# A Deep Reinforcement Learning Approach for Audio-based Navigation and Audio
Source Localization in Multi-speaker Environments

Petros Giannakopoulos, Aggelos Pikrakis, and Yannis Cotronis Research in this
paper was supported by the Hellenic Foundation for Research and Innovation
(H.F.R.I.) under the “1st Call for H.F.R.I. Research Projects to support
Faculty Members & Researchers and the Procurement of High-Cost Research
Equipment Grant” (Project Number: 3449).Petros Giannakopoulos is a PhD
candidate at the Department of Informatics and Telecommunications, National
and Kapodistrian University of Athens, Greece (e-mail:
petrosgk@di.uoa.gr).Aggelos Pikrakis is an Assistant Professor at the
Department of Informatics, University of Pireaus, Greece (e-mail:
pikrakis@unipi.gr).Yannis Cotronis is a Professor at the Department of
Informatics and Telecommunications, National and Kapodistrian University of
Athens, Greece (e-mail: cotronis@di.uoa.gr).

###### Abstract

In this work we apply deep reinforcement learning to the problems of
navigating a three-dimensional environment and inferring the locations of
human speaker audio sources within, in the case where the only available
information is the raw sound from the environment, as a simulated human
listener placed in the environment would hear it. For this purpose we create
two virtual environments using the Unity game engine, one presenting an audio-
based navigation problem and one presenting an audio source localization
problem. We also create an autonomous agent based on PPO online reinforcement
learning algorithm and attempt to train it to solve these environments. Our
experiments show that our agent achieves adequate performance and
generalization ability in both environments, measured by quantitative metrics,
even when a limited amount of training data are available or the environment
parameters shift in ways not encountered during training. We also show that a
degree of agent knowledge transfer is possible between the environments.

###### Index Terms:

audio-based navigation, sound source localization, reinforcement learning

##  I Introduction

We jointly address the problems of real-time audio-based navigation and real-
time audio source localization in virtual three-dimensional environments, for
the case where the sound sources are human speakers, using an approach based
on online Deep Reinforcement Learning. Our objectives are: a) for the problem
of audio-based navigation, to investigate whether an autonomous reinforcement
learning agent, in the form of an entity able to walk in all directions, can
successfully learn to navigate in a three-dimensional space, guiding itself
towards a target sound source (a given speaker) while avoiding other sound
sources (other speakers), b) for the problem of audio source localization, to
investigate whether an autonomous reinforcement learning agent, in the form of
a highly directional microphone that is capable of rotating in all directions,
can learn to locate sound sources in the surrounding environment and point the
microphone towards each detected sound source. In both applications, the agent
is trained via a reward-punishment scheme, which is designed to motivate it to
solve the environment and discourages it from spending too much time in doing
so. In both cases, the only information available to the agent is the raw
audio data from the environment, in the form of a two-channel (stereo) audio
signal, in an attempt to simulate the information that a human listener placed
inside the environment would receive. The raw audio signal is generated by the
Unity game engine to have pseudo-spatial three-dimensional information,
meaning that the audio engine employs DSP techniques to simulate a number of
channels higher than two and improve directionality of audio sources.

More specifically, for application (a) we created an environment in the Unity
game engine [1] which simulates an entity, controlled by the agent, moving in
a room where there also exist a number of sound sources in the form of human
speakers, placed in random locations in the room. The objective of the entity
is to move towards and eventually reach a specified target speaker in the
shortest possible amount of time, while at the same time avoiding coming into
contact with the other speakers or going outside the room boundaries. Our
experiments show that the entity controlled by the deep reinforcement learning
agent, rewarded when reaching the target speaker and punished when bumping
into other speakers or going outside the room boundaries, is able to solve the
aforementioned problem with a degree of success much higher than random
chance, i.e., when compared with an agent which moves the entity in a random
manner. The agent also outperforms a human in control of the entity. For
application (b) the environment consists, again, of a room inside which a
number of speakers are placed in random locations. However, in this case, a
virtual directional microphone is placed in the center of the room, capable of
rotating freely in every direction. An agent is in control of this microphone
and can rotate it by applying a corresponding torque to adjust its azimuth and
elevation. The objective is to point the microphone towards each speaker in
the room until all speakers are detected, in the shortest possible amount of
time and with the least possible microphone movement. Our experiments show
that the reinforcement learning agent is successful in solving this task,
achieving higher detection accuracy than an agent moving the microphone
randomly, as well as outperforming a human controlling the microphone.

In both environments, we observed that our reinforcement learning agent is
able to generalize well to speakers not seen during training and appears
robust to distortion effects not encountered during training, such as various
types of sound reverberation.

We also observed that a significant amount of acquired agent knowledge is
transferable between the two application environments. The ability to locate
sound sources based on environmental audio data was shown to be the most
important and difficult skill for the agent to learn. However, once acquired
for one application environment, it can be applied to another. The agent, for
the most part does not have to re-learn that skill in the new environment if
an appropriate transfer learning scheme is used.

The motivation behind this work was to investigate whether it is possible for
a reinforcement learning agent to detect the locations and identities of
speakers present in its environment and successfully perform tasks based on
that information. This has the potential to expand the range of stimulus
autonomous agents can use in order to be successful in a variety of
environments and tasks. For example, if there are auditory cues in the
environment that could be potentially helpful, then an agent capable of
incorporating them, possibly along with other types of information (e.g.
visual cues) could be more successful than an agent relying only on one type
of sensory inputs (e.g. only visual). One practical application could be the
development of AI in games which can respond to auditory cues, as well as
possibly other stimuli, with a minimal amount of engineering and hard-coded
behaviours. Furthermore, it was shown that behaviors acquired by reinforcement
learning agents in a simulated environment can, to a degree, transfer to the
real world [2], [3]. Behaviors learned in an appropriately crafted simulated
environment, which incorporate auditory sensory information also present in
the real world, could potentially be transferable to a real world setting.

##  II Related Work

To our knowledge there has been limited published work in the domains of
audio-based navigation and audio source localization for methods based on
reinforcement learning. The work in [4] combined deep reinforcement learning
with environmental audio information in the context of navigation of a virtual
environment by an autonomous agent. They discovered that an agent trained via
reinforcement learning and leveraging raw audio information could reach more
reliably a sound-emitting target within a maze compared to the case when only
visual information was used. In [5] the authors augmented the VizDoom
simulator [6] with the ability to provide audio from the environment to an
agent. They showed that an RL agent based on PPO was able to perform a number
of simulated tasks requiring audio sensory information. The agent was also
able to outperform an agent relying only on visual information in a duel
inside the VizDoom simulator. In [7] the authors employed deep reinforcement
learning for controlling the gaze of a robotic head based on audio and visual
data from a virtual environment. The work of [8] focused on development of a
framework for sound localization and tracking of sound sources in combination
with visual information, for navigation of an autonomous agent in a virtual
environment. Their system consists of a sound propagation model and sound
localization model based on classical (non-DNN) algorithms. Previous
preliminary work by the authors in [9] created an autonomous agent based on
deep reinforcement learning, able to navigate a virtual environment posing an
audio-based navigation challenge, where a fixed number of speakers are present
in a room, and guide itself towards a defined speaker while avoiding colliding
with the other speakers, using only raw audio information from the
environment. We extend upon this preliminary work by adding another
environment focused on the task of audio source localization, unifying the
training regime of the autonomous agent across both environments and studying
the degree of agent knowledge transfer between the environments.

It has to be noted that supervised deep learning methods have also enjoyed
success in recent years over classical methods in the domains of speaker
detection and identification [10, 11], speaker localization [12, 13, 14] and
more general audio source localization tasks [15, 16, 17]. Most of these types
of approaches largely rely on datasets created from prior recordings of sound
sources in specific environments made with special microphone types.

In our approach we are assessing the viability of methods based on deep
reinforcement learning to perform the tasks of detection and localization of
human speaker audio sources in three-dimensional spaces, while simultaneously
performing continuous control of autonomous entities. We experimentally show
that such deep reinforcement learning algorithms have the potential to achieve
these tasks while being solely dependent on the auditory sensory information
present in the environment. Due to the lack, to our knowledge, of an accepted
standard benchmark for reinforcement learning tasks in the audio domain
(contrary to the more mature field of reinforcement learning in the visual
domain, with standardized benchmarks such as ALE [18]) we created our own
virtual environments based on the Unity game engine, simulating the tasks
described above. We have the ability to customize several parameters about
each environment, such as number of sound sources, audio signal parameters of
each source, sources placement and realistic audio distortions due to
environment characteristics.

Figure 1: Left: Agent’s neural network architecture. The left and right channel of a stereo audio signal from the Unity environment are concatenated into a single vector. The resulting vector is then fed into a fully connected feed-forward neural network. The output of the network is either a) the velocity $(V_{x},V_{y})$ of the agent-controlled entity in the navigation application or b) the torque $(T_{\theta},T_{\phi})$ that should be applied to the agent-controlled microphone to adjust its azimuth $\theta$ and elevation $\phi$ respectively in the source localization application. Right: Alternative convolutional neural network architecture, where $f$ is the number of filters, $k$ is the size of the 1-D convolutional kernel, and $s$ is the stride. The features extracted from the concatenated audio input by this convolutional encoder are then averaged and passed to a fully-connected layer. For our experiments we found that this architecture did not provide significant performance benefits over the simpler architecture of only fully-connected layers. TABLE I: Performance comparison of DNN and CNN architectures Architecture | Reward | Success Rate
---|---|---
DNN | $0.497\pm 0.053$ | $96\%$
CNN | $0.502\pm 0.047$ | $96\%$
TABLE II: Performance scaling with number of samples Number of samples | Reward | Success Rate
---|---|---
256 | $0.317\pm 0.092$ | $91\%$
512 | $0.402\pm 0.069$ | $95\%$
1024 | $0.497\pm 0.053$ | $96\%$
2048 | $0.435\pm 0.072$ | $96\%$
4096 | $0.338\pm 0.098$ | $92\%$

Figure 2: Each one of the environments can be described as a Markov Decision
Process (MDP), where the state $s$ is the auditory information that an agent
is able to extract from the environment at a time instant $t$. The agent can
then perform an action $a$ based on that information and receive a reward $r$
from the environment based on the outcome of its action, before transitioning
to the next state.

Figure 3: Top-down views of the environment used for the audio-based
navigation experiments, for the case where $3$ speakers are present. The views
show two example state snapshots, one where agent $A$ has direct “line-of-
sight” to $TS$ (left) and one where it does not (right). Agent $A$ can move
along the $x$ and $y$ axes, while the speakers are stationary. $TS$ is the
target speaker and $S$ are other speakers in the room whose signals interfere
with the signal produced by speaker $TS$. The agent $A$ must reach speaker
$TS$ without running into the other speakers $S$.

Figure 4: Top and Side views of the room where the microphone (M) and speakers
(S) are located, for the case where $3$ speakers are present. The microphone
is located in the center of the room and it is free to point to any direction
by adjusting its azimuth and elevation. Sound sources can be located in a
$360$ degrees range around the microphone but they are constrained to be up to
$30$ degrees above or below the microphone.

##  III Overview of Agent and Environments

We start by providing detailed descriptions of our methods and the virtual
environments that we created for each one of the aforementioned problems in
order to assess our reinforcement learning approach.

###  III-A Agent Architecture

Our agent architecture is based on a combination of the Proximal Policy
Optimization (PPO) online reinforcement learning algorithm [19] with a small
neural network consisting of two fully connected hidden layers. Our choice of
reinforcement learning algorithm is not restrictive and other policy gradient
optimization algorithms can be potentially suitable as well. The input to the
neural network is the left and right channels of the stereo audio signal
produced by the Unity game engine, concatenated into a single one-dimensional
vector. In our experimental setup we retrieve the most recent $1024$ audio
samples from the Unity engine’s audio buffer for each channel. This means
that, for a sampling frequency of $48$kHz, approximately, the most recent
$21$ms of the audio signal produced by the environment is the input frame to
the agent at timestep $t$. Therefore, the resulting vector, after the
concatenation operation, has a length of $2048$ samples and it is fed as input
to the network (Figure 1). Each hidden layer of the network has $256$ neurons.
We selected this particular architecture and input scheme after a coarse grid
search over different fully-connected and 1D convolutional hidden-layer
architectures, lengths of audio buffer and sampling frequencies: 1) In Table I
we compare the performance on our audio-based navigation task of the current
DNN architecture and a CNN architecture based on a stack of 1D convolutional
layers (Figure LABEL:fig:conv_arch). We found that the agent achieves the same
performance, within margin of error, with both architectures, however the CNN
is more computationally demanding, therefore we decided to use a DNN
architecture for the rest of the experiments. However, there is always the
possibility that the CNN-based architecture may outperform the simpler DNN-
based architecture depending on the environment and task. 2) In Table II we
compare the performance of different lengths of audio buffer given as input to
the agent in number of samples. Initially, the agent’s performance improves as
the length of the audio input increases since there is more audio context
available to the agent. However, after a point, performance decreases as the
markovian property of the state space required by the reinforcement learning
algorithm is violated. Longer audio buffer lengths are more computationally
expensive as well. The output of the network is a fully connected layer
consisting of $2$ neurons. In the case of audio-based navigation, the network
outputs are the estimates for the velocity with which the entity should move
in the $x$ and $y$ directions, normalized in the $[-1,1]$ range. In the case
of audio source localization, the network outputs are the estimates of the
torque that should be applied to the microphone gimbal in order to adjust the
azimuth $\theta$ and elevation $\phi$ of where the microphone should be
pointing, also normalized in the $[-1,1]$ range. The desired outputs are the
only difference between the two environments, otherwise the agent’s
architecture is identical between them.

###  III-B Environments

In our study, we used the Unity editor to create custom virtual environments
that run on the Unity game engine. We chose to create our environments in
Unity after investigating various known frameworks and platforms for creating
virtual environments suitable for training reinforcement learning agents. The
reason we eventually selected Unity was that it provided the easiest to use
and most robust set of tools for creating environments containing audio
sources and entities with highly customizable properties and also allowed for
seamless capturing of audio data from the simulated environment without the
need for additional tools. We present below a detailed description of each the
two test environments:

####  III-B1 Audio-based navigation environment

Our environment focused on audio-based navigation (Figure 4) consists of the
following key components: a) A rectangular room, b) One up to five speakers
(denoted $S$) placed inside the room, one of which is the target speaker
(denoted $TS$) and c) The agent ($A$) inside the room. Each speaker is a
stationary Unity audio source which plays back an audio file containing an
utterance for that speaker, randomly selected from a pool of utterances. When
an utterance finishes, another one is randomly selected from the pool and
played back. All audio sources share the same properties and are accordingly
configured in the Unity editor. Specifically, the signal intensity (volume) of
the audio source decreases linearly with the distance from the source (linear
volume roll-off). The maximum distance from which an audio source can still be
heard is set so that all sources are audible across the entire room. The agent
($A$) can move inside the room along the $x$ and $y$ axes by adjusting its
velocity vectors $v_{x}$ and $v_{y}$ over each axis. Each training episode
begins with the agent appearing in a random location of the lower edge of the
room and the speakers show up in random locations inside the room. The audio
sources are placed at various heights above the floor of the room in order to
simulate speakers of different heights, e.g. one speaker can be $1.70$m tall
while another may be $1.85$m tall and their mouths are therefore located at
different heights. The agent receives a positive reward of $+1.0$ when it
reaches speaker $TS$ and a negative reward of $-1.0$ when it crashes onto
another speaker $S$ or when it moves outside the room boundaries. It also
receives a small negative reward of $-0.001$ for every step it takes to
discourage the agent from procrastinating. A training episode ends when the
agent reaches the target speaker, one of the other speakers or moves outside
the room boundaries.

####  III-B2 Audio source localization environment

The environment focused on audio source localization (Figure 4) consists of
three key components: a) A rectangular room, b) One up to five speakers, $S$,
placed inside the room, c) A highly directional microphone, $M$, placed on an
omnidirectional gimbal able to freely rotate in every direction. The
microphone apparatus is placed in the center of the room. The agent can adjust
the direction that the microphone ($M$) is pointing to by applying the
appropriate amount of torque to it, in order to change its azimuth ($\theta$)
and elevation ($\phi$) angles. Each training episode begins with the
microphone at its default starting rotation of $\theta=0\degree$ and
$\phi=0\degree$. A random number of sound sources, ranging from $1$ to $5$ on
each episode, are placed at random locations inside the room. Each source
represents a speaker located at a certain distance from the microphone and at
a certain height. Like in the audio-based navigation environment, each speaker
is an audio source playing back a randomly selected utterance, audible across
the entire room with a linear volume roll-off relative to the distance from
the source. The agent in control of the microphone receives a positive reward
of $+1.0$ when it has pointed it towards all of the speakers and a small
negative reward of $-0.001$ for every action it takes, in order to encourage
it to complete the task as soon as possible. A training episode ends either
after the agent has pointed the microphone towards every speaker or when a
predefined time limit has ran out.

For the sake of reproducibility, the Unity project files have been made
publicly available over the Internet 111https://github.com/petrosgk/AudioRL,
including all required components for running the experiments, i.e., the Unity
environments, the pre-trained agents and the datasets.

##  IV Training the Agent

Each environment can be formalized as a Markov Decision Process (Figure 2)
with the following characteristics: a) The state, $s_{t}$, at time $t$
consists of two 1-D vectors, representing the left and right channels of the
raw audio waveform of the environment, which are subsequently concatenated
into a single 1-D vector, b) Given $s_{t}$, the agent takes an action $a_{t}$,
which consists of a $1$-D vector made up from two scalars. For the audio-based
navigation environment, each scalar is representing, the normalized velocity
of the moving entity on the horizontal and vertical axis ($V_{x}$, $V_{y}$),
respectively, with $V_{x}\in[-1,1]$ and $V_{y}\in[-1,1]$. For the audio source
localization environment, each scalar is representing the torque
($T_{\theta}$, $T_{\phi}$), with $T_{\theta}\in[-1,1]$ and
$T_{\phi}\in[-1,1]$, which are applied to adjust the azimuth and elevation of
the agent-controlled microphone, respectively, c) As a result of action
$a_{t}$, the agent transitions to the next state $s_{t+1}$ and receives a
reward $r\in[-1,1]$.

We assume that there exists an optimal policy $\pi^{*}$, which, when followed
by the agent, maximizes the cumulative reward, $r$, achieved for a horizon of
$T$ time steps, where each timestep is an interaction with the environment.
The agent’s training objective is to find a close approximation,
$\pi_{\theta}\approx\pi^{*}$, to the optimal policy $\pi_{\theta}$, where
$\theta$ are the parameters of a neural network. To train the network
parameters, the policy runs for $T$ time steps and the collected samples are
used for updating the policy gradient. In order to do this we first need to
estimate the advantage at timestep $t$ of action $a_{t}$ in state $s_{t}$, as
it was formulated in [20]:

$A_{t}(a_{t},s_{t})=\delta_{t}+\gamma\delta_{t+1}+\dots+\gamma^{T-t+1}\delta_{T-1},$
(1)

with

$\delta_{t}=r_{t}+\gamma V^{\pi}(s_{t+1})-V^{\pi}(s_{t}),$ (2)

Figure 5: Left: Progression, in terms of cumulative reward achieved in an
episode, during training the agent in the audio-based navigation environment.
Right: Average number of steps that the agent needed to complete an episode as
training proceeded. The number of steps starts low in the beginning of
training as the agent tends to go outside the room boundaries or crashes into
other speakers, then rises sharply as the agent tends to procrastinate, then
drops sharply while the reward rises as the agent learns to reach the target
speaker.

Figure 6: Left: Progression, in terms of cumulative reward achieved in an
episode, during training the agent in the audio source localization
environment. Right: Average number of steps that the agent needed to complete
an episode as training proceeded. The number of steps decreases as the agent
learns to locate all the speakers present in the room faster.

where

$V^{\pi}(s)=\mathbb{E}[\sum_{k=0}^{\infty}\gamma^{k}r_{t+k}]$ (3)

is the value estimate of state $s$ under policy $\pi$ and it is the mean
expected reward, over $k$ time steps, for following policy $\pi$ from state
$s$. In the above, $t\in[0,T]$ is the timestep index, $r_{t}$ is the reward at
timestep $t$ and $\gamma\in(0,1]$ is the future reward discount factor. The
objective minimized by PPO is, according to [14]:

$L_{t}(\theta)=\mathbb{E}_{t}[L_{t}^{C}(\theta)-c_{1}L_{t}^{VF}(\theta)+c_{2}S[\pi_{\theta}](s_{t})],$
(4)

where

$L^{C}(\theta)=\mathbb{E}_{t}[min(r_{t}(\theta)A_{t},clip(r_{t}(\theta),1-\epsilon,1+\epsilon)A_{t})]$
(5)

and

$L_{t}^{VF}(\theta)=(V_{\theta}(s_{t})-V_{t}^{target})^{2}$ (6)

$c_{1}$, $c_{2}$ are coefficients, $S$ is an entropy bonus added to policy
estimation, which encourages exploration, and $\epsilon$ is a hyper-parameter.
In all our experiments we set $\gamma=0.99$, $c_{1}=0.95$, $c_{2}=0.001$ and
$\epsilon=0.2$.

##  V Experiments

In this section we outline our testing methodology and present our
experimental results for each of the two environments.

###  V-A Testing Methodology

The datasets used for the experiments for each environment were created from
$10$ publicly available audiobooks, each read by a different speaker. We first
split each audiobook in utterances in an automated way, using a Voice Activity
Detector (VAD) which labels the boundaries of each utterance. We then manually
verified the VAD results and corrected the boundaries of detected utterances
where needed. We assume that the definition of an utterance is the linguistic
one: ”An uninterrupted chain of spoken or written language.” [21]. This
procedure resulted into a dataset of approximately $600$ utterances for each
one of the speakers.

####  V-A1 Audio-based navigation environment

For the audio-based navigation environment, we pick $5$ speakers from the
dataset. We then randomly split the pool of utterances for each one of the $5$
speakers into a training partition and a testing partition. The resulting
training set consists of approximately $500$ utterances for each speaker,
while the test set consists of around $100$ utterances per speaker. We then
set one speaker as the target and train the agent for a total of $8$ million
steps in the created environment. Figure 5 shows: a) the progression of the
agent’s performance during training in terms of the cumulative reward achieved
during an episode relative to the number of training steps the agent has taken
in the environment and b) the number of steps it took to complete an episode,
by reaching the target speaker, another speaker, or going outside the room
boundaries, relative to the number of training steps. Upon completion of the
training stage, we evaluate the agent’s performance by switching to the test
set and letting the trained agent play $100$ episodes in the environment. For
each episode we keep a record of the cumulative reward achieved by the agent,
which is a quantitative measure of its performance. We mentioned in our
description of the environment that the agent receives a large positive reward
when it reaches the target speaker, a large negative reward when it collides
with another speaker or goes outside the room boundaries, and a small negative
reward for each step it takes. Therefore, the higher the cumulative reward,
the quicker the agent has reached the target speaker without running into
another speaker or going outside the room. After the $100$ testing episodes
are completed, we compute the average cumulative reward achieved over all
testing episodes and then compare it with the corresponding reward achieved by
a baseline agent performing a random action policy as well as a human in
control of the agent. The human player was first allowed to familiarize
himself with the environment and the task. He can move the agent using the
arrow keys on the keyboard and has to complete the task using only auditory
information (e.g. without looking at the screen). This training-testing
experiment is repeated five times by setting each time one of the five
speakers as the target speaker. In the end, we average the cumulative reward
over all target speakers. We also calculate the overall success rate
(successfully reaching the target speaker) and likewise compare it with the
success rate of the random and human baselines.

TABLE III: Results for Audio-based Navigation Environment

Agent Type | Reward | Success Rate
---|---|---
PPO | $0.497\pm 0.053$ | $96\%$
Human | $-0.910\pm 0.082$ | $16\%$
Random | $-1.052\pm 0.042$ | $4\%$
Training Utterances | Reward | Success Rate
500 | $0.497\pm 0.053$ | $96\%$
50 | $0.348\pm 0.046$ | $89\%$
1 | $0.274\pm 0.046$ | $67\%$
Reverberation Type | Reward | Success Rate
No reverberation | $0.497\pm 0.053$ | $96\%$
Room (low reverberation) | $0.391\pm 0.052$ | $93\%$
Auditorium (high reverberation) | $0.418\pm 0.063$ | $92\%$
Pitch Shift | Reward | Success Rate
No pitch shift | $0.497\pm 0.053$ | $96\%$
Pitch shift | $0.329\pm 0.077$ | $91\%$

TABLE IV: Results for Audio Source Localization Environment

Agent Type | Reward | Success Rate
---|---|---
PPO | $0.748\pm 0.021$ | $74\%$
Human | $0.105\pm 0.095$ | $10\%$
Random | $-0.158\pm 0.082$ | $2\%$
Training Utterances | Reward | Success Rate
500 | $0.748\pm 0.021$ | $74\%$
50 | $0.621\pm 0.055$ | $70\%$
1 | $0.427\pm 0.039$ | $59\%$
Reverberation Type | Reward | Success Rate
No reverberation | $0.748\pm 0.021$ | $74\%$
Room (low reverberation) | $0.691\pm 0.049$ | $71\%$
Auditorium (high reverberation) | $0.688\pm 0.034$ | $70\%$
Pitch Shift | Reward | Success Rate
No pitch shift | $0.748\pm 0.021$ | $74\%$
Pitch shift | $0.682\pm 0.083$ | $71\%$

####  V-A2 Audio source localization environment

For the audio source localization environment, we pick all $10$ speakers from
the dataset. We then split them into a training set and a test set, each
containing $5$ speakers. The training set contains $3$ male and $2$ females
speakers, while the test set contains $2$ male and $3$ female speakers. Each
speaker is unique to the training and test set. We subsequently train the
agent for a total of $8$ million steps in the created environment. Figure 6
shows: a) the progression of the agent’s performance during training in terms
of the cumulative reward achieved during an episode relative to the number of
training steps the agent has taken in the environment and b) the number of
steps it took to complete an episode by finding all speakers in the room
relative to the number of training steps. Upon completion of the training
stage, we evaluate the agent’s performance similarly to the audio-based
navigation environment, i.e., we compute the average cumulative reward and
success rate over $100$ testing episodes and then compare them with a random
baseline agent and a human controller. Similar to the audio-based navigation
environment, the human player can rotate the microphone using the arrow keys
on the keyboard and has to complete the task without visual information. In
this case, the agent receives a large positive reward when it finds all
speakers in the room and a small negative reward for each step it takes.
Therefore, the higher the cumulative reward, the quicker the agent has pointed
the microphone towards all speakers in the room. The agent’s success rate is
defined in this environment as having located all speakers in the room before
the predefined time limit has lapsed.

For both application environments, we also evaluate the agent’s ability to
learn when only a small amount of training data is available. To that end,
during training, we limit the number of utterances per speaker to $50$ and $1$
(from $500$). In this way, we can observe if the agent can still learn when
just a few training data are available per speaker.

Furthermore, we also evaluate the agent’s ability to generalize to environment
variables that it did not encounter during training. To that end, we add
reverberation to the sound sources, which is a step towards bringing the
virtual environments closer to realism, since in reality the speakers voices
would bounce off the walls, floor and ceiling of the rooms. We try out two
levels of reverberation, low and high, corresponding to different
environments. We only add reverberation in the testing phase, i.e.,
reverberation is not present during training. Additionally, we randomly shift
the pitch frequency of each speaker by $4$ to $8$ percent, during test time
only, and compute again the agent’s success rate.

###  V-B Results

####  V-B1 Audio-based navigation environment

Our findings for the audio-based navigation environment are summarized in
Table IV for the three aforementioned experiments. In more detail, the table
shows:

  1. a)

The average cumulative reward and success rate achieved over $100$ episodes by
our agent based on PPO, compared to the average cumulative reward and success
rate achieved by an agent randomly navigating the room and by a human player:
This result shows that our agent is able to navigate the room and reach the
target speaker in a short amount of time, with performance higher than with
random chance and a human player. This can be interpreted as an indication
that our proposed RL agent is capable of learning a representation of the
environment that leads it to identify the target speaker and its position with
a high degree of accuracy, based only on raw audio data. We observed that,
while a human is quickly able to infer the approximate locations of the
speakers, they have trouble doing so with enough precision relative to their
position to effectively navigate the environment, unless also using visual
information.

  2. b)

The reward achieved by our agent when trained on the full training set of
$500$ utterances per speaker, when it is only trained on $50$ utterances per
speaker and when it is trained on only $1$ utterance per speaker: Like before,
the agent is again evaluated on the same test set of $100$ utterances per
speaker. When trained on the small training set of $50$ utterances per speaker
we observe a drop of approximately $30\%$ in the reward achieved and a $7\%$
drop in success rate. This result shows that, while the agent now struggles
more to reach the target speaker, as evidenced by the lower reward, it can
still reach the target speaker with a high degree of success. When trained on
the absolute minimum training set of just $1$ utterance per speaker, the
achieved reward drops by $45\%$ and the success rate drops by $30\%$. While
the drop in success rate is now significant, the agent still outperforms the
human and random baselines. We can interpret these results as an indication
that the agent can generalize satisfactorily from very little training data.

  3. c)

The reward achieved by our agent when there is reverberation present in the
environment during the testing phase but not during the training phase. The
cumulative reward achieved by the agent suffers on average a $20\%$ drop with
both types of reverberation, compared to the case when no reverberation is
present, while the success rate drops by $4\%$ on average. We can conclude
from this result that the agent is not particularly affected by distortions
due to reverberation, even if those are not encountered during training.

  4. d)

The reward achieved by our agent when the pitch of the speakers is shifted
during the testing phase. The pitch remains unaltered during the training
phase. The cumulative reward achieved by the agent when the pitch is shifted
is lower by $10\%$, while the success rate drops is $5\%$. We can conclude
that the agent is mostly robust to changes in speakers pitch, even if those
were not encountered during training.

####  V-B2 Audio source localization environment

Our findings for the audio source localization environment are summarized in
Table IV, similarly to the results for the audio-based navigation environment.
We record the average cumulative reward and success rate achieved by the agent
over $100$ played episodes after:

  1. a)

training the agent on the full training set,

  2. b)

training the agent on a reduced training set,

  3. c)

including reverberation distortions in the environment at test time,

  4. d)

introducing random shifts in the speakers pitch at test time.

Then we compare with the performance achieved by random and human baselines.
We can observe that the performance of the agent far outperforms by far both a
random and a human baseline. It is also able to generalize well from a small
amount of training data, suffering a modest $4\%$ drop in success rate when
trained on the small training set of $50$ utterances per speaker, while
suffering a significant $15\%$ drop in success rate when trained on the
absolute minimum training set of $1$ utterance per speaker. The agent also
appears to be robust to different types of reverberation, added only during
the testing phase, suffering a modest $3-4\%$ drop in success rate depending
on the amount of reverberation. The agent also does not appear to be
significantly affected by random changes in the speakers pitch, added only
during the testing phase, since its success rate then drops by $3\%$.
Similarly to the audio-based navigation application, the poor human
performance can be explained by the fact that, while a human was easily able
to infer the general direction of incoming audio, they were not able to
precisely pin-point it, unlike the PPO agent, unless also using visual
information.

Figure 7: Left: Training progression on the audio-based navigation environment
of: a) an agent trained from scratch on the environment, b) an agent that was
pre-trained on the audio source localization environment. Right: Training
progression on the audio source localization environment of: a) an agent
trained from scratch on the environment, b) an agent that was pre-trained on
the audio-based navigation environment. We can observe that the ability to do
audio source localization is the most important and difficult skill to learn
in both environments.

###  V-C Transfer Learning

We investigate whether there exists a degree of transferable skills between
the audio-based navigation and audio source localization tasks, meaning that
an agent trained on one task might be able to perform better on the other task
compared to an agent who has not encountered either task before. For example,
in the audio-based navigation environment it is plausible to assume that the
agent also implicitly learns to perform audio source localization as part of
tackling the navigation task. We can make this assumption if we consider that,
in order for the agent to successfully navigate the environment and find the
target speaker, it would need to acquire implicit knowledge about the
locations of the speakers in the room from the environmental audio data. Then
it must distill that knowledge down to the information it deems necessary to
accomplish its task. To test this hypothesis, we first took an agent trained
on the audio source localization task and then fine-tuned it on the audio-
based navigation task. We observed that the pre-trained agent starts at a
higher performance level, reaches the final performance of an agent trained
from scratch in less training steps, but does not exceed it. Similarly, we
started with an agent trained on the audio-based navigation task and fine-
tuned it on the audio source localization task. We again observed that the
pre-trained agent performs better and reaches peak performance much faster
than an agent starting from scratch. Both experiments are illustrated in
Figure 7. An agent trained on the audio source localization task and then
evaluated on the audio-based navigation task, without any further training on
this task, achieves a success rate of $72\%$ compared to $96\%$ when trained
on this task (Table VI). An agent trained on the audio-based navigation task
and then evaluated on the audio source localization task achieves a success
rate of $61\%$ compared to $74\%$ when trained on this task specifically
(Table VI). These results show that the agent learns to perform implicit audio
source localization as part of the audio-based navigation task. Furthermore,
it appears that the ability to do audio source localization is the most
important component of the audio-based navigation task as well, while the
navigation ability component appears easier for the agent to learn. We can
conclude that the skill of audio source localization is what the agent spends
most of its time acquiring, as evident by the degree of this knowledge
transfer between the audio source localization and audio-based navigation
environments.

TABLE V: Transfer Learning Results for Navigation Environment

Trained for | Reward | Success Rate
---|---|---
Navigation | $0.497\pm 0.053$ | $96\%$
Localization | $0.135\pm 0.049$ | $72\%$

TABLE VI: Transfer Learning Results for Localization Environment

Trained for | Reward | Success Rate
---|---|---
Localization | $0.748\pm 0.021$ | $74\%$
Navigation | $0.558\pm 0.074$ | $61\%$

##  VI Conclusions

We examined the viability of applying deep reinforcement learning on the
problems of real-time audio-based navigation and audio source localization in
three-dimensional spaces. For this purpose we designed and constructed, using
the Unity game engine, two virtual environments, representative of the above
tasks. We also built an agent based on the PPO online reinforcement learning
algorithm which accepts raw stereophonic environmental audio data as the state
representation at a given time instant and maps it to actions that should be
performed given this representation. After training the agent in each
environment, we found that it is able to outperform both a random and a human
baseline. Furthermore, the agent was still able to achieve satisfactory
performance when trained on less training data. It also showed robustness to
distortions due to different types of sound reverberation not encountered
during training, suffering a small or zero performance impact. The agent was
also robust to random changes in the pitch of the speakers voice, which were
not encountered during training. Finally, we examined the degree with which
skills learned by the agent in one application environment can be transferred
to the other. We concluded that the audio source localization skill learned in
the respective environment can be, to a large degree, transferred to the
audio-based navigation task, reducing the number of time steps that the agent
must spend to learn the new environment and reach peak performance. We also
concluded that audio source localization is the most important and difficult
skill that the agent needs to acquire in order to succeed in both
environments.

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# Emergence of Cosmic Space and Horizon Thermodynamics from Kaniadakis Entropy

Pranav<EMAIL_ADDRESS>Sarath
<EMAIL_ADDRESS>Navaneeth
<EMAIL_ADDRESS>and Sreejith E
<EMAIL_ADDRESS>
1Department of Physics, Payyanur College, Payyanur-670327, India
1Department of Physics, Indian Institute of Technology, Kanpur-208016, India

###### Abstract

Utilizing Kaniadakis entropy associated with the apparent horizon of the
Friedmann-Robertson-Walker (FRW) Universe and applying the emergence of cosmic
space paradigm, we deduce the modified Friedmann equation for a non-flat
(n+1)-dimensional universe. Employing the first law of thermodynamics, we
arrive at the same modified Friedmann equation, showing the connection between
emergence of cosmic space and first law of thermodynamics. We also establish
the condition to satisfy the Generalized second law of thermodynamics within
the Kaniadakis framework. Our study illuminates the intricate connection
between the law of emergence and horizon thermodynamics, offering a deeper
insight through the lens of Kaniadakis entropy.

## Introduction

The deep connection between gravity and thermodynamics serves as the driving
force behind the emergent understanding of gravity. This intriguing connection
was first realized after the discovery of blackhole thermodynamics by
Bekenstein and Hawking[1, 2, 3, 4]. The idea that gravity and thermodynamics
are closely related is supported by evidence from other scenarios including
Hooft’s investigations into black hole physics and quantum field theory which
contributed to the understanding of thermodynamic properties of black holes,
establishing the connection between thermodynamics and gravity [5] and
Susskind’s holographic principle which suggests that entropy and other
thermodynamic property of a region of spacetime can be described by degrees of
freedom on the boundary of that region substantiating the connection [6].
Jacobson’s derivation of Einstein’s field equations from the fundamental
Clausius connection on a local Rindler causal horizon leads to a significant
advancement in this direction [7]. Following this, various schemes were put
forward for relating gravity with thermodynamics across a range of
gravitational theories [8, 9]. For instance, Verlinde proposed that gravity
emerges as an entropic force resulting from variations in entropy linked to
the positions of material objects [10]. The majority of research on the
emergent paradigm treats the gravitational field equations as an emergent
phenomenon that emerges from a pre-existing space-time background.

Padmanabhan proposed a new perspective on gravity, viewing spacetime as an
emergent structure [11, 12]. The idea of time emerging from some pre-geometric
variables is difficult to comprehend. Treating the space surrounding finite
gravitational systems as emergent is equally challenging. Padmanabhan
contended that these challenges are resolved within the cosmological framework
by selecting the proper time of geodesic observers as the time variable. Thus
the spatial expansion of the universe can be described as the emergence of
cosmic space with the progress of cosmic time. He could arrive at the
Friedmann equation using this new idea in the context of Einstein gravity.
Extending this procedure, Cai obtained the Friedmann equations for higher
dimensional Gauss-Bonnet, and Lovelock gravities for a spatially flat universe
[13]. Following [13], a successful generalization of Padmanabhan’s idea to a
non-flat universe was done by Sheykhi in ref.[14]. By making slight
adjustments to Cai’s proposal, Sheykhi could derive the Friedmann equations
within the frameworks of Einstein, Gauss-Bonnet, and Lovelock gravities for a
universe with any spatial curvatures. Following this, Ali arrived at the
Friedmann equations regarding a general form of entropy [15]. In references
[16, 17], a further generalization to Padmanabhan’s proposal was put forth
with the assumption that the change in Hubble volume is proportional to a more
general function $f(\Delta N,N_{surf})$, where $\Delta N=N_{surf}-N_{bulk}$,
rather than altering the degrees of freedom. Eune and Kim introduced an
alternative generalization of Padmanabhan’s concept, utilizing the proper
invariant volume to depict the rate of emergence within a non-flat
universe[18]. In Ref.[19], Padmanabhan’s conjuncture and its modified versions
were extracted from the Friedmann equations. The paper also addresses the
challenges involved in extending Padmanabhan’s proposal to a non-flat
universe. Subsequently, Sheykhi effectively implemented his suggestion in [20]
to obtain the modified Friedmann equations in several horizon entropy
situations, including Tsallis and Cirto entropy [21] and Barrow entropy [22].
Padamanabhans conjecture [11] has also been used recently to the modified
Renyl entropy scenario [23], Sharma-Mittal entropy scenario [24], blonic
systems[25], minimal length frameworks[17] and brane scenarios [20, 26, 27].

Kaniadakis entropy, which is a generalized form of entropy that accounts for
non-extensive statistical mechanics. Here, we aim to explore the effects of
Kaniadakis entropy on the cosmological field equations on a higher dimensional
spacetime. The effects of the generalized Kaniadakis entropy on the Friedmann
equations have been investigated in referrences [28, 29, 30, 31, 32, 33, 34].
In ref. [28], the authors extracted the modified Friedmann equations for a
$3$-dimensional non-flat universe, starting from the relation $-dE=TdS$, where
they assumed the apparent horizon radius $\tilde{r}_{A}$ to be fixed for an
infinitesimal time interval $dt$. New extra terms that constitute an effective
dark energy sector depending on the Kaniadakis parameter K were presented in
this approach. However, in ref. [34], Sheykhi deduced the modified Friedmann
equations corresponding to the generalized Kaniadakis entropy. The author
posited that since entropy is a geometric quantity, any modification on it
should modify the geometric part of the cosmological field equations. So this
approach is considered to be more logical than the first. The author also
stated that since our universe is expanding, the work term due to the volume
change must be included in the first law of thermodynamics, and modified it as
$dE=TdS+WdV$. In ref. [35], Salehi proposed that Kaniadakis entropy plays a
crucial role in studying the accelerated expansion of the universe without
relying on dark energy. The scope of this paper is to deduce the Friedmann
equations in higher dimensional non-flat FRW universe from the law of
emergence, and the first law of thermodynamics and to establish the
generalized second law of thermodynamics through the lens of Kaniadakis
entropy.

The remaining portion of the paper is organised as follows. The Sect. 1,
provides a brief description about the Kaniadakis statistics. In Sect. 2, we
extract the modified Friedmann equation for $(n+1)$-dimensions from the
Kaniadakis entropy employing the law of emergence. In Sect. 3, we present the
thermodynamic derivation of the first and second modified Friedmann equations.
In Sect. 4, we test the validity of the Generalized second law of
thermodynamics. Finally, we conclude in Sect. 5.

##  1 Kaniadakis statistics

In the realm of cosmology, the advent of Kaniadakis statistics by Giorgio
Kaniadakis [36], which is a further generlization of Tsallis statistics [37]
introduces a paradigm shift, offering a fresh perspective on the universe’s
evolution by incorporating additional parameters, particularly with heavy-
tailed distributions. At its core, Kaniadakis statistics extends the
principles of standard Boltzmann-Gibbs statistics [38] by introducing
parameters $K-$exponential and $K-$logarithm defined respecitively as

$\displaystyle exp_{K}(x)$ $\displaystyle=(\sqrt{1+K^{2}x^{2}}+Kx)^{{1}/{K}},$
(1) $\displaystyle\ln_{\\{K\\}}x$ $\displaystyle=\frac{x^{K}-x^{-K}}{2K}.$ (2)

Where $K$ is a dimensionless parameter, known as the Kaniadakis parameter
which ranges as $-1<K<1$. It signifies the deviation from standard statistical
mechanics and in the limit $K\rightarrow 0$, the standard mechanics is
restored. Studies on relativistic particle systems [39, 40] have indicated a
distribution function that deviates from exponential behavior, showing power
tails attributed to Kaniadakis entropy [41, 36, 42] is given as

$\displaystyle S_{K}=k_{B}\sum_{i}n_{i}\ln_{\\{K\\}}n_{i},$ (3)

where $k_{B}$ is the Boltzmann constant and $n_{i}$ is the generalized
Boltzmann factor for the $i$-th level of a system given as

$\displaystyle n_{i}=\alpha\exp_{K}[-\beta(E_{i}-\mu)],$ (4)

where the parameters takes the form $\alpha=[{(1-K)}/{(1+K)}]^{1/2K}$ and
$1/\beta=\sqrt{1-K^{2}}T$. Also, the Kaniadakis entropy is related to Tsallis
entropy in the limit when the non-extensive parameter $q$ of Tsallis entropy
approaches $1$, both reducing to classical Boltzmann-Gibbs entropy [43]. One
key benefit of Kaniadakis entropy is its ability to maintain the fundamental
principles of conventional statistical theory while also being able to recover
them in the limiting scenario [36, 42]. At low energies, the aforementioned
$K$-distribution behaves like the typical Boltzmann distribution, but at high
energies, it exhibits a power-law tail. Many researchers have been intrigued
by the statistical theory [41, 44, 36, 42, 45, 46, 47, 48, 49, 50] founded on
the distribution outlined in Eq.(4).

From the introduction of $K-$statistics, it has found significant applications
in blackhole thermodynamics. Using the definition of micro-canonical ensemble,
it has been argued that, for the case of black holes, the Kaniadakis entropy
in Eq.(3), can be written as [29, 28]

$\displaystyle S_{K}=\frac{1}{K}\sinh\left(KS_{BH}\right),$ (5)

where $S_{BH}=A/4G$ is the standard Bekenstein-Hawking entropy and
$A=4\pi\tilde{r}_{A}^{2}$ is the area of the hubble horizon. From Eq. (5), the
relation connecting Kaniadakis entropy $(S_{K})$ and apparent horizon area
($A_{n+1}$) in ($n+1$)-dimensions is

$\displaystyle S_{K}=\frac{1}{K}\sinh\left(K\frac{A_{n+1}}{4G_{eff}}\right),$
(6)

where $A_{n+1}=n\Omega_{n}\tilde{r}_{A}^{n-1}$ is the area of the horizon. We
extend this idea to cosmology and obtain the Friedmann equation within the
framework of law of emergence utilising the Kaniadakis entropy.

##  2 Law of emergence and Kaniadakis entropy

According to law of emergence paradigm introduced by Padmanabhan [11], the
spatial expansion of the universe can be interpreted as the emergence of
cosmic space ($dV$) with the progress of cosmic time ($dt$). It can be
expressed by a simple law

$\displaystyle\frac{dV}{dt}=L_{p}^{2}(N_{surf}-N_{bulk}),$ (7)

where $N_{surf}$ is the number of degrees of freedom in the boundary of the
apparent horizon of the universe and $N_{bulk}$ is the number of degrees of
freedom in the bulk, $V$ is the cosmic volume and $t$ is the cosmic time in
planck’s unit. However, the Eq. (7) failed to derive the Friedmann equations
for a non-flat (FRW) universe in alternative gravity theories [13]. The
modification of this proposal was done by Sheykhi [20], who suggested that in
a non-flat universe the expansion law should be generalized as

$\displaystyle\frac{dV}{dt}=L_{p}^{2}\frac{\tilde{r}_{A}}{H^{-1}}(N_{surf}-N_{bulk}).$
(8)

This suggests that in a non-flat universe, the increase in volume remains
directly related to the discrepancy between the degrees of freedom on the
apparent horizon and in the bulk. However, the factor of proportionality is
not merely a constant; it is determined by the ratio of the apparent horizon
to the Hubble radius. In the case of a spatially flat universe where
$\tilde{r}_{A}=H^{-1}$, the original proposal in Eq. (7) is restored. Now, our
aim is to derive the modified Friedmann equation from the expression presented
in (8), when the entropy associated with the apparent horizon get modified to
Kaniadakis entropy in ($n+1$)-dimensions. The Kaniadakis entropy is given by
Eq. (6), where the apparent horizon $\tilde{r}_{A}$ is [51]

$\displaystyle\tilde{r}_{A}=\frac{1}{\sqrt{H^{2}+\frac{k}{a^{2}}}}.$ (9)

We assume that the $S_{K}$ is proportional to an effective area $\tilde{A}$.
Hence the effective area can be expressed as

$\displaystyle\tilde{A}=\frac{4G^{eff}}{K}\sinh\left(\frac{Kn\Omega_{n}\tilde{r}_{A}^{n-1}}{4G_{eff}}\right).$
(10)

Here, we substituted for
${A}_{n+1}=\dfrac{n\Omega_{n}\tilde{r}_{A}^{n-1}}{4G_{eff}}$ where
$\tilde{r}_{A}$ is the radius of the apparent horizon and $\Omega_{n}$ being
the volume of unit $n$-sphere. The time progress of effective volume
($\tilde{V}$) in terms of $\tilde{r}_{A}$ and $\tilde{A}$ is expressed as

$\displaystyle\frac{d\tilde{V}}{dt}=\frac{\tilde{r}_{A}}{n-1}\frac{d\tilde{A}}{dt}.$
(11)

On substituting Eq. (10) in (11), we obtain the time progress of effective
volume as

$\displaystyle\frac{d\tilde{V}}{dt}=n\Omega_{n}\tilde{r}_{A}^{n-1}\cosh\left(K\frac{n\Omega_{n}\tilde{r}_{A}^{n-1}}{4G_{eff}}\right)\dot{\tilde{r}}_{A}.$
(12)

Expressing the right-hand side of Eq. (12) in terms of a total time derivative
yields a more convenient form

$\displaystyle\frac{d\tilde{V}}{dt}=-\frac{n\Omega_{n}\tilde{r}_{A}^{n+2}}{2}\frac{d}{dt}\left[\tilde{r}_{A}^{-2}cosh\left[K\frac{n\Omega_{n}\tilde{r}_{A}^{n-1}}{4G_{eff}}\right]-\int_{0}^{\tilde{r}_{A}}\dfrac{\sinh\left[K\dfrac{n\Omega_{n}\tilde{r}_{A}^{n-1}}{4G_{eff}}\right]}{\tilde{r}_{A}^{2}}d\left[K\frac{n\Omega_{n}\tilde{r}_{A}^{n-1}}{4G_{eff}}\right]\right].$
(13)

Motivated by Eq. (13), we propose the degrees of freedom on the apparent
horizon in ($n+1$)-dimensions as

$\displaystyle N_{surf}$
$\displaystyle=\alpha\frac{n\Omega_{n}}{G_{eff}}\tilde{r}_{A}^{n-1}\cosh\left[K\frac{n\Omega_{n}\tilde{r}_{A}^{n-1}}{4G_{eff}}\right]-\alpha\frac{n\Omega_{n}}{G_{eff}}\tilde{r}_{A}^{n+1}\int_{0}^{\tilde{r}_{A}}\dfrac{\sinh\left[K\dfrac{n\Omega_{n}\tilde{r}_{A}^{n-1}}{4G_{eff}}\right]}{{\tilde{r}_{A}^{2}}}d\left[K\frac{n\Omega_{n}\tilde{r}_{A}^{n-1}}{4G_{eff}}\right],$
(14)

where $\alpha=\dfrac{n-1}{2(n-2)}$. The bulk Komar energy in the case of
($n+1$)-dimensions [52][53] is given by

$\displaystyle E_{Komar}=\frac{(n-2)\rho+np}{n-2}V,$ (15)

where the volume $V=\Omega_{n}\tilde{r}_{A}^{n}$. Then the bulk degrees of
freedom becomes

$\displaystyle
N_{bulk}=-\frac{2E_{Komar}}{T}=-4\pi\Omega_{n}\tilde{r}_{A}^{n+1}\frac{(n-2)\rho+np}{n-2},$
(16)

where $T=\dfrac{1}{2\pi\tilde{r}_{A}}$ is the Hawking temperature. A negative
sign is introduced in the above equation in order to have $N_{bulk}>0$ in the
accelerating phase with $(n-2)\rho+np<0$. We also replace $L_{p}^{2}$ with
$G^{eff}$ and $V$ with $\tilde{V}$ in the expression (8) and write it down as

$\displaystyle\alpha\frac{d\tilde{V}}{dt}=G_{eff}\frac{\tilde{r}_{A}}{H^{-1}}(N_{surf}-N_{bulk}).$
(17)

Substituting Eq. (10), (14) and (16) in Eq. (17) yields,

$\displaystyle-\frac{8\pi
G_{eff}}{n(n-1)}((n-2)\rho+np)=\tilde{r}_{A}^{-2}\cosh$
$\displaystyle\left(K\frac{n\Omega_{n}\tilde{r}_{A}^{n-1}}{4G_{eff}}\right)-\tilde{r}_{A}^{-3}\dot{\tilde{r}}_{A}H^{-1}\cosh\left(K\frac{n\Omega_{n}\tilde{r}_{A}^{n-1}}{4G_{eff}}\right)$
$\displaystyle-\int_{0}^{\tilde{r}_{A}}\dfrac{\sinh\left(K\dfrac{n\Omega_{n}\tilde{r}_{A}^{n-1}}{4G_{eff}}\right)}{\tilde{r}_{A}^{2}}d\left(K\frac{n\Omega_{n}\tilde{r}_{A}^{n-1}}{4G_{eff}}\right).$
(18)

Using the ($n+1$)-dimensional continuity equation $\dot{\rho}+nH(\rho+p)=0$
and multiplying the
above equation with a factor of $2a\dot{a}$ followed by adding and subtracting
the R.H.S with a term

$a^{2}\tilde{r}_{A}^{-2}\sinh\left(K\dfrac{n\Omega_{n}\tilde{r}_{A}^{n-1}}{4G_{eff}}\right)\dfrac{d}{dt}\left(K\dfrac{n\Omega_{n}\tilde{r}_{A}^{n-1}}{4G_{eff}}\right),$

Eq. (2) takes the form

$\displaystyle\frac{16\pi G_{eff}}{n(n-1)}\frac{d}{dt}(\rho
a^{2})=\frac{d}{dt}$
$\displaystyle\left[a^{2}\tilde{r}_{A}^{-2}\cosh\left[K\frac{n\Omega_{n}\tilde{r}_{A}^{n-1}}{4G_{eff}}\right]-a^{2}\int_{0}^{\tilde{r}_{A}}\dfrac{\sinh\left[K\dfrac{n\Omega_{n}\tilde{r}_{A}^{n-1}}{4G_{eff}}\right]}{\tilde{r}_{A}^{2}}d\left[K\frac{n\Omega_{n}\tilde{r}_{A}^{n-1}}{4G_{eff}}\right]\right].$
(19)

Upon integration, we obtain

$\displaystyle\frac{16\pi
G_{eff}}{n(n-1)}\rho=\cosh\left[K\frac{n\Omega_{n}\tilde{r}_{A}^{n-1}}{4G_{eff}}\right]\left[H^{2}+\frac{k}{a^{2}}\right]-\int_{0}^{\tilde{r}_{A}}\dfrac{\sinh\left[K\dfrac{n\Omega_{n}\tilde{r}_{A}^{n-1}}{4G_{eff}}\right]}{\left(H^{2}+\frac{k}{a^{2}}\right)^{-1}}d\left[K\frac{n\Omega_{n}\tilde{r}_{A}^{n-1}}{4G_{eff}}\right].$
(20)

Here, we substituted $\tilde{r}_{A}$ from Eq. (9). Any integration constant
can be taken to the L.H.S of the equation so that the density $\rho$ is
modified (but we are keeping the same symbol $\rho$). The Eq. (20) is the
modified Friedmann equation in a non-flat ($n+1$)-dimensional FRW universe
deduced from the Kaniadakis entropy (6). For $n=3$ ($G_{eff}\rightarrow G$),
the Eq. (20) simplifies to

$\displaystyle\frac{8\pi
G}{3}\rho=\cosh\left(K\frac{\pi\tilde{r}_{A}^{2}}{G}\right)\left(H^{2}+\frac{k}{a^{2}}\right)-\frac{K\pi}{G}\text{shi}\left(K\frac{\pi}{G(H^{2}+\frac{k}{a^{2}})}\right),$
(21)

which is consistent with the result obtained in [28] using the first law of
thermodynamics. Here, shi($x$) is an entire mathematical odd function of $x$
with no branch discontinuities given by

$\text{shi}(x)=\int_{0}^{x}\dfrac{\sinh xdx}{x}.$

Also in the limit $K\rightarrow 0$, Eq. (21) reduces to the Friedmann
equations derived in the context of Bekenstein-Hawking entropy [20].

##  3 Kaniadakis entropy and first law of thermodynamics

In this section, we deduce the modified Friedmann equation in non-flat,
($n+1$)-dimensions by employing the first law of thermodynamics. Sheykhi in
his work [34] posited that since entropy is a geometric property, any changes
to it would influence the geometric nature of the field equations. This lends
a greater coherence to this approach than the one followed in ref. [28].
Furthermore, given the expanding nature of our universe, it becomes imperative
to account for the work term arising from changes in volume.
The associated temperature at the apparent horizon can be defined as [51, 54]

$\displaystyle
T=\frac{\kappa}{2\pi}=-\frac{1}{2\pi\tilde{r}_{A}}\biggl{(}1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\biggr{)},$
(22)

where $\kappa$ is the surface gravity. The work density associated with
expansion is defined as [55]

$\displaystyle W=\frac{1}{2}(\rho-p).$ (23)

The work density represents the work performed as a result of the volume
change of the universe caused by the variation in the apparent horizon radius.
The first law of thermodynamics at the apparent horizon is assumed to be
satisfied, and has the form

$\displaystyle dE=TdS+WdV.$ (24)

The energy $E=\rho V$ represents the total energy of the material enclosed
within the apparent horizon, which is actually direct ($n+1$) dimensional
generalization of the ($3+1$) dimensional one, given by Misner and Sharp [55,
56]. The energy ($E$) is different from the Komar energy, ($E_{Komar}$) used
in emergence of cosmic space. Here, $V$ is the volume enclosed by
$n$-dimensional sphere with area $A_{n+1}=n\Omega_{n}\tilde{r}_{A}^{n-1}$.
This law reduces to the conventional law $dE=TdS-pdV$, for a pure de Sitter
phase with $\rho=-p$. Taking the differential of $E$, we get

$\displaystyle dE=\rho
n\Omega_{n}\tilde{r}_{A}^{n-1}d\tilde{r}_{A}+\Omega_{n}\tilde{r}_{A}^{n}\dot{\rho}dt.$
(25)

Using the continuity equation $\dot{\rho}+nH(\rho+p)=0$, we obtain

$\displaystyle dE=\rho
n\Omega_{n}\tilde{r}_{A}^{n-1}d\tilde{r}_{A}-n\Omega_{n}\tilde{r}_{A}^{n}(\rho+p)Hdt.$
(26)

We further assume the entropy associated with the apparent horizon is in the
form of Kaniadakis entropy (6). Differentiating the modified entropy-area
relation (6), we get

$\displaystyle
dS=\frac{n(n-1)\Omega_{n}}{4G_{eff}}\tilde{r}_{A}^{n-2}\cosh\left(K\frac{n\Omega_{n}\tilde{r}_{A}^{n-1}}{4G_{eff}}\right)d\tilde{r}_{A}.$
(27)

Substituting Eq. (22), (23), (26) and (27) in Eq. (24), we obtain

$\displaystyle(\rho+p)Hdt=\frac{(n-1)}{4G_{eff}}\frac{1}{2\pi\tilde{r}_{A}^{3}}\cosh\left(K\frac{n\Omega_{n}\tilde{r}_{A}^{n-1}}{4G_{eff}}\right)d\tilde{r}_{A}.$
(28)

Using the continuity equation $\dot{\rho}+nH(\rho+p)=0$, the Eq. (27) reduces
to

$\displaystyle-\frac{8\pi
G_{eff}}{n(n-1)}\dot{\rho}dt=\tilde{r}_{A}^{-3}\cosh\left(K\frac{n\Omega_{n}\tilde{r}_{A}^{n-1}}{4G_{eff}}\right)d\tilde{r}_{A}.$
(29)

Upon integration , we get

$\displaystyle\frac{16\pi
G_{eff}}{n(n-1)}\rho=\cosh\left[K\frac{n\Omega_{n}\tilde{r}_{A}^{n-1}}{4G_{eff}}\right]\left[H^{2}+\frac{k}{a^{2}}\right]-\int_{0}^{\tilde{r}_{A}}\dfrac{\sinh\left[K\dfrac{n\Omega_{n}\tilde{r}_{A}^{n-1}}{4G_{eff}}\right]}{\left(H^{2}+\frac{k}{a^{2}}\right)^{-1}}d\left[K\frac{n\Omega_{n}\tilde{r}_{A}^{n-1}}{4G_{eff}}\right],$
(30)

which is exactly the modified Friedmann equation presented in Eq. (20),
obtained from the law of emergence in the section 2. This suggest that the
Kaniadakis entropy is consistent with the connection between the law of
emergence and laws of thermodynamics. It is important to highlight the efforts
to derive the expansion law from the first law of thermodynamics done in
various references [57, 58, 59, 60, 61, 62, 63, 64, 65]. The revised expansion
laws of the universe, as suggested by Cai [13], Sheykhi [20], and Yang [16],
can be traced from the the fundamental equation of the first law of
thermodynamics, which is expressed as $dE=TdS+WdV$. These studies demonstrate
that the first law of thermodynamics can be viewed as the fundamental basis
for understanding the expansion law of the universe.

Combining the continuity equation $\dot{\rho}+nH(\rho+p)=0$, with the first
modified Friedmann equation (30), we can deduce the second modified Friedmann
equation for a non-flat ($n+1$) dimensional FRW universe in the context
Kaniadakis entropy. Taking the time derivative of the first Friedmann equation
(30), we obtain

$\displaystyle\frac{8\pi
G_{eff}}{n(n-1)}\dot{\rho}=\cosh\left(K\frac{n\Omega_{n}\tilde{r}_{A}^{n-1}}{4G_{eff}}\right)H\left(\dot{H}-\frac{k}{a^{2}}\right).$
(31)

With the help of continuity equation $\dot{\rho}+nH(\rho+p)=0$, the Eq. (31)
becomes

$\displaystyle-\frac{8\pi
G_{eff}}{(n-1)}(\rho+p)=\cosh\left(K\frac{n\Omega_{n}\tilde{r}_{A}^{n-1}}{4G_{eff}}\right)\left(\dot{H}-\frac{k}{a^{2}}\right).$
(32)

Now utilizing the relation $\dot{H}={\ddot{a}}/{a}-H^{2}$ and substituting for
$\rho$ from the first Friedmann equation (30), Eq. (32) rearranges to

$\displaystyle-\frac{8\pi
G_{eff}}{(n-1)}p-\frac{n}{2}\cosh\left(K\frac{n\Omega_{n}\tilde{r}_{A}^{n-1}}{4G_{eff}}\right)\left[H^{2}+\frac{k}{a^{2}}\right]$
$\displaystyle-\frac{n}{2}\int_{0}^{\tilde{r}_{A}}\dfrac{\sinh\left[K\dfrac{n\Omega_{n}\tilde{r}_{A}^{n-1}}{4G_{eff}}\right]}{\left(H^{2}+\frac{k}{a^{2}}\right)^{-1}}d\left[K\frac{n\Omega_{n}\tilde{r}_{A}^{n-1}}{4G_{eff}}\right]$
$\displaystyle=\cosh\left(K\frac{n\Omega_{n}\tilde{r}_{A}^{n-1}}{4G_{eff}}\right)\left(\frac{\ddot{a}}{a}-H^{2}-\frac{k}{a^{2}}\right).$
(33)

Simplifying and rearranging Eq. (3), we arrive at

$\displaystyle\cosh\left[K\frac{n\Omega_{n}\tilde{r}_{A}^{n-1}}{4G_{eff}}\right]\left[2\frac{\ddot{a}}{a}+(n-2)\left[H^{2}+\frac{k}{a^{2}}\right]\right]-n\int_{0}^{\tilde{r}_{A}}\dfrac{\sinh\left[K\dfrac{n\Omega_{n}\tilde{r}_{A}^{n-1}}{4G_{eff}}\right]}{\left(H^{2}+\frac{k}{a^{2}}\right)^{-1}}d\left[K\frac{n\Omega_{n}\tilde{r}_{A}^{n-1}}{4G_{eff}}\right]$
$\displaystyle=-\frac{16\pi G_{eff}}{(n-1)}p,$ (34)

which is the second modified Friedmann equation governing the evolution of a
non-flat ($n+1$) dimensional FRW universe deduced from the Kaniadakis entropy.
For $n=3$ ($G_{eff}\rightarrow G$), the Eq. (3) simplifies to

$\displaystyle\cosh\left[K\frac{\pi\tilde{r}_{A}^{2}}{G}\right]\left[2\frac{\ddot{a}}{a}+\left[H^{2}+\frac{k}{a^{2}}\right]\right]-3\frac{K\pi}{G}\text{shi}\left(K\frac{\pi}{G(H^{2}+\frac{k}{a^{2}})}\right)=-8\pi
Gp,$ (35)

which is the second modified Friedmann equation for a $3$-dimensional non flat
FRW universe based on Kaniadakis entropy. In the limit $K\rightarrow 0$, Eq.
(35) reduces to the second Friedmann equation in standard cosmology

$\displaystyle 2\frac{\ddot{a}}{a}+H^{2}+\frac{k}{a^{2}}=-8\pi Gp.$ (36)

Combining the first and second modified Friedmann equations (30) and (3),
gives the second time derivative of the scale factor as

$\displaystyle\frac{\ddot{a}}{a}\cosh\left[K\frac{n\Omega_{n}\tilde{r}_{A}^{n-1}}{4G_{eff}}\right]-\int_{0}^{\tilde{r}_{A}}\dfrac{\sinh\left[K\dfrac{n\Omega_{n}\tilde{r}_{A}^{n-1}}{4G_{eff}}\right]}{\left(H^{2}+\frac{k}{a^{2}}\right)^{-1}}d\left[K\frac{n\Omega_{n}\tilde{r}_{A}^{n-1}}{4G_{eff}}\right]=-\frac{8\pi
G_{eff}}{n(n-1)}[(n-2)\rho+np].$ (37)

Using the equation of state parameter $\omega=p/\rho$, the Eq. (37) becomes

$\displaystyle\frac{\ddot{a}}{a}\cosh\left[K\frac{n\Omega_{n}\tilde{r}_{A}^{n-1}}{4G_{eff}}\right]-\int_{0}^{\tilde{r}_{A}}\dfrac{\sinh\left[K\dfrac{n\Omega_{n}\tilde{r}_{A}^{n-1}}{4G_{eff}}\right]}{\left(H^{2}+\frac{k}{a^{2}}\right)^{-1}}d\left[K\frac{n\Omega_{n}\tilde{r}_{A}^{n-1}}{4G_{eff}}\right]=-\frac{8\pi
G_{eff}}{n(n-1)}\rho[(n-2)+n\omega].$ (38)

The condition for the accelerated expansion of the universe ($\ddot{a}>0$),
requires

$\displaystyle(n-2)+n\omega<0\;\rightarrow\omega<-\frac{n-2}{n}.$ (39)

For $n=3$, we obtain the inequality $\omega<-\dfrac{1}{3}$ in Friedmann
cosmology. It typically indicates that the cosmic fluid has a negative
pressure relative to its energy density. This condition is often associated
with certain types of energy components such as dark energy, which exert a
repulsive gravitational force, causing the universe’s expansion to accelerate.
The equations (30) and (38) depicts the evolution of a ($n+1$) dimensional FRW
universe with any spatial curvature through the lens of Kaniadakis entropy.
The cosmological consequences of the above derived Friedmann equations are
left for future research.

##  4 Generalized Second Law of Thermodynamics

Our aim here is to establish the generalized second law of thermodynamics when
the entropy associated with the horizon is given by the Kaniadakis entropy
(6). The authors in Ref. [66, 67, 68], delved into the investigation of
generalized second law of thermodynamics within the framework of an
accelerating universe. Using Eq. (28), solving for $\dot{\tilde{r}}_{A}$ gives

$\displaystyle\dot{\tilde{r}}_{A}=\frac{8\pi
G_{eff}}{(n-1)}\frac{\tilde{r}_{A}^{3}H(\rho+p)}{\cosh\left[K\dfrac{n\Omega_{n}\tilde{r}_{A}^{n-1}}{4G_{eff}}\right]}.$
(40)

When the dominant energy condition holds, $\rho+p\geq 0$, implies
$\dot{\tilde{r}}_{A}\geq 0$. Next we calculate $T_{h}\dot{S}_{h}$.

$\displaystyle
T_{h}\dot{S}_{h}=\frac{1}{2\pi\tilde{r}_{A}}\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right)\frac{d}{dt}\left[\frac{1}{K}\sinh\left(K\frac{n\Omega_{n}\tilde{r}_{A}^{n-1}}{4G_{eff}}\right)\right],$
(41)

upon simplification, we get

$\displaystyle
T_{h}\dot{S}_{h}=n\Omega_{n}\tilde{r}_{A}^{n}H(\rho+p)\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right).$
(42)

For an accelerating universe, the equation of state parameter
$\omega=p/\rho<-1$, leads to a potential violation of the dominant energy
condition, $\rho+p<0$. Consequently, the second law of thermodynamics,
$\dot{S}\geq 0$ is no longer valid. To address this, the total entropy of
universe can be taken as $S=S_{h}+S_{m}$, where $S_{m}$ accounts for the
entropy of matter field inside the apparent horizon. Hence, it becomes
imperative to examine the time evolution of total entropy, $S$. So, if the
generalized second law of thermodynamics holds, the condition
$\dot{S}_{h}+\dot{S}_{m}\geq 0$, should hold true for the total entropy.
Employing the Gibbs equation [69], we have

$\displaystyle T_{m}dS_{m}=d(\rho V)+pdV=Vd\rho+(\rho+p)dV.$ (43)

Here, $T_{m}$ represents the temperature of matter fields. We propose that the
boundary of universe maintains thermal equilibrium with the matter field
contained in it. This implies that the temperatures of both components are
equal, $T_{h}=T_{m}$ [69]. If the local equilibrium assumption is not strictly
maintained, it would entail observing an energy transfer between the horizon
and bulk fluid, which is not deemed physically acceptable. Substituting
$V=\Omega_{n}\tilde{r}_{A}^{n}$ and the continuity equation
$\dot{\rho}+nH(\rho+p)=0$ in the Gibbs equation (43), we obtain

$\displaystyle
T_{h}\dot{S}_{m}=n\Omega_{n}\tilde{r}_{A}^{n-1}(\rho+p)\dot{\tilde{r}}_{A}-n\Omega_{n}\tilde{r}_{A}^{n}H(\rho+p)$
(44)

Now, we obtain the total time evolution of $S_{h}+S_{m}$, by combining
Equations (42) and (44) as

$\displaystyle
T_{h}(\dot{S}_{h}+\dot{S}_{m})=\frac{n}{2}\Omega_{n}\tilde{r}_{A}^{n-1}(\rho+p)\dot{\tilde{r}}_{A}$
(45)

Substituting for $\dot{\tilde{r}}_{A}$ from Eq. (40) in Eq. (45), we obtain

$\displaystyle T_{h}(\dot{S}_{h}+\dot{S}_{m})=\frac{4\pi
G_{eff}}{(n-1)}\frac{n\Omega_{n}\tilde{r}_{A}^{n+2}(\rho+p)^{2}H}{\cosh\left[K\dfrac{n\Omega_{n}\tilde{r}_{A}^{n-1}}{4G_{eff}}\right]},$
(46)

which is clearly a non-negative function during the evolution of universe.
This is the generalized second law of thermodynamics when the entropy
associated with the horizon takes the form of Kaniadakis entropy. For $n=3$
($G_{eff}\rightarrow G$), the Eq. (46) simplifies to

$\displaystyle
T_{h}(\dot{S}_{h}+\dot{S}_{m})=\frac{8\pi^{2}G\tilde{r}_{A}^{5}(\rho+p)^{2}H}{\cosh\left[K\dfrac{\pi\tilde{r}_{A}^{2}}{G}\right]}.$
(47)

In the limit $K\rightarrow 0$, Eq. (47) reduces to the generalized second law
of thermodynamics in standard cosmology

$\displaystyle
T_{h}(\dot{S}_{h}+\dot{S}_{m})=8\pi^{2}G\tilde{r}_{A}^{5}(\rho+p)^{2}H\geq 0.$
(48)

##  5 Conclusions

Kaniadakis entropy is an extension of the Boltzmann-Gibbs entropy to
accommodate systems exhibiting long-range interactions or non-extensive
properties. Unlike the Boltzmann-Gibbs entropy, which assumes extensive
properties (additivity), Kaniadakis entropy embraces non-extensivity, making
it suitable for systems where long-range correlations and non-additive
behaviors are present. Kaniadakis cosmology incorporates the Kaniadakis
entropy and extends the standard cosmological models by considering non-
trivial statistical effects and deviations from conventional thermodynamic
behavior, especially in extreme conditions such as those encountered in the
early universe or near black hole horizons. The Bekenstein entropy-area
relation is modified to Kaniadakis entropy inorder to extend the thermodynamic
description of blackholes in scenarios where standard thermodynamics may not
be applicable.

In this study, we successfully implemented Sheykhi’s modified law of
emergence, to extract the modified Friedmann equation for an
($n+1$)-dimensional, non-flat FRW universe utilizing the Kaniadakis entropy.
The emergent space paradigm suggests that the expansion of universe or the
emergence of cosmic space is driven by the departure from holographic
equipartition or it can be seen as a tendency to satisfy the holographic
discrepancy. This implies that the expansion of the universe is propelled by
the difference between the degrees of freedom existing on the horizon and
those within the bulk. This paradigm shifts the perspective from considering
spacetime as a fundamental entity to viewing it as an emergent phenomenon. The
integration of Kaniadakis cosmology provides a new perspective on cosmological
dynamics within the framework of emergent space paradigms. Further, we
employed the first law of thermodynamics $dE=TdS+WdV$ in conjunction with
Kaniadakis entropy to deduce the modified Friedmann equation, where we
considered the work term related to change in volume due to the expansion of
the universe. The Friedmann equations obtained from both perspectives
coincides. This coincidence supports the viability of Padmanabhan’s law of
emergence and reveals the intimate relationship of the emergent space paradigm
with horizon thermodynamics in Kaniadakis cosmology. Combining the continuity
equation and the first modified Friedmann equation, we extracted the second
modified Friedmann equation for a ($n+1$)-dimensional, non-flat FRW universe
within the Kaniadakis framework. Finally, we established the Generalized
second law (GSL) of thermodynamics incorporating the Kaniadakis entropy. Here,
we assumed that the boundary of the universe maintains thermal equilibrium
with the matter field contained in it. Also, considering the time-dependent
behaviour of total entropy of the universe, which is taken as the sum of
Kaniadakis horizon entropy and matter entropy, we established the condition
for the generalized second law of thermodynamics, $\dot{S}\geq 0$. In the
standard limit ($K\rightarrow 0$), our results obtained from the Kaniadakis
cosmology reduces to the standard cosmology.

In ref. [35], the author showed that the acceleration of the universe can be
modelled without relying on dark energy, within the Kaniadakis framework.
Hence, it is important to confront the model with the latest observational
probes to constraint the parameter space of the model and to test the evidence
of Kaniadakis cosmology over the standard cosmology.

## Acknowledgment

One of the authors Sarath Nelleri acknowledges Indian Institute of Technology
Kanpur for providing the Institute postdoctoral fellowship. The authors
acknowledge Dr.Vinodkumar Thekkeyil and, Dr. E.K Sreejith, Department of
Physics, Payyanur College, for fruitful discussion on the subject. The authors
thank Sreerag Radhakrishnan for suggestions on the manuscript.

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