# ALIFE2022 template

Stavros Anagnou, Daniel Polani  Christoph Salge

Adaptive Systems Research Group, School of Computer Science, University of
Hertfordshire

<EMAIL_ADDRESS>
# The Effect of Noise on the Emergence of Continuous Norms and its
Evolutionary Dynamics

Stavros Anagnou, Daniel Polani  Christoph Salge

Adaptive Systems Research Group, School of Computer Science, University of
Hertfordshire

<EMAIL_ADDRESS>
###### Abstract

We examine the effect of noise on societies of agents using an agent-based
model of evolutionary norm emergence. Generally, we see that noisy societies
are more selfish, smaller and discontent, and are caught in rounds of
perpetual punishment preventing them from flourishing. Surprisingly, despite
the detrimental effect of noise on the population, it does not seem to evolve
away. In fact, in some cases it seems the level of noise increases. We carry
out further analysis and provide reasons for why this might be the case.
Furthermore, we claim that our framework that evolves the noise/ambiguity of
norms is a new way to model the tight/loose framework of norms, suggesting
that despite ambiguous norms’ detrimental effect on society, evolution does
not favour clarity.

## Introduction

The social world is replete with norms, an important aspect of organising
societies. Social norms reduce the degrees of freedom in the actions of
individuals, making them more predictable and stabilising societies
(FeldmanHall and Shenhav,, 2019). Norms also enable unrelated agents to manage
shared resources (Mathew et al.,, 2013) , thereby extending cooperation beyond
genetic relatives (Richerson et al.,, 2016).

Norm emergence is usually studied with discrete behaviours. Game theory tends
to consider moral behaviour to be composed of discrete actions: cooperate and
defect (Axelrod,, 1986), hawk and dove (Smith,, 1982), stag and hare (Skyrms,,
2003). Other examples of discrete norms include political party affiliation,
coordinating or not coordinating (Lewis,, 1969; McElreath et al.,, 2003) or
adopting a given behaviour e.g. a possession norm (Epstein and Axtell,, 1996;
Flentge et al.,, 2001). We know, however, that norms are not always this
discrete, and a large number of norms exist on a continuous spectrum of
behaviour, e.g. what amount is acceptable to take from a shared resource, how
fast you walk, how close you stand next to someone during a conversation
(Kelly and Setman,, 2021). These have received much less attention in terms of
modelling (Le and Boyd,, 2007), with the exception of continuous opinions as
modelled in the closely related field of opinion dynamics (Flache et al.,,
2017).

Previous work using continuous behaviour includes continuous iterated
prisoners dilemma (Le and Boyd,, 2007) (on a scale of 0 complete defection, to
1, cooperation). In general, cooperation in continuous dilemmas is less stable
and it is harder for cooperation to invade a population of defectors (Le and
Boyd,, 2007). Bendor et al., (1991), investigated how noise affected the
success of fixed strategies in a continuous prisoners dilemma. They showed
that populations of generous strategies were more successful because generous
strategies avoided spiraling into rounds of mutual recrimination in noisy
environments. Going beyond continuous game theory, Aubert-Kato et al., (2015)
investigated the emergence of frugal and greedy behaviours in an embodied
version of a dilemma where agents varied in how long they exploited a food
source – the longer it exploits the food source, the more selfish the agent
is. Michaeli and Spiro, (2015) showed how “liberal” and “conservative”
punishment regimes can affect the polarisation of a continuous opinion.
Further, previous work on iterated prisoners dilemma by Ashlock et al., (2006)
showed that even small differences in implementation, e.g. representation
choice, can lead to significantly different dynamics.

We intend to combine these previous elements together to investigate the
effects of noise on the emergence of continuous social norms. We investigate
this in an evolutionary agent-based simulation, comparing agent societies with
deterministic and probabilistic behaviours to see if noise significantly
changes the dynamics of the society, i.e. norm emergence and other properties
of agent societies. To achieve this, we evolve three continuous norms.
Uniquely, we also evolve the level of noise on each of these properties. This
allows us to investigate the effect of noise on a continuous model of norm
emergence, which is of use to modelers considering whether or not to include
noise in their models. Further, by making the amount of noise a variable that
is available to evolution, it allows us to study the evolutionary dynamics of
noise.

We define criteria for norm emergence in a continuous system and show that our
deterministic societies obey these criteria. We find that deterministic
societies tend to be less selfish, less hypocritical and less discontented,
with agents sharing resources more effectively and sanctioning each other
less. In contrast, noisy societies tend to fall into perpetual punishment of
each other despite the abundance of resources.

This begs the question, if noise is detrimental to the agent society, why does
it not evolve away? We show that there does not seem to be an evolutionary
pressure to eliminate noise and offer some reasons as to why this may be the
case. Further, we suggest our model offers some insight into thinking about
the evolution of loose and tight societies. The tightness-looseness framework
looks at culture in terms of the strength of their cultural norms (number and
clarity) and strength of punishment when a norm is violated. Tight cultures
have stronger norms and punishments and loose cultures have more vague norms
with less harsh punishments. This framework provides insights into the
function of norms, with cultures tightening in response to threats, making
them better at dealing with them. Our model expands the existing work by
considering the noise inherent in looser societies that have more vague or
ambiguous social norms (Gelfand et al.,, 2006; Roos et al.,, 2015; Pan et
al.,, 2021).

## Model and Experiments

Figure 1: Flow diagram describing the stages of the agent-based simulation.

The following section introduces a multi-agent model we developed to study the
effect additive noise has on continuous norm emergence. We study two
experimental conditions in the model. In the deterministic case, the behavior
of each agent is defined by three internal, continuous variables (Bite Size
(B), Sanction Threshold (T) and Sanction Strength (S)). In the probabilistic
case, we add Gaussian noise to those variables each time they are used to
determine behavior. The strength (variance) of this added noise is defined for
each of the three internal values by another three agent-specific values,
respectively - i.e. BN, TN and SN. The simulation can be separated into
different steps, as visualized in Fig. 1, which are defined as follows:

Figure 2: The average value of each trait in the population plotted over time.
Deterministic (left) and probabilistic (right). Individual runs are plotted as
coloured lines and the average of those runs is plotted as a black line. N =
34 per condition. Figure 3: The population-level variance (the variance of a
trait in each population) of each trait plotted over time. Deterministic
(left) and probabilistic (right). Individual runs are plotted as coloured
lines and the average of those runs is plotted as a black line. N = 34 per
condition.

### Initialisation

At the beginning of the simulation, we create 100 agents and set the resource
level to 1000 units. Each agent’s internal values for B, S and T are
initialized to uniformly random values between $0.0$ and $1.0$. For the
probabilistic model, each agent’s noise values (BN, TN and SN) are initialized
between $0.0$ and $0.5$. Each agent’s energy level is set to 10. After
initialization, the simulation proceeds in rounds. Each round has a different,
randomized order of all agents, and each of the following steps is performed
in that order.

### Eat

When it is their turn, each agent tries to consume resources according to
their Bite Size (B). This value is added to their internal energy and removed
from the global resource level. If there are no resources left, the agent gets
no energy. The higher the value, the more greedy/selfish the agent is compared
to other agents. If all agents eat at a higher Bite Size, the environment will
not be able to support as many agents; thereby exhibiting ’tragedy of the
commons’ dynamics (Hardin,, 1968).

### Sanction

During their turn, each agent can observe the actually consumed resources of
the 10 previous agents. Each agent checks if the previous agents ate more than
its own internal Sanctions Threshold (T); if so, it sanctions them. In other
words, T is the amount of deviance an agent tolerates before punishing another
agent, i.e. what an agent finds acceptable. Sanctioning means the agent
reduces the other agent’s internal energy by its own Sanction Strength (S),
and it also pays a sanctioning cost of 0.1 $*$ S, which is subtracted from its
own energy level.

### Metabolise

Each agent has their energy level reduced by $0.01$ during each round.

### Death and Reproduction

After those steps, the simulation checks if any agents have an energy level
lower than 0.0, in which case they are removed from the simulation. Then any
agent with an energy level larger than 10 gets to reproduce. Reproduction
means we generate a copy of the agent, with the same 3 or 6 internal values,
mutated with a $0.1$ chance by adding Gaussian noise with $0$ mean and $0.1$
variance to each of those values. The energy of the child and parent are both
set at half the parent’s prior energy level. Before the next round starts, we
add 100 units of resources to the resource level.

### Creating Probabilistic Behaviour

To create the probabalistic behaviour, we used a Gaussian function each time
the agent consumed from the shared resource or punished another agent. Using
their value and accompanying standard deviation (noise parameter) to create a
value for the amount of food eaten B, how tolerant they are T and how harshly
they punish S, e.g. for each instance of sanctions we select the strength from
a normal distribution with a mean of S and a standard deviation of SN. This
added noise can be interpreted either as behavioral or perceptual error.

### Implementation Choices

The evolutionary dynamics in this simulation are created by differential
reproduction, rather than by defining an explicit fitness function and
selection process. Since the order agents eat and sanction each other is
sequential, it means agents at the front of the queue have an advantage, as
they get to consume from the resource first. Conversely, being last in the
queue also has an advantage as there is no one behind the agent to sanction
them. To minimise the impact of these asymmetries on the results, we randomise
this order in each round. We limit the initial values for the noise values to
$[0.0,0.5]$, because some initial simulation with the full range resulted in
very volatile dynamics that were hard to analyze. But after initialization, it
is possible for the noise parameters to have any value between 0.0 and 1.0 so
they can adapt in that direction. We explored further parameter settings not
reported here (varied agent metabolism, cost of sanctions), which produced
results similar to those in this paper. Note that the deterministic condition
can be seen as a special case of the probabilistic condition, where the three
noise parameters are very close to 0 for all agents.

Figure 4: Various agent and population properties plotted over time.
Deterministic (left) and probabilistic (right). Individual runs are plotted as
coloured lines and the average of those runs is plotted as a black line. N =
100 per condition.

## Results

### Continuous Norm Emergence

First we want to answer the questions, is this a model for the emergence of
continuous norms? Usually in discrete models there is an arbitrary threshold,
such as 80 percent of the population must possess that behaviour for it to be
considered a norm (Savarimuthu and Cranefield, (2011)). Since our behaviours
are continuous, and its not clear what it would mean for two agents to have
the same behaviour, we define criteria on how to assess norm emergence in a
continuous context.

  1. 1.

The behaviour converges and stabilises: Do traits decrease terms in variance
across the population from where they began, and do the average behaviours
stabilise? This would be indicated by a decreasing value for the variance of a
given behaviour across the agent population, and a lack of change of the
average behaviour over time.

  2. 2.

The behaviour the population stabilises at is arbitrary across runs to a
certain extent: This criterion ensures the resultant behaviour is not fully
due to environmental scaffolding, meaning when the behaviour is the only
rational/viable action given the environmental constraints, e.g. a population
level preference for walking over a bridge as opposed to walking across lava
would not be considered a norm (Westra and Andrews,, 2022). This would be
indicated by repeated simulations stabilizing at different average values.
Note that this requires some level of randomness in the simulation, with
different seeds.

Further to our stipulations, we clarify that we are talking about norms under
the general definition of normative regularities: "A socially maintained
pattern of behavioral conformity within a community."(Westra and Andrews,,
2022). We take this definition rather than a rule-based one, which requires
higher cognitive capacities such as language (required to express the rule)
(Kelly and Setman,, 2021). Further, we think it is a wider framework that
encompasses a broader range of phenomena that are of interest. This permissive
of "bottom-up" approach may help us reveal the building blocks of normative
cognition (Westra and Andrews,, 2022; de Waal and Ferrari,, 2010). In Fig. 2,
we see that in our deterministic simulation, after about 200 tics, all agent
traits manage to settle on a particular value. This value is arbitrary to a
certain extent with different runs settling at different values. This is
important as it means that the behaviour is indeed a norm and not just a
product of environmental scaffolding, i.e. the only rational action given the
environmental circumstances (Westra and Andrews,, 2022). In the probabilistic
cases, it seems that norms are a lot more volatile, with average values in
flux. Further, if we look at the population level variances of each trait in
Fig. 3 we see that Bite Size converges onto a much lower level than it began,
thereby satisfying our criteria for norm emergence as the population converges
on a shared behaviour. On the other hand, the population level variance of
Sanction Threshold and Sanction Strength does not always decrease, so it is
harder to make a case for the population to converge on a norm for the latter
two traits. This effect doesn’t happen as much in the noisy case (Fig. 3),
although there is some convergence for Sanction Threshold and Sanction
Strength. However, this could be explained by the fact that probabilistic
populations tend to have much lower populations than deterministic ones (Fig.
4), which could be decreasing the variance through random drift.

### Comparison Deterministic vs. Probabilistic Model

Probabilistic agent societies are generally more selfish and have less stable
cooperation. Fig. 4 shows that deterministic societies tend to have lower Bite
Sizes than probabilistic ones, meaning the societies tend to be less selfish
as they are all consuming less. Further, the norm seems to be more stable in
the deterministic condition, with many cases in the probabilistic condition
that initially settled on low Bite Sizes breaking out into higher Bite Sizes.
Particularly striking is that in 100 runs none of the deterministic runs broke
away, suggesting very stable norms in the deterministic population and
volatile in the noisy populations. Further, when this experiment was done with
an initial noise range of $[0.0,1.0]$ as opposed to $[0.0,0.5]$, the average
value of Bite Size went up to $0.9$, indicating very high levels of
selfishness. Values for the other two norm traits (Sanction Threshold and
Sanction Strength) are comparable between probabilistic and deterministic
societies, but as with Bite Size, the runs in the probabilistic version are
more volatile (Fig. 2). The probabilistic populations are dramatically smaller
(Fig. 4), with populations of 20 compared to thousands in the deterministic
case. Only a handful of the probabilistic populations manage to reach
comparable population levels to the deterministic populations. Probabilistic
societies are more hypocritical (Fig. 4). We defined hypocrisy as an agents
sanction threshold being lower than its own Bite Size ($T_{self}>B_{self}$).
After an initial increase at the beginning of the simulation, both
deterministic and probabilistic populations see a sharp decrease. However,
rates of hypocrisy are much lower in the deterministic case (around $0$)
compared to hypocrites in the probabilistic case (around $0.05$), and the
noisy runs are generally more volatile, with numerous runs breaking out into
high numbers of hypocrites.

Initially, probabilistic and deterministic societies also have similar levels
of norm convergence as measured by trait variance decreasing initially and
staying at around $0.1$. However, as in Fig. 2, this level of convergence is
less stable in the probabilistic condition. Further, since the populations are
much smaller in the probabilistic condition, this may bias the population
level variance metric, so we should not read too much into this result.
Further, the small populations mean genetic/cultural drift effects could
overpower subtle selection pressures. Despite the small populations, however,
there is still a significant amount of births/deaths (data not shown here),
suggesting selection is still happening. To address the effects of small
populations in the future, we will have a fixed population with replacement
instead of a dynamic one. But for now, since we are interested in seeing the
effect on noise on the size of the populations as well, we will keep it as is.

Strangely, the reason for low populations in the probabilistic population is
not a lack of resources. To see the real reason, we look at the amount of
energy lost due to punishment of the agents (Fig. 4). This plot shows that the
amount of energy was lost either as a result of damage from sanctions or cost
of executing sanctions. In the deterministic case, we see a spike in sanctions
at the onset if the simulation as agents punish each other due to the
diversity of norms. The society then converges toward a norm as the
sanctioning decreases. In the probabilistic case, it seems the societies do
not adequately manage to converge most of the time, resulting in this period
of adjustment never ending, leading to discontented society marred by
perpetual punishment. This may be the reason agents raise their Bite Size in
order to better protect themselves against punishment; in a high-noise society
has hypocrites, you will likely be punished anyway, so there is no rational
reason to be generous and have a low bite size. It is important here to
mention our simulation is not a case of limited resources; in fact, the
regrowth rate of the simulation is quite high to assess the dynamics of noise
in plentiful conditions.

In conclusion, probabilistic populations are more selfish, hypocritical,
discontented and less stable while the deterministic populations managing to
reach drastically higher populations with the same amount of resources.

Figure 5: The average standard deviation (noise) for each trait plotted over
time. Individual runs are plotted as coloured lines and the average of those
runs is plotted as a black line. N = 100 per condition.

### Noise Evolution

Given that noise in our model seems to be detrimental to a society overall,
and noting that the deterministic model is just a special case of the
probabilistic model, we would expect that our models evolve away the noise.
But if we look at the development of the average value for the three added
noise parameters, we do not see a decrease of noise (Fig. 5). On the contrary,
if we just look at the first 500 steps, the average noise seems to slowly
increase. This could in part be explained by a random walk, since we
initialize between $[0.0,0.5]$ but allow for noise to evolve to the full range
of $[0.0,1.0]$. If there was no evolutionary pressure, we would expect the
noise parameters to drift towards $0.5$.

To investigate the evolutionary dynamics of noise, we looked at how the
standard deviation evolved over time (Fig. 5). We see that average noise apart
from small deviations does not really seem to increase. With the individual
runs looking like random walks and the average not really changing.To further
investigate why this is happening and to make sure this lack of evolution is
not due to the shortness of the runs, we did the following. We first ran a
sample of 34 runs for each condition for 5000 time steps (10x longer than
previous experiments) and then compared noisy runs that were successful (i.e.
reach large populations larger than 5000, which is comparable to the
deterministic case) with those that were not successful (i.e. where the
population collapse or stayed at low population numbers).

Figure 6: The average standard deviation (noise) for each trait plotted over
time. All runs (left) and only runs with $populations>5000$ (right).
Individual runs are plotted as coloured lines and the average of those runs is
plotted as a black line. N = 34 per condition.

### Interpreting the Evolution of Noise

In the longer runs, we see that noise seems to slightly increase for Bite Size
for all runs Fig. 6 (left panels). But if we look at only the successful runs
(right panels), it seems that on average they don’t really change in terms of
their value over time. For some of the runs, this might not be so much due to
evolutionary adaptation, but the fact that the runs started with low noise to
begin with and happened to be better at surviving. This being said, a minor
proportion of successful runs had an initial increase in noise but eventually
settled at low noise values. It seems then that for Bite Size, successful runs
(those that reach higher populations) are runs that, by chance, started at the
low noise levels.

If we look at the noise for Sanction Threshold (tolerance) (TN), we see a
similar pattern: a slight increase in noise overall if we look at all of the
runs but the majority of the successful runs start at low noise and remain
there. So although the graphs show that we could reduce noise by selecting
populations by their overall performance, noise level does not seem to
decrease when selection occurs on a per agent basis.

In contrast to the other two traits, Sanction Strength noise (SN) seems to
increase in both successful and unsuccessful runs, in spite of the fact that
noise is initialised at [0, 0.5] (Fig. 6). Specifically, there seems to be a
region between 0.6 and 0.8 where the most successful runs seem to settle, with
unsuccessful runs above and below this region. One could interpret this as
being a selection pressure favouring an intermediate level of Sanction
Strength noise, however more experiments with runs starting at a wider range
of noise would be needed to conform this.

Taken together, this results seem to suggest that although noise makes agent
populations less successful, there doesn’t seem to be an evolutionary process
that reduces noise, in fact, in the case of Sanction Strength, evolution may
increase noise.

## Discussion

We present a model of continuous norm emergence to show the effects of noise
on the dynamics of norm emergence and the simulation more generally.

We first showed that deterministic societies satisfy our criteria for norm
emergence:

  * •

1: That they converge on behaviours compared to the start of the simulation;
and

  * •

2: That the norm is somewhat arbitrary, meaning not the only rational action
given the circumstances i.e. environmental scaffolding (Westra and Andrews,,
2022).

Second, we showed that deterministic societies compared to noisy ones are:
more able to settle on norms, distribute resources effectively (altruism),
less hypocritical, less discontent and are more stable in these properties
over time. In contrast, noisy societies do not prosper because they have high
levels of sanctioning not seen in deterministic societies, falling into rounds
of perpetual punishment. This result goes against the "mad man theory"
hypothesis, where an adversary in a negotiation is more likely to stand down
if they think their opponent (in our simulation, the agent sanctioning) is
unpredictable to avoid provoking them (McManus, (2019)). In our simulation, it
appears that noisy punishment only seems to incentivise agents to increase
their provoking behavior (breaking norms by increasing Bite Size). This occurs
because they will be punished whether they obey a norm or not, so they might
as well consume as much as they can to increase their chances of survival.
Although, the detrimental effects of noisy sanctions may be due to the fact
that both the punishment behaviours (S and T) and eating behaviour (B) are
noisy, further analysis by adding noise to only one of the behaviours e.g. B
or T would be needed to confirm this.

The widespread instability of noisy societies raised the question: if noise is
detrimental, wouldn’t it evolve away? Through further analysis, we showed that
although populations with low levels of noise ended up being more successful
(reaching higher populations), there wasn’t an evolutionary trend toward
reducing noise, in fact in some cases there seemed to be an evolutionary
pressure to increase noise.

We offer some explanations why this may be happening. Firstly, for noise to be
selected against there might need be group selection, as deterministic
societies tend to be much larger, they would be able to out compete smaller
noisy groups. Secondly, although high levels of noise are detrimental to the
group, there might be a benefit to the individual of having noise; perhaps it
enables them to avoid punishments. Further, one agent lowering their noise
would not necessarily benefit them enough to dominate the population if the
rest of society still has high levels of noise.

A further contribution of this paper is that it may offer a way to model/think
about the evolution of cultural tightness/looseness (Gelfand et al.,, 2006).
Which is defined as 1. strength of sanctioning (tolerance to deviance from
norms) and 2. strength of social norms (number and clarity). Tight cultures
have stronger norms and punishments and loose cultures have vaguer norms with
less harsh punishments (Gelfand et al.,, 2006; Roos et al.,, 2015). We refine
1. by differentiating between tolerance to deviation (i.e. sanction threshold)
and also the strength of punishment when someone deviates (i.e. sanction
strength). Further, our model could be a new way to model the evolution of
clarity of social norms using noise (2). Finally, computer models studying
tightness and looseness assume discrete behaviours, we relax this assumption
by grounding our study in a continuous modelling framework (Pan et al.,,
2021). Taking this lens on our simulation, we could claim that the clarity of
norms (tightness) can’t be evolved despite there being selective pressure
against it. This is a peculiar finding, as it would imply although societies
with vague rules are at a disadvantage, ambiguity persists.

### Future Work

In our future work we would like add the following extensions. Currently the
regrowth rate of the shared resource is static. We could vary this and make
the resource growth dynamic by having "seasons" and see the resultant
dynamics. Further, we only studied vertical cultural transmission but didn’t
include horizontal transmission, where individuals in the same generation copy
each other’s strategies. In contrast to other models of tightness and
looseness, our model has three norms instead of one, we could add more norms
and analyse the interplay between the "tightness" and "looseness" of different
norms. Finally, to further study the evolutionary dynamics of noise; we should
compare different combinations of traits with and without noise, e.g. have a
noisy punishment threshold but deterministic bite size and punishment
strength.

## Acknowledgements

We would like to thank Niki Papadogiannaki, Imran Khan and the anonymous
reviewers for their helpful comments and feedback. Stavros Anagnou is
supported by a studentship from the University of Hertfordshire.

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#  $P-v$ criticalities, phase transitions and geometrothermodynamics of
charged AdS black holes from Kaniadakis statistics

G. G. Luciano<EMAIL_ADDRESS>Applied Physics Section of
Environmental Science Department, Escola Politècnica Superior, Universitat de
Lleida, Av. Jaume II, 69, 25001 Lleida, Spain  E. N. Saridakis
<EMAIL_ADDRESS>National Observatory of Athens, Lofos Nymfon, 11852 Athens,
Greece  CAS Key Laboratory for Researches in Galaxies and Cosmology,
Department of Astronomy, University of Science and Technology of China, Hefei,
Anhui 230026, P.R. China  Departamento de Matemáticas, Universidad Católica
del Norte, Avda. Angamos 0610, Casilla 1280 Antofagasta, Chile

###### Abstract

Boltzmann entropy-based thermodynamics of charged anti-de Sitter (AdS) black
holes has been shown to exhibit physically interesting features, such as $P-V$
criticalities and van der Waals-like phase transitions. In this work we extend
the study of these critical phenomena to Kaniadakis theory, which is a non-
extensive generalization of the classical statistical mechanics incorporating
relativity. By applying the typical framework of condensed-matter physics, we
analyze the impact of Kaniadakis entropy onto the equation of state, the Gibbs
free energy and the critical exponents of AdS black holes in the extended
phase space. Additionally, we investigate the underlying micro-structure of
black holes in Ruppeiner geometry, which reveals appreciable deviations of the
nature of the particle interactions from the standard behavior. Our analysis
opens up new perspectives on the understanding of black hole thermodynamics in
a relativistic statistical framework, highlighting the role of non-extensive
corrections in the AdS black holes/van der Waals fluids dual picture.

##  I Introduction

It is widely believed that black hole (BH) physics provides a promising arena
to explore the quantum nature of gravity. After the pioneering discovery that
BHs behave as thermodynamic systems Hawking:1975vcx ; Bek1 , the study of
their properties has received a boost of new interest, and insights emerged
toward the unification of general relativity, quantum theory and statistical
physics Bek2 ; Hawking:1982dh . Yet, although BH dynamics can be fully
described by a small number of classical parameters (namely mass, angular
momentum and charge - _no hair theorem_), the microscopic degrees of freedom
responsible for the thermal behavior of BHs have not yet been adequately
identified Wald:1999vt .

The development of thermodynamic geometry (geometrothermodynamics) based on
Weinhold Wein1  and Ruppeiner Rupp1 ; Rupp2  formalisms is an effort to
extract, phenomenologically or qualitatively, the microscopic interaction
information of a given system from the axioms of thermodynamics. The core idea
is that, in ordinary thermodynamic systems, the curvature of Weinhold and
Ruppeiner metrics is related to the nature of interactions among the
underlying particles. For systems where the micro-structures interact
attractively, the curvature scalar carries a negative sign, whereas it is
positive for predominantly repulsive forces. Moreover, the metric is flat for
non-interacting systems - such as the ideal gas - or systems where
interactions are perfectly balanced. This scheme has been tested for a wide
number of statistical physical models Rupp2 . Interestingly enough, recent
studies have revealed that it is feasible for BHs too Cai:1998ep ; Wei:2015iwa
; Wei:2019uqg ; Guo:2019oad ; Xu:2020gud ; Ghosh:2020kba ; Xu:2020ftx ; Prom ;
Dehghani:2023yph ; Santos:2023eqp , providing an empirical tool to access the
microstructure of BHs from their macroscopic knowledge, despite the absence of
a quantum gravitational theory.

Black holes in anti-de Sitter (AdS) spacetimes have been thoroughly studied in
the last decades due to their applications in holography Witt . The
observation that asymptotically AdS BHs can be described by dual thermal field
theory has motivated a parallel study with condensed matter systems. This has
led to the discovery of first order phase transitions in BHs Chamb1 ; Chamb2 ;
Niu  that resemble in many aspects the liquid-gas change of phase of van der
Waals fluids Sahay1 ; Sahay2 ; MannPT . A constitutive ingredient of this
picture is the (negative) cosmological constant $\Lambda$, which is identified
as pressure and included in the first law of BH thermodynamics alongside its
conjugate quantity - the thermodynamic volume Kastor ; Dolan . The ensuing
_extended phase space_ allows to formulate the $P=P(V,T)$ equation of state
and study the critical behavior of AdS BHs Dolan ; Dolan2 .

A subtle concept in BH physics is thermodynamic entropy. According to the
holographic principle tHooft:1993dmi ; Susskind:1994vu  BHs could store
information at the event horizon like holograms. In the standard Boltzmann-
Gibbs statistics, this behavior is encoded by the Bekenstein-Hawking area law,
which states that BH entropy scales like the surface area

$S_{BH}=\frac{A_{bh}}{A_{0}}\,,$ (1)

where $A_{0}=4$ is the Planck area111We adopt geometric units
$\hslash=c=k_{B}=G=1$.. Clearly, this is an unconventional scaling. Indeed, if
BHs are physically identified with their event horizon surface, then they can
be regarded as genuine $(2+1)$-dimensional systems and $S_{BH}$ is with the
correct (extensive) thermodynamic entropy. However, if BHs are to be
considered as $(3+1)$-dimensional objects (as arguably more natural in a
$(3+1)$-dimensional description of the spacetime background), then the area
scaling would violate thermodynamic extensivity. Thus, Boltzmann-Gibbs theory
may not be the appropriate framework for studying the thermodynamics of BHs,
and a generalized non-additive entropy notion Cirto  or a quasi-homogeneous
black hole thermodynamics Quevedo  could be needed for such non-standard
systems.

To better understand the intimate nature of BH entropy, several extensions of
Boltzmann-Gibbs statistics have been considered in literature, motivated by
either gravitational considerations (Tsallis Tsallis ; Cirto , Barrow Barrow
and more generalized Nojiri  entropies) or information theory (Rényi Reny  and
Sharma-Mittal Sharma  entropies). Predictions of these models have been tested
in cosmology SarBar ; SarTs ; SheTs ; LucPRD ; LucPLB ; ShePert  and quantum
physics Shaba ; LucTsGup ; Luciano:2021mto ; JizbaLamb . Recently, a non-
extensive generalization inspired by the symmetries of the relativistic
Lorentz group has been proposed by Kaniadakis Kania0 ; Kania1 ; Kania2 ;
Scarf1 ; Scarf2  based on the modified entropy

$S_{\kappa}\,=\,-\sum_{i}n_{i}\ln_{\kappa}n_{i}\,,$ (2)

where the $\kappa$-deformed logarithm is defined by

$\ln_{\kappa}x\,\equiv\,\frac{x^{\kappa}-x^{-\kappa}}{2\kappa}\,.$ (3)

The generalized Boltzmann factor for the $i$-th microstate is

$n_{i}\,=\,\alpha\exp_{\kappa}\left[-\beta\left(E_{i}-\mu\right)\right],$ (4)

where

$\displaystyle\exp_{\kappa}(x)$ $\displaystyle\equiv$
$\displaystyle\left(\sqrt{1+\kappa^{2}\,x^{2}}\,+\,\kappa\,x\right)^{1/\kappa}\,,$
(5) $\displaystyle\alpha$ $\displaystyle=$
$\displaystyle\left[(1-\kappa)/(1+\kappa)\right]^{1/2\kappa}\,,$ (6)
$\displaystyle 1/\beta$ $\displaystyle=$
$\displaystyle\sqrt{1-\kappa^{2}}\,\hskip 0.56905ptT\,,$ (7)

with $T$ and $\mu$ being the temperature and chemical potential of the system,
respectively.

Deviations from Boltzmann-Gibbs statistics are quantified by the dimensionless
parameter $-1<\kappa<1$. The classical framework is, however, recovered in the
$\kappa\rightarrow 0$ limit. Besides theoretical arguments, we emphasize that
phenomenological evidences for Kaniadakis statistics come from the high-
quality agreement between the modified distribution (4) and the observed
power-law tailed spectrum of cosmic rays Kania1 .

One can show that, for the case of BHs, Kaniadakis entropy (2) can be cast as
Luciano:2022eio ; Morad ; LympKan ; Drepanou:2021jiv ; She

$S_{\kappa}\,=\,\frac{1}{\kappa}\sinh\left(\kappa\,S_{BH}\right)\,.$ (8)

We mention here that, since the above expression is an even function of
$\kappa$, i.e. $S_{\kappa}=S_{-\kappa}$, in the following we shall restrict to
the $\kappa\geq 0$ domain.

Kaniadakis entropy in the form (8) has been mostly used for holographic
applications and, in particular, to infer corrections brought about in the
Friedmann equations Morad ; LympKan ; Drepanou:2021jiv ; She ; Hernandez-
Almada:2021aiw ; Hernandez-Almada:2021rjs ; Lambiase:2023ryq  (see also
Luciano:2022eio  for a recent review). In the light of the gravity-
thermodynamic conjecture, preliminary studies in BH thermodynamics have been
considered in Cimi  by computing the $\kappa$-deformed temperature and heat
capacity in the context of generalized Heisenberg relations Kempf ; Buoninf .
Nevertheless, to the best of our knowledge, a dedicated analysis of BH
geometrothermodynamics and critical phenomena in Kaniadakis statistics has not
yet been conducted.

Starting from the above premises, in this work we address the thermodynamics
of AdS BHs from the Kaniadakis entropy perspective. We investigate the impact
of Eq. (8) on small-large BH phase transitions and critical exponents by
exploiting the language of condensed matter physics. In this sense, the main
effort here is to lay the foundation towards formulating BH thermodynamics in
a fully relativistic statistical context. We then examine the underlying
microstructure of BHs in Ruppeiner geometry, which reveals predominantly
repulsive intermolecular forces. In line with the discussion of NRGS1 ; NRGS4
; NRGS5 ; NRGS6 ; NRGS8 ; Inl1 ; Inl2 , our analysis shows that the
development of BH thermodynamics based on a non-extensive entropy notion
involves a consistent redefinition of all other thermodynamic quantities,
including the Hawking temperature and thermodynamic energy.

The structure of the work is as follows: for later comparison with physics of
BHs, the next Section is devoted to review phase transitions and critical
phenomena of van der Waals fluids. In Sec. III we study thermodynamics of
charged AdS BHs in Kaniadakis statistics, while Sec. IV concerns
geometrothermodynamic analysis. Conclusions and perspectives are finally
discussed in Sec. V.

##  II $P-V$ criticality of van der Waals fluids

Van der Waals model provides an effective description of real interacting
fluids and liquid-gas phase transitions. The characteristic equation is

$\left(P+\frac{a}{v^{2}}\right)\left(v-b\right)=T\,,$ (9)

where $v=V/N$, $N$, $V$, $P$ and $T$ denote the specific volume, number of
constituents, global volume, pressure and temperature of the van der Waals
system, respectively. The positive constant $a$ and $b$ quantify the
attraction and finite size of the molecules in the fluid.

Figure 1: $P-v$ diagram of van der Waals fluids. The red dot-dashed line
indicates the critical isotherm at $T=T_{c}$. We have set $a=b=1$ (online
colors).

The qualitative behavior of $P-V$ isotherms is displayed in Fig. 1. It can be
seen that the _critical point_ of the liquid-gas phase transition occurs when
$P(v)$ has an inflection point, which is obtained by imposing

$\left(\frac{\partial P}{\partial
v}\right)_{T}=\left(\frac{\partial^{2}P}{\partial v^{2}}\right)_{T}=0\,.$ (10)

In this way, we obtain

$v_{c}=3b\,,\quad\,T_{c}=\frac{8a}{27b}\,,\quad\,P_{c}=\frac{a}{27b^{2}}\,,$
(11)

for the critical volume, temperature and pressure, respectively. It is
immediate to check that

$P_{c}v_{c}/T_{c}=3/8\,,$ (12)

which is a universal number predicted for all fluids (independently of the
constants $a$ and $b$).

To gain more insights on the phase transitions of van der Waals fluids, let us
introduce the (specific) Gibbs free energy, $G=G(P,T)$. For fixed $N$, this is
given by MannPT

$G(T,P)=-T\left\\{1+\log\left[\frac{\left(v-b\right)T^{\frac{3}{2}}}{\lambda}\right]\right\\}-\frac{a}{v}+Pv\,,$
(13)

where $v$ is to be understood as a function of pressure and temperature
through Eq. (9), while $\lambda$ is a (dimensional) specific constant of the
gas. The behavior of $G$ versus $T$ is shown in Fig. 2 for different $P$.
Below the critical pressure (yellow dotted curve), it exhibits the “swallow-
tail” shape characteristic of first order phase transitions from liquid to
gas. Such a feature disappears for $P>P_{c}$ (black solid line).

Figure 2:  Gibbs free energy $G$ versus temperature $T$ for various pressures
$P$, for van der Waals fluids. The red dot-dashed line indicates the critical
isobar at $P=P_{c}$ (online colors).

The behavior of the physical variables near the critical point is
quantitatively described by the critical exponents. Following MannPT , we
introduce

$t=\frac{T-T_{c}}{T_{c}}=\tau-1\,,\,\quad\,\,\phi=\frac{v-v_{c}}{v_{c}}=\nu-1\,.$
(14)

The basic critical exponents $\alpha,\beta,\gamma$ and $\delta$ are then
defined as follows (for computational details, see Gold ):

  * -

$\alpha$ governs the dynamics of the specific heat at constant volume $C_{v}$
according to $C_{v}=T\left(\frac{\partial S}{\partial
T}\right)_{v}\propto|t|^{-\alpha}$ . By explicit computation, one sees that
$C_{v}$ does not depend on $t$, which implies $\alpha=0$.

  * -

$\beta$ describes the behavior of the order parameter $\eta=v_{g}-v_{l}$ for a
given isotherm as $\eta\propto|t|^{\beta}$ , where $v_{g,l}$ denote the volume
of the gas and liquid phases, respectively. From the equation of corresponding
states for van der Waals fluids and the Maxwell’s equal area law, it follows
that $\beta=1/2$.

  * -

$\gamma$ measures the isothermal compressibility $\kappa_{T}$ of the fluid in
compliance with $\kappa_{T}=-\frac{1}{v}\left(\frac{\partial v}{\partial
P}\right)_{T}\propto|t|^{-\gamma}$ . By using again the equation of
corresponding states, one finds $\gamma=1$.

  * -

$\delta$ controls the difference $|P-P_{c}|$ on the critical isotherm
$T=T_{c}$ according to $|P-P_{c}|\propto|v-v_{c}|^{\delta}$. The study of the
shape of the critical isotherm gives $\delta=3$.

The above considerations provide the basics of our next analysis.
Specifically, we elaborate on the correspondence between phase transitions of
BHs and van der Waals fluids within Kaniadakis framework, with focus on the
$\kappa$-deformed analogues of Eq. (9)-(14).

##  III Kaniadakis thermodynamics of charged AdS black holes

The general static and spherically symmetric metric that describes
$(3+1)$-dimensional charged AdS BHs in the Schwarzschild coordinates
$(t,r,\theta,\phi)$ is given by Re

$ds^{2}=-f(r)dt^{2}+f(r)^{-1}dr^{2}+r^{2}d\Omega^{2}\,,$ (15)

where $d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\phi^{2}$ is the angular part
of the metric on the two-sphere. Here, we have defined222Following the study
of Saridakis:2020lrg ; Rani:2022xza ; Jawad:2022 ; Luciano:2023fyr ;
Basilakos:2023kvk  in Tsallis and Barrow frameworks, here we assume that
Kaniadakis model only modifies BH entropy, while leaving the field equations
of the theory unaffected. A comprehensive analysis of BH thermodynamics
involving an ab initio derivation of a lagrangian driven by the Kaniadakis
entropic index is reserved for the future.

$f(r)=1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}+\frac{r^{2}}{l^{2}}\,,$ (16)

where $M,Q$ are the mass and electric charge of the BH, respectively, while
$l$ is the AdS radius related to the (negative) cosmological constant by

$\Lambda=-\frac{3}{l^{2}}\,.$ (17)

In our setup, the parameter $M$ shall be associated with the enthalpy of the
BH conceived as a thermodynamic system. Clearly, for $Q=0$ and $l\gg r$, Eq.
(15) reduces to the well-known Schwarzschild metric. Additionally, the event
horizon $r_{+}$ of the geometry (15) corresponds to the largest root of
$f(r)=0$. One can use this solution to express the BH mass as

$M(r_{+})=\frac{r_{+}}{2}+\frac{Q^{2}}{2r_{+}}+\frac{r^{3}_{+}}{2l^{2}}\,.$
(18)

On the other hand, the surface area $A_{bh}$ of the BH horizon reads

$A_{bh}=\int_{r=r_{+}}\sqrt{g_{\theta\theta}\,g_{\phi\phi}}d\theta d\phi=4\pi
r_{+}^{2}\,.$ (19)

Accordingly, the Boltzmann-Gibbs-based Bekenstein-Hawking entropy obeys the
area law

$S_{BH}=\frac{A_{bh}}{4}=\pi r_{+}^{2}\,.$ (20)

As discussed in the Introduction, due to the area scaling of BH entropy,
arguments from multiple perspectives suggest that Boltzmann-Gibbs statistics
may not be the appropriate context for studying the thermodynamics of BHs. In
particular, in a relativistic scenario Eq. (20) is expected to be generalized
to Kaniadakis entropy (8), which we rewrite here by dropping for simplicity
the index $\kappa$

$S=\frac{1}{\kappa}\sinh\left(\kappa\,S_{BH}\right)\,.$ (21)

Two comments are in order: first, we observe that $S$ is a monotonically
increasing function of $S_{BH}$ and, thus, of the horizon radius $r_{+}$.
Furthermore, in order to provide analytical solutions, it proves sometimes
convenient to perform Taylor expansions for small $\kappa$ LympKan ; LucBar .
This assumption is substantiated by the best agreement between theoretical
predictions and phenomenological implications of Kaniadakis model, which is
obtained for $\kappa=0.2165$ in the physics of cosmic rays Kania1 . On the
other hand, observational Cosmology constrains Kaniadakis parameter around
zero Luciano:2022eio , while the interval $0<\kappa<1$ has been considered in
connection with the Bekenstein bound conjecture in Schwarzschild BHs
Abreu:2022pil . In what follows, we shall retain the exact expression of $S$
as far as possible, resorting to the small $\kappa$-approximation when
strictly necessary.

The thermodynamic picture of AdS BHs is completed by the introduction of an
extended phase space, where the pressure is identified with the cosmological
constant and the thermodynamic volume with its conjugate quantity, i.e.

$\displaystyle P$ $\displaystyle=$
$\displaystyle-\frac{\Lambda}{8\pi}=\frac{3}{8\pi l^{2}}\,,$ (22)
$\displaystyle V$ $\displaystyle=$ $\displaystyle\left(\frac{\partial
M}{\partial P}\right)_{S,Q}=\frac{4}{3}\pi r_{+}^{3}\,,$ (23)

respectively. Equipped with these new definitions, it is easy to check that
BHs still obey the first law of thermodynamics MannPT

$dM=TdS+\varphi dQ+VdP\,,$ (24)

and Smarr relation

$M=2\left(TS-VP\right)+\Phi Q\,,$ (25)

where

$T=\left(\frac{\partial M}{\partial
S}\right)_{P,Q}\,,\quad\,\Phi=\left(\frac{\partial M}{\partial
Q}\right)_{S,P}\,,$ (26)

are the temperature and electric potential, respectively.

Using the standard thermodynamic machinery, we now have all the ingredients to
compute the necessary BH thermodynamic variables. Since BH phase transitions
have been shown to occur in the canonical (fixed charge) ensemble Chamb1 ;
Chamb2 , we conduct our analysis in this framework. As a first step, we
express the mass parameter (18) in terms of the Kaniadakis entropy, using the
relation (8). This gives

$M(S)=\frac{\left(\pi lQ\kappa\right)^{2}+\pi
l^{2}\kappa\operatorname{ash}\left(\kappa
S\right)+\operatorname{ash}^{2}\left(\kappa
S\right)}{2\pi^{\frac{3}{2}}l^{2}\kappa^{\frac{3}{2}}\operatorname{ash}^{\frac{1}{2}}\left(\kappa
S\right)}\,,$ (27)

where we have introduced the shorthand notation

$\operatorname{ash}\left(x\right)\equiv\mathrm{arcsinh}\left(x\right)\,.$ (28)

One can verify that the limit for $\kappa\rightarrow 0$ of Eq. (27) reproduces
the standard expression of $M$ for charged AdS BHs (solid black lines in Fig.
3), namely

$M_{\kappa\rightarrow 0}(S)=\frac{S^{2}+\pi l^{2}\left(\pi
Q^{2}+S\right)}{2\pi^{\frac{3}{2}}l^{2}S^{\frac{1}{2}}}\,.$ (29)

Figure 3: The mass parameter $M$ versus the entropy $S$ for various values of
$\kappa$, and for $l=l_{c}$ fixed through the critical condition (40) (upper
panel), $l=2l_{c}$ (middle panel) and $l=0.5l_{c}$ (lower panel) (online
colors).

The behavior of the $\kappa$-deformed mass (27) versus $S$ is shown in Fig. 3
for various $\kappa,l$ and fixed $Q=0.25$ as in Soroushfar . According to our
previous considerations on the expected smallness of Kaniadakis exponent, we
restrict to the range $\kappa\leq 0.2$. Note that in cosmological
applications, $\kappa$ is found to be much closer to zero Hernandez-
Almada:2021aiw ; Hernandez-Almada:2021rjs , however this does not need to be
the case in BH applications (nevertheless, even in the present analysis one
could use arbitrarily small values of the entropic parameter). As we can see,
$M$ remains positive and shows an initially decreasing behavior (evaporation
phase of the BH), followed by a later growth (absorption process). While
leaving the initial stage of the evolution nearly unaffected, Kaniadakis
entropy influences the final growth rate of $M$. In particular, the higher
$\kappa$, the faster the growth, and vice-versa. For the sake of comparison
with recent literature, we emphasize that a similar result has been found in
the context of quantum gravity-induced deformations of Boltzmann-Gibbs entropy
Jawad:2022  and in non-linear electrodynamics and the Einstein-massive gravity
Nam .

Figure 4: The temperature $T$ versus the entropy $S$ for various values of
$\kappa$, and for $l=l_{c}$ fixed through the critical condition (40) (upper
panel), $l=2l_{c}$ (middle panel) and $l=0.5l_{c}$ (lower panel). In order to
reveal all features of the $T-S$ diagrams, in the middle panel we have
additionally considered smaller values of $\kappa$. The vertical lines in the
middle panel separate Region II \- Intermediate Black Hole (IBH) - from Region
I - Small Black Hole (SBH) - and Region III - Large Black Hole (LBH) - see
text (online colors).

The usage of Eq. (27) along with the first law of thermodynamics (24) allows
us to derive the $\kappa$-temperature of the thermal radiation emitted by BHs
as

$\displaystyle T(S)$ $\displaystyle=$ $\displaystyle\left(\frac{\partial
M}{\partial S}\right)_{P}$ $\displaystyle=$ $\displaystyle\frac{-\left(\pi
lQ\kappa\right)^{2}+\pi l^{2}\kappa\operatorname{ash}\left(\kappa
S\right)+3\operatorname{ash}^{2}\left(\kappa
S\right)}{4\pi^{\frac{3}{2}}l^{2}\left[\left(1+\kappa^{2}S^{2}\right)\kappa\operatorname{ash}^{3}\left(\kappa
S\right)\right]^{\frac{1}{2}}}\,,$

which, in the limit of $\kappa\rightarrow 0$, still recovers the usual result
(solid black line in Fig. 4)

$T_{\kappa\rightarrow 0}(S)=\frac{3S^{2}+\pi l^{2}\left(S-\pi
Q^{2}\right)}{4\pi^{\frac{3}{2}}l^{2}S^{\frac{3}{2}}}\,.$ (31)

We further notice that the above relation has the well-defined Schwarzschild
limit $T=1/(4\pi r)$ for $Q\rightarrow 0$ and $l\rightarrow\infty$.

Equation (III) is plotted as a function of $S$ in Fig. 4 for various
$\kappa,l$ and fixed $Q$ as before. The points where the slope of the $T-S$
graphs vanishes are of special interest, as they signal a potentially critical
behavior of BHs (see the next section for more quantitative discussion). From
the upper and lower panel of Fig. 4, we observe that $T$ increases
monotonically and has one (upper panel) or no (lower panel) stationary point,
depending on the value of $l$ (or, equivalently, of $P$).

On the other hand, the middle panel shows that $T$ increases for small (Region
I - Small Black Hole (SBH)) and large (Region III - Large Black Hole (LBH))
values of the entropy, while it decreases in the intermediate domain (Region
II - Intermediate Black Hole (IBH)). These regions are separated by two
stationary points, which have been marked by vertical lines for better visual
clarity. Effects of Kaniadakis entropy manifest through a variation of the
width of the IBH region, with higher $\kappa$ yielding a larger IBH domain and
vice-versa. Below, we shall see this behavior of $T-S$ graph is peculiar to a
first-order phase transition between SBH and LBH, which resembles in many
aspects the liquid-gas change of phase of van der Waals fluids. In this
picture, the larger amplitude of the IBH domain for higher $\kappa$ can be
understood in terms of the non-extensive character of Kaniadakis entropy by
looking at the BH structure on a molecular level (see Sec. IV). For higher
$\kappa$, indeed, the repulsive forces among BH microstructures tend to be
weaker, at least in the first stage of BH evolution, which implies a slowed
small-to-large phase transition of BH. To carry on the similarity with van der
Waals-like systems (where, however, interparticle forces are mostly
attractive), one can think of a Kanidakis BH as a fluid with stronger
attraction for larger $\kappa$. In this case, more heat is necessary to the
internal molecules to overcome these attractive interactions, which results in
a delayed liquid-gas change of phase.

Finally, regardless of the value of $l$, the condition $T(S_{0})=0$ gives the
_physical limitation point of BHs_. Indeed, for $S<S_{0}$ the temperature
becomes negative, which means this region is physically inaccessible.

Before moving on, we remark that the employment of generalized (non-extensive)
entropies like that in Eq. (8) leads to multiplicity in the temperature value
of BHs. In NRGS8  it has been observed that three different scenarios may
occur, depending on the assumed energy and temperature definitions. In
compliance with Rani:2022xza ; Jawad:2022 , here we are considering the energy
definition of GR, the first law of thermodynamics and the thermodynamic
temperature definition as fundamental. An alternative viewpoint has been
adopted in NRGS8 , based on the assumption that the Hawking temperature must
be kept unaffected. It is interesting to explore whether, and if so, how the
present results get modified in such a complementary approach. Investigation
along this direction is left for future work.

###  III.1 Heat capacity and critical point

With the help of the temperature (III), one can obtain the heat capacity at
constant pressure as

$\displaystyle C_{p}(S)=T\left(\frac{\partial S}{\partial T}\right)_{P}$
$\displaystyle=$
$\displaystyle-\frac{2}{\kappa}\left(1+\kappa^{2}S^{2}\right)^{\frac{3}{2}}\operatorname{ash}\left(\kappa
S\right)\left[\left(\pi lQ\kappa\right)^{2}-\operatorname{ash}\left(\kappa
S\right)\left(\pi l^{2}\kappa+3\operatorname{ash}\left(\kappa
S\right)\right)\right]$ (32) $\displaystyle\times\bigg{\\{}3\left(\pi
lQ\kappa\right)^{2}\left(1+\kappa^{2}S^{2}\right)-\operatorname{ash}\left(\kappa
S\right)\Big{\\{}\pi l^{2}\kappa\left[1+\kappa^{2}S\left(S-2\pi
Q^{2}(1+\kappa^{2}S^{2})^{\frac{1}{2}}\right)\right]$
$\displaystyle+\operatorname{ash}\left(\kappa
S\right)\left[-3+\kappa^{2}S\left(-3S+2\pi
l^{2}(1+\kappa^{2}S^{2})^{\frac{1}{2}}\right)+6\kappa
S\operatorname{ash}\left(\kappa
S\right)\left(1+\kappa^{2}S^{2}\right)^{\frac{1}{2}}\right]\Big{\\}}\bigg{\\}}^{-1}\,,$

which for $\kappa\rightarrow 0$ consistently reduces to (solid black line in
Fig. 5)

$C_{p,\kappa\rightarrow 0}(S)=\frac{2S\left[3S^{2}+\pi l^{2}\left(S-\pi
Q^{2}\right)\right]}{\pi l^{2}\left(3\pi Q^{2}-S\right)+3S^{2}}\,.$ (33)

Figure 5: The heat capacity at constant pressure $C_{p}$ versus the entropy
$S$ for various $\kappa$ values, and for $l=l_{c}$ fixed through the critical
condition (40) (upper panel), $l=2l_{c}$ (middle panel) and $l=0.5l_{c}$
(lower panel). For visual clarity, we have only considered two values of
$\kappa$ in the middle panel (online colors).

It is important to note that $C_{p}>0$ corresponds to local stability of BHs,
while for $C_{p}<0$ even small perturbations may cause BH disappearance. Also,
discontinuities potentially indicate a critical behavior of BHs.

Equation (32) is plotted as a function of $S$ in Fig. 5 for various $\kappa,l$
and fixed $Q$ as before. In compliance with the discussion below Eq. (III), it
is observed that $C_{p}$ has one (upper panel), two (middle panel) or no
(lower panel) discontinuity, depending on the value of $l$. The SBH, IBH and
LBH regions are clearly distinguishable from the middle panel (the vertical
lines correspond to the two stationary points of of the $T-S$ graphs in the
middle panel of Fig. 4). While SBH and LBH are thermodynamically stable
($C_{p}>0$), IBH is unstable ($C_{p}<0$). As discussed, for instance, in
MannPT  this gives rise to a transition between the SBH and LBH “phases”. As
$l$ progressively decreases to a certain critical value (upper panel), the IMB
range reduces to a point, which in turn corresponds to the single stationary
point of the $T-S$ graph in the upper panel of Fig. 4. By further decreasing
$l$ (lower panel), $C_{p}$ is always continuous and keeps positive values. In
this case, BHs are locally stable and do not undergo any transition (see also
the corresponding $T-S$ graphs in the lower panel of Fig. 4).

To better understand the origin of this critical value of $l$, we now derive
the analogue of the equation of state (9) for BHs in the extended phase space.
By using Eq. (22) and (III), after some algebra we get

$P(r_{+})=\cosh\left(\pi\kappa r_{+}^{2}\right)\hskip
1.13809pt\frac{T}{2r_{+}}+\frac{Q^{2}}{8\pi r_{+}^{4}}-\frac{1}{8\pi
r_{+}^{2}}\,,$ (34)

which is now straightforward to match with the $\kappa\rightarrow 0$ limit
MannPT

$P_{\kappa\rightarrow 0}(r_{+})=\frac{T}{2r_{+}}+\frac{Q^{2}}{8\pi
r_{+}^{4}}-\frac{1}{8\pi r_{+}^{2}}\,.$ (35)

To directly compare Eq. (34) with Eq. (9), let us identify the horizon radius
$r_{+}$ with the specific volume of van der Waals fluid as MannPT

$v=2r_{+}\,.$ (36)

In this way, we obtain

$P(v)=\cosh\left(\frac{\pi\kappa v^{2}}{4}\right)\hskip
1.13809pt\frac{T}{v}+\frac{2Q^{2}}{\pi v^{4}}-\frac{1}{2\pi v^{2}}\,.$ (37)

As discussed for van der Waals fluids, the critical point of phase transitions
can be derived from the conditions (10). However, analytical expressions for
the critical specific volume, temperature and pressure can only be obtained to
the leading order in $\kappa$. In this approximation, we are allowed to write
down

$\displaystyle v_{c}$ $\displaystyle=$ $\displaystyle
2\sqrt{6}Q+144\sqrt{6}\pi^{2}Q^{5}\kappa^{2}+\mathcal{O}(\kappa^{3})\,,$ (38)
$\displaystyle T_{c}$ $\displaystyle=$ $\displaystyle\frac{1}{3\sqrt{6}\pi
Q}+3\sqrt{6}\pi Q^{3}\kappa^{2}+\mathcal{O}(\kappa^{3})\,,$ (39)
$\displaystyle P_{c}$ $\displaystyle=$ $\displaystyle\frac{1}{96\pi
Q^{2}}+2\pi Q^{2}\kappa^{2}+\mathcal{O}(\kappa^{3})\,,\quad
l_{c}^{2}=\frac{3}{8\pi P_{c}}\,,$ (40)

all reducing to the standard critical expressions for $\kappa\rightarrow 0$
MannPT .

From Eq. (38), we can also infer the thermodynamic volume corresponding to the
critical volume $v_{c}$, which is

$\displaystyle V_{c}$ $\displaystyle=$ $\displaystyle\left(\frac{\partial
M}{\partial P}\right)_{S}\bigg{|}_{r_{+}=r_{c}}=\frac{4}{3}\pi r_{c}^{3}$ (41)
$\displaystyle=$ $\displaystyle 8\sqrt{6}\pi
Q^{3}+1728\sqrt{6}\pi^{3}Q^{7}\kappa^{2}+\mathcal{O}(\kappa^{3})\,.$

It is worth noting that $v_{c}\rightarrow 0$, while
$T_{c},P_{c}\rightarrow\infty$ for $Q\rightarrow 0$ regardless of $\kappa$,
which means that the critical transition described above is characteristic of
charged BHs also in Kaniadakis entropy model (on the other hand, in GB  it has
been found that AdS BHs in Gauss-Bonnet gravity undergo small-large
transitions in the uncharged case too).

Interestingly enough, the critical parameters (38)-(40) satisfy the relation

$\frac{P_{c}v_{c}}{T_{c}}=\frac{3}{8}+\frac{315}{4}\pi^{2}Q^{4}\kappa^{2}+\mathcal{O}(\kappa^{3})\,.$
(42)

By comparison with Eq. (12), we see that Kaniadakis entropy-based BHs slightly
deviate from pure van der Waals behavior, due to the $\kappa$-dependent
correction. The latter behavior is, however, recovered for $\kappa\rightarrow
0$.

Now, by introducing the re-scaled variables

$p=\frac{P}{P_{c}}\,,\quad\nu=\frac{v}{v_{c}}\,,\quad\tau=\frac{T}{T_{c}}\,,$
(43)

the equation of state (37) can be rearranged as

$8\tau=3\nu\left(Ap+\frac{2B}{\nu^{2}}\right)-\frac{D}{\nu^{3}}\,,$ (44)

where

$\displaystyle A$ $\displaystyle=$
$\displaystyle\frac{8P_{c}v_{c}}{3T_{c}\cosh\left(\frac{\pi\kappa
v^{2}}{4}\right)}$ (45) $\displaystyle=$ $\displaystyle
1+6\pi^{2}Q^{4}\left(35-3\nu^{4}\right)\kappa^{2}+\mathcal{O}(\kappa^{3})\,,$
$\displaystyle B$ $\displaystyle=$ $\displaystyle\frac{2}{3\pi
T_{c}v_{c}\cosh\left(\frac{\pi\kappa v^{2}}{4}\right)}$ (46) $\displaystyle=$
$\displaystyle
1-18\pi^{2}Q^{4}\left(7+\nu^{4}\right)\kappa^{2}+\mathcal{O}(\kappa^{3})$
$\displaystyle D$ $\displaystyle=$ $\displaystyle\frac{16Q^{2}}{\pi
T_{c}v_{c}^{3}\cosh\left(\frac{\pi\kappa v^{2}}{4}\right)}$ (47)
$\displaystyle=$ $\displaystyle
1-18\pi^{2}Q^{4}\left(15+\nu^{4}\right)\kappa^{2}+\mathcal{O}(\kappa^{3})\,.$

Equation (44) has the same structure as the law of corresponding states for
fluids MannPT

$8\tau=3\nu\left(p+\frac{2}{\nu^{2}}\right)-\frac{1}{\nu^{3}}\,.$ (48)

It is easy to check that this equation is correctly restored for
$\kappa\rightarrow 0$, since $A=B=D=1$ in this limit.

###  III.2 $P-v$ diagram

Figure 6: $P-v$ diagrams for $\kappa=0.1$ (top panel) and $\kappa=0.15$
(bottom panel). In each panel, the temperature of the isotherms decreases from
top to bottom. The red dot-dashed line indicates the critical isotherm at
$T=T_{c}$ (online colors).

We proceed by investigating the $P-v$ diagrams of AdS BHs as given in Eq.
(37). These diagrams are displayed in Fig. 6 for various $\kappa$, $T$ and
fixed $Q$ as before. By comparison with Fig. 1, we see that the isotherms at
$T<T_{c}$ (green and yellow curves) have van der Waals-like oscillations with
a local minimum and maximum. As $T$ increases to $T_{c}$ (red curve), the
oscillating branch squeezes and the two stationary points converge into the
inflection point $(P_{c},v_{c},T_{c})$ (see Eqs. (38)-(40)). This behaviour is
reminiscent of the van der Waals fluid transition. Though not changing the
qualitative behavior of the isotherms, Kaniadakis entropy non-trivially
affects the critical pressure and temperature at which such transition occurs.
For $T>T_{c}$ (blue and black curves), there are no more stationary points and
$P$ decreases monotonically along each isotherm.

###  III.3 Gibbs free energy

Let us now explore the global stability of charged AdS BHs in Kaniadakis
thermodynamics. For this purpose, we compute the Gibbs free energy as MannPT ;
Deng:2018wrd

$\displaystyle G(T,P)$ $\displaystyle=$ $\displaystyle
M-TS\,=\,\frac{3\left(Q^{2}+r_{+}^{2}\right)+8\pi Pr_{+}^{4}}{6r_{+}}$
$\displaystyle+\,\frac{\left[Q^{2}-r_{+}^{2}\left(1+8\pi
Pr_{+}^{2}\right)\right]}{4\pi\kappa r_{+}^{3}}\tanh\left(\pi\kappa
r_{+}^{2}\right)\,,$

where $r_{+}$ is to be regarded as a function of $P$ and $T$ through the
equation of state (34). Once more, one can check that the $\kappa\rightarrow
0$ limit gives back the classical Gibbs free energy for charged AdS BHs

$G_{\kappa\rightarrow
0}(T,P)=\frac{3Q^{2}}{4r_{+}}+\frac{r_{+}}{4}-\frac{2}{3}\hskip 0.56905pt\pi
Pr_{+}^{3}\,.$ (50)

Figure 7: The Gibbs free energy $G$ versus the temperature $T$, for
$\kappa=0.1$ (the same qualitative behavior is obtained for other values of
$\kappa$). The red dot-dashed line indicates the critical isobar at $P=P_{c}$.
SBH denotes Region I - Small Black Hole, IBH denotes Region II - Intermediate
Black Hole and LBH denotes Region III - Large Black Hole, see text (online
colors).

The behavior of Eq. (III.3) as a function of $T$ is shown in Fig. 7, to be
compared with Fig. 2. Consistently with the previous discussion, it is
observed that, below the critical pressure (dotted yellow line), $G$ exhibits
has the swallow tail behavior typical of first order phase transitions.
Specifically, in the first branch BHs are in the SBH domain. As $T$ increases
to the critical point $O$, SBH and LBH phases coexist, since they have the
same Gibbs free energy. As noted in MannPT , the coexistence line in the $P-T$
plane can be derived by using Maxwell’s equal area law or finding a curve for
which $G$ and $T$ coincide for SBH and LBH. This line is plainly visible from
the 3D plot in Fig. 8. Above the critical temperature, LBH becomes the
preferred thermodynamic state because of its lower Gibbs free energy.
Therefore, there is a first-order small-large phase transition at the point
$O$. Clearly, owing to the definition (8) of entropy, different horizon areas
for the SBH and LBH during this transition correspond to a discontinuity in
the entropy (and also in the thermodynamic volume, see Eq. (23)) and, thus, to
the release of latent heat.

###  III.4 Behavior near the critical point

For quantitative discussion of the behavior of BHs approaching the critical
point, we now calculate the critical parameters as defined at the end of Sec.
II. First, we introduce the free energy

$\displaystyle F(T,V)$ $\displaystyle=$ $\displaystyle G-PV\,.$ (51)

By using Eqs. (23) and (III.3), we get

$F=\frac{1}{2}\left[\frac{Q^{2}}{r_{+}}+r_{+}-\frac{2T\sinh\left(\pi\kappa
r_{+}^{2}\right)}{\kappa}\right]\,.$ (52)

Accordingly, the entropy is

$S(T,V)=-\left(\frac{\partial F}{\partial
T}\right)_{V}=\frac{\sinh\left(\pi\kappa r_{+}^{2}\right)}{\kappa}\,,$ (53)

consistently with Eq. (8). In turn, from the definition below Eq. (14), we
find that $C_{V}=0$, which yields $\alpha=0$.

Figure 8: 3D plot of the Gibbs free energy $G$ versus pressure the $P$ and
temperature $T$, for $\kappa=0.1$ (the same qualitative behavior can be
obtained for other values of $\kappa$). The coexistence line between the SBH
and LBH phases in the $P-T$ plane is visible.

In order to compute $\beta$, we approximate Eq. (44) around a critical point.
We use the re-scaled coordinates (14), here rewritten for convenience as

$t=\frac{T}{T_{c}}-1\,,\qquad\omega=\frac{V}{V_{c}}-1\,.$ (54)

In the approximation of small $\kappa$, we obtain

$\displaystyle p$ $\displaystyle=$ $\displaystyle
1+\left(\frac{8}{3}-512\pi^{2}Q^{4}\kappa^{2}\right)t+\left(-\frac{8}{3}+704\pi^{2}Q^{4}\kappa^{2}\right)t\omega$
(55)
$\displaystyle+\left(-\frac{4}{3}+1120\pi^{2}Q^{4}\kappa^{2}\right)\omega^{3}+\mathcal{O}\left(t\omega^{2},\omega^{4}\right)\,,$

where the various terms have been grouped together in this specific way for a
direct comparison with MannPT . It is worth noting that the re-scaled pressure
as appears in Eq. (55) contains terms of order higher than the qudratic in
$\kappa$, due to the implicit $\kappa$-dependence of $t,\omega$. However, for
our purpose of computing Kaniadakis corrections to critical exponents, it is
useful to present Eq. (55) et seq. in their current form and restore the
leading order at the end. For more details on the validity of the series
expansion respect to $t,\omega$, see MannPT .

Now, differentiation of Eq. (55) respect to $\omega$ at a fixed $t<0$ gives

$\displaystyle dp$ $\displaystyle=$ $\displaystyle
P_{c}\left[\left(-\frac{8}{3}+704\pi^{2}Q^{4}\kappa^{2}\right)t+\left(-4+3360\pi^{2}Q^{4}\kappa^{2}\right)\omega^{2}\right]d\omega\,.$

By employing Maxwell’s equal area law

$\oint vdP=0\,,$ (57)

along with the knowledge that there is no pressure variation during the phase
transition, we obtain

$\displaystyle
1+\left(\frac{8}{3}-512\pi^{2}Q^{4}\kappa^{2}\right)t+\left(-\frac{8}{3}+704\pi^{2}Q^{4}\kappa^{2}\right)t\omega_{l}+\left(-\frac{4}{3}+1120\pi^{2}Q^{4}\kappa^{2}\right)\omega_{l}^{3}$
$\displaystyle=1+\left(\frac{8}{3}-512\pi^{2}Q^{4}\kappa^{2}\right)t+\left(-\frac{8}{3}+704\pi^{2}Q^{4}\kappa^{2}\right)t\omega_{s}+\left(-\frac{4}{3}+1120\pi^{2}Q^{4}\kappa^{2}\right)\omega_{s}^{3}$
(58)

and

$0=\int_{\omega_{l}}^{\omega_{s}}\omega\left[\left(-\frac{8}{3}+704\pi^{2}Q^{4}\kappa^{2}\right)t+\left(-4+3360\pi^{2}Q^{4}\kappa^{2}\right)\omega^{2}\right]d\omega\,,$
(59)

where $\omega_{s,l}$ denote the “specific volume” of the small and large
phases of BHs, respectively. One can verify that the only non-vanishing
solution that reduces to the standard one for $\kappa\rightarrow 0$ is MannPT

$\omega_{s}=-\omega_{l}=\sqrt{-2t}\left(1+288\pi^{2}Q^{4}\kappa^{2}\right)\,.$
(60)

Thus, from the definition of the critical exponent $\beta$ below Eq. (14), it
follows that

$\eta=V_{c}\left(\omega_{l}-\omega_{s}\right)=2V_{c}\omega_{l}\propto\sqrt{-t}\,\,\,\Longrightarrow\,\,\,\beta=\frac{1}{2}\,.$
(61)

As concerns the exponent $\gamma$, we need to differentiate Eq. (55) as

$\left(\frac{dV}{dP}\right)_{T}=\frac{V_{c}}{P_{c}}\left(\frac{d\omega}{dp}\right)_{T}=-\frac{3}{8}\frac{V_{c}}{P_{c}}\frac{1}{t}\left(1+264\pi^{2}Q^{4}\kappa^{2}\right).$
(62)

Hence, the isotherm compressibility $\kappa_{T}$ of BHs takes the form

$\kappa_{T}=-\frac{1}{V}\hskip 0.56905pt\left(\frac{\partial V}{\partial
P}\right)_{T}\propto\frac{1}{t}\,,$ (63)

which implies $\gamma=1$.

Lastly, the shape of the critical isotherm $t=0$ and the related
$\delta$-exponent are given by

$p-1=\left(-\frac{4}{3}+1120\pi^{2}Q^{4}\kappa^{2}\right)\omega^{3}\,\,\,\Longrightarrow\,\,\,\delta=3\,.$
(64)

In spite of the non-trivial modifications induced by the $\kappa$-deformed
entropy to the critical pressure, volume and temperature, the basic critical
exponents remain unaffected. This allows to conclude that the qualitative
similarity between Kaniadakis BHs and van der Waals fluids near the critical
point holds at quantitative level too.

###  III.5 Sparsity of black hole radiation

Although BHs nearly behave like a black body and spontaneously emit particles
at a temperature proportional to their surface gravity, the flow of Hawking
radiation exhibits some peculiar features. For instance, it is known to be
more sparse than black body radiation. Quantitatively speaking, such a
difference can be estimated through the computation of the so-called sparsity,
which is a measure of the average time-gap between the emission of successive
quanta defined by

$\tilde{\eta}=\frac{C}{g}\left(\frac{\lambda_{t}^{2}}{A_{eff}}\right),$ (65)

(we have used the symbol $\tilde{\eta}$ instead of the traditional $\eta$ to
avoid confusion with the critical exponent (61)). Here, the constant $C$ is
dimensionless, while $g$, $\lambda_{t}=2\pi/T$ and $A_{eff}=27A_{bh}/4$ denote
the spin degeneracy factor of the emitted quanta, the thermal wavelength and
the effective horizon area of the BH, respectively. For Schwarzschild BHs and
emission of massless bosons, one has $\lambda_{t}=2\pi/T_{H}=8\pi^{2}r_{h}$,
which entails

$\tilde{\eta}_{H}=\frac{64\pi^{3}}{27}\approx 73.49\gg 1\,.$ (66)

For comparison, we remind that $\tilde{\eta}\ll 1$ in the case of a black
body.

Effects of deformed entropies on sparsity of Schwarzschild BHs have been
recently considered in literature (see Cimi ; Alonso ; Alonsobis ; Rainb  and
references therein). For instance, in Cimi  it has been shown that generalized
models of Heisenberg relation combined with non-extensive (Rényi, Tsallis-
Cirto, Kaniadakis, Sharma Mittal and Barrow) entropies lead to substantial
modifications of the sparsity, which turns out to be mass dependent. A similar
statement has been claimed in Rainb  in rainbow gravity. The question arises
as to how such results appear for charged AdS BHs in Kaniadakis framework.

Figure 9: The sparsity $\tilde{\eta}_{\kappa}$ versus the entropy $S$, for
various $\kappa$ and $l=2$. The vertical lines represent the physical
limitation entropy $S_{0}$ for each curve. For comparison, we have also
depicted the sparsity of Schwarzschild BHs with emission of massless bosons
(yellow curve) (online colors).

To compute the $\kappa$-deformed sparsity, we resort to the definition (65)
equipped with Eq. (III). Straightforward calculations yield

$\tilde{\eta}_{\kappa}=\tilde{\eta}_{H}\frac{\pi^{2}l^{4}\kappa^{2}\left(1+\kappa^{2}S^{2}\right)\operatorname{ash}^{2}\left(\kappa
S\right)}{\left[-\left(\pi lQ\kappa\right)^{2}+\pi
l^{2}\kappa\operatorname{ash}\left(\kappa
S\right)+3\operatorname{ash}^{2}\left(\kappa S\right)\right]^{2}}.$ (67)

We notice that in the Schwarzschild limit (i.e. $Q=0$ and sufficiently large
AdS radius $l$), this expression reduces to

$\tilde{\eta}_{\kappa}(Q=0,l\rightarrow\infty)=\tilde{\eta}_{H}\left(1+\kappa^{2}S^{2}\right),$
(68)

which coincides with the result of Cimi  for Schwarzschild BHs. By further
imposing $\kappa\rightarrow 0$, we have
$\tilde{\eta}_{\kappa}=\tilde{\eta}_{H}$, as expected.

The behavior of the $\kappa$-modified sparsity (67) for various $\kappa$ and
fixed $l=2$, $Q=0.25$ is plotted in Fig. 9, which shows an apparent divergence
for a certain ($\kappa$-dependent) value of $S$. This singularity, however,
lies at the physical limitation point $S_{0}$ (see the discussion below Eq.
(III)), as it is easy to understand from the definition (65). Thus, it is
unphysical and we only have to consider the region $S>S_{0}$ delimited by the
vertical line. We can see that the $\kappa$-deformed sparsity lies always
above the $\kappa=0$ (black) curve, in such a way that increasing the value of
$\kappa$ directly results in sparser Kaniadakis BH radiation. This is in line
with the result of Cimi . On the other hand, $\tilde{\eta}_{\kappa}$ is
greater than the sparsity $\tilde{\eta}_{H}$ of Hawking radiation of
Schwarzschild BHs (yellow curve) for sufficiently small and large entropies,
where it significantly departs from the black body-like behavior, while it
falls below in the intermediate range.

##  IV Geometrothermodynamics of charged AdS black holes

Since it is possible to define a temperature for BHs, it is natural to think
of an associated substructure. Recently, special care has been devoted to
analyze the microscopic constituents and underlying interactions of BHs
Cai:1998ep ; Wei:2015iwa ; Wei:2019uqg ; Guo:2019oad ; Xu:2020gud ;
Ghosh:2020kba ; Xu:2020ftx ; Prom ; Dehghani:2023yph ; Santos:2023eqp , which
can be described in the same fashion as the molecules of a non-ideal fluid.

To investigate phenomenologically the nature of interactions among BH
microstructures, the common technique consists in studying the thermodynamic
geometry of the whole macroscopic system. In this perspective, the analysis of
Weinhold Wein1  and Ruppeiner Rupp1 ; Rupp2  geometries has proved to give
qualitative insights on the internal dynamics of ordinary thermodynamics
systems via exploring the sign of the corresponding metric curvature.
Specifically, negative (positive) scalar curvatures emerge for prevailing
attractive (repulsive) microinteractions, while flatness characterizes non-
interacting systems, such as the ideal gas, or systems where interactions are
perfectly balanced.

In the effort to probe the character of BH microinteractions, Weinhold and
Ruppeneir formalisms have been adapted to BH thermodynamics. This kind of
study has been first developed for Banados, Teitelboim and Zanelli (BTZ) BHs
Cai:1998ep  and later extended to Reissner-Nordström, Kerr and Reissner-
Nordström-AdS BHs Shen:2005nu . In the plethora of results obtained so far,
there is general consensus that the scalar curvature of BH systems with
charged molecules should be positive, revealing a repulsive behavior of
microinteractions Wei:2015iwa ; Guo:2019oad ; Wei:2019uqg ; Ghosh:2020kba .

In order to figure out to what extent Kaniadakis’ prescription (8) affects the
above conclusion, let us compute Weinhold and Ruppeneir scalar curvature in
Kaniadakis entropy-based thermodynamics. Toward this end, we remind that
Weinhold metric is defined as the second derivative of internal energy of the
system with respect to given thermodynamic variables Wein1 . For the case of
BHs, by identifying the internal energy with the mass, we obtain

$\displaystyle
g_{ij}^{w}=-\partial_{i}\partial_{j}M(S,P,Q)\,\,\,\,\Longrightarrow\,\,\,\,ds^{2}_{w}=g^{w}_{ij}dx^{i}dx^{j}\,,$
(69)

where we have we have generically denoted the independent fluctuation
coordinates by $x^{i}$.

Similarly, in Ruppeiner formalism one considers the entropy as basic
thermodynamic potential, i.e.

$g_{ij}^{Rup}=-\partial_{i}\partial_{j}S\,.$ (70)

From Eqs. (III), (69) and (70), it follows that the line elements of Weinhold
and Ruppeiner are connected each other via the conformal transformation
Mrugala

$ds^{2}_{R}=\frac{ds^{2}_{w}}{T}\,.$ (71)

We focus our next geometrothermodynamic analysis on Eq. (71). Considering the
entropy and pressure as the fluctuation coordinates, while keeping $Q$ fixed,
we obtain the following expression for the Ruppeiner scalar curvature:

$\displaystyle R^{Rup}(S,P)$ $\displaystyle=$
$\displaystyle\left(1+\kappa^{2}S^{2}\right)^{-\frac{1}{2}}\left[\operatorname{ash}\left(\kappa
S\right)\right]^{-1}$ (72) $\displaystyle\times\kappa^{2}\left[-2\pi
Q^{2}\kappa+\operatorname{ash}\left(\kappa S\right)\right]$
$\displaystyle\,\times\left\\{\pi\left(Q\hskip
0.56905pt\kappa\right)^{2}-\operatorname{ash}\left(\kappa
S\right)\left[\kappa\\!+\\!8P\operatorname{ash}\left(\kappa
S\right)\right]\right\\}^{-1}\,,$

which reduces to the standard curvature for charged AdS BHs in the
$\kappa\rightarrow 0$ limit Guo:2019oad

$R^{Rup}_{\kappa\rightarrow 0}(S,P)=\frac{2\pi Q^{2}-S}{S\left[-\pi
Q^{2}+S\left(1+8PS\right)\right]}\,.$ (73)

Figure 10: The Ruppeiner scalar curvature $R^{Rup}$ versus the entropy $S$,
for various values of $\kappa$. The parameter $l$ is fixed to
$l=\sqrt{2}l_{c}$ defined through the critical condition (40). The vertical
lines represent the physical limitation entropy $S_{0}$ for each curve (online
colors).

The curvature (72) versus $S$ is displayed in Fig. 10 for various $\kappa$ and
fixed $Q=0.6$, $P=0.5P_{c}$. As discussed for the sparsity above, the physical
region is delimited by $S>S_{0}$ (vertical lines). Although Eq. (72) is non-
trivially modified comparing to the classical curvature (73), Kaniadakis
entropy does not affect the overall sign of $R^{Rup}$, which is still positive
and indicates prevailing repulsive interactions among BH microstructures. As
$S$ increases, $R^{Rup}$ gradually decreases, which means that the repulsion
progressively fades, possibly due to thermal fluctuations and/or molecular
collisions. Kaniadakis corrections here manifest through a variation of the
rate of decrease, with higher $\kappa$ corresponding to faster decrease for
sufficiently small $S$, and vice-versa. This tendency is reversed for $S$
large enough. The former behavior resembles the physics of composite systems
with non-extensive (and, in particular, superadditive) entropy. Indeed, for
such systems the single constituents tend to merge more strongly than the
classical extensive case Vedral , thus balancing swiftly the effects of
internal repulsive forces. Asymptotically (i.e. for large BH horizon radii),
the internal microstructures end up being so far apart that
$R^{Rup}\rightarrow 0$, which reveals that BHs behave as effectively non-
interacting.

Finally, Fig. 11 shows the 3D plot of $R^{Rup}$ versus $S$ and $P$ for
$\kappa=0.2$ and fixed $Q$ as before. We can see that the scalar curvature
remains positive even for varying $P$, which supports previous arguments on
the repulsive nature of BH micro-interactions.

Figure 11: 3D plot of the Ruppeiner scalar curvature $R^{Rup}$ versus entropy
$S$ and pressure $P$, for $\kappa=0.2$ and $Q=0.6$.

##  V Conclusions and Discussion

Geometrothermodynamics and phase transitions of charged AdS BHs have been
addressed within the framework of Kaniadakis theory, which arises from a self-
consistent relativistic generalization of the classical statistical mechanics.
The latter is coherently recovered by setting the deformation parameter
$\kappa$ to zero. We would like to stress that the highlight of the present
analysis is to deepen our knowledge of BH thermodynamics in a fully
relativistic statistical scenario. As far as we know, this is the first work
where this scenario is addressed.

Following the standard literature, the study has been conducted by identifying
the cosmological constant and its conjugate quantity with the thermodynamic
pressure and volume, respectively. In the ensuing extended phase space, we
have examined the impact of Kaniadakis entropy on the formal duality black-
hole/fluid, showing that Kaniadakis BHs still exhibit a van der Waals-like
first order phase transition.

Although Kaniadakis corrections do not affect the qualitative behavior of
$P-v$ diagrams and the basic critical exponents, the critical volume, pressure
and temperature are non-trivially modified, even at the leading order in the
deformation parameter $\kappa$ (see Eqs. (38)-(40)). Should we have access to
the phenomenology of AdS BHs and measure such quantities, we could elaborate
more on the role of Kaniadakis entropy in BH physics and possibly constrain
$\kappa$-corrections. We have finally probed the nature of interactions among
BH micro-structures. Using the picture of fluid-like interacting molecules, we
have applied Ruppeiner geometrothermodynamic formalism and computed the scalar
curvature of the associated metric. The investigation of the sign of the
Ruppeiner scalar curvature $R^{Rup}$ reveals that these micro-interactions are
prevailing repulsive and tend to vanish for sufficiently large BH horizon
radii, with the $\kappa$-parameter ruling the rate of decrease. In passing, we
mention that a possible explanation for this behavior can be provided based on
the physics of the two fluid model, where the dominant character of
interactions is determined by the relative number densities of the molecules
of the two fluids Guo:2019oad .

As future prospects, it would be interesting to enrich the above analysis by
considering the presence of global monopoles, which are known to have non-
trivial effects on BH physics Jing:1993np ; Yu:1994fy ; Li:2002ku ;
Jiang:2005xb ; Carames:2017ngt . Furthermore, one can study Kaniadakis
entropy-based thermodynamics of other BHs, such as rotating or exotic BTZ BHs,
and additionally examine its effect on the primordial black holes and
stochastic gravitational waves Papanikolaou:2022did ; Basilakos:2023xof . On
the other hand, inspired by Cimi , it is suggestive to understand how BH
critical phenomena appear in the context of modified uncertainty principles
Mignemi:2009ji ; LucEUP  combined with nonextensive entropies, and possibly
connect the two frameworks. The study of these aspects is under active
consideration and will be presented elsewhere.

###### Acknowledgements.

GGL would like to thank Jaume Giné for useful comments on the original
manuscript. He is also grateful to the Spanish “Ministerio de Universidades”
for the awarded Maria Zambrano fellowship and funding received from the
European Union - NextGenerationEU. The authors acknowledge the contribution of
the LISA CosWG and of COST Actions CA18108 “Quantum Gravity Phenomenology in
the multi-messenger approach” and CA21136 “Addressing observational tensions
in cosmology with systematics and fundamental physics (CosmoVerse)”.

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