# Modeling of Network Based Digital Contact Tracing and Testing Strategies for
the COVID-19 Pandemic

Daniel Xu

(November 2020)

###### Abstract

With more than 1.7 million COVID-19 deaths, identifying effective measures to
prevent COVID-19 is a top priority. We developed a mathematical model to
simulate the COVID-19 pandemic with digital contact tracing and testing
strategies. The model uses a real-world social network generated from a high-
resolution contact data set of 180 students. This model incorporates
infectivity variations, test sensitivities, incubation period, and
asymptomatic cases. We present a method to extend the weighted temporal social
network and present simulations on a network of 5000 students. The purpose of
this work is to investigate optimal quarantine rules and testing strategies
with digital contact tracing. The results show that the traditional strategy
of quarantining direct contacts reduces infections by less than 20% without
sufficient testing. Periodic testing every 2 weeks without contact tracing
reduces infections by less than 3%. A variety of strategies are discussed
including testing second and third degree contacts and the pre-exposure
notification system, which acts as a social radar warning users how far they
are from COVID-19. The most effective strategy discussed in this work was
combined the pre-exposure notification system with testing second and third
degree contacts. This strategy reduces infections by 18.3% when 30% of the
population uses the app, 45.2% when 50% of the population uses the app, 72.1%
when 70% of the population uses the app, and 86.8% when 95% of the population
uses the app. When simulating the model on an extended network of 5000
students, the results are similar with the contact tracing app reducing
infections by up to 79%.

##  1 Introduction

More than 1.7 million people have died of COVID-19 [12]. With such
catastrophic loss of life at risk, it is a top priority to identify effective
measures to prevent the spread of COVID-19. Preventing the spread of COVID-19
has several challenges. Firstly, 44% of COVID-19 transmissions occur during
the asymptomatic stage with infected individuals being most infectious 1 to 2
days before symptom onset [18]. Secondly, according to the CDC Planning
Scenarios report [8], 40% of all COVID-19 cases remain asymptomatic and this
number could be as high as 79% for those under 20 [10]. The CDC Planning
Scenario also estimates that asymptomatic individuals are 75% as infectious as
symptomatic individuals. Finally, the false-negative rate of RT-PCR tests on
presymptomatic individuals ranges from 67% to 100% [22]. All of these factors
make it very difficult to prevent COVID-19 with symptom-based measures.

Contact tracing has existed for decades, helping to reduce tuberculosis [3],
sexually transmitted diseases [4], and Ebola [27]. However, manual contact
tracing relies on substantial human labor. According to a recent survey by
NPR, 39 states do not have enough contact tracers [35]. Additionally, of the
contacts reported, only around 50% were reachable by contact tracers [24].
Health officials estimate an additional $12 billion dollars are needed to fund
the 180,000 manual tracers needed [37]. Manual contact tracing is also
susceptible to errors with the United Kingdom’s Test and Trace service losing
15,000 positive COVID-19 cases between September 25 and October 2 [11].

Digital contact tracing is a relatively new method of fighting pandemics.
Although the Bluetooth technology for digital contact tracing was first
validated in 2014 [14], contact tracing apps have only recently been
implemented to fight the COVID-19 pandemic. Since digital contact tracing apps
require a critical mass of a population in order to be effective, a key goal
is to convince enough people to use the app. In [15], it is suggested that
targeting small communities like universities first will allow the app to be
used by enough people within that community. This is more feasible than
expecting a significant proportion of the population at large to use the app.
Many universities have experienced outbreaks [39] and are looking for ways to
prevent further spread. Currently, Georgia Tech, Carnegie Mellon, Grand Valley
State University, and even the city of Santa Fe are beginning to adopt contact
tracing apps like NOVID [29]. Currently, there are not enough users within
these communities to prove the effectiveness of contact tracing apps. We will
use simulation to evaluate the effectiveness of digital contact tracing.

Efforts to model diseases date back to the 1920s with the creation of the SIR
(Susceptible, Infectious, Recovered) model [21] where people in a fully mixed
population are modeled as being Susceptible, Infectious, or Recovered. The
SEIR (Susceptible, Infectious, Exposed, Recovered) model [1] is a variant of
the SIR framework used for diseases with longer incubation periods. It
includes the Exposed stage which are infected individuals who do not show
symptoms. More recently, network based models, where humans are vertices and
contacts are edges, have been adopted. Epidemiological models have been
instrumental in encouraging preventative measures in the COVID-19 pandemic
such as masking [20, 13], social distancing [36, 17], testing [30, 31]. In
particular, [20, 30] are network based models.

Since digital contact tracing relies on the exact contacts that occur in a
population, it is best modeled with a contact network where people are
vertices and contacts are edges. Recent efforts to simulate digital contact
tracing [19, 2] have been done using synthetic networks where households and
communities are constructed with random processes such as full mixing, and
each pair of people have the same probability of contact. However, very little
is known about the exact structure of human contacts and how these structures
affect digital contact tracing. Additionally, those models assume perfect
COVID-19 tests which changes optimal strategies of digital contact tracing and
testing.

This paper presents an enhanced network based SEIR model of the COVID-19
pandemic with digital contact tracing and testing strategies that incorporates
variations in infectivity and test sensitivity. In contrast to previous work
on digital contact tracing, the networks in the model are generated from a
real world data set of interactions among 180 students of a high school in
France [16]. Since the data set was only recorded over 7 days, we use a MUNGE-
like heuristic to generate additional days for the model. We present a new
method to extend temporal weighted graphs in order to perform simulations on a
larger population of 5000 people.

Our model incorporates test sensitivity and shows that it has a significant
impact on digital contact tracing strategies. The model simulates a new
digital contact tracing strategy developed by NOVID [29] called the pre-
exposure notification system. The pre-exposure notification system acts like a
”social radar” telling app users how close they are to COVID-19, allowing them
to take extra precautionary measures. In contrast with traditional SEIR
models, test sensitivity is modeled to change over time depending on the time
of symptom onset [22]. The incubation period is sampled from the COVID-19
incubation period distribution [25]. The infectiousness of an individual is
modeled to change over the infectious period and is fitted to the function in
[18].

The model shows that the traditional strategy of quarantining direct contacts
reduces infections by less than 10% when more than half the population is
asymptomatic. Testing second and third degree contacts reduces infections by
up to 40% when 70% of the population uses the app. The pre-exposure
notification system reduces infections by an additional 43% and reduces the
number of quarantines required by 51%. Quarantining second degree contacts
reduces infections but leads to a high number of quarantines. If large
proportions of the population are asymptomatic, periodic testing reduces
infections by an additional 41%. However, periodic testing without tracing
reduces infections by only 3%. The most effective strategy discussed in this
work was combined the pre-exposure notification system with testing second and
third degree contacts. This strategy reduces infections by 18.3% when 30% of
the population uses the app, 45.2% when 50% of the population uses the app,
72.1% when 70% of the population uses the app, and 86.8% when 95% of the
population uses the app. When simulating the model on an extended network of
5000 students, the results are very similar with the contact tracing app
reducing infections by up to 79%.

###  1.1 Paper Outline

In Section 2, we present the contact network generation process. Section 3
outlines the SEIR based model of COVID-19 spread. In Section 4, we create the
model of the contact tracing app. In Section 5 we present the results of 8
simulated scenarios. In Section 5.1, the traditional strategy of quarantine of
first degree contacts is investigated. In Section 5.2, testing of second and
third degree contacts is incorporated. In section 5.3, the pre-exposure
notification system is investigated. Section 5.4 investigates periodic
testing. Section 5.5 simulates the model on an extended version of the graph.
In Section 5.6, we estimate the economic value of the contact tracing app. In
Section 6, we provide a summary and concluding remarks.

##  2 Model of Contact Network

The first component of the model is the contact network which encapsulates the
interactions between individuals in the simulation. The model iterates through
discrete timestamps each representing a day in the simulation. Each day in the
model, individuals come into contact, and these contacts ultimately determine
how the virus transmits. The contact network on day $t$, $G_{t}$ is defined to
be a graph where each vertex $v_{1},...,v_{n}$ represent an individual in the
model and the edge weight $e_{ij}$ between $v_{i}$ and $v_{j}$ represents the
total amount of time person $i$ and person $j$ have spent in contact with each
other during this time step. We do not distinguish between times of the day of
the contact or number of contacts between the same individuals. This is in
accordance with a recent CDC policy change that defines a close contact to be
measured using cumulative contact time over the course of a day [6].

Traditional SEIR models assume uniform mixing where each individual has the
same probability of coming into contact with every other individual. Since
digital contact tracing involves the exact contacts that occur in a
population, a realistic contact network is needed. Temporal networks in the
model are generated using a publicly available high-resolution data set [16]
which used RFID (Radio Frequency Identification Devices) to record all
contacts between 180 students from Lycée Thiers high school in France over the
course of 7 days.

Since the simulation is over longer periods of time, we present a heuristic
for generating additional days. Define $H_{a}$ to be the contact network in
day $a$ of the data set where $1\leq a\leq 7$. The edge weight $H_{a,i,j}$ is
the duration of contact between people $i$,$j$ on day $a$ from the data set.
The heuristic is similar to the MUNGE algorithm [5] which generates synthetic
training data. The algorithm picks an initial data value, and for each
feature, with a certain probability, swaps that feature with the feature of
its nearest neighbor. In this heuristic, we use a random day rather than the
nearest neighbor. Initially, 2 distinct days $a$,$b$ are chosen at random from
the data set. Starting with the contact graph for day $a$, for the contact
duration between $v_{i}$ and $v_{j}$ with probability $0.5$, we replace that
contact duration with $H_{b,i,j}$. On average, the generated contact network
$G_{t}$ has half of its edges equal to the corresponding edges in $H_{a}$, and
the other half equal to $H_{b}$.

To simulate the app on larger networks, we present a method to generate larger
networks from the original data set. We create a modified version of the
Albert-Barabási process that is adjusted for constructing weighted temporal
graphs and maintains the average degree of the vertices.

The principal idea of the Albert-Barabási process is that nodes of high degree
are more likely to interact with new nodes. Given a temporal graph $G$ with
$n$ vertices where the graph on timestamp $t$ is $G_{t}$, extra people are
added one at a time with the following process:

A new vertex $v$ is added, and for each existing $w$, a random third vertex
$u$ is chosen. The temporal edge weight between $v$ and $w$ is determined to
be the same as the edge weight between $w$ and $u$: $G_{t,v,w}=G_{t,u,w}$ for
all $t$. This step is the same as in Albert-Barabási except adjusted to fit a
weighted temporal graph. To preserve the average degree of the vertices, each
of the original edges of $G$ are deleted with probability $\frac{1}{n-1}$.
Note that the generated graph no longer has the scale-free property.

The extended network is initialized to be the original graph in the data set
and extended in accordance to the above procedure to a total of 5000
individuals.

##  3 Model of COVID-19 Spread

The SEIR framework organizes the people in the simulation into one of the
following four states: Susceptible (S), Exposed (E), Infectious (I), or
Recovered (R). Individuals in the Susceptible stage have not been infected and
therefore are susceptible to the virus. Exposed individuals have been infected
with the virus but do not yet show symptoms. The Infectious stage begins when
infected individuals show symptoms. Finally, the Recovered stage contains
individuals who have recovered or otherwise removed from the model and are now
immune from further spread. Individuals in both the Exposed and Infectious
stage can infect others. An individual moves onto the next stage according to
the following rules:

  * •

$S\rightarrow E$: Individuals in the susceptible stage can only be exposed if
they come into contact with an infected individual. Infections will be
discussed in later sections.

  * •

$E\rightarrow I$: After being exposed to a virus, the period of time before
symptom onset is called the Incubation period, typically within 2 to 14 days.
According to [25], the distribution of incubation periods is approximately the
log-normal distribution with parameters $\mu=1.621$, $\sigma=0.418$. The log-
normal distribution is defined as follows: Let $Z$ be a random variable with
normal distribution with mean $\mu$ and deviation $\sigma$, then $X=e^{Z}$
where $X$ is the random variable with log-normal distribution with parameters
$\mu,\sigma$. To determine the incubation period of an individual, we take a
random sample from this distribution rounded to the nearest positive integer.

  * •

$I\rightarrow R$:Individuals with moderate symptoms stop being infectious
around 10 days from symptom onset [9]. As in traditional SEIR models, we
assume that every time stamp after symptom onset, there is a probability
$\lambda=0.11$ which is $\frac{1}{9}$ that an Infectious person recovers or is
removed from the model.

As in traditional SEIR models, the functions $S(t),E(t),I(t),R(t)$ are the
number of individuals in the compartments susceptible, exposed, infections,
and recovered respectively at time $t$. We defined $Q(t)$ to be the number of
individuals quarantined at time $t$ but have not received a positive test
result and $T(t)$ to be the number of individuals who have received a positive
before or during time $t$ but have not recovered. Individuals counted in
$T(t)$ are in quarantine as well. Quarantines will be discussed in Section 4.
In the rest of the paper, for each individual $v_{i}$, we will refer to
$E_{v_{i}}$, $I_{v_{i}}$, and $R_{v_{i}}$ as the time of exposure, symptom
onset, and recovery respectively of $v_{i}$. Additionally, $T_{v_{i}}$ is the
time of the first positive test of $v_{i}$.

In realistic scenarios, individuals can infect others before symptom onset
with the level of infectiousness changing throughout of the infection. The
relative infectiousness of an individual is calculated in [18]. We will call
this function $ID(t)$ where the input $t$ is the number of days since symptom
onset. $ID(t)$ is taken to be the function in [18] where infectiousness was
assumed to start 5 days before symptom onset. Individuals are shown to be more
infectious before symptom onset than after. We assume the infectiousness of an
individual, which is defined to be the probability of infection given a
20-second contact, is some constant multiple, $p$, of this function.

Consider the graph $G_{t}$ to be the contact network during timestamp $t$ in
the model, with vertices $v_{1},...,v_{n}$ representing people and edge weight
$e_{ij}$ to be the total contact duration between $v_{i},v_{j}$ during day
$t$. Assuming infection to be an event that can happen during the course of a
contact, if $v_{i}$ is in either the exposed or infectious compartments, and
$v_{j}$ is susceptible, we model the probability that $v_{j}$ becomes exposed
to be

$1-(1-pID(t-I_{v_{i}}))^{d}$

where $t-I_{v_{i}}$ is the number of days since symptom onset of $v_{i}$.

The basic reproduction number, or $R_{0}$, is the expected number of secondary
infections caused by a single infectious individual. Estimates of the value of
$R_{0}$ range from $1.5$ to $6.7$ with the median value being $2.8$ [28] and
the methods used to estimate $R_{0}$ varies widely between studies [41].
Through simulation, we calculate the value of $R_{0}$ as a function of $p$ as
shown in Figure 1. The value of $p$ is ranged in increments of $0.0025$ from
$0$ to $0.225$ for a total of 100 values. For each value of $p$, we run the
simulation 1,800 times where each individual is the seed infection 10 times
and the $R_{0}$ value is measured as the average number of secondary
infections caused by the seed infection. As shown in Figure 1, the scenario
when $p=0.10$ yields an $R_{0}$ value of $2.8$. In the rest of this work, the
value of $p$ will be set to $0.10$

Figure 1: Probability of infection vs individual reproduction number

According to [8], 40% of the population remain asymptomatic throughout their
infectious periods. Again, estimates vary widely and it is noted that this
value could range from 10% to 70%. In [32], it is estimated that over 80% of
young individuals are asymptomatic. The possibilities of when 20%, 40%, 60%,
and 80% are asymptomatic are tested in the results. Asymptomatic individuals
have similar viral loads to symptomatic individuals [26]. Thus, we will assume
that asymptomatic individuals are as infectious as symptomatic individuals.

Testing is a key component of contact tracing. The contact tracing app relies
on positive tests to identify cases of COVID-19. We adjust test sensitivities
based on days from symptom onset to the distribution calculated in [22]. The
false-negative rate of RT-PCR COVID-19 tests are between $67\%$ to $100\%$
before symptom onset, and fall to $20\%$ to $40\%$ after symptom onset [22].
The median number of days from symptom onset to taking a test is 3 days with
interquartile range 1 to 6 [8]. We approximate this distribution by assuming
that each day there is a probability of 19% that symptomatic individuals are
tested. This preserves the median of 3 days and interquartile range 1 to 6. We
do not account for false-positive tests or individuals symptomatic with a
disease unrelated to COVID-19. Test results are received after a delay of 1
day. This is similar to the delay on university campuses [40]. After receiving
a positive COVID-19 test result, we assume the person will remain in
quarantine until recovery.

##  4 Model of Contact Tracing App

The contact tracing app has incomplete information about the contact network
and the states of the individuals. Ultrasound apps such as NOVID can measure
distances to the resolution of inches and detect contacts with accuracy over
99.6% [29]. We will assume all contacts between individuals with the app are
sensed. Thus, the contact tracing app can detect the subgraph of the contact
map induced by the set of app users. Contact tracing apps support the
consideration of degree $k$ contacts rather than just direct contacts. This is
not possible with manual contact tracing. A contact is defined to be a tuple
$(p_{1},p_{2},t,d)$ that contains the following four pieces of information:
$p_{1},p_{2}$ are the people involved in the contact, $t$ is the day of
contact, and $d$ is the duration of the contact measured in units of 20
seconds. Since the digital contact tracing app cannot determine the time of
symptom onset, the estimated transmission probability of the contact is
$(1-(1-p^{\prime})^{d_{i}})$. The value $p^{\prime}=\widehat{ID}p$ is
substituted for $pID(t)$ where $\widehat{ID}=0.11$ is the average value of
$ID(t)$ during the 6 most infectious days. In addition to contacts, users
report all positive test results. If a user has tested positive, we assume
that this user will report recovery. In the contact tracing app, we define a
degree $k$ contact to be a sequence of contacts $c_{1},…,c_{k}$ that satisfy
the following properties:

  1. 1.

The individuals $p_{1},...,p_{i+1}$ involved in the contacts form a chain:
$c_{i}=(p_{i},p_{i+1},t_{i},d_{i})$.

  2. 2.

$p_{1}$ has reported a positive test: $t_{1}\leq T_{p_{1}}\leq t_{1}+10$.
Through simulation of 1800 outbreaks without interventions, 95% of all
symptomatic individuals took a test within 8 days of symptom onset. Since
infectiousness becomes significant 2 days before symptom onset, the contacts
from 10 days before the test must be recalled.

  3. 3.

$p_{i},p_{i+1}$ have not reported recovery and thus can transmit or catch the
virus: $R_{p_{i}},R_{p_{i+1}}>t_{i}$ for each $i$.

  4. 4.

The serial interval is between 1 and 10 days: $t_{i}+1\leq t_{i+1}\leq
t_{i}+10$. The serial interval is defined to be the duration from the exposure
time of the infector($t_{i}$) to the exposure time of the infected($t_{i+1}$).
Through simulation of the model 1800 time, 94% of all serial intervals are
within 10 days.

The weight of this contact chain is the product of the estimated transmission
probabilities of each contact:

$\prod_{i=1}^{k}(1-(1-p^{\prime})^{d_{i}}).$

First and second degree contacts are computed using contact data from previous
days directly after $p_{1}$ reports a positive test. Larger degree contacts
are computed recursively. For each person $v_{i}$ in the model, we keep track
of a matrix $M_{i}$ where $M_{i,l,t}$ is the total sum of contact chains of
length $l$ that affect person $v_{i}$ at time $t$. Using the contacts, the app
can calculate $M_{i}$ recursively.

On day $t$, if $v_{i}$ has contacts with $v_{a_{1}},...,v_{a_{k}}$ with
durations $d_{1},...,d_{k}$ respectively, where
$v_{i},v_{a_{1}},...,v_{a_{k}}$ have not reported recovery by day $t$, then:

$M_{i,l,t}=\sum_{s=1}^{k}\left(\left(1-(1-p^{\prime})^{d_{s}}\right)\sum_{r=1}^{10}M_{s,l-1,t-r}\right).$

For all of $v_{i}$’s contacts, we take the sum of all contact chains of length
$l-1$ and multiply by the estimated transmission probability
$1-(1-p^{\prime})^{d_{s}}$ to obtain the sum of all contact chains of length
$l$.

A person $x$ is a degree $k$ contact on day $t$ if the sum of all weights of
their contacts with degree $\leq k$ is above $10\%$:

$\sum_{i=1}^{k}\sum_{j=1}^{10}M_{x,i,t-j}\geq C.$

The contact cutoff value, $C$, only affects the quarantine rules and not
simulating the disease transmission. Essentially, direct contacts that are
calculated to be infected with probability at least $C$ are quarantined. The
value of $C$ is set to $10\%$ by default. By default, first degree contacts
are quarantined.

The pre-exposure notification system, as implemented in NOVID [29], acts like
a ”social radar” telling users how close they are to COVID-19 by showing the
number of positive cases at each distance in their social network. App users
can see the number of neighbors of distance $d$ that are COVID-19 positive. We
assume that first, second, and third degree neighbors will take precautionary
measures. For second degree contacts, we assume a 75% reduction in contacts.
For third degree contacts, we assume a 50% reduction in contacts.

##  5 Results

At the beginning of the simulation, exactly one individual is exposed while
the rest are susceptible. The simulation runs for 120 days. There are 180
individuals and for each simulation we run 1800 trials where each individual
starts as the seed infection 10 times. We simulate the cases when $0\%$,
$30\%$, $50\%$, $70\%$, and $95\%$ of the population use the app.

We simulate the following 5 scenarios:

  1. 1.

Quarantine of first degree contacts

  2. 2.

Quarantine of first degree contacts with followup testing of second and third
degree contacts

  3. 3.

Pre-exposure notification system

  4. 4.

Pre-exposure notification system with periodic testing

  5. 5.

Scenario 4 on the extended graph

We will measure the contact tracing app and COVID-19 testing configuration by
3 metrics: Total infected, total days spent in quarantine, and total tests
used. In the tables below, the App Proportion column shows the proportion of
individuals in the model that use the app. The Asymptomatic column show the
proportion of individuals that are asymptomatic. The Infected column shows the
average number of individuals infected after 120 days. Quarantines are split
into 3 categories: false, true, tested. False quarantines are quarantined
individuals who do not have COVID-19. Individuals in the True quarantine
category are infected with COVID-19 but have not received a positive test
result. Tested quarantines are those who have tested positive. We distinguish
the Tested quarantines because these infections have been confirmed by test
while True and False quarantines are predicted by the quarantine rules. Each
of the quarantine columns show the total number of days spent in quarantine
across all individuals over 120 days. The Tests Used column shows the number
of tests used in the simulation after 120 days.

In scenario 1, the most basic strategy of quarantining first degree contacts
is not effective at high asymptomatic levels, suggesting that a more
comprehensive testing strategy is required. Scenario 2 shows that testing
second and third degree contacts greatly increases the effectiveness of the
app. The pre-exposure notification system, which warns second and third degree
contacts to take extra precautionary measures, can reduce infections by an
additional 40% while also reducing the number of quarantines by up to 50%. IN
Scenario 4, it is shown that periodic testing without contact tracing reduces
infections by less than 3%. Each app user decreases the economic cost of
COVID-19 by $2,841 at 50% app usage and $4,185 at 70% app usage.

###  5.1 Scenario 1 — Quarantine of first degree contacts (Traditional
Strategy)

This is the most basic strategy where only first degree contacts are
quarantined. Every day, symptomatic individuals have a 19% chance of getting a
test as discussed in Section 3. The simulation is performed on the original
network of 180 students.

Table 1: Quarantine direct contacts, no followup testing Situation | Infected | Quarantine | Tests Used
---|---|---|---
App Proportion | Asymptomatic | False | True | Tested
0% | 40% | 106.05 | 0 | 0 | 291.87 | 53.57
30% | 40% | 101.20 | 78.78 | 39.73 | 278.42 | 50.97
50% | 40% | 97.15 | 195.71 | 94.06 | 265.96 | 48.97
70% | 40% | 87.48 | 329.17 | 149.91 | 242.71 | 44.41
95% | 40% | 70.25 | 477.30 | 200.56 | 194.77 | 35.39
0% | 80% | 110.43 | 0 | 0 | 101.43 | 18.43
30% | 80% | 107.99 | 33.53 | 19.23 | 98.83 | 18.09
50% | 80% | 108.12 | 88.47 | 52.20 | 97.12 | 17.96
70% | 80% | 101.82 | 159.04 | 91.53 | 94.11 | 17.04
95% | 80% | 99.73 | 282.78 | 156.70 | 90.58 | 16.76

(a)

(b)

Figure 2: Quarantine direct contacts, no followup testing

From Table 1, we can find out that the effectiveness of the app depends on the
percentage of the people use the app. At 40% asymptomatic ratio, the contact
tracing app reduces infections by 8.4% at 50% app usage and 17.5% at 70% app
usage. The effectiveness of the app increases doubles from adding the extra
20% of app users. This shows that it is very important to have a majority of
app users.

A limitation of this strategy is that it is ineffective at high asymptomatic
ratios. At 80% asymptomatic ratio the contact tracing app reduces infections
by less than 10% in all cases. This is because in this scenario, since
COVID-19 testing relies on symptomatic individuals, the effectiveness of the
app is greatly reduced at high asymptomatic levels. Less than 20 tests are
used while over 100 students are infected which indicates that the majority of
infections are undetected. The results in this scenario suggest that more
comprehensive testing strategies are needed, especially when large proportions
of the population are asymptomatic.

The cost of using the app is that quarantines increase as more people use the
app. In all cases, the number of false quarantines is approximately double the
number of true quarantines. The time of peak infections is similar for each
app usage. In Figure 1, infections rise dramatically, peaking at around 30
days after the initial infection and then falls as herd immunity is reached.
Because quarantine rules require a history of contacts to be recalled, the
initial 10 days show no difference between the graphs of each app usage level.
At 60% and 80% asymptomatic ratio, the app is unable to reduce infections
significantly.

###  5.2 Scenario 2 — Testing of second and third degree contacts

In this scenario, first, second, and third degree contacts are tested every 3
days. All other parameters are the same as in scenario 1. The simulation is
performed on the original network of 180 students.

Table 2: Quarantine direct contacts, test second, third degree contacts Situation | Infected | Quarantine | Tests Used
---|---|---|---
App Proportion | Asymptomatic | False | True | Tested
0% | 40% | 104.61 | 0 | 0 | 286.37 | 52.68
30% | 40% | 98.53 | 113.23 | 32.84 | 325.58 | 107.60
50% | 40% | 83.02 | 259.09 | 73.70 | 340.81 | 234.49
70% | 40% | 63.33 | 395.14 | 101.05 | 315.52 | 470.46
95% | 40% | 27.87 | 314.2 | 77.42 | 172.38 | 670.94
0% | 80% | 111.52 | 0 | 0 | 100.72 | 18.71
30% | 80% | 106.79 | 88.73 | 23.59 | 156.34 | 62.29
50% | 80% | 95.37 | 253.73 | 71.30 | 250.42 | 188.05
70% | 80% | 78.03 | 418.75 | 110.40 | 304.72 | 400.50
95% | 80% | 44.10 | 423.31 | 105.73 | 238.87 | 645.00

(a)

(b)

Figure 3: Test second, third degree contacts

As in scenario 1, the effectiveness of the app increases greatly as the number
of app users increases. In Table 1, at 40% asymptomatic ratio, the contact
tracing app reduces infections by 20.6% at 50% app usage and 39.5% at 70% app
usage. These reductions are over 2 times greater than in scenario 1. Notably,
at 95% app usage, infections are reduced by 73.4%, which is almost twice the
reduction at 70% app usage. Compared to scenario 1, the number of true
quarantines decreases while the number of tested quarantines increases at 40%
asymptomatic ratio. At 80% asymptomatic ratio, the contact tracing app reduces
infections by 14.5% at 50% app usage and 30.0% at 70% app usage. Again, this
strategy is significantly more effective than scenario 1. However, the number
of quarantines increases significantly in all categories compared to scenario
1. In particular, the number of tested quarantines rises by 224% at 70% app
usage. This is to be expected since the number of tests used rises
significantly, from less than 20 to over 400. Again, the app is significantly
more effective at 95% app usage, with infections being reduced by 60.05%.
Although results for 40% and 80% asymptomatic ratio are similar for app
proportions less than 70%, at 95% app proportion, the case of 80% asymptomatic
yields 58% more infections.

Similar to scenario 1, the graphs are similar during the first 10 days and
then diverge significantly. However, at high app usage, infections peak before
20 days while no app usage yields peak infections at around 30 days. In
conclusion, testing second and third degree contacts greatly increases the
effectiveness of the app especially at higher asymptomatic proportions. Thus,
the increase in test usage and quarantines is justified.

###  5.3 Scenario 3 — Pre-exposure notification system

In this scenario, we simulate the pre-exposure notification system. For second
degree contacts, we assume a 75% reduction in contacts. For third degree
contacts, we assume a 50% reduction in contacts. All other parameters are the
same as those in scenario 2. The simulation is performed on the original
network of 180 students.

Table 3: Quarantine direct contacts, pre-exposure notification system Situation | Infected | Quarantine | Tests Used
---|---|---|---
App Proportion | Asymptomatic | False | True | Tested
0% | 40% | 106.99 | 0 | 0 | 295.84 | 54.24
30% | 40% | 93.48 | 92.80 | 27.40 | 303.44 | 101.73
50% | 40% | 67.25 | 162.16 | 44.36 | 254.02 | 161.40
70% | 40% | 36.21 | 185.80 | 45.24 | 162.85 | 213.24
95% | 40% | 16.61 | 190.22 | 41.60 | 98.10 | 279.80
0% | 80% | 110.90 | 0 | 0 | 101.08 | 18.69
30% | 80% | 103.19 | 87.01 | 22.46 | 150.39 | 64.83
50% | 80% | 81.68 | 187.45 | 48.33 | 189.58 | 142.75
70% | 80% | 54.06 | 251.14 | 61.42 | 179.84 | 220.68
95% | 80% | 29.99 | 287.72 | 65.62 | 156.12 | 309.38

As in the previous scenarios, the effectiveness of the app increases as the
number of app users increases. In Table 3, at 40% asymptomatic ratio, the
contact tracing app reduces infections by 37.1% at 50% app usage and 66.1% at
70% app usage. At 95% app usage, infections are reduced by 84.5% which is an
additional 40.2% reduction compared to scenario 2. Compared to scenario 2, the
total number of quarantines is reduced by 51.4% at 70% app usage. There is a
54.7% reduction in the number of tests used which is caused by the reduced
interactions for second and third degree contacts. This strategy is more
effective even though the number of true and tested quarantines are reduced by
55.2% and 48.4% respectively.

At 80% asymptomatic ratio, the contact tracing app reduces infections by 14.5%
at 50% app usage and 30.0% at 70% app usage. Again, this strategy is
significantly more effective than scenario 1. At 95% app usage, the 80%
asymptomatic case yields 80.6% more infections than the 40% case. This is an
even more pronounced gap than in scenario 2.

(a)

(b)

Figure 4: Quarantine direct contacts, pre-exposure notification system

###  5.4 Scenario 4 — Pre-exposure notification system with periodic testing

In scenario 4, we simulate the Pre-Exposure Notification System with periodic
testing every 14 days is investigated. As in scenario 3, second and third
degree contacts take extra precautionary measures. Individuals in the model
are tested every 14 days. The simulation is performed on the original network
of 180 students.

At 40% asymptomatic ratio, the contact tracing app reduces infections by 45.2%
at 50% app usage and 72.1% at 70% app usage. At 95% app usage, infections are
reduced by 86.8%. Scenario 3 and 4 show similar results at higher app usage.
In this scenario, the app is effective even at 30% app usage. The 30% app
usage case reduces infections by 18.1% which is much more effective than in
previous scenarios. Additionally, results are similar at all asymptomatic
ratios. At 80% asymptomatic ratio, the contact tracing app reduces infections
by 37.8% at 50% app usage and 66.4% at 70% app usage. Thus, periodic testing
is especially helpful when larger proportions of individuals are asymptomatic.

In Figure 5(b), the number of infected but not tested individuals drop sharply
during days when the population is tested. The graph displays bumps that come
from testing the new second and third degree contacts every 3 days. Since the
graphs are averaged over thousands of trials, the jaggedness is significant.
It is likely caused by many second and third degree contacts being detected at
the same time, and thus, the followup testing becomes synchronized.
Surprisingly, periodic testing at 0% app usage reduces infections from 106.05
in scenario 1 to 102.83, only a 3.0% decrease. This is an insignificant change
given that the population of 180 students. Thus, periodic testing without
contact tracing is not effective.

Table 4: Pre-exposure notification system with periodic testing Situation | Infected | Quarantine | Tests Used
---|---|---|---
App Proportion | Asymptomatic | False | True | Tested
0% | 40% | 102.83 | 0 | 0 | 465.62 | 1021.34
30% | 40% | 84.05 | 94.78 | 27.03 | 403.80 | 1125.57
50% | 40% | 56.32 | 153.39 | 37.74 | 288.68 | 1286.85
70% | 40% | 28.73 | 166.27 | 38.08 | 159.21 | 1440.73
95% | 40% | 13.61 | 173.85 | 35.31 | 85.44 | 1571.49
0% | 80% | 103.95 | 0 | 0 | 375.20 | 1044.10
30% | 80% | 90.64 | 104.25 | 28.62 | 358.05 | 1117.67
50% | 80% | 64.69 | 174.51 | 44.14 | 281.18 | 1266.00
70% | 80% | 34.95 | 195.32 | 45.23 | 172.00 | 1426.44
95% | 80% | 17.66 | 208.92 | 43.91 | 105.50 | 1562.05

(a)

(b)

Figure 5: Pre-exposure notification system with periodic testing

###  5.5 Graph Extension

In this section, We simulate the strategy in scenario 4 on an extended version
of the graph. By the process described in Section 2, the original weighted
temporal graph is extended to 5000 individuals. This population size is closer
to larger communities such as schools or universities.

Table 5: Graph Extension Situation | Infected | Quarantine | Tests Used
---|---|---|---
App Proportion | Asymptomatic | False | True | Tested
0% | 40% | 2429.94 | 0 | 0 | 10991.94 | 31001.67
30% | 40% | 1941.70 | 5858.34 | 869.33 | 10054.09 | 36557.43
50% | 40% | 1753.64 | 15410.51 | 1686.31 | 9847.13 | 45625.01
70% | 40% | 1301.11 | 26188.07 | 2111.04 | 7902.39 | 62003.10
95% | 40% | 501.21 | 22809.78 | 1438.89 | 3319.28 | 81969.96
0% | 80% | 2433.38 | 0 | 0 | 8759.07 | 31576.44
30% | 80% | 2253.48 | 6640.62 | 1017.38 | 10021.54 | 36332.13
50% | 80% | 1888.32 | 16593.59 | 1851.73 | 9563.09 | 45640.46
70% | 80% | 1341.07 | 26338.16 | 2191.90 | 7671.17 | 61131.04
95% | 80% | 591.61 | 26708.16 | 1723.15 | 3861.01 | 86490.97

At 40% asymptomatic ratio, the contact tracing app reduces infections by 27.8%
at 50% app usage and 46.5% at 70% app usage. This is less effective than
scenario 4. At 95% app usage, infections are reduced by 79.4% which is similar
to in scenario 4. Since the average degree is preserved on the network of 5000
students, the similar results are not surprising. As in scenario 4, the
results are similar for all asymptomatic ratios. At 80% asymptomatic ratio,
the contact tracing app reduces infections by 22.4% at 50% app usage and 44.9%
at 70% app usage. At 95% app usage, infections are reduced by 75.7%. These
numbers are very similar to the corresponding values of 40% asymptomatic
ratio.

As in scenario 4, Figure 6(b) shows noticeable drops in infections on days
when the population is tested. Interestingly, at 95% app usage, the number of
infected individuals never rises to more than 100 people, or 2% of the
population. The graph does not exhibit an obvious peak but rather a sustained
amount of infections. Thus is a reflection of the greater population size.

(a)

(b)

Figure 6: Graph Extension

###  5.6 Economic Impact

In this section, we estimate the economic impact of COVID-19 and preventative
measures. According to a report by the Kaiser Family Foundation, most COVID-19
tests range between $100 to $200 with the median cost being $127 [23].
Universities are charging students up to $1000 to quarantine for 2 weeks [34,
38], around $71 a day. For students, we estimate a 20 hour work week with an
hourly wage of $20 to put daily income at $57. The average tuition cost at
universities can be as high as $41,411 [33]. Assuming 180 days in a school
year and that students miss classes and coursework, the average tuition per
day is $230. We roughly estimate the social cost of a single day in quarantine
to be $358 which is the sum of the previous numbers.

The economic cost of a single COVID-19 infection, or the societal willing-to-
pay threshold of avoiding one COVID-19 infection, is estimated to be $8,500
[30]. We calculate the total economic cost of COVID-19 for Scenario 3 and 4.
The decrease in economic cost is computed in comparison to total economic cost
at 0% app usage in scenario 1. The average economic impact per app user is the
decrease in the total economic cost of COVID-19 divided by the number of app
users.

Table 6: Average Economic Impact Per App User (Scenario 3) Situation | Cost of Infection
---|---
App Proportion | Asymptomatic
30% | 40% | $991
50% | 40% | $2,841
70% | 40% | $4,260
95% | 40% | $4,198
30% | 80% | -$19
50% | 80% | $1,251
70% | 80% | $2,678
95% | 80% | $2,928
Table 7: Economic Value of Contact Tracing App Per User (Scenario 4) Situation | Cost of Infection
---|---
App Proportion | Asymptomatic
30% | 40% | -$607
50% | 40% | $2,208
70% | 40% | $3,614
95% | 40% | $3,461
30% | 80% | -$2,052
50% | 80% | $974
70% | 80% | $2,788
95% | 80% | $2,927

The economic benefit per app user can reach as high as $4,260 in scenario 3.
In the majority of cases, the strategy in scenario 3 yields a higher benefit
to user ratio. This shows that testing the entire population every 2 weeks is
not cost effective. The economic impact per user is actually negative at low
app usage in scenario 4. This is because the periodic testing creates
significant cost while, as mentioned in Section 5.4, is not effective without
sufficient contact tracing.

##  6 Discussion and Conclusion

Given the significant loss of life at risk, finding effective measures to
prevent the spread of COVID-19 is a top priority. Challenges in COVID-19
prevention include significant pre-symptomatic transmission, high proportions
of asymptomatic cases, and inaccurate tests during the pre-symptomatic stage.
This work focuses on solving these challenges by presenting effective
strategies in digital contact tracing and testing.

The parameters in this model can be easily changed as more precise values of
COVID-19 parameters are measured. As more data on human social patterns are
collected, larger social networks can be constructed leading to more accurate
predictions in the model. Additionally, as more accurate COVID-19 tests are
developed, model results and optimal strategies could change. Finally, this
model only considers infections based on close contacts. Although COVID-19
mainly spreads through close contacts [7], the significance of spread through
indirect contacts is unclear and could be investigated in future models.

In conclusion, by simulating a variety of tracing and testing strategies, we
found that digital contact tracing can be very effective when combined with
testing. In scenario 1, the most basic strategy of quarantining first degree
contacts is not effective at high asymptomatic levels, suggesting that a more
comprehensive testing strategy is required. Testing second and third degree
contacts greatly increases the effectiveness of the app. The pre-exposure
notification system, which warns second and third degree contacts to take
extra precautionary measures, can reduce infections by an additional 40% while
also reducing the number of quarantines by up to 50%. While periodic testing
with contact tracing is effective, periodic testing without contact tracing
reduces infections by less than 3%. We find the results on the extended
network of 5000 students to be similar.

##  7 Acknowledgements

I would like to thank my mentor Dr. Jesse Geneson for his invaluable advice
and guidance throughout the project. This would not be possible without him.
Thank you to Dr. Tanya Khovanova and Alexander Vitanov for reviewing the paper
draft. This work was completed under the MIT PRIMES-USA program.

##  8 Appendix A: App Usage

In the following, we present figures for the number of infections as a
function of the app usage. Simulations are run with the proportion of app
users ranging from 0% to 100% in increments of 5%. The app becomes much more
effective as more users use the app. As shown in Section 5.4, scenario 4 shows
very close results for all asymptomatic levels. Scenarios 3 and 4 shows that
the effectiveness of the app begins to level off after more than 80% of the
population uses the app.

(a)

(b)

(c)

(d)

Figure A.1: App Usage vs Infections

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# Restricted phase space thermodynamics for
AdS black holes via holography

Zeyuan Gao and Liu Zhao
School of Physics, Nankai University, Tianjin 300071, China
email<EMAIL_ADDRESS>and<EMAIL_ADDRESS>Correspondence
author.

###### Abstract

A new formalism for thermodynamics of AdS black holes called the restricted
phase space thermodynamics (RPST) is proposed. The construction is based on
top of Visser’s holographic thermodynamics, but with the AdS radius fixed as a
constant. Thus the RPST is free of the $(P,V)$ variables but inherits the
central charge and chemical potential as a new pair of conjugate thermodynamic
variables. In this formalism, the Euler relation and the Gibbs-Duhem equation
hold simultaneously with the first law of black hole thermodynamics, which
guarantee the appropriate homogeneous behaviors for the black hole mass and
the intensive variables. The formalism is checked in detail in the example
case of 4-dimensional RN-AdS black hole in Einstein-Maxwell theory, in which
some interesting thermodynamic behaviors are revealed.

##  1 Introduction

Black hole thermodynamics is a subject which is under intensive study for
about half a century. Roughly speaking, the study of black hole thermodynamics
can be subdivided into two major stages. The first and the initial stage is
pioneered by Bekenstein [1, 2], Bardeen, Carter and Hawking [3, 4] et al. The
formalism established in this stage may be referred to as the traditional
black hole thermodynamics (TBHT), of which the most important accomplishment
is the four fundamental laws of black hole physics, which hold for most known
black hole solutions in Einstein as well as higher curvature gravities. In
this context, the black hole mass is interpreted as the internal energy, while
the surface gravity and the area of the event horizon are respectively
proportional to the temperature and the entropy. However, the nature of
microscopic degrees of freedom which contribute the entropy of the black holes
remains mysterious. According to the holographic principle [5, 6], or better
known as the AdS/CFT duality [7], an AdS black hole state in the bulk is
equivalent to a thermal state of the dual field theory, for which the
microscopic degrees of freedom can be more easily understood. A parallel line
of research on black hole thermodynamics is proposed by Wald [8], which
recovers the thermodynamic relations for black holes starting from various
symmetries of the corresponding gravity models and interprets thermodynamic
quantities of the black holes as Noether charges. Another important
development in this stage is the discovery of Hawking-Page phase transition
[9] for AdS black holes. This is a phase transition between a black hole state
and a pure thermal gas state, which is characterized by a vanishing Gibbs free
energy.

The second major stage is the so-called extended phase space thermodynamics
(EPST) [10, 11, 12, 13, 14, 15, 16], also known as black hole chemistry [16].
This formalism is initiated by Kastor, Ray and Traschen [10], and subsequently
developed by a great number of researchers. To name a few of them, we refer to
[11, 12, 13, 14, 15, 16, 17, 18, 19]. The major innovation made in this stage
is the inclusion of the negative cosmological constant (which is associated to
the pressure), together with a conjugate variable known as the thermodynamic
volume, as a new pair of thermodynamic variables. The sacrifice paid for this
change is the reinterpretation of the mass parameter as the enthalpy. In this
formalism, very rich thermodynamic behaviors have been found for various black
holes in a diverse number of gravity models. In particular, phase transitions,
either of the normal type (e.g. like in the Van der Waals liquid-gas system
[14]) or of some exotic types (e.g. with multiple critical points, allowing
for reentrant (zero-th order) and/or supercritical phase transitions, etc.
[17, 18, 19]), were studied extensively. Such studies also triggered the
motivation for including other model parameters as novel thermodynamic
parameters [15, 17]. Meanwhile, the inclusion of the $(P,V)$ variables makes
it possible to consider AdS black holes as heat engines [20, 21]. Most
recently, Visser [22] made a novel contribution to the study of EPST by
including of the central charge $C$ and the chemical potential $\mu$ as a new
pair of conjugate thermodynamic variables. In this way, the first law of black
hole thermodynamics is further extended, and the study of the corresponding
central charge criticality has just emerged [23, 24]. Please note that the
idea of introducing a chemical potential which is conjugate to the central
charge is explored earlier in [25]. Similar ideas are also introduced in [26,
27, 28, 29], however the conjugate variable of the chemical potential is not
the central charge but the square of number of colors in the dual theory,
which is related but not always equal to the central charge [27, 30].

Despite the aforementioned significant progresses, there are still some
aspects in black hole thermodynamics which remain poorly understood, and some
of the issues are important enough to raise concerns about the overall
completeness or consistency of the existing formalisms.

In this paper, we will provide a new formalism for black hole thermodynamics
which avoids most of the issues that appear in previous formalisms. In order
to clarify our motivation and lay down the problem, we will briefly outline
the major structure of the EPST and introduce Visser’s improvement above it in
Section 2. In this process, the existing issues and their seriousness will be
made clear. Then, in Section 3, we develop the new formalism which we call the
restricted phase space thermodynamics (RPST) for AdS black holes and examine
its structure and merits in great detail in the example case of Reissner-
Nordström anti-de Sitter (RN-AdS) black hole in 4-dimensional Einstein-Maxwell
theory. Finally, in Section 4, we summarize the results and discuss about some
possible future developments.

##  2 Outline of the EPST and related issues

We begin by briefly outline the structure of the EPST and clarify its major
issues. For this purpose, it suffices to consider a rotating charged AdS black
hole solution in $d$-dimensional Einstein-Maxwell theory as an illustrative
example. The action of this theory is given by

$\displaystyle\mathcal{A}$
$\displaystyle=\frac{1}{2\kappa_{d}}\int_{\mathcal{M}}\mathrm{d}^{d}x\sqrt{-g}(R-2\Lambda)-\frac{1}{4\mu_{0}}\int_{\mathcal{M}}\mathrm{d}^{d}x\sqrt{-g}(F_{\mu\nu}F^{\mu\nu}),$
(1)

where $\kappa_{d}$ is the $d$-dimensional Einstein constant, $\mathcal{M}$
represents the spacetime manifold, and
$F_{\mu\nu}=\nabla_{\mu}A_{\nu}-\nabla_{\nu}A_{\mu}$ is the Maxwell field
strength tensor.

The EPST works only in the case with a negative cosmological constant
$\Lambda$. It extends the TBHT in that the pressure $P=-\frac{\Lambda}{8\pi
G}$ associated with $\Lambda$ is introduced as a new macro state parameter,
and a conjugate variable, $V=r_{H}^{d-1}\mathcal{A}_{d-1}$, where $r_{H}$
represents the radius of the event horizon and $\mathcal{A}_{d-1}$ is the area
of the unit $(d-1)$-sphere, commonly referred to as the thermodynamic volume,
is also included, so that the first law in the EPST takes the form

$\displaystyle\mathrm{d}M=T\mathrm{d}S+\Omega\mathrm{d}J+\Phi\mathrm{d}Q+V\mathrm{d}P,$
(2)

where $M$ is the black hole mass, $T,S$ respectively represent the temperature
and entropy, $\Omega,J$ are the angular velocity and angular momentum, and
$\Phi,Q$ are the Coulomb potential and electric charge. Besides the first law,
another important relation in the EPST is the Smarr relation,

$\displaystyle M$ $\displaystyle=\frac{d-2}{d-3}\left(TS+\Omega J\right)+\Phi
Q-\frac{2}{d-3}PV.$ (3)

The first issue of the EPST comes about by allowing $P$ to vary. Since the
cosmological constant is among the model-defining parameters, varying $P$
implies changing the underlying gravity model. Therefore, the corresponding
ensemble does not describe the collection of black holes from the same gravity
model which are in the same macro state (i.e. ensemble of macro systems), but
rather describes the collection of gravity models which admit the same or
similar black hole solutions (i.e. ensemble of gravity theories). The standard
excuse for this “ensemble of theories” picture is that there is some
underlying fundamental theory of which the model parameters arise as moduli
parameters. Given that the existence of the assumed fundamental theory is
still unjustified, it feels more comfortable and safe to consider only the
ensemble of black holes from the same gravity model instead of the ensemble of
gravity models admitting the same black hole solution.

Several other issues will arise by simply looking at eq.(2). For instance, the
appearance of the term $V\mathrm{d}P$ implies that the mass $M$ is to be
understood as enthalpy rather than internal energy. However, this
understanding seems to be in contradiction with the definition of the black
hole mass as the total energy of the black hole spacetime, or as the conserved
charge associated with the timelike Killing vector field. Meanwhile, the
meaning of the thermodynamic volume $V$ remains obscure, except that $V$ is
required in order to write down the first law in the EPST.

The last and the most important issue can be revealed by making comparison
between eqs.(2) and (3). From eq.(2) one obtains

$T=\left(\frac{\partial M}{\partial
S}\right)_{J,Q,P},\quad\Omega=\left(\frac{\partial M}{\partial
J}\right)_{S,Q,P},\quad\Phi=\left(\frac{\partial M}{\partial
Q}\right)_{S,J,P},\quad V=\left(\frac{\partial M}{\partial P}\right)_{S,J,Q}.$

Inserting these relations into eq.(3) yields

$M=\frac{d-2}{d-3}\left[S\left(\frac{\partial M}{\partial
S}\right)_{J,Q,P}+J\left(\frac{\partial M}{\partial
J}\right)_{S,Q,P}\right]+Q\left(\frac{\partial M}{\partial
Q}\right)_{S,J,P}-\frac{2}{d-3}V\left(\frac{\partial M}{\partial
P}\right)_{S,J,Q}.$

This equation implies that $M$ is a homogeneous function of $S$ and $J$ with
order $\frac{d-2}{d-3}$, of $Q$ with order $1$, and of $-P$ with order
$\frac{2}{d-3}$. However, in standard thermodynamics, the thermodynamic
potentials must be homogeneous functions of the related additive extensive
variables of the first order, which is best exemplified by the Euler relation
for the internal energy $E$ of an ordinary gaseous system consisting of a
single constituent,

$E=TS-PV+\mu N.$

It is very important that the absolute values of the coefficients appearing on
the RHS of the Euler relation must be uniformly $+1$. This characteristic
property has been realized neither in the traditional formalism nor in the
EPST formalism 111However it has indeed been realized in a particular
formalism developed specifically for AdS black holes in [31, 32], see the
notes added in the end of this paper..

Recently, Visser [22] made an important step towards the resolution of the
last issue. The key idea is to take the central charge $C=L^{d-2}/G$ of the
holographic dual theory as a new extensive variable, and consider the mass $M$
as a function of the variables $A,J,Q,\Lambda,G$, where $A$ represents the
area of the event horizon. It is observed that

$\displaystyle\mathrm{d}M$
$\displaystyle=\frac{\kappa}{2\pi}\mathrm{d}\bigg{(}\frac{A_{d-2}}{4G}\bigg{)}+\Omega\mathrm{d}J+\frac{\Phi}{L}\mathrm{d}(QL)-\frac{M}{d-2}\frac{\mathrm{d}L^{d-2}}{L^{d-2}}$
$\displaystyle\quad+\left(M-\frac{\kappa A_{d-2}}{8\pi
G}-\frac{\Phi}{L}QL-\Omega J\right)\frac{\mathrm{d}(L^{d-2}/G)}{L^{d-2}/G},$
(4)

where $L^{2}=-\frac{(d-1)(d-2)}{2\Lambda}$ is the AdS radius, $G$ is Newton’s
constant in $d$-dimensions and $\kappa$ represents the surface gravity. By
identifying222There is a slight difference between the conventions of [22] and
the present work: $\tilde{\Phi}$, $\tilde{Q}$ in eq.(5) (which is cited from
[22]) should be changed into $\hat{\Phi}=\frac{\Phi\sqrt{G}}{L}$,
$\hat{Q}=\frac{QL}{\sqrt{G}}$ under our convention. In contrast, the
convention used in [23] agrees with ours.

$\displaystyle E=M,\quad T=\frac{\kappa}{2\pi},\quad
S=\frac{A_{d-2}}{4G},\quad\tilde{\Phi}=\frac{\Phi}{L},\quad\tilde{Q}=QL,\quad
V=L^{d-2}$ (5)

and introducing the equation of states (EOS)

$\displaystyle E=(d-2)PV$ (6)

for the dual CFT333For $d$-dimensional AdS black hole spacetimes, the dual CFT
lives in a $(d-1)$-dimensional boundary spacetime, for which the EOS takes the
form (6). One can imagine that eq.(6) is analogous to the EOS $E=3PV$ for a
photon gas in 4-dimensional spacetime, where 4 corresponds to $d-1$., eq.(2)
can be rearranged in the form

$\displaystyle\mathrm{d}E=T\mathrm{d}S+\Omega\mathrm{d}J+\tilde{\Phi}\mathrm{d}\tilde{Q}-P\mathrm{d}V+\mu\mathrm{d}C,$
(7)

where $\mu$ is defined using the following identity,

$\displaystyle E=TS+\tilde{\Phi}\tilde{Q}+\Omega J+\mu C.$ (8)

The rescaled electric charge $\tilde{Q}$ and Coulomb potential $\tilde{\Phi}$
can be understood as the corresponding quantities of the dual CFT. Since we
encounter two charges $\tilde{Q}$ and $C$ in this context, we shall call them
the $e$-charge and the $C$-charge for short.

It is remarkable that eq.(8) looks very similar to the Euler relation in
standard thermodynamics, except that the $(P,V)$ variables are absent on the
RHS. The meaning of $P,V$ has changed in this holographic description: $V$ is
now a measure of the spatial size of the dual CFT (which has one less spatial
dimension comparing to the thermodynamic volume in the EPST), and $P$ is
introduced in eq.(6) in order that the corresponding equation of states indeed
describes a CFT. Since the $C$-charge is a measure for the number of
microscopic degrees of freedom in CFT, its conjugate $\mu$ naturally acquires
an explanation as the chemical potential. Notice that, since $V$ and $C$ are
regarded as independent extensive variables, varying $C$ implies varying
Newton’s constant $G$. Another point to be noticed is the appearance of the
term $-P\mathrm{d}V$ in eq.(7). This term reflects the fact that $E$ restored
its role as internal energy, and thus the first issue mentioned above is
avoided. However, since $V$ is now defined in terms of the AdS radius $L$,
varying $V$ implies varying $\Lambda$. Therefore, Visser’s formalism still
suffers from the “ensemble of theories” issue.

##  3 RPST for RN-AdS black hole

Recall that the AdS/CFT duality is a one to one correspondence between a given
gravity model and the dual CFT. When considering a concrete AdS/CFT duality,
it is important to fix a gravity model in the first place, rather than
considering a collection of different gravity models altogether. This
requirement forbids the appearance of the term $V\mathrm{d}P$ in eq.(2) and
the term $-P\mathrm{d}V$ in eq.(7). The formalism for black hole
thermodynamics to be described below is free from the $(P,V)$ extension, and
hence we call it the RPST. This formalism is free of all the issues mentioned
in the last section.

The simplest and most straightforward realization of the RPST is given by
fixing $L$ in eqs.(7) and (8), which results in

$\displaystyle\mathrm{d}M$
$\displaystyle=T\mathrm{d}S+\Omega\mathrm{d}J+\tilde{\Phi}\mathrm{d}\tilde{Q}+\mu\mathrm{d}C,$
(9) $\displaystyle M$ $\displaystyle=TS+\tilde{\Phi}\tilde{Q}+\Omega J+\mu C.$
(10)

These equations bear the standard form for the first law and Euler relation in
traditional thermodynamics. Since the variables $P,V$ no longer appear, there
can be no volume work involved in black hole thermodynamic processes, hence
the black hole cannot act as heat engine in this formalism. Even though, being
a thermodynamic system with multiple thermodynamic degrees of freedom, the
black hole can still have interesting thermodynamic behaviors in the RPST, as
will be exemplified below by the RN-AdS black hole in 4 dimensions.

Since the RPST formalism still allows $G$ to vary, it seems necessary to pay
some words in explaining the different roles of variable $G$ and variable
$\Lambda$. Basically, $\Lambda$ is a part of the Lagrangian density, varying
$\Lambda$ leads to a different set of gravitational field equations. On the
contrary, $G$ can be arranged to become an overall factor in front of the
total action, the only sacrifice to be paid is a rescaling of the integration
constant $Q$ in the case of Einstein-Maxwell theory. This not only explains
the necessity of introducing a rescaling for the $e$-charge (and consequently
also for the electric potential) in the subsequent analysis, but also ensures
that varying $G$ does not lead to different field equations. That is why we
think of our formalism to be free of the “ensemble of theories” issue. Another
difference between variable $G$ and $\Lambda$ lies in that varying $\Lambda$
changes the geometry, while varying $G$ does not. We thank Visser for pointing
out this last difference in private communications.

Before presenting the detailed study of the RN-AdS example, it is interesting
to mention that an alternative restricted version of Visser’s formalism [22]
with fixed $C$ but varying $V$ is studied in detail in [30], wherein the
varying cosmological constant is attributed to varying the curvature radius of
the spacetime on which the dual CFT resides. Here, our concentration is on the
black hole side, with emphasis on the correct Euler relation and the
homogeneity behaviors of the black hole mass and other state variables.

###  3.1 Realization of the RPST for RN-AdS black hole

To be more concrete, let us write down the metric of 4-dimensional RN-AdS
black hole together with its accompanied electromagnetic potential as follows,

$\displaystyle\mathrm{d}s^{2}=-f(r)\mathrm{d}t^{2}+f(r)^{-1}\mathrm{d}r^{2}+r^{2}(\mathrm{d}\theta^{2}+\sin^{2}\theta\mathrm{d}\phi^{2}),$
$\displaystyle A^{\mu}=\left(\Phi(r),0,0,0\right),$

where

$f(r)=1-\frac{2GM}{r}+\frac{GQ^{2}}{r^{2}}+\frac{r^{2}}{\ell^{2}},\qquad\Phi(r)=\frac{Q}{r}.$

Here we work in units $\mu_{0}=4\pi,\epsilon_{0}=1/4\pi,c=\hbar=k_{\rm B}=1$,
however keep $G$ untouched. $\ell$ is related to the cosmological constant via
$\displaystyle\ell^{2}=-\frac{3}{\Lambda}$. When $M>Q/\sqrt{G}$, $f(r)$ has
two distinct real zeros $r=r_{\pm}$, and the bigger one $r_{+}$ corresponds to
the event horizon. The black hole mass $M$ can be described as a function of
$r_{+},G,Q$ by solving the equation $f(r_{+})=0$, i.e.

$\displaystyle
M=\frac{r_{+}}{2G}\left(1+\frac{GQ^{2}}{r_{+}^{2}}+\frac{r_{+}^{2}}{\ell^{2}}\right).$
(11)

In the RPST for RN-AdS black hole, the macro states are characterized by the
following three pairs of conjugate variables: $(S,T)$, $(\hat{Q},\hat{\Phi})$
and $(C,\mu)$. The values for each pair of variables are given as follows.
First, the entropy $S$ and its conjugate, the temperature $T$, are given as

$\displaystyle S$ $\displaystyle=\frac{A}{4G}=\frac{\pi r_{+}^{2}}{G},\qquad
T=\frac{f^{\prime}(r_{+})}{4\pi}=\frac{1}{2\pi
r_{+}}\bigg{(}\frac{GM}{r_{+}}-\frac{GQ^{2}}{r_{+}^{2}}+\frac{r_{+}^{2}}{\ell^{2}}\bigg{)}.$
(12)

Next, the (rescaled) $e$-charge $\hat{Q}$ and it conjugate, the rescaled
Coulomb potential $\hat{\Phi}$ on the event horizon, are given by

$\displaystyle\hat{Q}=\frac{Q\ell}{\sqrt{G}},\qquad\hat{\Phi}=\frac{\Phi(r_{+})\sqrt{G}}{\ell}=\frac{Q\sqrt{G}}{r_{+}\ell}.$
(13)

Finally, the $C$-charge $C$ and its conjugate, the chemical potential $\mu$,
are given by

$\displaystyle
C=\frac{\ell^{2}}{G},\qquad\mu=\frac{M-TS-\hat{\Phi}\hat{Q}}{C}.$ (14)

Inserting eq.(11) into eq.(12) and then into (14), it becomes evident that the
state variables $S,\hat{Q},C$ are also functions of $(r_{+},G,Q)$. By
straightforward calculations, it can be verified that

$\displaystyle\mathrm{d}M=T\mathrm{d}S+\hat{\Phi}\mathrm{d}\hat{Q}+\mu\mathrm{d}C.$
(15)

Moreover, the definition for $\mu$ given in eq.(14) indicates that

$\displaystyle M=TS+\hat{\Phi}\hat{Q}+\mu C.$ (16)

Eqs.(15), (16) are nothing but a special case of eqs.(9), (10) with
$\Omega=J=0$. Using eqs.(15) and (16), we can easily get the Gibbs-Duhem
relation

$\displaystyle\mathrm{d}\mu=-\hat{\mathcal{Q}}\mathrm{d}\hat{\Phi}-\mathcal{S}\mathrm{d}T,$
(17)

where $\hat{\mathcal{Q}}=\hat{Q}/C$ and $\mathcal{S}=S/C$ are respectively the
$e$-charge per unit $C$-charge and the entropy per unit $C$-charge. This
equation is absent in both the TBHT and the EPST formalisms for AdS black
holes.

###  3.2 Equations of states and extensivity of $S$ and $\hat{Q}$

In order to describe RN-AdS black hole in the RPST, we need to rewrite the
mass $M$ as a function of the variables $S,\hat{Q}$ and $C$. To do so, we
first solve $G$ and $r_{+}$ in terms of $S,C$ using the first equalities in
eqs.(12) and (14),

$\displaystyle G=\frac{\ell^{2}}{C},\qquad r_{+}=\ell\left(\frac{S}{\pi
C}\right)^{1/2}.$ (18)

Then, substituting eq.(18) into (11), we get the desired result,

$\displaystyle M(S,\hat{Q},C)=\frac{{S}^{2}+\pi
SC+{\pi}^{2}{\hat{Q}}^{2}}{2{\pi}^{3/2}{\ell}(SC)^{1/2}}.$ (19)

The first law (15) allows us to write down the equations of states directly as
partial derivatives of $M$,

$\displaystyle T$ $\displaystyle=\left(\frac{\partial M}{\partial
S}\right)_{\hat{Q},C}=\frac{3{S}^{2}+\pi
SC-\pi^{2}\hat{Q}^{2}}{4{\pi}^{3/2}\ell S(SC)^{1/2}},$ (20)
$\displaystyle\hat{\Phi}$ $\displaystyle=\left(\frac{\partial
M}{\partial\hat{Q}}\right)_{S,C}=\left(\frac{\pi}{SC}\right)^{1/2}\frac{\hat{Q}}{\ell},$
(21) $\displaystyle\mu$ $\displaystyle=\left(\frac{\partial M}{\partial
C}\right)_{S,\hat{Q}}=-\frac{S^{2}-\pi
SC+\pi^{2}\hat{Q}^{2}}{4{\pi}^{3/2}\ell{C}(SC)^{1/2}}.$ (22)

If $S,\hat{Q},C$ are rescaled as $S\to\lambda
S,\hat{Q}\to\lambda\hat{Q},C\to\lambda C$, then eq.(19) tells that $M$ will
also rescale as $M\to\lambda M$, while eqs.(20)-(22) indicate that
$T,\hat{\Phi},\mu$ will not get rescaled. Thus the first order homogeneity of
$M$ and the zeroth order homogeneity of $T,\hat{\Phi},\mu$ are clear as
crystal. Please note that, the zeroth order homogeneous functions are, by
definition, intensive.

It remains to check that the variables $S$ and $\hat{Q}$ are extensive. This
is a necessary step because the area dependence of the black hole entropy is
often missinterpreted as nonextensivity.

First we need to make clear what is meant by the term extensivity here.
According to the first law (15), the $C$-charge plays the role like the
particle number, therefore it is automatically extensive. Other extensive
variabes are defined such that they are proportional to $C$, with the
coefficients of proportionality depending only on the intensive variables.

Now let us prove that $S$ is extensive in the above sense. To do so, let us
first solve $\hat{Q}$ from eq.(21),

$\displaystyle\hat{Q}=\ell\hat{\Phi}\left(\frac{SC}{\pi}\right)^{1/2}.$ (23)

Substituting eq.(23) into eq.(20), we get

$\displaystyle T$ $\displaystyle=\frac{3{S}+\pi
C\left(1-\ell^{2}\hat{\Phi}^{2}\right)}{4{\pi}^{3/2}\ell(SC)^{1/2}}.$ (24)

In order that $T$ takes a non-negative value, it is necessary to constrain the
value of $\hat{\Phi}$, so that for any $S\geq 0$, we have $T\geq 0$. This
leads to the following bound

$\displaystyle\hat{\Phi}\leq\frac{1}{\ell}.$ (25)

Assuming that the above bound is unsaturated, we can solve $S$ out of eq.(24).
The result reads

$\displaystyle S$
$\displaystyle=\mathcal{S}C,\qquad\mathcal{S}=\frac{\pi^{3}\ell^{2}}{9}\left[8\tau^{2}-3\pm
4\tau\left(4\tau^{2}-3\right)^{1/2}\right]T_{0}^{2},$ (26)

where

$\displaystyle\tau=\frac{T}{T_{0}},\qquad
T_{0}\equiv\frac{\left(1-\ell^{2}\hat{\Phi}^{2}\right)^{1/2}}{\pi\ell}.$ (27)

We see that $S$ is indeed proportional to $C$, with the coefficient
$\mathcal{S}$ (which already appeared in eq(17)) depending only on the
intensive variables $T$ and $\hat{\Phi}$. This proves the extensivity of $S$.
By the way, the double-valuedness of $S$ indicated by eq.(26) will be
explained in the next subsection.

The proof for extensivity of $\hat{Q}$ is accomplished by inserting eq.(26)
into (23). The result reads

$\displaystyle\hat{Q}=\mathcal{\hat{Q}}C,\qquad\mathcal{\hat{Q}}=\ell\hat{\Phi}\left(\frac{\mathcal{S}}{\pi}\right)^{1/2},$

where $\mathcal{\hat{Q}}$ depends solely on the intensive variables
$T,\hat{\Phi}$.

###  3.3 Thermodynamic processes and phase transitions

Eqs.(20)-(22) provide three constraint conditions over six variables.
Therefore, a black hole macro state is characterized by only three of the
thermodynamic variables $T,\hat{\Phi},\mu,S,\hat{Q}$ and $C$. If any of the
three chosen variables varies, a macroscopic process takes place. Notice that
in eqs.(19), (20) and (22), $\hat{Q}$ always appear in squared form. The only
exception appears in eq.(21), which indicates that $\hat{\Phi}$ and $\hat{Q}$
always take the same signature. Therefore, it suffices to consider only the
cases with $\hat{Q}\geq 0$.

Since every intensive variables depends on three extensive ones in the
equations of states, the allowed thermodynamic processes can be extremely
complicated. For simplicity, we only consider processes along certain simple
curves by fixing two of the independent variables.

First let us consider the $T-S$ curves at fixed $\hat{Q},C$. The extremal
black hole state with $T=0$ is approached if

$\hat{Q}=\hat{Q}_{\rm max}=\frac{\left[S(\pi C+3S)\right]^{1/2}}{\pi}.$

This gives an upper bound for the value of $\hat{Q}$ at each choices of $S,C$.
Assume that the bound $\hat{Q}\leq\hat{Q}_{\rm max}$ is unsaturated. We wish
to find the inflection point on the $T-S$ curve with fixed $\hat{Q},C$ which
is characterized by the equations

$\displaystyle\left(\frac{\partial T}{\partial
S}\right)_{\hat{Q},C}=0,\qquad\left(\frac{\partial^{2}T}{\partial
S^{2}}\right)_{\hat{Q},C}=0.$

It turns out that the inflection point exists only when $\hat{Q}$ takes the
critical value

$\displaystyle\hat{Q}=\hat{Q}_{c}\equiv\frac{{C}}{6},$

wherein $C$ can take any positive value. The critical values for $S$ and $T$
are given by

$\displaystyle S_{c}=\frac{\pi C}{6},\qquad T_{c}=\frac{\sqrt{6}}{3\pi\ell}.$

Introducing the relative parameters

$\displaystyle t=\frac{T}{T_{c}},\quad s=\frac{S}{S_{c}},\quad
q=\frac{\hat{Q}}{\hat{Q}_{c}},$

eq.(20) can be rewritten as

$\displaystyle t=\frac{3s^{2}+6s-q^{2}}{8s^{3/2}}.$ (28)

It is interesting to notice that the $T-S$ EOS rewritten in the form (28) does
not depend on the value of $C$. This is typical in the study of thermodynamics
of ordinary matter, known as the law of corresponding states.

We can also introduce the Helmholtz free energy by a Legendre transform of
eq.(19),

$\displaystyle F(T,\hat{Q},C)=M(S,\hat{Q},C)-TS,$

which, in terms of the relative parameter $f=F/F_{c}$ with
$F_{c}=\frac{\sqrt{6}C}{18\ell}$, is given as

$\displaystyle f(t,q)=\frac{{q}^{2}+{s}^{2}+6s-4\,t{s}^{3/2}}{4{s}^{1/2}}.$
(29)

We see once again that the free energy equation can be written in a form which
is independent of $C$. This shows the power of scaling properties in
thermodynamics. Please note that, in eq.(29), the variable $s$ appearing in
eq.(29) must be understood as being implicitly determined by $t$ and $q$ via
eq.(28).

Figure 1: $T-S$ and $F-T$ curves in the iso-$e$-charge processes

The $T-S$ and $F-T$ curves corresponding to the iso-$e$-charge processes can
be depicted using eqs.(28) and (29), as shown in Fig.1. It can be seen that,
at any subcritical value of $\hat{Q}$, the $T-S$ curve becomes non-monotonic,
correspondingly, the $F-T$ curve contains a swallow tail with the root located
at some supercritical temperature $T>T_{c}$. This behavior indicates that
there is a Van der Waals like first order phase equilibrium appearing in the
iso-$e$-charge processes provided $0<\hat{Q}<\hat{Q}_{c}$. Since we did not
associate the black hole with the concept of a volume, the above process is
not accompanied by any volume work. If $\hat{Q}=\hat{Q}_{c}$, the phase
transition can still occur, but with the order shifted into the second.

That the first order phase transition occurs at supercritical temperatures is
something to be noticed. Such phenomena also appear in the EPST formalism for
black hole thermodynamics, but never appear in the thermodynamics of ordinary
matter. Another point to be noticed in Fig.1 is that, for any $\hat{Q}>0$, the
iso-$e$-charge $T-S$ curve intersects with the $S$ axis at some nonvanishing
$S=S_{0}$. This means that, when the black hole carries out a (not necessarily
spontaneous) iso-$e$charge process and get its temperature decreased, the
process must stop at $S=S_{0}$, which corresponds to a black hole remnant,
from where the entropy cannot be further decreased. If $\hat{Q}=0$, the
iso-$e$-charge $T-S$ curve will not intersect with the $S$ axis. Instead, $T$
acquires a single extremum which is a minimum at some nonvanishing value of
$S$, which indicates that the phase transition becomes one of a different
type, but we will not expand on this special case because similar phenomenon
will also happen in the isovoltage processes regardless of the $\hat{Q}$
values, as will be discussed below.

Let us consider the isovoltage processes by depicting the $T-S$ curve at fixed
$\hat{\Phi},C$, assuming that the bound (25) is unsaturated. It can be seen
that, at fixed $\hat{\Phi}$ and $C$, eq.(24) regarded as a function
$T=T(S)|_{\hat{\Phi},C}$ does not have any inflection point. Rather, there is
a single extremum which corresponds to the minimum of $T$ located at

$S=S_{\rm min}=\frac{\pi C}{3}(1-\ell^{2}\hat{\Phi}^{2}),\qquad T=T_{\rm
min}=\frac{\sqrt{3}}{2}\frac{\left(1-\ell^{2}\hat{\Phi}^{2}\right)^{1/2}}{\pi\ell}=\frac{\sqrt{3}}{2}T_{0}.$

Defining

$\tilde{t}=\frac{T}{T_{\rm min}},\qquad\tilde{s}=\frac{S}{S_{\rm min}},$

eq.(24) can be recast in the simple form

$\displaystyle\tilde{t}=\frac{\tilde{s}+1}{2\tilde{s}^{1/2}}.$

Figure 2: $T-S$ curve in the isovoltage processes

Fig.2 depicts the $T-S$ curve in the isovoltage process. Unlike the
iso-$e$-charge $T-S$ curves, the isovoltage $T-S$ curve does not discriminate
different $\hat{\Phi}$ values, because curves with different $\hat{\Phi}$ are
scaled into a single one by use of the relative variables $\tilde{s}$,
$\tilde{t}$. This is the law of corresponding states at an enhanced level. The
isovoltage process does not allow the temperature to be decreased below
$T_{\rm min}$. At $T>T_{\rm min}$, there are two black hole states with the
same $T,\hat{\Phi},C$ but with different entropies (i.e. different horizon
radii). This fact has already been reflected by eq.(26). According to the
principle of maximum entropy, the state with bigger $S$ should be
thermodynamically preferred. In other words, if the black hole in the state
$(T,\hat{\Phi},C)$ happens to have the smaller radius, it will jump to the one
with the bigger radius under any small perturbation. This phenomenon should
also be considered to be a phase transition, though there is not a specific
transition temperature. Therefore, such phase transitions should be regarded
as being non-equilibrium transitions. The same conclusion can be achieved from
a different perspective. Since $T$ decreases as $S$ increases for $S<S_{\rm
min}$, the heat capacity $C_{\hat{\Phi},C}=T\left(\frac{\partial T}{\partial
S}\right)_{\hat{\Phi},C}<0$ for $S<S_{\rm min}$, therefore the states with
$S<S_{\rm min}$ are unstable, whereas $C_{\hat{\Phi},C}>0$ for $S>S_{\rm
min}$, thus the states with $S>S_{\rm min}$ are stable.

Remember that the above analysis works only when the bound (25) is
unsaturated. If, instead, the bound (25) is saturated, the minimum on the
$T-S$ curve will disappear, and the curve extends to the origin of the $(T,S)$
plane. In this case, $T$ increases monotonically with $S$, the heat capacity
$C_{\hat{\Phi},C}$ is positive for any $S>0$, and the non-equilibrium phase
transitions no longer appear.

It is tempting to consider also the thermodynamic processes along the
$\hat{\Phi}-\hat{Q}$ and $\mu-C$ curves. However, according to eq.(21),
$\hat{\Phi}$ is proportional to $\hat{Q}$ for fixed $S,C$. Therefore, the
adiabatic iso-$C$-charge process on the $(\hat{\Phi},\hat{Q})$ plane is
trivial: there is neither inflection point nor extremum on the
$\hat{\Phi}-\hat{Q}$ curve.

On the other hand, from eq.(22) we find that there is a maximum for $\mu$ at
fixed $S,\hat{Q}$ which is located at

$C=C_{\rm max}\equiv\frac{3(S^{2}+\pi^{2}\hat{Q}^{2})}{\pi
S},\qquad\mu=\mu_{\rm
max}\equiv\frac{\sqrt{3}}{18\ell}\frac{S}{\left(S^{2}+\pi^{2}\hat{Q}^{2}\right)^{1/2}}.$

Introducing the dimensionless parameters

$c=\frac{C}{C_{\rm max}},\qquad m=\frac{\mu}{\mu_{\rm max}},$

the $\mu-C$ equation of states can be rescaled into

$\displaystyle m=\frac{3c-1}{2c^{3/2}}.$ (30)

Naturally, for the above rescaling of variables to make sense, the conditions
$S\neq 0$ and $S^{2}+\pi^{2}\hat{Q}^{2}\neq 0$ must hold.

Figure 3: $\mu-C$ curve in the adiabatic iso-$e$-charge/isovoltage processes

Fig.3 depicts the $\mu-C$ curve at fixed $S,\hat{Q}$. Just like the isovoltage
$T-S$ curve, the $\mu-C$ curves for different choices of $S,\hat{Q}$ are all
scaled into a single curve by use of the relative parameters $c,m$. It is
remarkable that at $C=\frac{1}{3}C_{\rm max}$, we have $\mu=0$, hence the
Gibbs free energy $G=\mu C$ vanishes. For $C>\frac{1}{3}C_{\rm max}$, $\mu$
becomes positive but not too positive to exceed $\mu_{\rm max}$. Therefore, at
$C>\frac{1}{3}C_{\rm max}$, the microscopic degrees of freedom for the RN-AdS
black hole are repulsive, while at $C<\frac{1}{3}C_{\rm max}$, they become
attractive.

One may also be interested in studying the $\mu-C$ processes at fixed $S$ and
$\hat{\Phi}$. This can be done by replacing $\hat{Q}$ in eq.(22) by use of
eq.(21). Omitting the details, we find that the $\mu-C$ relation with fixed
$S,\hat{\Phi}$ can be reduced into the same equation (30), but with the
maximal values $C_{\rm max}$ and $\mu_{\rm max}$ represented in terms of
$\hat{\Phi}$,

$C_{\rm max}=\frac{3S}{\pi(1-\ell^{2}\hat{\Phi}^{2})},\qquad\mu_{\rm
max}=\frac{\sqrt{3}}{18\ell}\left(1-\ell^{2}\hat{\Phi}^{2}\right)^{3/2}.$

Therefore, Fig.3 can also be viewed as the adiabatic isovoltage $\mu-C$ curve.
As expected, the existence of the maximum of $\mu$ relies on the fact that the
bound (25) is unsaturated, or equivalently on the fact that the black hole is
non-extremal.

Besides isocharge $\mu-C$ processes, we may also consider the isothermal
$\mu-C$ processes. By use of eqs.(20) and (22), we can find that

$\displaystyle\mu=\frac{TS}{C}-\frac{S^{2}}{\pi^{3/2}\ell(SC)^{1/2}}.$ (31)

This equation is free of both the $e$-charge and the $C$-charge, but involves
$T$ and $S$. Therefore, we can make use of the above equation to study the
adiabatic isothermal processes. It is easy to see that, at fixed $T,S$,
eq.(31) has a single maximum which is located at

$\tilde{C}_{\rm max}=\frac{9S}{4\pi^{3}\ell^{2}T^{2}},\qquad\tilde{\mu}_{\rm
max}=\frac{4\pi^{3}\ell^{2}T^{3}}{27}.$

Using the dimensionless parameters

$\tilde{c}=\frac{C}{\tilde{C}_{\rm
max}},\qquad\tilde{m}=\frac{\mu}{\tilde{\mu}_{\rm max}},$

eq.(31) becomes

$\tilde{m}=\frac{3\tilde{c}^{1/2}-2}{\tilde{c}^{3/2}}.$

The corresponding plot is given in Fig.4, which looks very similar to Fig.3,
though it corresponds to different processes.

Figure 4: $\mu-C$ curve in the adiabatic isothermal processes

In either Fig.3 or Fig.4 one can see zeros of the chemical potential. In the
absence of $e$-charge, such zeros correspond to the famous Hawking-Page (HP)
transition, which corresponds to the transition from an AdS black hole state
to a pure thermal AdS gas. Therefore, it is also interesting to analyze the HP
transition in the present context.

An important concept related to HP transition is the HP temperature $T_{\rm
HP}$, i.e. the temperature at which HP transition occurs. For the present
case, $T_{\rm HP}$ can be worked out exactly. To solve $T_{\rm HP}$, we simply
set $\hat{Q}=0$ in (22), yielding

$\displaystyle\mu$ $\displaystyle=-\frac{S^{1/2}(S-\pi
C)}{4{\pi}^{3/2}\ell{C}^{3/2}}.$ (32)

Then, substituting eq.(26) into eq.(32), we get

$\displaystyle{\mu(T)}$
$\displaystyle=-\frac{\pi^{3}\ell^{2}}{27}(T_{0})^{3}\left[8\tau^{2}-3\pm
4\tau\left(4\tau^{2}-3\right)^{1/2}\right]^{1/2}\left[2\tau^{2}-3\pm\tau\left(4\tau^{2}-3\right)^{1/2}\right],$
(33)

where $\tau$ is given in eq.(27) with $T_{0}$ replaced by $\displaystyle
T_{0}=\frac{1}{\pi\ell}$. It is evident that the positive branch of $\mu(T)$
has a zero at $\tau=1$, which means that the sought-for HP temperature $T_{\rm
HP}$ is precisely $T_{0}$,

$T_{\rm HP}=T_{0}=\frac{1}{\pi\ell}.$

Moreover, according to eq.(32), the entropy at the HP transition point is
$S=\pi C$, which, combined with the original Bekenstein-Hawking entropy
expression given in eq. (12) and the defining relation (14) for $C$, reveals
that the HP transition occurs precisely when the horizon radius $r_{+}$
reaches the AdS radius $\ell$.

We close this subsection by plotting the isovoltage $\mu-T$ curve in Fig.5.
Please notice that the coexistence point of the two branches for $\mu(T)$ is
located at a temperature below $T_{\rm HP}$. The HP transition occurs at the
crossing point of the lower branch of $\mu(T)$ with the $\mu=0$ line.

Figure 5: The isovoltage $\mu-T$ curve

##  4 Concluding Remarks

A new formalism for thermodynamics of asymptotically AdS black holes with the
correct Euler relation and Gibbs-Duhem equation is established. The resulting
RPST is based on top of Visser’s construction made in [22], but with an
important restriction, i.e. the AdS radius must be fixed. This new formalism
is not only consistent with the usual logic of AdS/CFT duality, but also
removes the un-necessary volume work in the description of the first law. The
mass parameter restores its role as internal energy, and the thermodynamic
processes do not lead the black hole solution of one underlying theory to that
of another theory.

In spite of the simple algebra used while analyzing the example case of
4-dimensional RN-AdS black holes in Einstein-Maxwell theory, the significance
of the results is highly nontrivial. It is shown that the entropy and the
$e$-charge are both extensive in the sense that they are both proportional to
the $C$-charge with the coefficients of proportionality depending only on the
intensive variables, and that the mass is a homogeneous function of the first
order in all the extensive variables.

It is also shown that there can be a first order Van der Waals like phase
transition in the iso-$e$-charge processes on the $(T,S)$ plane if the
$e$-charge $\hat{Q}$ takes some nonvanishing subcritical value
$0<\hat{Q}<\hat{Q}_{c}=\frac{C}{6}$. If $\hat{Q}$ approaches the critical
value, the phase transition becomes a second order one. However, when
considering the isovoltage processes, the $T-S$ curve acquires a single
minimum, which signifies a non-equilibrium phase transition. The constant
$S,C$ process on the $(\hat{\Phi},\hat{Q})$ plane is trivial, and the constant
$(S,\hat{Q})$, constant $(S,\hat{\Phi})$ and constant $(T,S)$ processes on the
$\mu-C$ plane contains a single maximum and a Hawking-Page transition point.
The study for the phase structure is far from being complete yet, but what we
have done already is sufficient to indicate that there are very rich
thermodynamic structures in the RPST which worths further explorations.

As a final remark, let us stress that, since Visser’s construction is fairly
general, it is highly expected that the RPST should work for AdS black holes
in generic gravity models. Exploring the applicability of the RPST and
studying the corresponding thermodynamic behaviors in other gravity models are
subjects of immediate interests. We hope to come back on these topics very
soon.

Notes added:

After the original version of this manuscript has been submitted, we became
aware that, an alternative formalism for AdS black hole thermodynamics, which
is intensively dependent on a holographic interpretation, has been introduced
in Refs.[31, 32], in which the Euler relation is also established, but with a
different set of extensive variables. The relationship between the formalism
presented in [31, 32] and ours remains unclear. However, further studies
reveals that our formalism works also for non-AdS black holes without a
holographic interpretation [33, 34], which justifies its wider applicability.

## Acknowledgement

This work is supported by the National Natural Science Foundation of China
under the grant No. 11575088.

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