratio of two particle positions. More
specifically, in order to obtain a better comparison between the two types of
forces, we put the two curves (Fig. 8 and Fig. 10) in the same coordinates.

Figure 11: The comparison between the $F(x)-x$ graph with $a=0$, $\phi=0.01$,
${\beta_{c}}=0.1$, $\omega=-2/3$ (dashed line) and the $F(y)-y$ graph with
$\frac{{3\times{2^{10}}{\phi_{\min}}}}{{{r_{0}}}}=300$ (solid line).

The results shown in Fig. 11, again, confirm the above conclusion. In
particular, when $x$ approaches one $x\to 1$, the entropy force of the two
horizons reaches its maximum value and thus generates the cosmic acceleration;
when the ratio of the horizon positions reaches to a certain value, the
entropy force of the two horizons becomes negative and then gravity should
lead to a slowing of the expansion; at the critical point $x={x_{c}}$, the
Universe will evolve into a new stage of de Sitter phase and exhibits the
slowing expansion due to gravity; finally, when $x$ approaches zero $x\to 0$,
the entropy force of the two horizons reaches zero and thus generates constant
expansion rate. Therefore, the exploration of the entropy force between the
black hole horizon and the cosmological horizon will not only contribute to
deep understanding the evolution of our Universe (inflation in the early
universe and cosmic acceleration in the late time), but also provide a new
window to differentiate the three typical modes of cosmic expansion.

##  V Conclusion and discussion

It is well known that black hole is an ideal system to understand the nature
and behavior of quantum gravity. More importantly, black hole thermodynamics
continues to be one of great importance in gravitational physics. In this
paper, in the framework of a Reissner-Nordström-de Sitter black hole, we
discuss the combined effects of the cloud of strings and quintessence on its
thermodynamic properties, by considering the interaction between the black
hole horizon and the cosmological horizon. Here we summarize our main
conclusions in more detail:

1\. Focusing on the derivation of the equivalent thermodynamic quantities, we
obtain the effective temperature, the effective pressure, and the total
entropy of the RN-dSSQ space-time, and furthermore quantify the effects of
relevant state parameters ($a$, $\phi$, $\beta_{c}$, $\omega$). One can
clearly see that the maximum value of the effective temperature will
significantly decrease with the value of $a$ and $\phi$, which is quite
similar to the behavior of effective pressure.

2\. The effective isovolumetric heat capacity of RN-dSSQ space-time is not
equal to zero, which is quite different from with the case of in AdS space-
time. However, such finding in RN-dSSQ space-time agrees very well with the
previous result obtained in the framework of ordinary thermodynamic systems,
which strongly implies the strong consistency between the two types of
thermodynamic systems.

3\. In the framework of RN-dSSQ space-time, our results show that the entropy
S is an explicit function of the horizon position (the black hole horizon and
the cosmological horizon), instead of the function of the electric charge $Q$
and cosmological constant $l^{2}$. Meanwhile, the function $f(x)$ , which
represents the extra contribution from the correlations of the two horizons,
is also well consistent with the form in the RN-dS space-time.

4\. In the analysis of the thermodynamic study of critical phenomena, our
findings prove that similar to the case in AdS space-time, second-order phase
transitions could take place under certain conditions, with the absence of
first-order phase transition in the charged de Sitter black holes with cloud
of string and quintessence. More importantly, a quantitative analysis
indicates that the criticality does not vary with the two parameters of
$\beta_{c}$ and $\omega$, while the position of the critical point may
dramatically change with different values of a and $\phi$. Therefore, our
results could provide a new approach to study the thermodynamic qualities of
unstable dS space-time, as well as a theoretical foundation for investigating
the phase transition and phase structure of RN-dSSQ black hole systems.

5\. Working on the entropy force generated by the interaction between the
black hole horizon and the cosmological horizon, we find that the relation
between the entropy force of the two horizons and the ratio of the horizon
positions follows the similar law between the Lennard-Jones force and the
ratio of two particle positions. Therefore, the exploration of the entropy
force between the black hole horizon and the cosmological horizon will not
only contribute to the construction of the overall evolution history of the
Universe (three typical modes of cosmic expansion), but also shed light on the
possible interactions between the particles in black holes.

## Acknowledgments

We thank Prof. Z. H. Zhu for useful discussions.

This work was supported by the National Natural Science Foundation of China
(Grant Nos. 11847123, 11475108, 11705106, 11705107, 11605107), the Natural
Science Foundation of Shanxi Province, China (Grant No. 201601D102004). The
Initial Foundation of Mianyang Teachers’ College (Grant No. QD 2016A002),
Natural Science Foundation of Education Department in Sichuan Province (Grant
No. 17ZB0210).

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# Nonideal self-gravity and cosmology: the importance of correlations in the
dynamics of the large-scale structures of the Universe

P. Tremblin  Maison de la Simulation, CEA, CNRS, Univ. Paris-Sud, UVSQ,
Université Paris-Saclay, F-91191 Gif-sur-Yvette, France

###### Abstract

Inspired by the role of correlations in the statistical mechanics of nonideal
self-interacting fluids, we suggest that unresolved sub-structures (i.e.
correlations) have to be taken into account in the Virial theorem of self-
gravitating astrophysical systems. We demonstrate that their omission leads to
a missing mass problem by using the semi-analytic polytropic solutions of the
Lane-Emden equation. This problem suggests to extend the Friedmann equations
to the nonideal regime by taking into account correlations in the dynamics of
the expansion. The increase of correlations induced by the formation of the
large-scale structures could explain naturally the accelerated expansion of
the Universe in such a paradigm.

##  1 The role of correlations in statistical mechanics

The statistical approach of the classical problem of a system of N-interacting
particles (with $N\rightarrow\infty$) of mass $m$ in a volume $V$, is based on
the so-called BBGKY hierarchy [1, 2]. In this hierarchy, the probability
function to find one particle at $\vec{r}$ is defined by $P_{1}(\vec{r})$
while the probability to find one particle at $\vec{r}$ and another one at
$\vec{r}^{\prime}$ is given by $P_{2}(\vec{r},\vec{r}^{\prime})$. These
probabilities yield the definition of the radial distribution function as, for
a homogeneous, isotropic fluid:

$g(||\vec{r}-\vec{r}^{\prime}||)=V^{2}P_{2}(\vec{r},\vec{r}^{\prime}),$ (1)

or equivalently the correlation function, $\xi(||\vec{r}-\vec{r}^{\prime}||)$:

$\displaystyle\xi(||\vec{r}-\vec{r}^{\prime}||)$ $\displaystyle=$
$\displaystyle
V^{2}(P_{2}(\vec{r},\vec{r}^{\prime})-P_{1}(\vec{r})P_{1}(\vec{r}^{\prime}))$
(2) $\displaystyle=$ $\displaystyle g(||\vec{r}-\vec{r}^{\prime}||)-1.$ (3)

For an ideal fluid, in the absence of correlation,
$P_{2}(\vec{r},\vec{r}^{\prime})=P_{1}(\vec{r})P_{1}(\vec{r}^{\prime})$, i.e.
$\xi(||\vec{r}-\vec{r}^{\prime}||)=0$ and
$g(||\vec{r}-\vec{r}^{\prime}||)=1,\,\forall(\vec{r},\vec{r}^{\prime})$. In
contrast, in the presence of correlations,
$\xi(||\vec{r}-\vec{r}^{\prime}||)\neq 0$ and
$g(||\vec{r}-\vec{r}^{\prime}||)\neq 1$. The presence of correlations yields a
non-ideal contribution impacting all thermodynamic quantities in the fluid,
including also interaction energies whose definition from the N-body dynamics
is the sum over all distinct pairs of particles:

$\langle
H_{\mathrm{int,Nbody}}\rangle=\sum_{i<j}\phi(||\vec{r}_{i}-\vec{r}_{j}||),$
(4)

which corresponds exactly to an integral over the two-point probability
distribution function:

$\langle
H_{\mathrm{int,Nbody}}\rangle\approx\frac{N^{2}}{2}\iint_{V,V}P_{2}(\vec{r},\vec{r}^{\prime})\phi(||\vec{r}-\vec{r}^{\prime}||)dVdV^{\prime},$
(5)

with $\phi(r=||\vec{r}-\vec{r}^{\prime}||)$ the pair potential.

##  2 The missing-mass problem induced by the omission of correlations

The Virial theorem in astrophysics is usually applied in a simplified way,
ignoring correlations and sub-structures by using only the total mass $M$
inside a given volume $V=4\pi R^{3}/3$ with a gravitational interaction energy
per unit mass proportional to $-GM/R$.

In order to show the importance of inhomogeneities in the virial theorem, we
apply it as a simple semianalytical example to the case of a polytropic
structure. In this case, we can explicitly compute
$P_{2}(\vec{r},\vec{r}^{\prime})$ by semianalytically computing an
inhomogeneous density field and compare the result with a homogeneous
assumption on the virial theorem. A polytropic stellar structure can be
calculated by the Lane-Emden equation,

$\frac{1}{z^{2}}\frac{d}{dz}\left(z^{2}\frac{dw}{dz}\right)+w^{n}=0,$ (6)

with $\rho=\rho_{c}w^{n}$, $P=K\rho^{\gamma}$, $n=1/(\gamma-1)$ the polytropic
index, $z=Ar$, and $A^{2}=(4\pi G/(n+1))\rho_{c}^{1-1/n}/K$. We choose a
dimensionless unit system in which $\rho_{c}=K=G=1$. By solving the Lane-Emden
equation for different polytropic indexes $n<5$, we can compute inhomogeneous
density profiles that have a finite radius $R$ (see the left panel of Fig. 1).
For a polytrope, the following virial theorem (per unit mass) reads

$\displaystyle H/M$ $\displaystyle=2H_{\mathrm{int}}/M,$ (7) $\displaystyle H$
$\displaystyle=\int_{V}\frac{\gamma}{\gamma-1}PdV,$ (8) $\displaystyle M$
$\displaystyle=\int_{V}\rho dV,$ (9)

where $H$ is the total enthalpy, $M$ is the total mass, and $H_{\mathrm{int}}$
is the gravitational interaction energy with the zero of potential energy
defined at the surface of the star. For a homogeneous density profile, with
$\rho=M/(4\pi R^{3}/3)$, we obtain

$\frac{H}{M}=\frac{GM}{5R}.$ (10)

We now assume that an observer has only access to the total enthalpy, mass,
and radius of different objects and does not know whether these objects have
substructures. The naive assumption, as explained above, is to ignore the
possible inhomogeneities and assume that the density is constant,
$\rho=M/(4\pi R^{3}/3)$, hence using Eq. 10 for the virial equation. This
assumption, however, is only correct for a polytropic index $n=0$. As shown in
the right panel of Fig. 1, for profiles $n\neq 0$, the error on the total
energy is significant and can be up to a factor 25 for $n=4$. Therefore, the
assumption of a homogeneous profile within the structures will lead to a
missing-mass problem. For a polytropic structure, however, we can calculate
the inhomogeneous density profile semianalytically and thus calculate exactly
the contribution of inhomogeneities to the total energy. We characterize this
contribution by an inhomogeneous amplification factor $\alpha_{\mathrm{ih}}$ ,
which can be calculated exactly,

$\alpha_{\mathrm{ih}}=\frac{N^{2}\iint_{V,V}P_{2}(\vec{r},\vec{r}^{\prime})\phi(|\vec{r}-\vec{r}^{\prime}|)dVdV^{\prime}}{N^{2}\iint_{V,V}1/(4\pi
R^{3}/3)^{2}\phi(|\vec{r}-\vec{r}^{\prime}|)dVdV^{\prime}},$ (11)

with $V=4\pi R^{3}/3$. In our calculation,
$N^{2}P_{2}(\vec{r},\vec{r}^{\prime})=N^{2}P_{1}(\vec{r})P_{1}(\vec{r}^{\prime})=\rho(\vec{r})\rho(\vec{r}^{\prime})$
for the inhomogeneous structure at the numerator, while the homogeneous
calculation corresponds to
$N^{2}P_{2}(\vec{r},\vec{r}^{\prime})=N^{2}P_{1}(\vec{r})P_{1}(\vec{r}^{\prime})=N^{2}/V^{2}$,
with $P_{1}(\vec{r})=1/V$ at the denominator. We can now correct Eq. 10 to
account for inhomogeneities,

$\frac{H}{M}=\frac{GM}{5R}\alpha_{\mathrm{ih}}.$ (12)

As shown in the bottom panel of Fig. 1, when this factor is taken into
account, we exactly recover the correct enthalpy and the correct gravitational
interaction energy for the different polytropic structures (the green and blue
crosses in the figure are indistinguishable). This simple example demonstrates
that it is mandatory to account for the contribution of substructures to
correctly evaluate interaction energies in astrophysical structures.

Figure 1: Polytropic solutions. Left: Density profiles solutions to the Lane-
Emden equation with different polytropic indexes $n$. Right: Total enthalpy
per unit mass compared to gravitational interaction energy per unit mass
assuming either a homogeneous density profile or accounting for
inhomogeneities. Blue and green crosses are indistinguishable.

##  3 The nonideal Friedmann equations

As suggested in Tremblin et al.[4], Einstein equations may not be suited for
the study of correlated fluids for two reasons. First, the Newtonian limit
gives back the Poisson equation that is a mean-field approximation ignoring
the presence of correlations. Second, it can be explicitly shown that the
equivalence principle for a fluid implicitly assumes the absence of
correlations. We therefore propose to rely on the Newtonian demonstration of
the Friedmann equations in order to get an intuition of how to design a
nonideal version that takes into account correlations. Following Peacock[3],
we assume an ensemble of particles of mass $m$ in a volume $V$ of radius $R$,
whose fluid density is assumed to be homogeneous and isotropic at very large
scales, but can be correlated at smaller scales, that is $P_{1}(\vec{r})=1/V$
but $P_{2}(\vec{r},\vec{r}^{\prime})\neq
P_{1}(\vec{r})P_{1}(\vec{r}^{\prime})$. The total mechanical energy of this
system under its N-body or fluid form is given by

$\displaystyle E_{m}=\langle\ mv^{2}/2\rangle+\langle
H_{\mathrm{int,Nbody}}\rangle.$ (13)

Similarly to $\alpha_{\mathrm{ih}}$ in Sect. 2, we define
$\alpha_{\mathrm{ni}}$ by

$\alpha_{\mathrm{ni}}=\frac{\iint_{V,V}\phi(|\vec{r}-\vec{r}^{\prime}|)P_{2}(\vec{r},\vec{r}^{\prime})dVdV^{\prime}}{\iint_{V,V}\phi(|\vec{r}-\vec{r}^{\prime}|)P_{1}(\vec{r})P_{1}(\vec{r}^{\prime})dVdV^{\prime}}.$
(14)

The ideal (uncorrelated) fluid hypothesis corresponds to
$\alpha_{\mathrm{ni}}=1$. Eq. 13 can then be rewritten as

$\displaystyle E_{m}$ $\displaystyle=$ $\displaystyle\langle\
mv^{2}/2\rangle+\alpha_{\mathrm{ni}}\langle H_{\mathrm{int,ideal}}\rangle.$
(15)

Following Peacock[3], we take $\langle\ mv^{2}/2\rangle=M\dot{R}^{2}/2$ and
$\langle H_{\mathrm{int,ideal}}\rangle=-GM^{2}/R$ and rewrite this equation as

$\displaystyle
E_{m}/M=\frac{\dot{R}^{2}}{2}-\frac{4\pi}{3}\alpha_{\mathrm{ni}}G\rho_{m}R^{2}.$
(16)

Rearranging the different terms, we obtain

$\displaystyle\frac{\dot{R}^{2}}{R^{2}}-\frac{2E_{m}/M}{R^{2}}=\frac{8\pi
G}{3c^{2}}\alpha_{\mathrm{ni}}\rho_{m}c^{2}.$ (17)

In the uncorrelated case, $\alpha_{\mathrm{ni}}=1$, and with the substitutions
$R\rightarrow a$ and $-E_{m}/M\rightarrow K$, we recognize the first Friedmann
equation. Therefore, Eq. 17 indicates that correlations should be accounted
for through a multiplicative factor $\alpha_{\mathrm{ni}}$ of the energy
density for a proper nonideal generalization of the Friedmann equation (this
could also be seen as a multiplicative factor of the gravitational constant).
The last step in obtaining a nonideal first Friedmann equation is to take the
thermodynamic limit in $\alpha_{\mathrm{ni}}$, $(N,V)\rightarrow\infty$
keeping $N/V=\rho$ constant. This can be done by introducing a near-field
approximation on a scale $\lambda_{H}$, that is, by replacing $\phi$ by
$\exp(-r/\lambda_{H})\phi$ in Eq. 14, or equivalently, introducing a cutoff of
the integrals at a radius $r_{\mathrm{bound}}$ of approximately $\lambda_{H}$.
In the latter case, and assuming a large-scale flat, homogeneous, and
isotropic Universe, we obtain the following first nonideal Friedmann equation:

$H_{\mathrm{ni}}^{2}=\frac{8\pi
G}{3c^{2}}\alpha_{\mathrm{ni}}\rho_{b},\quad\mathrm{with}\quad\alpha_{\mathrm{ni}}=\frac{\int_{0}^{r_{\mathrm{bound}}}g(r)rdr}{\int_{0}^{r_{\mathrm{bound}}}rdr}.$
(18)

We can now study the acceleration of the expansion. Because we can link the
value of the Hubble parameter to the nonideal amplification induced by bound
substructures, $\alpha_{\mathrm{ni}}$, it is natural to expect an acceleration
of the expansion linked to an increase in the densities of these structures,
due to the ongoing gravitational collapse. In order to get the second nonideal
Friedmann equation, we use the first law of thermodynamics in an expanding
universe with the derivation of the first Friedmann equation. The two nonideal
Friedmann equations (for $K=0$) in our formalism are given by

$H_{\mathrm{ni}}^{2}=\frac{8\pi
G}{3c^{2}}\alpha_{\mathrm{ni}}\rho_{b},\quad\mathrm{and}\quad
q_{\mathrm{ni}}=-1-\frac{\dot{H}_{\mathrm{ni}}}{H_{\mathrm{ni}}^{2}}\approx\frac{1}{2}-\frac{\dot{\alpha_{\mathrm{ni}}}}{2H_{\mathrm{ni}}\alpha_{\mathrm{ni}}}.$
(19)

It is then natural to obtain an acceleration of the expansion induced by the
formation of the large scale structures of the Universe which naturally
produces $\dot{\alpha}_{\mathrm{ni}}>0$ with an increase of the absolute value
of the gravitational interaction energy even if the mean density $\rho_{b}$ is
decreasing. Assuming the nonideal amplification factor $\alpha_{\mathrm{ni}}$
entirely explains the observed present-day expansion, that is, it varied from
$\alpha_{\mathrm{ni}}=1$ to $\alpha_{\mathrm{ni}}=\rho_{c}/\rho_{b}\approx 20$
over the age of the universe, $t_{u}\sim$14 Gyr, we can obtain an order-of-
magnitude analytical expression,

$q_{\mathrm{ni}}\approx\frac{1}{2}-\frac{\ln(\rho_{c}/\rho_{b})}{2t_{u}H},$
(20)

which gives a deceleration parameter $q_{\mathrm{ni}}=-1.06$ that is
compatible with the value based on type Ia supernovae $q=-1.0\pm 0.4$ [5].

## References

## References

  * [1] J. Yvon in La théorie statistique des fluides et l’équation d’état, ed. Actualités Scientifiques et Industrielles: Théories Mécaniques (Hermann & cie, 1935).
  * [2] J. G. Kirkwood and E. M. Boggs, J. Chem. Phys. 10, 394 (1942).
  * [3] J. A. Peacock in Cosmological Physics, ed. Cambridge Astrophysics (Cambridge University Press, 1999).
  * [4] P. Tremblin et al., A&A 659, A108 (2022).
  * [5] A. G. Riess et al., AJ 116, 1009 (1998).