# A Heuristic Informative-Path-Planning Algorithm for Autonomous Mapping of
Unknown Areas

{mo365<EMAIL_ADDRESS>Mobolaji O. Orisatoki, Mahdi Amouzadi, Arash
M. Dizqah  Smart Vehicle Control Laboratory (SVeCLab), School of Engineering
and Informatics, University of Sussex, Brighton, UK

###### Abstract

Informative path planning algorithms are of paramount importance in
applications like disaster management to efficiently gather information
through a priori unknown environments. This is, however, a complex problem
that involves finding a globally optimal path that gathers the maximum amount
of information (e.g., the largest map with a minimum travelling distance)
while using partial and uncertain local measurements. This paper addresses
this problem by proposing a novel heuristic algorithm that continuously
estimates the potential mapping gain for different sub-areas across the
partially created map, and then uses these estimations to locally navigate the
robot. Furthermore, this paper presents a novel algorithm to calculate a
benchmark solution, where the map is a priori known to the planar, to evaluate
the efficacy of the developed heuristic algorithm over different test
scenarios. The findings indicate that the efficiency of the proposed
algorithm, measured in terms of the mapped area per unit of travelling
distance, ranges from 70% to 80% of the benchmark solution in various test
scenarios. In essence, the algorithm demonstrates the capability to generate
paths that come close to the globally optimal path provided by the benchmark
solution.

###### Index Terms:

Non-Convex Areas, Completely Unknown Areas, Coverage Path Planning, Voronoi
partitioning, Estimation of Information

##  I Introduction

Mapping an unknown area by a robot is increasingly important in an ever-
growing number of industries operating in environments detrimental to humans
health, for example, mining, contaminated areas, fire and forestry, just to
name a few [1, 2]. These industries need robots that can autonomously plan
their motion to explore unknown dangerous areas. Moreover, in such precarious
environments, robots must be able to complete the given information-gathering
task in the shortest possible time or distance due to their limited storage
capacity of energy. Such path planning within unknown and complex (e.g., non-
convex) areas remains an open research challenge.

Path planning for mapping has been studied by researchers in the past,
however, a majority of these studies focus on cases where the map is known or
partially known and the problem is to identify a priori unknown obstacles
within the map. [3] provides a survey on these studies.

For example, the authors in [1, 4, 5] provided a solution to such a path
planning problem where the robots meet a given set of way-points on the known
map with a minimum travelling distance while following the rules of the
environment (i.e., no collision with the obstacles and walls). The works in
[4] and [5] also propose a coverage path planning (CPP) algorithm to determine
these way-points ensuring the robot visits all areas of the map without
overlapping paths. A CPP-led robot must generate a path that maximises the
coverage of an area while avoiding obstacles. [6] provides a survey on
different CPP algorithms.

The work in [4] also introduces a technique to optimally move a robot within a
set of predetermined points starting from an initial location to the
neighbouring closest way-point one after another so that the total coverage
path is traversed, moving the robot in a boustrophedon pattern [7].
Alternatively, the authors in [8] proposed a solver to the CPP problems by
generalising the travelling salesperson problem (TSP) in which the salesperson
wishes to find the shortest path that visits a list of given clients within
neighbourhoods where they are willing to meet. This technique is explored
further in [9].

Moreover, the authors in [10] proposed a solution to a different but still
relevant path planning problem where the robot needs to find a path on a graph
with a minimum expected cost until success. Each node of the graph is assumed
to carry a probability of failure and the mission success is defined as
reaching a terminal node with zero probability of failure. The work also
covers the cases where there are no such terminal nodes and the objective is
to find a non-revisiting path with the minimum expected cost that traverses
all the nodes carrying a non-unity probability of failure.

The aforementioned methods are non-adaptive and do not update the covering
partitions to recalculate the travelling paths of robots as new obstacles are
identified. The authors in [11] and [12], on the other hand, proposed an
informative path planning (IPP) where the robots continuously re-partition the
known area based on newly identified obstacles. By solving an IPP problem, a
robot generates an efficient path based on maximising the information that can
be gathered. However, these works still focus on the efficient mapping problem
of the partially known environment where the boundary of the area and the
initial and final points are known.

Alternatively, the authors in [13] proposed an information-theoretic measure
to estimate the mapping gains of each perceived cell, i.e., the rate of
expanding the map if the robot moves to each cell. They proposed an algorithm
to calculate this mapping gain and showed that a path-planning strategy which
maximises this mapping gain eventually moves the robot to unexplored areas.

The authors in [14] refined the measure in [13] using the Cauchy-Schwarz
quadratic information (CSQMI), instead of the odds ratio of events, which they
showed is more computationally effective for 3D mapping. The authors also
proposed a path-planning algorithm to find out the next action that maximises
the CSQMI normalised by the execution time of the action.

The works in [15] combine the already applied frontier exploration planning
(FEP) and receding horizon next-best-view planning (RH-NBVP) algorithms to
explore unknown environments. The works show that such a combination recovers
the disadvantages of the algorithms which are the slowness of FEP and the high
risk of RH-NBVP getting stuck in large areas.

Similar to [13, 14, 15], the authors in [16] also considered the problem where
the map is fully unknown in advance and proposed a sampling-based approach to
explore such environments. In particular, they proposed a RRT*-based online
path planning algorithm which is shown more effective as compared to other
recent algorithms to explore more complex environments.

Unlike the other research, the works in [13, 14, 15, 16] address the problem
with no prior knowledge of the boundaries and obstacles. However, they do not
investigate the optimality of the generated path in terms of travelling
distance or time. Since each segment of the path is generated online using the
partially available measurements, an analysis showing how the set of locally
generated paths is close to the global optimum solution is of paramount
importance. Moreover, they do not consider the error between the measured
distance of a point and its real location which causes the mapping of points
beyond walls. This error requires embedding an explicit obstacle-avoidance
mechanism into the path-planning algorithm.

In summary, there is extensive prior research on path planning for gathering
information. Many of the proposed algorithms only deal with partially unknown
maps where only the location of obstacles is not known. The papers dealing
with fully unknown environments, on the other hand, do not investigate the
optimality of their solutions. This paper addresses these gaps by the
following novel contributions to the knowledge:

  * •

A heuristic informative path planning (HIPP) algorithm that uses narrow-beam
range detectors to create a map of the unknown non-convex areas with the
shortest travelling distance. The proposed algorithm novelly approximates the
mapping gain of all the identified cells online to locally find the next move
of the robot which maximises the mapping gain with the shortest possible
travelling distance.

  * •

An algorithm to approximate a near-global-optimal solution to the posterior
problem as a benchmark to evaluate the results of the developed HIPP
algorithm. In the posterior problem, the map is known to the planner which
finds the shortest path for re-mapping of the known map. This algorithm can be
used to calculate a benchmark for other IPP algorithms for mapping unknown
environments.

  * •

An optimality investigation of the resulting HIPP’s solutions, indicating a
linear increase of the reward over the majority of the travelling time which
matches the performance of the benchmark solution.

The remainder of this paper is organised as follows: Section II provides a
description of the involved components and their models. Section III
introduces the proposed HIPP algorithm including the problem formulation and
design of a solver. Section IV presents an algorithm to generate a benchmark
solution which is then used in Section V to evaluate the efficacy of the
proposed HIPP algorithm. This section provides the simulation results and
discussions which is followed by a conclusion in Section VI.

Figure 1: The single-axle robot in this research along with the measurement
uncertainty of the LiDAR sensor which is used to develop the proposed HIPP.
$D$ is the distance of the real obstacle but due to the sensor uncertainty,
$\tilde{D}$ is measured for the presumed obstacle. $\alpha$ is the angle
between the scanning ray and the x-axis of the robot and $\theta$ is the
heading angle of the robot. ${R}$ represents the nominal range of the sensor.

##  II System description

Fig.1 shows a differential single-axle planar robot which is used throughout
this paper to generate results. ${V_{L}}\,(m/s)$ and ${V_{R}}\,(m/s)$ are the
linear speeds of the left and right wheels on the x-axis of the inertial
frame. Any difference between these two speeds lays an inertial yaw rate on
the vehicle, which is assumed to be the same as the variation of heading angle
${\theta}$ in the no-slip conditions.

Ignoring tyre slips, the kinematics of wheels are as follows:

$\begin{split}{V_{L}}={\omega_{L}r};\;\;{V_{R}}={\omega_{R}r}.\end{split}$ (1)

where ${r}\,(m)$ is the radius of the wheels and
${\omega_{i},\,i\in\\{L,R\\}}\,(rad/s)$ is the rotational speed of the left
and right wheels.

The kinematics of the whole robot is then approximated as follows:

$\begin{split}{V_{c}}={\frac{V_{R}+V_{L}}{2}};\;\;\dot{\psi}={\frac{V_{R}-V_{L}}{d}}.\end{split}$
(2)

where $V_{c}\,(m/s)$ is the longitudinal speed of the robot in the x-axis of
the moving frame of reference and $\psi\,(rad)$ is the yaw angle and hence
$\dot{\psi}\,(rad/s)$ is yaw rate. As mentioned, it is assumed that
$\dot{\psi}=\dot{\theta}$, ignoring the slips.

It is accepted in this paper that the robot is able to estimate its global
location $[X_{r}\,\,Y_{r}]^{T}$ within the map using a simultaneous
localisation and mapping (SLAM) algorithm.

The robot is equipped with a 2D narrow-beam light detection and ranging
(LiDAR) sensor which is located in its centre of gravity. This sensor emits 72
rays as per rotation and hence the angular scanning resolution is
$\frac{360^{\circ}}{72}$ or 5∘. The reflected pulses are used to measure the
distance of the obstacle, which is then summed up with the robot’s location to
estimate the obstacle’s location. If there is no reflection, it assumes no
obstacle within the nominal range ${R}$ of sensor [17].

This paper uses different models of the LiDAR sensor for the HIPP and
posterior problems. While the sensor model must simulate an uncertain
behaviour for the HIPP problem with respect to the range and direction of
scans, the sensor model in the posterior problem, on the other hand, simulates
laser scans with a certain range and direction.

Fig.1 shows an example measurement by an uncertain LiDAR sensor. While the
location of an obstacle is measured as $[X_{o}\,\,Y_{o}]^{T}$, an additive
white Gaussian noise (AWGN) is added to each measurement to imitate the
uncertainty of the sensor:

$\begin{bmatrix}\tilde{X_{o}}\\\
\tilde{Y_{o}}\end{bmatrix}=\begin{bmatrix}X_{o}\\\
Y_{o}\end{bmatrix}+\gamma;\;\;\gamma\sim\mathcal{N}(\begin{bmatrix}0\\\
0\end{bmatrix},\begin{bmatrix}\sigma_{XX^{2}}&0\\\
0&\sigma_{YY^{2}}\end{bmatrix}).$ (3)

where $\gamma$ is a multivariate random number with normal distribution. The
uncertainty in the X and Y directions are independent with a variance of
respectively $\sigma_{xx}$ and $\sigma_{yy}$. The difference between these two
values is a model depicting the uncertain deflection of the rays.

If there is no obstacle, the reading $[X_{o}\,\,Y_{o}]^{T}$ will be
$[R\cos(\theta)\,\,R\sin(\theta)]^{T}$ which is then summed up by an AWGN
$\gamma$.

Fig.2 compares sensor measurements for the ideal and uncertain cases. As
mentioned, the sensor model for the posterior problem represents an ideal
sensor with a certain range of $R$ (which equals $1\times$ cell width in Fig.
2). In this case, a cell is considered as scanned if the middle point of the
cell, noted as green dots, is within the range $R$ of the sensor.

The scanned environment is depicted as an occupancy grid map (OGM) [18]. Fig.2
illustrates an example map $\mathcal{M}$ consisting of $N=36$ cells
$C_{j},\,j\in\\{1..N\\}$ which are numbered in a boustrophedon pattern.

Each cell in map $\mathcal{M}$ carries an integer $n_{ray}$ indicating the
number of times the sensor rays pass through that cell. In other words,
$n_{ray}$ increases every time a pulsed light passes through the cells in any
direction. $n_{ray}$ is capped by $72$ since the cells where the robot is
located and hence known as empty are ideally scanned 72 times when the sensor
resolution is $5^{\circ}$. Using $n_{ray}$, a probability $P(C_{j})$ is
calculated for each cell as the likelihood of the cell not being empty, i.e.,
occupied by an obstacle. For the details, the reader is referred to Section
III.

##  III The proposed HIPP algorithm

This section presents the theoretical principles and design of the proposed
HIPP algorithm that navigates a robot from an initial position to generate a
2D map of an unknown non-convex environment with the minimum possible
travelling distance (i.e., consumed energy). It is important to note that the
final destination of the robot is unspecified (as the environment is unknown)
and exploration is completed when either a map with a given certainty is
created or time reaches a threshold.

Algorithm 1 summarises the proposed HIPP algorithm as a pseudo-code, including
four major sections: i) generating an occupancy grid map in lines 3-5; ii)
estimating the information gain of each cell and finding the cells with the
highest rewards in lines 6-7; iii) finding the shortest paths from the current
robot position passing through all the identified cells and select the first
element of the path as the next destination, as in line 8-9(iv); and moving
the robot along this path for one sampling time considering the potential
obstacles in lines 7-13. These sections are elaborated on in the following
paragraphs.

Figure 2: A 2-D occupancy grid map $\mathcal{M}$ with 36 cells and their
numbering convention. $p(C_{j})=0$ means that cell $C_{j}$ is definitely
empty. This figure also shows an example measurement of the ideal and
uncertain sensors. A robot located in $C_{15}$ measures the darker grey cells
when it uses an ideal sensor with the radius of $R=1\times$ cell width and
scans the lighter grey cells using the uncertain version of the sensor. In the
ideal case, Only the cells whose centre points (indicated by green dots)
within the circle are assumed to be scanned due to the quantization of the
map.

###  III-A Map generation

The uncertain reading of a LiDAR sensor at line 3 of Algorithm 1 is used to
generate $\mathcal{M}_{k}$ containing the accumulative number of rays
$n_{ray}$ passing through each cell up to sampling time $k$. The generated map
is assumed to be a stochastic 2D occupancy grid map where the odds ratio
(i.e., the robot’s belief) for cell $C_{j}$ at each time $k$ is implicitly
calculated as follows:

$\displaystyle p(C_{j};k)$
$\displaystyle=1-\frac{\sum_{l=1}^{k}\min(n_{ray}(C_{j};l),72)}{\min(\max(\mathcal{M}_{k}),72)+\epsilon};\>\forall
j\in{1..N}.$ (4)

where a small positive $\lim{\epsilon}\to 0$ is added to prevent the
singularity in numerical calculations. Also, as explained in Section II, using
a sensor with a resolution of $5^{\circ}$, a cell $C_{j}$ is considered as a
certainly empty cell once $72$ rays passed through it. Since the denominator
of the robot’s belief in (4) is nonlinear, the iterative nature of (4) is
implicit and it is not trivial to decompose the odds ratio of the inverse
sensor model.

Figure 3: Mapping gain of a sample cell $C_{j}$ of map $\mathcal{M}_{k}$ based
on the proposed formulation of belief in this paper (as red solid curve) as
compared to the one in [13] (as dashed grey curves). The belief of cell
$C_{j}$ is the odds ratio of the posterior of $C_{j}$ in [13] and is a number
in $[0,\infty]$ where $0$ means that the cell is certainly empty and $\infty$
means that it is certainly occupied. The information gain [13] is dynamically
adjusted based on the new reading guesses. This paper, on the other hand,
suggests (4) that gives a number in $[0,1]$ where $1$ and $0$ mean that there
is no information and the cell is certainly empty, respectively.

Apparently, $p(C_{j})$ is a positive value between $[0,1]$ where $p(C_{j})=1$
means that there is no information about cell $C_{j}$ (or the cell is
identified by all measurements to be occupied by an obstacle, which is unusual
due to the uncertain nature of the sensor measurements). $p(C_{j})=0$, on the
other hand, shows that cell $C_{j}$ is certainly an empty cell. If the truth
is that the cell $C_{j}$ is empty, $p(C_{j};k\to\infty)$ converges to $0$
because the numerator always grows faster than the denominator of (4) till
they are equal. If the truth is that the cell $C_{j}$ is occupied, on the
contrary, the values of $p(C_{j})$ will be close to $1$ and will not change
except due to the measurement noise.

Algorithm 1 A pseudo-code of the proposed HIPP algorithm

1  Initialise map $\mathcal{M}$ of arbitrary size $N$ with $n_{ray}(C_{j})=-1$
and mutual information gain $\mathcal{I}$ of the map with $i(C_{j})=0$ (i.e.,
all cells are unknown). Initialise the initial location of robot
$z_{orig}=[X_{r,0},Y_{r,0}]$. Initialise sampling time $T_{s}$, mission time
$T_{f}$, map reliability threshold $p_{th}$ and $k=1$.

2  while $k<T_{f}/T_{s}$ or $\exists\,j;\,p(C_{j})\leq p_{th}$ do

3  $tmpMap$ $\leftarrow$ depthScan($z_{orig}$) returns a $360^{\circ}$ sensor
reading

4  $\mathcal{M}\leftarrow$ element-wise $\mathcal{M}+tmpMap$

5  $p(C_{j})$ $\leftarrow$ updateProbability($p(C_{j})$) $\forall j\in{1..N}$
using (4)

6  $\mathcal{I}$ $\leftarrow$ updateInformationGain($\mathcal{I}$,
{$p(X_{0})..p(X_{N})$}) penalising travelling distance too, as in (8)

7  $cells$ $\leftarrow$ highestInformation($\mathcal{I}$) using an
$\epsilon$-greedy algorithm to select a predefined number of cells with the
highest surrounding information

8  $\mathcal{P}_{k}$ $\leftarrow$ shortestPath($z_{orig}$, $cells$)

9  $z_{dest}\leftarrow\mathcal{P}_{k}(1)$

10  if isObstacle($z_{orig}$, $z_{dest}$) then

11  $z_{dest}$ $\leftarrow$ RRT*($z_{orig}$, $z_{dest}$)

12  end if

13  $z_{orig}$ $\leftarrow$ moveRobot($z_{orig}$, $z_{dest}$)

14  $k$ $\leftarrow$ $k+1$

15  end while

16  returns $\mathcal{M}$ and $p(\mathcal{M})$

###  III-B Approximation of information

The algorithm needs to estimate a mapping gain per each already mapped cell,
which represents the potential expansion of the map if the robot moves to that
cell. Having the previous travelled path $\mathbf{Z}_{1:k}$, this mapping gain
at any time $k$ is equivalent to the mutual information
$\mathcal{I}(\mathcal{M}_{k};{C}_{k+1}\,|\,\mathbf{Z}_{1:k})$ between the map
$\mathcal{M}_{k}$ and the next destination of robot ${{C}}_{k+1}$ from the
current location of robot $z_{k}=[X_{r}\,Y_{r}]^{T}(k)$, which is defined as
follows [13]:

$\displaystyle\mathcal{I}(\mathcal{M}_{k};{{C}}_{k+1}\,|\,\mathbf{Z}_{1:k})=$
$\displaystyle\mathcal{H}(\mathcal{M}_{k}\,|\,\mathcal{O}({\mathbf{Z}_{1:k},{{C}}_{k+1})})-$
(5)
$\displaystyle\mathcal{H}(\mathcal{M}_{k}\,|\,\mathcal{O}(\mathbf{\mathbf{Z}}_{1:k}))$

where $\mathcal{H}(\mathcal{M}_{k}\,|\,\mathcal{O}(\mathbf{Z}_{1:k}))$ is the
conditional entropy of map $\mathcal{M}_{k}$ knowing the previous observation
$\mathcal{O}$ over the path $\mathbf{Z}_{1:k}$, and
$\mathcal{H}(\mathcal{M}_{k}\,|\,\mathcal{O}(\mathbf{Z}_{1:k}:{{C}}_{k+1}))$
is a similar entropy when the next step destination at time $k$ (i.e.,
${{C}}_{k+1}$) is also known.

The authors in [13] formulated the belief that a cell is occupied (i.e., odds
ratio) as a number between zero and infinity. They proposed an information
gain function which is monotonically increasing before a peak where it
switches to monotonically decreasing. It is shown in [13] that a path planning
algorithm which maximises such an information gain eventually moves the robot
to unexplored areas.

Moreover, the authors in [13] introduced a mechanism to adjust the information
gain function by leaning its peak towards the low values of the belief or the
other way around, as in Figure 3, depending on the guess of the next sensor
reading for the cell. This helps to moderately correct an occupied belief by a
new empty reading for a cell, and sharply strengthen an empty belief by a new
reading which confirms the emptiness of the cell.

Conversely, this paper proposes (4) to calculate belief as a positive number
less than one. This helps to apply the Shannon formula to heuristically
approximate the mutual information gain at each time $k$ between map
$\mathcal{M}_{k}$ and moving the robot to any cell $C_{j}$ as follows:

$\mathcal{I}(\mathcal{M}_{k};{C}_{k+1}=C_{j}\,|\,\mathbf{Z}_{1:k})\approx\mathcal{H}_{s}(C_{j};k);\>\forall
j\in\\{1..N\\}.$ (6)

where the Shannon entropy $\mathcal{H}_{s}$ is as follows:

$\displaystyle\mathcal{H}_{s}(C_{j};k)=-\sum_{x\in\\{0,1\\}}p(C_{j}=x;k)\log
p(C_{j}=x;k)$ (7)

Figure 3 illustrates the resulting curve (i.e., the solid red curve) of the
proposed information gain with respect to the current belief of cell $C_{j}$.
This formulation suggests that the cells which currently are the most
uncertain (i.e., their belief is $0.5$) receive the maximum gain if the robot
moves to them.

To consider the potential gain of the surrounding environment of cell $C_{j}$,
the proposed algorithm applies a $3\times 3$ integrator kernel to magnify the
high-information surroundings. The travelling distance is also taken into
account as an additive penalty. The resulting information gain is then
formulated as follows:

$\displaystyle\mathcal{I}_{f}($
$\displaystyle\mathcal{M}_{k};{C}_{k+1}=C_{j}\,|\,\mathbf{Z}_{1:k})=$ (8)
$\displaystyle\Bigr{[}\mathcal{I}_{f}(\mathcal{M}_{k};{C}_{k+1}=C_{j}\,|\,\mathbf{Z}_{1:k})-\beta\left\lVert
z_{k+1}-z_{k}\right\rVert\Bigr{]}*\begin{bmatrix}1&1&1\\\ 1&1&1\\\
1&1&1\end{bmatrix}$

where $*$ represents the matrix convolution operator and $\beta\in[0,1]$ is
the penalising factor of direct distance. $z_{k+1}$ is the location of centre
of the cell $C_{j}$.

###  III-C Path generation

This research proposes a path planning strategy that maximises
$\mathcal{I}_{f}$ in (8). Whilst $I_{f}$ is an approximation of (5), it is
already shown in [13] and section V of this paper that such a path planning
strategy eventually moves the robot to unexplored areas. At each sampling time
$k$, having the information gain (8) for each cell of the map, The proposed
strategy selects a predefined number of the cells with the highest gain. An
$\epsilon$-greedy algorithm is used for this selection to add elements of
exploration as well. A travelling salesperson problem with the return is,
then, formulated and solved to find the shortest path $\mathcal{P}_{k}$
starting from the current location of the robot and passing through all the
selected cells before returning back.

####  III-C1 Robot control and potential obstacle avoidance

The proposed HIPP algorithm uses (2) to calculate velocities of the left and
right wheels to move the robot from its current position to the first
destination of the calculated path $\mathcal{P}_{k}$ (i.e.,
$\mathcal{P}_{k}(1)$). The velocities are applied to the robot for a sampling
time $T_{s}$.

The proposed HIPP is equipped with a rapid exploring random tree star (RRT*)
algorithm for obstacle avoidance. HIPP examines if there is an obstacle
between the current location of the robot and the desired destination
$\mathcal{P}_{k}(1)$. If an obstacle is detected, the RRT* is applied to find
a path which avoids collision. Then, the first element of this path is
considered the next destination of the robot. RRT* is an efficient sample-
based algorithm that can search non-convex and high-dimensional environments
by constructing spatial filling trees randomly during exploration [19, 20].

##  IV The Proposed Algorithm to Calculate Benchmark Solution

This section develops a novel benchmark solution to evaluate the performance
of the proposed HIPP in section III. Principally, this benchmark is the
optimal solution of the posterior mapping problem where the robot remaps a
known environment using a limited-range ideal sensor as in Fig. 2.

The solution to the posterior problem is the shortest path to remap the area,
in which the robot is given an initial position but no final destination. A
formulation of the posterior problem is provided in (9) as a minimum-time
optimal control problem which is strongly non-convex.

$\displaystyle\mathbf{U}^{*}(z_{init},G,R)=\text{arg}\>\underset{\begin{subarray}{c}\mathbf{U},N\end{subarray}}{\text{maximise}}\quad
J(map,\mathbf{X},\mathbf{U}):=$
$\displaystyle\quad\quad\quad\quad\sum_{k=1}^{N}\Big{[}\alpha
f_{I}\big{(}\mathcal{M}_{k},u_{k}\,|\,\mathbf{Z}_{1:k}\big{)}-\beta
f_{d}\big{(}z_{k},u_{k}\big{)}\Big{]}-$ (9a)
$\displaystyle\quad\quad\quad\qquad\mu N+\gamma\big{|}\mathcal{M}_{N}\big{|}$
where: $\displaystyle\quad\quad\quad
f_{I}\big{(}\mathcal{M}_{k};u_{k}\,|\,\mathbf{Z}_{1:k}\big{)}=\left|\mathcal{M}_{k}\cup
sur\big{(}u_{k},R\big{)}\right|-\left|\mathcal{M}_{k}\right|$ (9b)
$\displaystyle\quad\quad\quad f_{d}\big{(}z_{k},u_{k}\big{)}=\left\lVert
z_{k}-u_{k}\right\rVert^{2}$ (9c) s.t.:
$\displaystyle\quad\quad\quad\quad\mathcal{M}_{k+1}=\mathcal{M}_{k}\cup
sur\big{(}u_{k},R\big{)};\>\>\mathcal{M}_{1}=\emptyset$ (9d)
$\displaystyle\quad\quad\quad\quad z_{k+1}=u_{k};\>\>z_{1}=z_{init}$ (9e)
$\displaystyle\quad\quad\quad\quad\mathbf{V}_{k+1}=\mathbf{V}_{k}\cup u_{k}$
(9f) $\displaystyle\quad\quad\quad\quad
u_{k}\in\mathbf{U}_{k}(G)\backslash\mathbf{V}_{k}$ (9g)

where $G(\mathbf{U},\mathbf{E})$ is a graph of all feasible locations
$\mathbf{U}$ of the robot on the map while every two locations with a line-of-
sight are connected by an edge $e\in\mathbf{E}$ representing their Euclidean
distance. $\mathbf{U}_{k}(G)=\\{u\mid u\in\mathbf{U}$; there is a direct link
from $z_{k}$ to $u$ in the graph $G$ }, $V_{k}$ ={ $u\mid u\in U$; $u$ is an
already visited vertex up to time $k$}, $sur(u_{k},R)$ is the set of the
surrounding cells of $u_{k}$ with a sensor of range $R$. $\alpha$ and $\beta$
are arbitrary convex coefficients. $|.|$ returns the cardinality of a set.

The objective of the problem (9) implies that the control strategy must find
the shortest path that maximises the size of the generated map $\mathcal{M}$
in a minimum time $N$. It is worth noting that the final destination is not
known which makes the problem even more complex.

To solve the problem (9), this paper proposes a heuristic solver that finds a
near-global-optimal solution. Algorithm 2 summarises the proposed solver as a
pseudo-code. As shown, the solver combines a CPP (Algorithm 2, lines 4-6) and
a TSP (Algorithm 2, lines 13-17) algorithm. The CPP algorithm optimally
locates a given number of way-points on the map to maximise the coverage with
a given sensor range $R$. The TSP algorithm, then, finds the shortest path
from the initial point to an unknown destination that passes all the dispersed
way-points. The details of these algorithms are provided in the remainder of
this section.

Algorithm 2 A pseudo-code of the proposed heuristic solver to find the near-
to-optimal solution of the posterior problem (9), i.e, a benchmark solution.

1  Initialise the map $\mathcal{M}$, number of way-points $N_{wp}$, initial
location of robot $z_{orig}=[X_{r,0},Y_{r,0}]$, required number of generations
$N_{gen}$ and k=1.

2  $\mathcal{S}_{0}$ $\leftarrow$ randomSeedsOnMap($\mathcal{M}$, $N_{wp}$)

3  while $k\leq N_{gen}$ do

4  $\mathcal{V}\leftarrow$
createVoronoiPartitions($\mathcal{M},\mathcal{S}_{k-1}$)

5  for $i=1:N_{wp}$ do

6  $\mathcal{S}_{k}(i)\leftarrow$ updateWayPoints($\mathcal{S}_{k-1}(i)$,
$\mathcal{V}$) using the modified version of the algorithm in [2], proposed in
section IV-A

7  if isObstacle($\mathcal{S}_{k}(i)$, $\mathcal{S}_{k-1}(i)$) then

8  $\mathcal{S}_{k}(i)$ $\leftarrow$ RRT*($\mathcal{S}_{k-1}(i)$,
$\mathcal{S}_{k}(i)$)

9  end if

10  $\mathcal{S}_{k}(i)$ $\leftarrow$ moveRobot($\mathcal{S}_{k-1}(i)$,
$\mathcal{S}_{k}(i)$)

11  end for$k$ $\leftarrow$ $k+1$

12  end while

13  $G(\mathcal{S}_{N_{gen}},E)\leftarrow$
generateGraph($\mathcal{S}_{N_{gen}}$) with the edges connecting way-points
with line-of-sight by their Euclidean distance

14  if there is no connected graph then

15  increase $N_{wp}$ and run the algorithm again

16  end if

17  $\mathcal{P}\leftarrow$ shortestPath($z_{orig}$, $\mathcal{S}_{N_{gen}}$)
calculates exhaustively the shortest path from $z_{orig}$ passes all the way-
points as in section IV-B

18  returns $\mathcal{P}$

###  IV-A The Proposed CPP Algorithm

This section proposes a modified version of the CPP algorithm in [2] and [21]
to maximise the coverage of a non-convex map, such as in Fig. 4, with a finite
number of robots equipped with sensors of range $R$.

The CPP algorithm in [2] and [21] randomly seeds a finite number of moving
robots across a map. Then, it creates Voronoi partitions [22] for the seeds
based on a geodesic proximity measure and hence decomposes the coverage
problem among the finite number of robots. Unlike the original algorithm, this
paper assumes that the sensors are of the line-of-sight type (e.g., LiDAR) and
therefore the original geodesic technique is replaced by the range-limited
visibility disc.

After partitioning the area, the CPP algorithm moves the robots using a free-
arc technique iteratively to minimise the overlap between the sensing range of
the robots. The details of these steps are explained in this section.

(a)

(b)

(c)

Figure 4:  Different disc techniques can be used to represent the ideal
sensing region within a non-convex map for calculating the benchmark solution.
These techniques are a) Euclidean, b) range-limited visibility and c)
geodesic. The black dots show the location of the sensor and the shaded areas
depict the scanned areas. While the geodesic disc represents the area that is
scanned by a non-line-of-sight radio sensor, the range-limited visibility
technique is suitable for modelling the camera. As shown in (a), Euclidean
discs are not suitable for non-convex areas because of including non-reachable
points. Figure 5: The proposed CPP algorithm, as part of Algorithm 2, uses the
free-arc method in [21] to calculate the desired movement $k\hat{m}$ of a
robot within a non-convex map to maximize its sensing coverage. The sensor
readings are modelled as a range-limited visibility disc (indicated by red
arcs). $k\hat{m}$ is the sum of $k_{j}\hat{m_{j}}$ vectors which are
calculated by applying (10) to each arc of the disc. Arcs are segmented with
an equal length $dq$. $\hat{n_{i}}$ indicates a normal vector at the centre
point $p_{i}$ of segment $i$.

(a)

(b)

Figure 6: The results generated by Algorithm 2, i.e., a benchmark solution,
for a sample known map, including (a) the final location of nodes (in blue),
derived iteratively from the initial random positions (in red) by the proposed
CPP algorithm to maximize the coverage. The black discs represent the coverage
of the ideal measuring sensor of the nodes; and (b) the shortest path derived
by the proposed TSP algorithm which starts from the initial location of the
robot (indicated by a red circle) and passes through all the nodes without
prior knowledge of the destination node.

####  IV-A1 Proximity measurements

It is important to consider the non-convexity of the map during proximity
measurement between nodes. Euclidean distance measure is not suitable for non-
convex areas since it may include non-reachable parts of the map within a
partition, as in Fig. 4(a).

To address this issue, the authors in [21] used a geodesic disc technique, as
in Fig. 4(b), where the proximity is measured by a non-direct but still
reachable distance between points. This technique fits well to the radio
sensors which are able to sense non-line-of-sight points.

As mentioned before, this paper uses a LiDAR sensor and hence employs a range-
limited visibility disc technique as the proximity measure, where only the
visible points within the sensor range $R$ belong to the partition [23] (Fig.
4(c)).

(a)

(b)

(c)

Figure 7: Three different scenarios, including a $20\times 20$ cells map and
an initial location of the robot, is used in this study to test the
performance of the proposed HIPP algorithm as compared to the benchmark
solution; where (a) is Test Scenario 1 which is an open space without
obstacles; (b) is Test Scenario 2 consists of five different obstacles and 355
available cells; and (c) is Test Scenario 3 which includes 346 free cells with
an increased complexity by adding narrow passages.

####  IV-A2 Free arc method

This paper uses the proposed free-arc method in [21] to calculate the vector
of movement of the way-points in a non-convex environment. The method creates
Voronoi partitions for the way-points. It draws a circle around the way-points
with the radius of the sensor range $R$. Parts of the circle’s edge outside of
the Voronoi partition of the way-point are discarded and only the parts inside
the partition are used to calculate the velocity (speed and its direction) of
movement of the way-point.

Fig.5 illustrates the details of how the free-arc method calculates the
velocity of a way-point for the case when the way-point has three eligible
arcs (red arcs). As shown, the movement vector $k\hat{m}$ of each way-point is
the vector sum of the resulting velocity $k\hat{m}_{j}$ of each arc $j$. The
velocity due to each arc is calculated by splitting the arc into segments with
an equal length of $dq$ and a centre point of $p_{i}$. Therefore, the velocity
of the required movement of a way-point is calculated as follows:

$k\hat{m}=\sum_{j=1}^{N_{arcs}}\int_{p_{i}\in arc_{j}}\phi\,\hat{n_{i}}dq$
(10)

where $N_{arcs}$ is the number of eligible arcs, $\hat{n}_{i}$ is the unit
normal vector to the segment with the centre $p_{i}$, $dq$ is the length of
segments and $\phi$ is a gain indicating the importance of each segment (that
can be one for all segments). In other words, each segment $p_{i}$ affects the
way-point velocity by a factor $\phi dq$ in the direction of $\hat{n_{i}}$.

Fig. 6(a) shows the results of applying the free-arc algorithm to move the
initially positioned red way-points to the better-positioned (i.e., smaller
overlapped sensor ranges or equivalently larger covered area) blue points.

###  IV-B The Proposed TSP Algorithm

The optimally relocated way-points by the free-arc algorithm are used to
create a graph connecting all the points with line-of-sight through their
Euclidean distance. In other words, only the way-points which can be connected
by a line without having an obstacle in between are connected in the resulting
graph.

The constructed graph is then used to exhaustively find the destination to
which the path from the robot’s original location passing through all the way-
points is the shortest amongst the similar paths to all other destinations on
the graph. A TSP is formulated and solved for each vertex of the graph [24]
and the shortest is chosen as a close-to-global solution of the original IPP
problem.

Fig. 6(b) illustrates an example optimum path generated by the proposed TSP
algorithm knowing the initial location of a robot (red point) and using the
relocated blue way-points in Fig. 6(a).

##  V Simulation Results and Discussions

Three dissimilar maps in Fig.7 are used as test scenarios to evaluate the
adaptation and efficacy of the proposed HIPP algorithm to different
environments. Fig. 7(a) is an empty map, whilst Fig. 7(b) is a complex map
with multiple rooms and obstacles. Fig. 7(c), as compared with Fig. 7(b),
contains narrow dead-end passages. The results and performance of the proposed
HIPP algorithm are evaluated and benchmarked against the solutions to the
proposed posterior problem.

Test scenarios are labelled as 1 to 3 which are aligned with Fig.7(a) to
Fig.7(c). Without loss of generality, the initial position of the robot in all
scenarios is assumed identical, as the red circles in the left-bottom corner
of Fig. 7.

The comparison measure is the number of identified cells on the OGM map
normalised by the travelling distance. However, to make a realistic
comparison, an equivalent number of identified cells is introduced in this
section for the results of HIPP, which is the number of identified cells
calculated by placing the ideal sensor of Fig.2 at the generated way-points by
HIPP. Hence, a realistic comparison measure is the equivalent number of
identified cells per unit of travelling distance.

Moreover, this section provides an optimality analysis where the accumulative
performance of the proposed HIPP over the travelling time is benchmarked
against the solution of the proposed posterior problem. In other words, it
shows whether the number of identified cells increases linearly by the
travelling time.

The confidence of the mapped cells is imperative to the performance of the
HIPP algorithm as the sensor’s range and direction are uncertain. This
confidence is measured by the number of times a cell is sensed. For the map to
have high confidence, this paper only considers a cell empty if the
probability of being sensed as empty is equal to or more than $20\%$. It means
that a cell must be scanned at least 14 times as an empty cell to be
considered so.

The HIPP and solver of the posterior problem run ten times for each test
scenario. Table I summarises the statistics of the resulting total travelling
time (TTD), the number of mapped cells, the number of identified cells per
unit of travelling distance(TD), the equivalent number of mapped cells, and
the equivalent number of mapped cells per unit of travelling distance(TD). As
explained before, for efficiency, the normalised equivalent number of
identified cells (i.e., the latter column of the table) by Algorithm 1 is
compared to the normalised number of identified cells (i.e., the third column
of the table) by Algorithm 2.

Fig.8 visualises this comparison as the box plots. As shown, the average
performance of the proposed HIPP reaches $80\%$ and $70\%$ of the proposed
benchmark for test scenarios 1 or 2 and the more complex test scenario 3,
respectively. It also shows how the proposed solver finds a sub-optimal (if
not close to global optimal) solution to the posterior problem. It is worth
noting that in test scenario 1 the HIPP outperformed on one occasion which is
due to an empty map and uncertain sensor. The results also show consistency in
HIPP results, where mapping cells per travelling distance decreases in more
complex test scenarios.

Fig. 9 compares the best-generated paths by the proposed HIPP algorithm
against the best result by solving the posterior problem out of ten runs of
the algorithms. The resulting maps of HIPP are also illustrated in Fig.
9(d)-9(f) indicating the unmapped parts as red cells. The readers are also
referred to the prepared video on https://youtu.be/wnBgTZxwG_o where the
performance of HIPP for all three test scenarios is demonstrated.

Fig. 10 visualizes the variation of the generated paths by HIPP over ten
different runs which shows the consistency of the algorithm. For example, it
shows that HIPP ignores exploring the central areas of rooms and the areas
close to the walls.

TABLE I: The statistics of total travelling distance (TTD), number of identified cells and that per each unit of travelling distance (TD) for ten times running of Algorithms 1 and 2 for test scenarios 1 to 3. An equivalent number of identified cells, calculated by placing the ideal sensor of Fig.2 at the generated way-points by HIPP, provides a realistic comparison, as explained in section V. |  |  TTD (m) |  No. of identified cells |  No. of identified cells/TD (cells/m) |  Equivalent no. of identified cells |  Equivalent no. identified cells/TD (cells/m)
---|---|---|---|---|---|---
Test Scenario 1 | Algorithm 1 (HIPP) results
Average |  48.9 |  384.5 |  8.1 |  233.5 |  4.9
Std. Deviation |  12.21 |  13.6 |  2.4 |  15.8 |  1.1
Median |  49.3 |  388.0 |  7.7 |  231.0 |  4.7
Algorithm 2 (Posterior) results, i.e., a benchmark solution
Average |  58.1 |  362.0 |  6.2 |  - |  -
Std. Deviation |  1.0 |  6.4 |  0.1 |  - |  -
Median |  58.4 |  364.0 |  6.2 |  - |  -
Test Scenario 2 | Algorithm 1 (HIPP) results
Average |  78.8 |  340.0 |  4.4 |  262.0 |  3.4
Std. Deviation |  8.8 |  7.4 |  0.6 |  10.0 |  0.4
Median |  80.6 |  343.0 |  4.3 |  262.0 |  3.3
Algorithm 2 (Posterior) results, i.e., a benchmark solution
Average |  83.8 |  353.0 |  4.2 |  - |  -
Std. Deviation |  2.9 |  1.7 |  0.1 |  - |  -
Median |  84.4 |  354.0 |  4.2 |  - |  -
Test Scenario 3 | Algorithm 1 (HIPP) results
Average |  80.7 |  305.5 |  3.8 |  241.0 |  3.0
Std. Deviation |  7.6 |  27.2 |  0.6 |  18.7 |  0.4
Median |  80.0 |  313.5 |  3.9 |  244.0 |  3.1
Algorithm 2 (Posterior) results, i.e., a benchmark solution
Average |  72.5 |  311.6 |  4.3 |  - |  -
Std. Deviation |  2.8 |  8.7 |  0.2 |  - |  -
Median |  72.7 |  310.0 |  4.3 |  - |  -
Figure 8: A graphical representation of the provided results in Table I
showing that in average HIPP, with no a priori knowledge of the map,
effectively follows the benchmark solution, where the map in known in advance
for different test scenarios in Figure 7.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 9:  (a)-(c) A sample of the generated path by the proposed HIPP (the
res curves) as compared to the benchmark solution (the blue curves), in three
test scenarios. These paths are those which are generated the best results (as
the maximum equivalent number of identified cells per unit of travelling
distance) out of the ten runs of test scenarios. (d)-(f) the cells (in red)
that HIPP-generated paths are not confidently scanned as empty cells.

(a)

(b)

(c)

Figure 10: Variation of generated paths by ten runs of the proposed HIPP
algorithm for the three test scenarios. These results show the similarities
between different runs of each test scenario in terms of the travelling paths
of the robot.

The instantaneous behaviour of the proposed HIPP algorithm over the travelling
time is compared in Fig. 11 to such results of the benchmark solution. These
results are for test scenario 2, however, the other scenarios follow a similar
trend.

As shown in Fig.11(a), the accumulative number of identified cells by HIPP
follows an almost linear increase by travelling time before it starts being
saturated at around $60$ sampling times $T_{s}$. In other words, HIPP follows
the performance of the benchmark solution over a wide range of operating time
before being outperformed.

However, it is shown in Fig.11(b) that within the same period of $60$ sampling
times, the robot travels less by HIPP than the benchmark solution. This means
that over the first half of the travelling time, HIPP outperforms the
benchmark solution in terms of the normalised identified cells by travelling
distance. Fig. 11(c) illustrates this comparison measure clearer where the
number of identified cells per unit of travelling distance is plotted against
the travelling time. This means that HIPP implicitly prioritises the wider
areas where the mapping gains are higher than narrow passages.

In addition, it is observed in Fig. 11(c) that there is a sharp rise in the
number of identified cells per travelling distance by the solution of the
posterior problem before reaching a saturated state after around $15$ seconds.
This is because of scanning multiple cells with a small movement. The
saturated $4$ cells per unit of travelling distance match the range of the
sensor which is $1.5$ cells or equivalently $75\,$cm which is an indicator of
finding a near-to-global optimal solution.

The normalised number of identified cells by HIPP, on the other hand,
continuously decreases over travelling time before reaching $2$ cells per unit
of travelling distance at the end of the test scenario.

(a)

(b)

(c)

Figure 11: Comparison of the (a) accumulative number of identified cells up to
each travelling time (in terms of the number of sampling times $T_{s}$), (b)
accumulative travelling distance and (c) the instantaneously identified cells
per variation of travelling distance for the Test Scenario 2.

##  VI Conclusion

This paper presents a novel heuristic algorithm (which is called HIPP) for
planning the exploration path of a single robot to efficiently map an unknown
non-convex environment and make a map out of it. The robot has a LiDAR sensor
and is equipped with the SLAM algorithm, however, it does not have knowledge
about the map or the final destination. The efficiency of the algorithm is
measured by the number of identified cells per unit of travelling distance.

A posterior problem (i.e., where the robot remaps a known map) is formulated
and its solution is considered as the benchmark to evaluate the performance of
HIPP. It is shown that such a problem is a complex discontinuous problem. This
paper also proposes a novel solver for the posterior problem to find a near-
optimal solution to remapping a known non-convex area where the final
destination is not given.

The simulation results show that the formulated IPP problem has potentially
multiple close sub-optimal solutions which are probably close to the globally
optimal path.

We could not find a direct solution to the posterior problem, however, we
could develop a heuristic solver to this problem. The results show that this
solver finds statistically similar results with a narrow distribution.
However, it is observed that the posterior problem has potentially multiple
close optimal solutions.

Overall, the simulation results for different test scenarios show that the
proposed HIPP achieves $70-80\%$ of the benchmark solution in terms of the
number of identified cells per travelling distance. Moreover, whilst the
target map is expanded linearly and with a similar rate as the benchmark
solution over the first $60\%$ of the exploration time, the robot travels less
using HIPP for this first period of exploration. This behaviour is desired for
such information-gathering algorithms indicating that the proposed HIPP
creates a significant portion of the map in the shortest possible time (or
travelling distance) by an implicit prioritisation which is of significant
importance in hazardous applications

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# Effects of Gauss-Bonnet gravity on thermodynamics of (3+1)$-D$ AdS black
holes

Neeraj Kumara111e-mail: neerajkumar@bose.res.in,Sunandan
Gangopadhyaya222e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>aDepartment of Theoretical Sciences,
S.N.Bose National Centre for Basic Sciences,
JD Block, Sector III, Salt Lake, Kolkata, 700106, India

## Abstract

We studied the thermodynamics of the black holes with Gauss-Bonnet correction
term in $3+1-$ dimensional AdS spacetime. It is known that the term has no
effect on the equation of motion, however, it modifies the entropy formula of
Wald as it has an effect of the Gauss-Bonnet parameter term. We studied
charged black holes, namely, Reissner-Nordström and Born-Infeld under this
regime. We encountered divergences in heat capacity. After eliminating the
possibility of first order phase transition, we applied two well trusted
methods from standard thermodynamics, namely, Ehrenfest scheme and Ruppeiner’s
state space geometry analysis to ensure the second order nature of phase
transition points. Effects of Gauss-Bonnet and Born-Infeld parameter are also
discussed.

##  I Introduction

Geometric definition of the black hole entropy is at the heart of black hole
thermodynamics as it seems to resolve the discrepancy between thermodynamics
and gravity. It was proposed by Bekenstein in bek1 , where entropy was
associated to black holes using Hawking’s area theorem hawk  as an analogy to
preserve the second law of thermodynamics. Mere simplicity of black holes and
their analogous behaviour to thermodynamic objects was shown in bhm  in the
form of black hole mechanics. The analogy however was made robust by Hawking
in hr  showing thermal nature of black holes using quantum effects of matter
fields in curved spacetime. A lot of research has been going on since then to
understand the thermodynamics of these objects with an aim to ultimately get
to the quantum nature of gravity.

There are many modifications to Einstein’s general relativity theory even at
classical level with a fair amount of motivation as general relativity has not
yet been verified in strong limit. Lovelock theory lov  is one such higher
curvature correction theory with higher curvature terms in the lagrangian, and
when the action is modified up to second term it is named as Gauss-Bonnet
gravity. Black hole solutions exist for these theories with the same symmetry
as that of Schwarzchild black hole in Einstein’s theory. Gauss-Bonnet term has
no effect on the equation of motion for $D=4$ spacetime dimensions and the
effects only appear for $D\geq 5$. Wald has shown that in general
diffeomorphism invariant theories, the entropy associated with a black hole
turns out to be the Noether charge associated with the general coordinate
transformations rm . Wald’s entropy for the black hole is given by the
following expression

$\displaystyle S=2\pi\int_{\sum}d\phi\sqrt{g_{\phi\phi}}\dfrac{\partial
L}{\partial R_{\mu\nu\lambda\rho}}\epsilon_{\mu\nu}\epsilon_{\lambda\rho}~{}.$
(1)

For Gauss-Bonnet black hole, entropy has a correction term apart from the area
law depending on Gauss-Bonnet parameter as calculated in ent  and recently
used in ent1 . It is interesting to see the effects of this correction to
entropy on black hole thermodynamic properties. We wish to study charged black
holes in Gauss-Bonnet gravity. We discuss two cases where black hole is
charged due to Maxwell theory and Born-Infeld theory as it is always important
to check the effects of non-linear electromagnetic theory on black holes. The
thermodynamics of these black holes when cosmological constant treated as a
thermodynamic variable is discussed in ent1  and gives rise to a pressure-
volume term in the first law as first proposed in roy . It is also interesting
to check the behaviour when cosmological constant is not a thermodynamic
variable and its value is fixed to some constant negative number. Results are
known for black holes in Einstein gravity and it is known that the black holes
undergo second order phase transition as discussed in r1 , r2 . So it is worth
checking the effects of Gauss-Bonnet parameter in general on the nature of
phase transition.

This paper is divided in two main parts. First part involves thermodynamic
analysis of Reissner-Nordstr$\ddot{o}$m black holes. We calculated the
thermodynamic quantities and find infinite divergences in heat capacity of the
black hole when plotted with entropy. The Hawking temperature of the black
hole is smooth function of entropy hence singular nature of heat capacity
hints some phase transition of order higher than one. We analysed these
singularities further using Ehrenfest scheme which has been used for black
holes extensively in (r1  \- n2 ) and many more. Prigogine-Defay ratio as
discussed in pr  is a parameter in standard thermodynamics giving deviation
from second order phase transition is also calculated. Further, we also used
Ruppeiner state space geometry analysis which again is a well trusted
technique in standard thermodynamics as well as in black hole thermodynamics
as discussed in ru1 , ru2 . Second part of the paper involves analysing same
black hole in non-linear electromagnetic theory. We discussed Born-Infeld
theory with an aim to understand the effects of non-linear parameter on
thermodynamic properties as well as on phase transition. Along the way, we
also calculated Smarr relation in both the cases keeping all dimensionful
variables in the first law.

This paper is organised in the following form. Section-II discusses Gauss-
Bonnet-Reissner-Nordstr$\ddot{o}$m black hole involving calculations of
thermodynamic quantities followed by Ehrenfest analysis and geometric
analysis. Section-III contains the same analysis for Gauss-Bonnet-Born-Infeld
black hole. We summarize our results in section-IV. The paper also contains an
Appendix.

##  II Gauss-Bonnet-Reissner-Nordstr$\ddot{O}$m black hole

### Thermodynamics and first law

In this section, we define the black hole spacetime system we are interested
in and calculate its thermodynamic properties. We consider charged black hole
in $3+1-$ dimensional Gauss-Bonnet corrected gravity theory in AdS spacetime.
We work in units where $G=h=c=1$. We start from the following action

$\displaystyle I=\dfrac{1}{16\pi}\int d^{4}x\sqrt{-g}[R-2\Lambda+\alpha
L_{GB}+L(F)]$ (2)

where Gauss-Bonnet lagrangian
$L_{GB}=R^{2}-4R_{\gamma\delta}R^{\gamma\delta}+R_{\gamma\delta\lambda\sigma}R^{\gamma\delta\lambda\sigma}$
and charge is due to Maxwell term $L(F)=F^{\mu\nu}F_{\mu\nu}$. Here $\alpha$
is Gauss-Bonnet parameter and $\Lambda=-\dfrac{3}{l^{2}}$ ($l$ being AdS
radius).

The solution of the black hole spacetime in this theory reads

$\displaystyle ds^{2}=-f(r)dt^{2}+\dfrac{1}{f(r)}dr^{2}+r^{2}d\Omega^{2}_{2}$
(3)

with $f(r)$ being

$\displaystyle f(r)=1-\dfrac{2M}{r}+\dfrac{Q^{2}}{r^{2}}+\dfrac{r^{2}}{l^{2}}$
(4)

where $M$ is the mass and $Q$ is the charge of the black hole. Note that the
Gauss-Bonnet term does not show any effect on the equation of motion in $3+1-$
spacetime dimensions.

Mass of the black hole can be written in terms of horizon radius ($r_{+}$) by
using $f(r_{+})=0$, so

$\displaystyle
M=\dfrac{r_{+}}{2}+\dfrac{Q^{2}}{2r_{+}}+\dfrac{r_{+}^{3}}{2l^{2}}~{}.$ (5)

Also the Hawking temperature of the black hole can be calculated as

$\displaystyle T$ $\displaystyle=$
$\displaystyle\dfrac{1}{4\pi}\left(\dfrac{\partial f}{\partial
r}\right)_{r_{+}}$ (6) $\displaystyle=$
$\displaystyle\dfrac{1}{4\pi}\left(\dfrac{1}{r_{+}}-\dfrac{Q^{2}}{r_{+}^{3}}+\dfrac{3r_{+}}{l^{2}}\right)~{}.$

In the rest of our analysis, the AdS radius ($l$) will be kept unity unless
stated otherwise.

Now the first law of thermodynamics for black hole is

$\displaystyle dM=TdS+\Phi dQ~{}$ (7)

which is analogous to the first law in standard thermodynamics

$\displaystyle dU=TdS-PdV$ (8)

with the identification of the pressure $P$ to the negative of the
electrostatic potential $\Phi$, the volume to the charge $Q$ and the internal
energy $U$ to the mass of the black hole $M$.

Therefore, the potential can be calculated as

$\displaystyle\Phi=\left(\dfrac{\partial M}{\partial
Q}\right)_{S}=\dfrac{Q}{r_{+}}~{}.$ (9)

Although the Gauss-Bonnet term does not affect the equation of motion, it
changes the form of entropy and introduces an extra term apart from the area
term. As calculated by using Wald formula in ent , the entropy takes the form

$\displaystyle S=\pi(r_{+}^{2}+4\alpha)~{}.$ (10)

So the Hawking temperature can be written in terms of entropy as

$\displaystyle
T=\dfrac{1}{4\pi}\left(\sqrt{\dfrac{\pi}{S-4\alpha\pi}}-Q^{2}\left(\dfrac{\pi}{S-4\alpha\pi}\right)^{3/2}+3\sqrt{\dfrac{S-4\alpha\pi}{\pi}}\right)~{}.$
(11)

Also the potential becomes

$\displaystyle\Phi=Q\sqrt{\dfrac{\pi}{S-4\pi\alpha}}~{}.$ (12)

We now plot the temperature with respect to entropy in Fig.(1a) for fixed
values of charge and Gauss-Bonnet parameter. Temperature behaves smoothly with
entropy and its behaviour in Einstein-Reissner-Nordstr$\ddot{o}$m limit (that
is, $\alpha\rightarrow 0$) is also plotted along to see the effects of the
Gauss-Bonnet parameter.

(a)

Figure 1: Temperature vs. Entropy

Solid line (Q=0.2, $\alpha$=0.02)
Dashed line (Q=0.2, $\alpha$=0)

Next, we calculate the heat capacity at constant potential $(C_{\Phi})$ and
analyse the system thermodynamically. Considering temperature of the black
hole to be a function of entropy and charge ($T\equiv T(S,Q)$), we can write

$\displaystyle\left(\dfrac{\partial T}{\partial
S}\right)_{\Phi}=\left(\dfrac{\partial T}{\partial
S}\right)_{Q}-\left(\dfrac{\partial T}{\partial
Q}\right)_{S}\left(\dfrac{\partial\Phi}{\partial
S}\right)_{Q}\left(\dfrac{\partial Q}{\partial\Phi}\right)_{S}~{}$ (13)

where we use the thermodynamic identity

$\displaystyle\left(\dfrac{\partial Q}{\partial
S}\right)_{\Phi}\left(\dfrac{\partial
S}{\partial\Phi}\right)_{Q}\left(\dfrac{\partial\Phi}{\partial
Q}\right)_{S}=-1$ (14)

to write down eq.(14).

Using the thermodynamic expression in eq.(13), the heat capacity at constant
potential $C_{\Phi}=T\left(\dfrac{\partial S}{\partial T}\right)_{\Phi}$ can
be calculated using eq.(s)(11, 12)

$\displaystyle
C_{\Phi}=2(S-4\pi\alpha)\left[\dfrac{\pi(S-4\pi\alpha)-Q^{2}\pi^{2}+3(S-4\pi\alpha)^{2}}{-\pi(S-4\pi\alpha)+Q^{2}\pi^{2}+3(S-4\pi\alpha)^{2}}\right]~{}~{}.$
(15)

This reduces to the expression given in r1  when $\alpha=0$.

(a)

Figure 2: Heat Capacity vs. Entropy

Solid line (Q=0.2, $\alpha$=0.02)
Dashed line (Q=0.2, $\alpha$=0)

We plot the heat capacity with entropy for same fixed parameters in Fig.(2a).
It is apparant from the Figure that the black hole undergoes phase transition
and exists in three different phases. One of the three phases also exhibit
negative heat capacity. Since the Hawking temperature behaves smoothly with
entropy so this eliminates the existence of first order phase transition and
hints toward the possibility of second order phase transition. We shall be
analysing this further. The effects of Gauss-Bonnet term on the phase
transition points are also apparent from the plot. It shifts the phase
transition points to higher entropy values.

To end the discussion in this section, we calculate the Smarr relation for the
black hole. In extended phase space thermodynamics, AdS radius $(l)$ has been
shown to behave like thermodynamic pressure in roy  and is related to pressure
as

$\displaystyle p=-\dfrac{\Lambda}{8\pi}=\dfrac{3}{8\pi l^{2}}~{}.$ (16)

Using eq.(s)(10, 16) in eq.(5), we get the mass M to be

$\displaystyle M\equiv
M(S,p,Q,\alpha)=\dfrac{1}{2}\sqrt{\dfrac{S-4\alpha\pi}{\pi}}+\dfrac{Q^{2}}{2}\sqrt{\dfrac{\pi}{S-4\alpha\pi}}+\dfrac{4\pi
p}{3}\left(\dfrac{S-4\alpha\pi}{\pi}\right)^{3/2}~{}.$ (17)

Keeping all these dimensionful parameters in mind, the general form of first
law of thermodynamics will take the form

$\displaystyle dM=TdS+vdp+\Phi dQ+\Omega d\alpha$ (18)

where $\Omega$ is some conjugate variable of Gauss-Bonnet parameter $\alpha$
and $v$ corresponds to $p$. These can be calculated from the expression of
mass as $\Omega=\dfrac{\partial M}{\partial\alpha}$ and $v=\dfrac{\partial
M}{\partial p}$. The dimensions of the quantities involved in eq.(17) are
$[M]=[L],~{}[S]=[L]^{2},~{}[p]=[L]^{-2},~{}[Q]=[L]$ and $[\alpha]=[L]^{2}$. So
the Smarr relation for the black hole can be calculated and it becomes

$\displaystyle M=2TS-2pv+Q\Phi+2\Omega\alpha~{}.$ (19)

Here we can explicitly see that the Gauss-Bonnet parameter enters into the
Smarr formula in $3+1-$ spacetime dimensions.

We shall now look at the phase transition points of the black hole indicated
in the heat capacity plots. We shall analyse the order of phase transition
using Ehrenfest scheme and Ruppeiner’s state space geometry analysis in the
following sections.

### Analysis of phase transition using Ehrenfest scheme

Ehrenfest’s formalism of studying phase transitions is the standard technique
in thermodynamics to determine the nature of phase transitions for various
thermodynamical systems. It simply says that the order of the phase transition
corresponds to the discontinuity in the order of the derivative of Gibb’s
potential. For second order phase transition the second derivative encounters
discontinuities. However, first derivative and Gibb’s potential at those
points are continuous. These conditions with Maxwell’s relations give two
equations which have to be satisfied for second order phase transition.

The first and second Ehrenfest equations in thermodynamics are given by

$\displaystyle\left(\dfrac{\partial P}{\partial T}\right)_{S}$
$\displaystyle=$
$\displaystyle\dfrac{1}{VT}\dfrac{C_{P_{2}}-C_{P_{1}}}{\beta_{2}-\beta_{1}}=\dfrac{\Delta
C_{P}}{VT\Delta\beta},$ (20) $\displaystyle\left(\dfrac{\partial P}{\partial
T}\right)_{V}$ $\displaystyle=$
$\displaystyle\dfrac{\beta_{2}-\beta_{1}}{\kappa_{2}-\kappa_{1}}=\dfrac{\Delta\beta}{\Delta\kappa}$
(21)

where subscripts $1$ and $2$ denote two distinct phases of the system. Now we
use the correspondence between the pressure ($P$) to the negative of the
electrostatic potential difference ($\Phi$) and the volume ($V$) to the charge
($Q$) of the black hole. These identifications lead to the following equations

$\displaystyle-\left(\dfrac{\partial\Phi}{\partial T}\right)_{S}$
$\displaystyle=$
$\displaystyle\dfrac{1}{QT}\dfrac{C_{{\Phi}_{2}}-C_{\Phi_{1}}}{\beta_{2}-\beta_{1}}=\dfrac{\Delta
C_{\Phi}}{QT\Delta\beta}$ (22)
$\displaystyle-\left(\dfrac{\partial\Phi}{\partial T}\right)_{Q}$
$\displaystyle=$
$\displaystyle\dfrac{\beta_{2}-\beta_{1}}{\kappa_{2}-\kappa_{1}}=\dfrac{\Delta\beta}{\Delta\kappa}~{}.$
(23)

Note that $\beta$ is the volume expansion coefficient and $\kappa$ is the
isothermal compressibility of the system and are defined as

$\displaystyle\beta=\dfrac{1}{Q}\left(\dfrac{\partial Q}{\partial
T}\right)_{\Phi}~{},~{}~{}~{}~{}\kappa=\dfrac{1}{Q}\left(\dfrac{\partial
Q}{\partial\Phi}\right)_{T}~{}.$ (24)

Now we proceed to check whether the black hole phase transition satisfies the
Ehrenfest equations (22, 23). In other words, we investigate the validity of
the Ehrenfest equations at the points of discontinuities $S_{i}$, (i=1, 2).
Here we denote the critical values of temperature by $T_{i}$ and charge by
$Q_{i}$. The left hand side of the first equation for the black hole spacetime
takes the form

$\displaystyle-\left[\left(\dfrac{\partial\Phi}{\partial
T}\right)_{S}\right]_{S=S_{i}}=\dfrac{2\pi}{Q_{i}}\left(\dfrac{S_{i}-4\alpha\pi}{\pi}\right)~{}.$
(25)

In getting this result, we have used

$\displaystyle\left[\left(\dfrac{\partial\Phi}{\partial
Q}\right)_{S}\right]_{S=S_{i}}=\sqrt{\dfrac{\pi}{S_{i}-4\pi\alpha}}~{},$ (26)
$\displaystyle\left[\left(\dfrac{\partial T}{\partial
Q}\right)_{S}\right]_{S=S_{i}}=-\dfrac{Q_{i}}{2\pi}\left(\dfrac{\pi}{S_{i}-4\pi\alpha}\right)^{3/2}$
(27)

and eq.(64) in the Appendix. Also, the right hand side becomes

$\displaystyle\dfrac{\Delta
C_{\Phi}}{T_{i}Q_{i}\Delta\beta}=\dfrac{2\pi}{Q_{i}}\left(\dfrac{S_{i}-4\alpha\pi}{\pi}\right)~{}$
(28)

which is computed using eq.(26),

$\displaystyle\left[\left(\dfrac{\partial\Phi}{\partial
S}\right)_{Q}\right]_{S=S_{i}}=-\dfrac{\sqrt{\pi}Q}{2}\left(\dfrac{1}{S_{i}-4\pi\alpha}\right)^{3/2}$
(29)

and eq.(67) in the Appendix. Eq.(s)(25, 28) show the validity of first
Ehrenfest’s equation for the black hole spacetime under consideration. We now
proceed to check the second equation.

Using eq.(69), the left hand side of second Ehrenfest equation becomes

$\displaystyle-\left[\left(\dfrac{\partial\Phi}{\partial
T}\right)_{Q}\right]_{S=S_{i}}=\dfrac{2\pi}{Q_{i}}\left(\dfrac{S_{i}-4\alpha\pi}{\pi}\right).$
(30)

The right hand side of the equation becomes

$\displaystyle\dfrac{\Delta\beta}{\Delta\kappa}=-\dfrac{4\pi^{2}Q_{i}(S_{i}-4\alpha\pi)}{-\pi(S_{i}-4\alpha\pi)+3Q_{i}^{2}\pi^{2}+3(S_{i}-4\alpha\pi)^{2}}~{}.$
(31)

Here we used eq.(29), the following expression

$\displaystyle\left[\left(\dfrac{\partial T}{\partial
S}\right)_{Q}\right]_{S=S_{i}}=\dfrac{1}{4\pi}\left(-\sqrt{\pi}\left(\dfrac{1}{S_{i}-4\pi\alpha}\right)^{3/2}+\dfrac{3\pi^{3/2}Q_{i}^{2}}{2}\left(\dfrac{1}{S_{i}-4\pi\alpha}\right)^{5/2}+\dfrac{3}{2\sqrt{\pi}}\sqrt{\left(\dfrac{1}{S_{i}-4\pi\alpha}\right)}\right)$
(32)

and eq.(73) in the Appendix. Since the heat capacity has infinite
discontinuities, therefore, eq.(15) at points of phase transition gives the
relation

$\displaystyle-\pi(S_{i}-4\alpha\pi)+Q_{i}^{2}\pi^{2}+3(S_{i}-4\alpha\pi)^{2}=0$
(33)

Using above condition in eq.(31), we obtain the right hand side of second
Ehrenfest equation to be

$\displaystyle\dfrac{\Delta\beta}{\Delta\kappa}=-\dfrac{2\pi}{Q_{i}}\left(\dfrac{S_{i}-4\alpha\pi}{\pi}\right)~{}.$
(34)

Eq.(s)(30, 34) show the validity of second Ehrenfest equation at the critical
points.

We calculate the Prigogine-Defay (PD) ratio pr  for the system. Using
eq.(s)(22, 69), we have

$\displaystyle\left[\left(\dfrac{\partial\Phi}{\partial
T}\right)_{Q}\right]_{S=S_{i}}=\left[\left(\dfrac{\partial\Phi}{\partial
T}\right)_{S}\right]_{S=S_{i}}=-\dfrac{\Delta
C_{\Phi}}{T_{i}Q_{i}\Delta\beta}$ (35)

and eq.(72) with above equation gives

$\displaystyle\Pi=\dfrac{\Delta
C_{\Phi}\Delta\kappa}{T_{i}Q_{i}(\Delta\beta)^{2}}=1~{}.$ (36)

These results show the exact second order nature of phase transition. Further
we apply Ruppeiner’s state space geometry analysis to reassure second order
nature of phase transition.

## State space geometry analysis

Ruppeiner’s state space geometry technique is another trusted mathematical
formalism to study thermodynamic systems. The idea is to write the abstract
manifold with the notion of distance from thermodynamic variables and study
the curvature in order to study phase transition. Singularities in the
manifold correspond to phase transition points. Definitions of metric
coefficients are quite standard ru1 , ru2 .

The Ruppeiner metric coefficients for the manifold are given by

$\displaystyle g^{R}_{ij}=-\dfrac{\partial^{2}S(x^{i})}{\partial x^{i}\partial
x^{j}}$ (37)

where $x^{i}=x^{i}(M,Q)$; i=1,2 are extensive variables of the manifold. The
calculation of the Weinhold metric coefficients is convenient for
computational purpose. These are given by

$\displaystyle g^{W}_{ij}=\dfrac{\partial^{2}M(x^{i})}{\partial x^{i}\partial
x^{j}}$ (38)

where $x^{i}=x^{i}(S,Q)$; i=1,2. It is to be noted that Weinhold geometry is
connected to the Ruppeiner geometry through the following map

$\displaystyle dS^{2}_{R}=\dfrac{dS^{2}_{W}}{T}~{}.$ (39)

By putting AdS radius to unity (that is, $p=\dfrac{3}{8\pi}$ from (16)) in
eq.(17), we get

$\displaystyle
M=\dfrac{1}{2}\sqrt{\dfrac{S-4\alpha\pi}{\pi}}+\dfrac{Q^{2}}{2}\sqrt{\dfrac{\pi}{S-4\alpha\pi}}+\dfrac{1}{2}\left(\dfrac{S-4\alpha\pi}{\pi}\right)^{3/2}~{}.$
(40)

Ruppeiner metric coefficients are calculated from the above equation and these
take the form

$\displaystyle g_{QQ}$ $\displaystyle=$
$\displaystyle\dfrac{1}{T}\sqrt{\dfrac{\pi}{S-4\alpha\pi}}$ $\displaystyle
g_{SQ}$ $\displaystyle=$
$\displaystyle-\dfrac{1}{T}\dfrac{Q}{2(S-4\alpha\pi)}\sqrt{\dfrac{\pi}{S-4\alpha\pi}}$
$\displaystyle g_{SS}$ $\displaystyle=$
$\displaystyle\dfrac{1}{8T}\sqrt{\dfrac{\pi}{S-4\alpha\pi}}\left[-\dfrac{1}{\pi(S-4\alpha\pi)}+\dfrac{3Q^{2}}{(S-4\alpha\pi)^{2}}+\dfrac{3}{\pi^{2}}\right]~{}.$
(41)

With these metric coefficients, we now calculate the standard curvature of the
two dimensional manifold. Singularities in the curvature indicate second order
phase transitions. The expression for the curvature in terms of the metric
coefficients read fobi

$\displaystyle R=-\dfrac{1}{\sqrt{g}}\left[\dfrac{\partial}{\partial
S}\left(\dfrac{g_{SQ}}{\sqrt{g}g_{SS}}\dfrac{\partial g_{SS}}{\partial
Q}-\dfrac{1}{\sqrt{g}}\dfrac{g_{QQ}}{\partial
S}\right)+\dfrac{\partial}{\partial Q}\left(\dfrac{2}{\sqrt{g}}\dfrac{\partial
g_{SQ}}{\partial Q}-\dfrac{1}{\sqrt{g}}\dfrac{\partial g_{SS}}{\partial
Q}-\dfrac{g_{SQ}}{\sqrt{g}g_{SS}}\dfrac{\partial g_{SS}}{\partial
S}\right)\right]~{}.$ (42)

where $g=g_{SS}~{}g_{QQ}-g_{SQ}^{2}$. The analytical expresion is too large to
write. We plot it with entropy for fixed value of other parameters in
Fig.(3a).

(a)

Figure 3: Ruppeiner curvature vs. Entropy

Solid line (Q=0.2, $\alpha$=0.02)
Dashed line (Q=0.2, $\alpha$=0)

From the graph we find that the Ruppeiner curvature diverges for all values
where heat capacity is singular. This confirms the second order nature of
phase transition. Same behaviour of Gauss-Bonnet parameter is observed through
Ruppeiner curvature as well, as the points of phase transition are observed to
shift to the right (that is, towards higher entropy value). In the following
sections, we shall be doing the same analysis for Gauss-Bonnet black hole with
Born-Infeld term in order to observe the non-linear effects of
electromagnetism on the thermodynamic quantities.

##  III Gauss-Bonnet-Born-Infeld Black hole

### Thermodynamics

In this section, we shall replace the Maxwell term from eq.(2) with Born-
Infeld term
$L(F)=4b^{2}\left(1-\sqrt{1+\dfrac{F^{\mu\nu}F_{\mu\nu}}{2b^{2}}}\right)$,
where $b$ is Born-Infeld paramter. This lagrangian term reduces to Maxwell
electrodynamics in the limit $b\rightarrow\infty$. Again, the Gauss-Bonnet
term is not going to affect the equation of motion. In this case, the function
$f(r)$ takes the form

$\displaystyle
f(r)=1-\dfrac{2M}{r}+\dfrac{r^{2}}{l^{2}}+\dfrac{2b^{2}r^{2}}{3}\left(1-\sqrt{1+\dfrac{Q^{2}}{b^{2}r^{4}}}\right)+\dfrac{4Q^{2}}{3r^{2}}{}_{2}F_{1}\left[\dfrac{1}{4},\dfrac{1}{2},\dfrac{5}{4},-\dfrac{Q^{2}}{b^{2}r^{4}}\right]$
(43)

where ${}_{2}F_{1}\left[\mu,\nu,\rho,x\right]$ is hypergeometric function.

Mass of the black hole in terms of horizon radius ($r_{+}$) (calculated using
$f(r_{+})=0$) is

$\displaystyle
M=\dfrac{r_{+}}{2}+\dfrac{r_{+}^{3}}{2l^{2}}+\dfrac{b^{2}r_{+}^{3}}{3}\left(1-\sqrt{1+\dfrac{Q^{2}}{b^{2}r_{+}^{4}}}\right)+\dfrac{2Q^{2}}{3r_{+}}{}_{2}F_{1}\left[\dfrac{1}{4},\dfrac{1}{2},\dfrac{5}{4},-\dfrac{Q^{2}}{b^{2}r_{+}^{4}}\right]$
(44)

and the Hawking temperature is

$\displaystyle
T=\dfrac{1}{4\pi}\left[\dfrac{1}{r_{+}}+\dfrac{3r_{+}}{l^{2}}+2b^{2}r_{+}\left(1-\sqrt{1+\dfrac{Q^{2}}{b^{2}r_{+}^{4}}}\right)\right]~{}.$
(45)

Using first law of black hole thermodynamics from eq.(7), we calculate the
electrostatic potential at horizon to be

$\displaystyle\Phi=\left(\dfrac{\partial M}{\partial
Q}\right)=\dfrac{Q}{r_{+}}{}_{2}F_{1}\left[\dfrac{1}{4},\dfrac{1}{2},\dfrac{5}{4},-\dfrac{Q^{2}}{b^{2}r_{+}^{4}}\right]~{}.$
(46)

Since, Wald’s entropy formula does not depend on the matter lagrangian, so it
is independent of the Born-Infeld term and its expression is same as the one
given in eq.(10). Again, we shall put AdS radius $l$ to unity.

The Hawking temperature and potential in terms of entropy read

$\displaystyle
T=\dfrac{1}{4\pi}\left[\sqrt{\dfrac{\pi}{S-4\alpha\pi}}+3\sqrt{\dfrac{S-4\alpha\pi}{\pi}}+2b^{2}\sqrt{\dfrac{S-4\alpha\pi}{\pi}}\left(1-\sqrt{1+\dfrac{Q^{2}\pi^{2}}{b^{2}(S-4\alpha\pi)^{2}}}\right)\right]$
(47)

$\displaystyle\Phi=Q\sqrt{\dfrac{\pi}{S-4\alpha\pi}}{}_{2}F_{1}\left[\dfrac{1}{4},\dfrac{1}{2},\dfrac{5}{4},-\dfrac{Q^{2}\pi^{2}}{b^{2}(S-4\alpha\pi)^{2}}\right]~{}.$
(48)

We plot the Hawking temperature with entropy in Fig.(4a) in order to see its
thermodynamic behaviour. It is a smooth function of entropy.

(a)

Figure 4: Temperature vs. Entropy

Solid line(Q=0.2, $\alpha$=0.02, b=10)
Dashed line (Q=0.2, $\alpha$=0.02, b$\rightarrow\infty$)

The effect of Born-Infeld parameter can also be seen from the plot for same
value of Gauss-Bonnet parameter. Black hole with non-linear electromagnetic
term in Gauss-Bonnet gravity will have higher temperature than with Maxwell
term for same entropy value and this effect is eminent at low entropy values.

Next, we calculate the heat capacity from eq.(13):

$\displaystyle C_{\Phi}$ $\displaystyle=$ $\displaystyle
T\left(\dfrac{\partial S}{\partial T}\right)_{\Phi}$ (49) $\displaystyle=$
$\displaystyle\dfrac{T\left(\dfrac{\partial\Phi}{\partial
Q}\right)_{S}}{\left(\dfrac{\partial\Phi}{\partial
Q}\right)_{S}\left(\dfrac{\partial T}{\partial
S}\right)_{Q}-\left(\dfrac{\partial T}{\partial
Q}\right)_{S}\left(\dfrac{\partial\Phi}{\partial S}\right)_{Q}}~{}.$

These derivatives can be calculated using eq.(s)(47, 48) as

$\displaystyle\left(\dfrac{\partial\Phi}{\partial
S}\right)_{Q}=-\dfrac{Q}{2(S-4\alpha\pi)}\sqrt{\dfrac{\pi}{S-4\alpha\pi}}\left(1+\dfrac{Q^{2}\pi^{2}}{b^{2}(S-4\alpha\pi)^{2}}\right)^{-1/2}~{},$
(50) $\displaystyle\left(\dfrac{\partial\Phi}{\partial
Q}\right)_{S}=\dfrac{1}{2}\sqrt{\dfrac{\pi}{S-4\alpha\pi}}\left[\left(1+\dfrac{Q^{2}\pi^{2}}{b^{2}(S-4\alpha\pi)^{2}}\right)^{-1/2}+{}_{2}F_{1}\left[\dfrac{1}{4},\dfrac{1}{2},\dfrac{5}{4},-\dfrac{Q^{2}\pi^{2}}{b^{2}(S-4\alpha\pi)^{2}}\right]\right]~{},$
(51) $\displaystyle\left(\dfrac{\partial T}{\partial
Q}\right)_{S}=-\dfrac{Q}{2(S-4\alpha\pi)}\sqrt{\dfrac{\pi}{S-4\alpha\pi}}\left(1+\dfrac{Q^{2}\pi^{2}}{b^{2}(S-4\alpha\pi)^{2}}\right)^{-1/2}~{}$
(52)

and

$\displaystyle\left(\dfrac{\partial T}{\partial
S}\right)_{Q}=\dfrac{1}{4\pi}\sqrt{\dfrac{\pi}{S-4\alpha\pi}}\left[-\dfrac{1}{2(S-4\alpha\pi)}+\dfrac{3}{2\pi}+\dfrac{b^{2}}{\pi}\left(1-\sqrt{1+\dfrac{Q^{2}\pi^{2}}{b^{2}(S-4\alpha\pi)^{2}}}\right)\right.$
$\displaystyle\left.+\dfrac{2Q^{2}\pi}{(S-4\alpha\pi)^{2}}\left(1+\dfrac{Q^{2}\pi^{2}}{b^{2}(S-4\alpha\pi)^{2}}\right)^{-1/2}\right]~{}.$
(53)

We plot the heat capacity in Fig.(5a).

(a)

Figure 5: Heat Capacity vs. Entropy

Solid line(Q=0.2, $\alpha$=0.02, b=10)
Dashed line(Q=0.2, $\alpha$=0.02, b$\rightarrow\infty$)

Black hole undergoes phase transition at two points of entropy and exists in
three different phases. Smooth T-S graph in Fig.(4a) excludes any first order
phase transition, so there is again the possibility of second order phase
transition. The qualitative behaviour of heat capacity is same as discussed
for Einstein-Born-Infeld black holes in r2 . Non-linear electromagnetic term
shifts the phase transition points to left. This effect is prominent at lower
entropy values and it dies down as entropy increases as observed for the
Hawking temperature as well.

We now calculate the Smarr relation of the black hole. Using eq.(s)(10,16) in
eq.(44), we can write the mass as

$\displaystyle M\equiv
M(S,p,Q,b,\alpha)=\dfrac{1}{2}\sqrt{\dfrac{S-4\alpha\pi}{\pi}}+\dfrac{1}{2l^{2}}\left(\dfrac{S-4\alpha\pi}{\pi}\right)^{3/2}+\dfrac{b^{2}}{3}\left(\dfrac{S-4\alpha\pi}{\pi}\right)^{3/2}\left(1-\sqrt{1+\dfrac{Q^{2}\pi^{2}}{b^{2}(S-4\alpha\pi)^{2}}}\right)$
$\displaystyle+\dfrac{2Q^{2}}{3}\sqrt{\dfrac{\pi}{S-4\alpha\pi}}{}_{2}F_{1}\left[\dfrac{1}{4},\dfrac{1}{2},\dfrac{5}{4},-\dfrac{Q^{2}\pi^{2}}{b^{2}(S-4\alpha\pi)^{2}}\right]~{}.$
(54)

General first law of thermodynamics for the system becomes

$\displaystyle dM=TdS+vdp+\Phi dQ+\Omega d\alpha+Bdb$ (55)

where $B$ is conjugate variable of $b$ and can be calculated as
$B=\dfrac{\partial M}{\partial b}$. By dimensional analysis, we see the
dimensions of the variables involved in the mass expression in eq.(46) are
$[M]=[L],~{}[S]=[L]^{2},~{}[p]=[L]^{-2},~{}[Q]=[L],~{}[b]=[L]^{-1}$ and
$[\alpha]=[L]^{2}$. Smarr relation for the black hole becomes

$\displaystyle M=2TS-2pv+Q\Phi-bB+2\Omega\alpha~{}.$ (56)

Again we can see that the Gauss-Bonnet and the Born-Infeld parameter enter
into the Smarr formula in $3+1-$ spacetime dimensions.

Next, we turn to phase transition points and their order. We check the status
of Ehrenfest equations for the system and also apply Ruppeiner state space
geometry analysis.

### Ehrenfest Scheme of phase transition for the system

We check the validity of Ehrenfest equations at the critical points for this
black hole system too. Left hand side of the first Ehrenfest equation in
eq.(22) at critical points $S_{i}$ can be calculated using eq.(s)(64, 51, 52)
as

$\displaystyle-\left[\left(\dfrac{\partial\Phi}{\partial
T}\right)_{S}\right]_{S=S_{i}}=\dfrac{S_{i}-4\alpha\pi}{Q_{i}}\left[1+\left(1+\dfrac{Q_{i}^{2}\pi^{2}}{b^{2}(S_{i}-4\alpha\pi)^{2}}\right)^{1/2}{}_{2}F_{1}\left[\dfrac{1}{4},\dfrac{1}{2},\dfrac{5}{4},-\dfrac{Q_{i}^{2}\pi^{2}}{b^{2}(S_{i}-4\alpha\pi)^{2}}\right]\right]$
(57)

and right hand side is calculated using eq.(s)(67, 50, 51) as

$\displaystyle\dfrac{\Delta
C_{\Phi}}{T_{i}Q_{i}\Delta\beta}==\dfrac{S_{i}-4\alpha\pi}{Q_{i}}\left[1+\left(1+\dfrac{Q_{i}^{2}\pi^{2}}{b^{2}(S_{i}-4\alpha\pi)^{2}}\right)^{1/2}{}_{2}F_{1}\left[\dfrac{1}{4},\dfrac{1}{2},\dfrac{5}{4},-\dfrac{Q_{i}^{2}\pi^{2}}{b^{2}(S_{i}-4\alpha\pi)^{2}}\right]\right]~{}$
(58)

which agree with each other.

Using eq.(69), the left hand side of second Ehrenfest equation at critical
points becomes

$\displaystyle-\left[\left(\dfrac{\partial\Phi}{\partial
T}\right)_{Q}\right]_{S=S_{i}}=\dfrac{S_{i}-4\alpha\pi}{Q_{i}}\left[1+\left(1+\dfrac{Q_{i}^{2}\pi^{2}}{b^{2}(S_{i}-4\alpha\pi)^{2}}\right)^{1/2}{}_{2}F_{1}\left[\dfrac{1}{4},\dfrac{1}{2},\dfrac{5}{4},-\dfrac{Q_{i}^{2}\pi^{2}}{b^{2}(S_{i}-4\alpha\pi)^{2}}\right]\right]~{}.$
(59)

The right hand side of second Ehrenfest equation is calculated using
eq.(s)(73, 50, 53) as

$\displaystyle\dfrac{\Delta\beta}{\Delta\kappa}=\dfrac{S_{i}-4\alpha\pi}{Q_{i}}\left[1+\left(1+\dfrac{Q_{i}^{2}\pi^{2}}{b^{2}(S_{i}-4\alpha\pi)^{2}}\right)^{1/2}{}_{2}F_{1}\left[\dfrac{1}{4},\dfrac{1}{2},\dfrac{5}{4},-\dfrac{Q_{i}^{2}\pi^{2}}{b^{2}(S_{i}-4\alpha\pi)^{2}}\right]\right]~{}.$
(60)

This proves the validity of second Ehrenfest equation too. This validates the
second order nature of phase transtion points $S_{i}$. Prigogine-Defay ratio
for the black hole can also be calculated using eq(s)(58, 60) as

$\displaystyle\Pi=\dfrac{\Delta
C_{\Phi}\Delta\kappa}{T_{i}Q_{i}(\Delta\beta)^{2}}=1~{}.$ (61)

We further study Ruppeiner’s state space geometry analysis for the black hole
as studied in section-II.

### State space geometry analysis

We calculate the Ruppeiner curvature for this black hole spacetime as done in
section-II. Mass relation at unit AdS radius from eq.(54) is

$\displaystyle
M=\dfrac{1}{2}\sqrt{\dfrac{S-4\alpha\pi}{\pi}}+\dfrac{1}{2}\left(\dfrac{S-4\alpha\pi}{\pi}\right)^{3/2}+\dfrac{b^{2}}{3}\left(\dfrac{S-4\alpha\pi}{\pi}\right)^{3/2}\left(1-\sqrt{1+\dfrac{Q^{2}\pi^{2}}{b^{2}(S-4\alpha\pi)^{2}}}\right)$
$\displaystyle+\dfrac{2Q^{2}}{3}\sqrt{\dfrac{\pi}{S-4\alpha\pi}}{}_{2}F_{1}\left[\dfrac{1}{4},\dfrac{1}{2},\dfrac{5}{4},-\dfrac{Q^{2}\pi^{2}}{b^{2}(S-4\alpha\pi)^{2}}\right]~{}.$
(62)

Ruppeiner metric coefficients are calculated from above equation and these
take the form

$\displaystyle g_{QQ}$ $\displaystyle=$
$\displaystyle\dfrac{1}{2T}\sqrt{\dfrac{\pi}{S-4\alpha\pi}}\left[\left(1+\dfrac{Q^{2}\pi^{2}}{b^{2}(S-4\alpha\pi)^{2}}\right)^{-1/2}+{}_{2}F_{1}\left[\dfrac{1}{4},\dfrac{1}{2},\dfrac{5}{4},-\dfrac{Q^{2}\pi^{2}}{b^{2}(S-4\alpha\pi)^{2}}\right]\right]~{}$
$\displaystyle g_{SQ}$ $\displaystyle=$
$\displaystyle-\dfrac{1}{2T}\sqrt{\dfrac{\pi}{S-4\alpha\pi}}\dfrac{Q}{(S-4\alpha\pi)}\left(1+\dfrac{Q^{2}\pi^{2}}{b^{2}(S-4\alpha\pi)^{2}}\right)^{-1/2}~{}$
$\displaystyle g_{SS}$ $\displaystyle=$
$\displaystyle\dfrac{1}{8T}\sqrt{\dfrac{\pi}{S-4\alpha\pi}}\left[-\dfrac{1}{\pi(S-4\alpha\pi)}+\dfrac{3}{\pi^{2}}+\dfrac{4b^{2}}{\pi^{2}}\left(1-\sqrt{1+\dfrac{Q^{2}\pi^{2}}{b^{2}(S-4\alpha\pi)^{2}}}\right)+\dfrac{4Q^{2}}{(S-4\alpha\pi)^{2}}\left(1+\dfrac{Q^{2}\pi^{2}}{b^{2}(S-4\alpha\pi)^{2}}\right)^{-1/2}\right]~{}.$
(63)

We can now calculate the Ruppeiner curvature and plot it with entropy for same
parameters which shows same discontinuities.

(a)

Figure 6: Ruppeiner curvature vs. Entropy

Solid line (Q=0.2, $\alpha$=0.02, b=10)
Dashed line (Q=0.2, $\alpha$=0.02, b$\rightarrow\infty$)

This conforms the second order nature of the phase transition. Effects of non-
linear electromagnetism are again certified showing same nature as in heat
capacity plot.

##  IV Conclusion

In this work, we analysed the black holes with Gauss-Bonnet correction term to
the curvature in AdS spacetime. We studied these black holes with charge due
to Maxwell and Born-Infeld modified non-linear electromagnetism. Gauss-Bonnet
lagrangian does not change the dynamics in $3+1-$ spacetime dimensions,
however, black hole entropy gets modified. We calculated the thermodynamic
properties associated with the spacetime and calculated heat capacity which
showed infinite discontinuities when plotted with entropy at constant other
parameters. We further analysed these singularities. Smoothness of $T-S$ plots
ruled out any first order phase transition in both the cases. We employed
Ehrenfest scheme and Ruppeiner geometry analysis to identify the order of
phase transition. Considered systems followed two Ehrenfest equations
certifying second order nature of phase transition. We reconfirmed this using
Ruppeiner curvature calculations. We calculated and plotted Ruppeiner
curvature with entropy for same fixed parameters. Singularities are observed
exactly at those points where heat capacity diverged. Both these methods
reconfirms second order phase transition. We conclude that the effects of
Gauss-Bonnet parameter on the nature of thermodynamic quantities as well as on
phase transition points are merely quantitative as compared to black holes in
Einstein gravity. The Born-Infeld parameter also shifts the points of phase
transitions, however, the effect dominates only in low entropy values.

## Appendix: Simplified Ehrenfest equations

In this Appendix, we shall recast the Ehrenfest equations in a simplified form
whose validity we have checked in the main text to establish the second order
nature of the black hole phase transitions.

The left hand side of the first Ehrenfest equation (22) at the critical point
can be written as

$\displaystyle-\left[\left(\dfrac{\partial\Phi}{\partial
T}\right)_{S}\right]_{S=S_{i}}=-\dfrac{\left[\left(\dfrac{\partial\Phi}{\partial
Q}\right)_{S}\right]_{S=S_{i}}}{\left[\left(\dfrac{\partial T}{\partial
Q}\right)_{S}\right]_{S=S_{i}}}~{}.$ (64)

Also, using eq.(24) and the definition of heat capacity
$C_{\Phi}=T(\frac{\partial S}{\partial T})_{\Phi}$, the right hand side of
eq.(22) becomes

$\displaystyle Q_{i}\beta=\left[\left(\dfrac{\partial Q}{\partial
T}\right)_{\Phi}\right]_{S=S_{i}}=\left[\left(\dfrac{\partial Q}{\partial
S}\right)_{\Phi}\right]_{S=S_{i}}\left(\dfrac{C_{\Phi}}{T_{i}}\right)$ (65)

which then yields

$\displaystyle\dfrac{\Delta
C_{\Phi}}{T_{i}Q_{i}\Delta\beta}=\left[\left(\dfrac{\partial S}{\partial
Q}\right)_{\Phi}\right]_{S=S_{i}}~{}.$ (66)

Using the identity (14), the above equation can be written in the form

$\displaystyle\dfrac{\Delta
C_{\Phi}}{T_{i}Q_{i}\Delta\beta}=-\dfrac{\left[\left(\dfrac{\partial\Phi}{\partial
Q}\right)_{S}\right]_{S=S_{i}}}{\left[\left(\dfrac{\partial\Phi}{\partial
S}\right)_{Q}\right]_{S=S_{i}}}~{}.$ (67)

The second Ehrenfest equation can also be recast in a form such that both
sides of the equation can be computed easily to check its validity. For that
we use the thermodynamic relation $T\equiv T(S,\Phi)$, which gives

$\displaystyle\left(\dfrac{\partial
T}{\partial\Phi}\right)_{Q}=\left(\dfrac{\partial T}{\partial
S}\right)_{\Phi}\left(\dfrac{\partial
S}{\partial\Phi}\right)_{Q}+\left(\dfrac{\partial
T}{\partial\Phi}\right)_{S}~{}.$ (68)

Since heat capacity diverges at the critical points, hence
$\left[\left(\dfrac{\partial T}{\partial S}\right)_{\Phi}\right]_{S=S_{i}}=0$,
and $\left(\dfrac{\partial S}{\partial\Phi}\right)_{Q}$ is finite at the
critical point, therefore the above equation becomes

$\displaystyle\left[\left(\dfrac{\partial
T}{\partial\Phi}\right)_{Q}\right]_{S=S_{i}}=\left[\left(\dfrac{\partial
T}{\partial\Phi}\right)_{S}\right]_{S=S_{i}}$ (69)

Now from eq.(24), at the critical points

$\displaystyle\kappa Q_{i}=\left[\left(\dfrac{\partial
Q}{\partial\Phi}\right)_{T}\right]_{S=S_{i}}~{}.$ (70)

Using the thermodynamic identity $\left(\dfrac{\partial
Q}{\partial\phi}\right)_{T}\left(\dfrac{\partial\Phi}{\partial
T}\right)_{Q}\left(\dfrac{\partial T}{\partial Q}\right)_{\Phi}=-1$ and the
definition of $\beta$, we find

$\displaystyle\kappa Q_{i}=\left[\left(\dfrac{\partial
T}{\partial\Phi}\right)_{Q}\right]_{S=S_{i}}Q_{i}\beta~{}.$ (71)

Therefore, the right hand side of the second Ehrenfest equation (23) reduces
to

$\displaystyle\dfrac{\Delta\beta}{\Delta\kappa}=-\left[\left(\dfrac{\partial\Phi}{\partial
T}\right)_{Q}\right]_{S=S_{i}}~{}.$ (72)

This can further be written as

$\displaystyle-\left[\left(\dfrac{\partial\Phi}{\partial
T}\right)_{Q}\right]_{S=S_{i}}=-\dfrac{\left[\left(\dfrac{\partial\Phi}{\partial
S}\right)_{Q}\right]_{S=S_{i}}}{\left[\left(\dfrac{\partial T}{\partial
S}\right)_{Q}\right]_{S=S_{i}}}~{}.$ (73)

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