060003 2014 G. Martinez Mekler F. Bagnoli, Dipartimento di Fisica ed
Astronomia,
Universita degli Studi di Firenze, Italy 060003

We consider a class of models of opinion formation where the dissemination of
individual opinions occurs through the spreading of local consensus and
disagreement. We study the emergence of full collective consensus or maximal
disagreement in one- and two-dimensional arrays. In both cases, the
probability of reaching full consensus exhibits well-defined scaling
properties as a function of the system size. Two-dimensional systems, in
particular, possess nontrivial exponents and critical points. The dynamical
rules of our models, which emphasize the interaction between small groups of
agents, should be considered as complementary to the imitation mechanisms of
traditional opinion dynamics.

# Critical phenomena in the spreading of opinion consensus and disagreement

A. Chacoma  [inst1] D. H. Zanette[inst1, inst2] E-mail:
<EMAIL_ADDRESS>
(11 March 2014; 1 August 2014)

††volume: 6

99 inst1 Instituto Balseiro and Centro Atómico Bariloche, 8400 San Carlos de
Bariloche, Río Negro, Argentina. inst2 Consejo Nacional de Investigaciones
Científicas y Técnicas, Argentina.

##  1 Introduction

The remarkable regularities observed in many human social phenomena —which, in
spite of the disparate behavior of individual human beings, emerge as a
consequence of their interactions— have since long attracted the attention of
physicists and applied mathematicians. Collective manifestations of human
behavior have been mathematically modeled in a variety of socioeconomic
processes, such as opinion formation, decision making, resource allocation,
cultural and linguistic evolution, among many others, often using the tools
provided by statistical physics [1]. The stylized nature of these models
emphasizes the identification of the generic mechanisms at work in human
interactions, as well as the detection of broadly significant features in
their macroscopic outcomes. They provide the key to a deep insight into the
common elements that underlie those processes.

Models of opinion formation constitute a central paradigm in the mathematical
description of social processes from the viewpoint of statistical physics.
Starting in the seventies and eighties [2, 3, 4, 5], much work —which we
cannot aim at inventorying here, but which has been comprehensibly reviewed in
recent literature [1]— has exploited the formal resemblance between opinion
spreading and spin dynamics in order to apply well-developed statistical
techniques to the analysis of such models.

The key mechanism driving most agent-based models of opinion formation is
imitation. For instance, in the voter model —to which we refer several times
in the present paper— the basic interaction event consists in an agent copying
the opinion of another agent chosen at random from a specified neighborhood.
At any given time, the opinion of each agent adopts one of two values,
typically denoted as $\pm 1$. The voter model can be exactly solved for
populations of agents distributed over regular (hyper)cubic arrays in any
dimension [6]. For infinitely large populations, it is characterized by the
conservation of the average opinion. In one dimension, a finite population
always reaches an absorbing state of full collective consensus, all agents
sharing the same opinion. The probability of final consensus on either opinion
coincides with the initial fraction of agents with that opinion, and the time
needed to reach the absorbing state is of the order of the population size
squared [1].

In this paper, we present an introductory analysis of a class of models where
opinion dynamics is driven by the spreading of consensus and disagreement,
rather than by the dissemination of individual opinions. The basic concept
behind these models is that agreement of individual opinions in a localized
portion of the population may promote the emergence of consensus in the
neighborhood while, in contrast, local disagreement may inhibit the growth of,
or even decrease, the degree of consensus in the surrounding region. In real
social systems, the mechanism of consensus and disagreement spreading should
be complementary to the direct transmission of opinions between individual
agents. In our models, however, we disregard the latter to focus on the
dynamical effects of the former.

Since the degree of consensus can only be defined for two or more agents, the
spreading of consensus and disagreement engages groups of agents rather than
individuals. Such groups are, thus, the elementary entities involved in the
social interactions [7, 8, 9, 10, 11]. We stress that several other social
phenomena — related, notably, to decision making [10] and resource allocation
[12]— are also based on group interactions that cannot be reduced to two-agent
events. In the class of models analyzed here, each interaction event is
conceived to occur between two groups: an active group $G$ and a reference
group $G^{\prime}$. As a result of the interaction, the agents in $G$ change
their individual opinions in such a way that the level of consensus in $G$
approaches that of $G^{\prime}$. This generic mechanism extends dynamical
rules where the opinion of each single agent changes in response to the
collective state of a reference group [1, 8, 13, 14]. The size and internal
structure of the interacting groups, as well as the precise way in which
opinions are modified in the active group with respect to the reference group,
defines each model in this class. For the sake of concreteness, we limit the
analysis to systems where, as in the voter model, individual opinions can
adopt two values ($\pm 1$). In the next section, we analyze the case where
both the active group and the reference group are formed by two agents, and
the population is structured as a one-dimensional array. In this case, the
system admits stationary absorbing states of full consensus and maximal
disagreement, with simple scaling laws with the population size. In Section 3,
we study a two-dimensional version of the same kind of model with larger
groups, where nontrivial critical phenomena —not present in the one-
dimensional case— emerge. Results and perspectives are summarized in the final
section.

##  2 Two-agent groups on one-dimensional arrays

We begin by considering the simple situation where each of the two groups
involved in each interaction event is formed by just two agents. The situation
within each group, thus, is one of either full consensus (when the two agents
bear the same opinion, either $+1$ or $-1$) or full disagreement (when their
opinions are different). We take a population where agents are distributed on
a one-dimensional array, and consecutively labeled from $1$ to $N$. Periodic
boundary conditions are applied at the ends. At each time step, we choose four
contiguous agents, say, $i-1$ to $i+2$. The central pair $i$, $i+1$ acts as
the reference group $G^{\prime}$. If they are in disagreement, the agents
$i-1$ and $i+2$ respectively adopt the opinions opposite to those of $i$ and
$i+1$ with probability $p_{D}$, while with the complementary probability
$1-p_{D}$ nothing happens. If, on the other hand, $i$ and $i+1$ agree with
each other, $i-1$ and $i+2$ copy the common opinion in $G^{\prime}$ with
probability $p_{C}$, while with probability $1-p_{C}$ nothing happens. In this
way, both consensus and disagreement spread from $G^{\prime}$ outwards, to the
left and right. The probabilities $p_{C}$ and $p_{D}$ control the relative
frequency with which consensus and disagreement are effectively transmitted.
The left panel of Fig. 1 illustrates the states of the four consecutive agents
in the two possible outcomes of the interaction (up to opinion inversions).

Figure 1: Left: The two possible outcomes of the interaction, up to opinion
inversions, for four consecutive agents along the one-dimensional array. The
active and the reference groups, $G$ and $G^{\prime}$, are respectively formed
by the outermost and innermost agents. Right: Time evolution of a 200-agent
array with $n_{+}(0)=0.5$ and $p_{D}=p_{C}=1$. Black and white dots
correspond, respectively, to opinions $+1$ and $-1$. At time $t=1534$, an
absorbing state of maximal disagreement is reached.

It is not difficult to realize that, for $p_{D}=p_{C}=1$, our one-dimensional
array is equivalent to two intercalated subpopulations —respectively occupying
even and odd sites— each of them evolving according to the voter model. The
dynamical rules are reduced in this case to binary interactions between
agents. In fact, whatever the opinions in group $G^{\prime}$ at each
interaction event, agent $i-1$ and $i+2$ respectively copy the opinions of
$i+1$ and $i$. Now, since the voter model always leads a finite population to
an absorbing state of full consensus, the final state of our system can be one
of full consensus on either opinion, or a state of maximal disagreement where
opposite opinions alternate over the sites of the one-dimensional array. In
the latter, the two neighbors of each agent with opinion $+1$ have opinion
$-1$ and vice versa. The right panel of Fig. 1 shows the evolution of a
200-agent array for $n_{+}(0)=0.5$ and $p_{D}=p_{C}=1$, black and white dots
respectively corresponding to opinions $+1$ and $-1$. At any given time, the
population is divided into well-defined domains either of consensus in one of
the opinions or disagreement. Note that the domain boundaries show the typical
diffusive motion found in stochastic coarsening processes [15, 1].

Figure 2: Numerical results for consensus and disagreement spreading on a one-
dimensional array with $p_{D}=p_{C}=1$, obtained from $10^{3}$ realizations
for each parameter set (see text for details). Upper panel: Probability of
reaching full consensus, $P_{\rm cons}$, as a function of the initial fraction
of agents with opinion $+1$, $n_{+}(0)$, for four values of the population
size $N$. Lower panel: Total time $T$ needed to reach the final absorbing
state, normalized by the squared population size $N^{2}$. Since both $P_{\rm
cons}$ and $N^{-2}T$ are symmetric with respect to $n_{+}(0)=1/2$, only the
lower half of the horizontal axis is shown.

Taking into account that, in the voter model, the probability of ending with
full consensus on opinion $+1$ is given by the initial fraction of agents with
that opinion, $n_{+}(0)$, and assuming that the initial distribution of
opinions is homogeneous over the array, the probability that our system ends
in a state of full consensus on either opinion is $P_{\rm
cons}=n_{+}^{2}(0)+n_{-}^{2}(0)=1-2n_{+}(0)+2n_{+}^{2}(0)$. Note that this
coincides with the probability that, in the initial state, any two contiguous
agents are in consensus. Moreover, we know that the time needed to reach an
absorbing state in the one-dimensional voter model is proportional to $N^{2}$,
a result that should also hold in our case.

The upper panel of Fig. 2 shows numerical results for the probability of final
full consensus $P_{\rm cons}$, determined as the fraction of realizations that
ended in full consensus out of $10^{3}$ runs, as a function of $n_{+}(0)$ and
for several population sizes $N$. The curve is the analytic prediction given
above. The result is analogous to the probability of final consensus found in
Sznajd-type models [13]. The lower panel shows the total time $T$ needed to
reach the final absorbing state (of either consensus or disagreement),
averaged over $10^{3}$ realizations and normalized by $N^{2}$. As expected,
both $P_{\rm cons}$ and $N^{-2}T$ are independent of the population size.

Figure 3: Probability of reaching full consensus, $P_{\rm cons}$, as a
function of the probability $p_{D}$, with $p_{C}=1$ and for four values of the
system size $N$. Results were obtained averaging over $10^{3}$ realizations
for each parameter set. Insert: Width of the variation range of $P_{\rm cons}$
as a function of $N$. The straight line has slope $-1$.

When $p_{D}\neq p_{C}$, the two intercalated subpopulations cannot be
considered independent of each other any more. If $p_{D}<p_{C}$, for instance,
an opinion prevailing in one of the subpopulations will invade the other
subpopulation faster than the opposite opinion, thus favoring the
establishment of collective consensus. To analyze this asymmetric situation,
we first fix $p_{C}=1$ and let $p_{D}$ vary in $(0,1)$, so that the spreading
of consensus is more probable than that of disagreement. The main plot in Fig.
3 shows numerical results for $P_{\rm cons}$, measured as explained above, as
a function of $p_{D}$ and for four values of $N$. In all the realizations,
$n_{+}(0)=0.5$, and the two opinions are homogeneously distributed over the
population. As $p_{D}$ decreases below $1$, the probability of reaching full
consensus grows rapidly, approaching $P_{\rm cons}=1$. As $N$ grows, moreover,
the change in $P_{\rm cons}$ is more abrupt. Fitting of a sigmoidal function
to the data of $P_{\rm cons}$ vs. $p_{D}$ near $p_{D}=1$ makes it possible to
assign a width to the range where $P_{\rm cons}$ changes between $1$ and
$0.5$. The insert of Fig. 3 shows this width as a function of the system size
$N$ in a log-log plot. The slope of the linear fitting is $-1.00\pm 0.02$.
Therefore, the width is inversely proportional to $N$.

The facts that $P_{\rm cons}=0.5$ for $p_{D}=1$ and for all $N$, and that the
width of the range where $P_{\rm cons}$ changes decreases as $N^{-1}$, make it
possible to conjecture the existence of a function $\Phi(u)$, with
$\Phi(0)=0.5$ and $\Phi(u)\to 1$ for large $u$, such that $P_{\rm
cons}=\Phi[N(1-p_{D})]$. To test this hypothesis, we have plotted our
numerical data for $P_{\rm cons}$ against $N(1-p_{D})$ in Fig. 4. The results
are those in the upper half of the plot (“varying $p_{D}$”). The collapse of
the data for different $N$ on the same curve confirms the conjecture.

Analogous results were obtained when fixing $p_{D}=1$ and $p_{C}$ was varied.
Now, $P_{\rm cons}$ drops to $0$ in a narrow interval for $p_{C}$ just below
$1$, indicating the prevalence of disagreement. Again, the width of the
interval is proportional to $N^{-1}$. The results in the lower half of Fig. 4
(“varying $p_{C}$”) illustrate the collapse of the corresponding values of
$P_{\rm cons}$ when plotted against $N(1-p_{C})$.

Figure 4: Probability of reaching full consensus, $P_{\rm cons}$, as a
function of $N(1-p_{D})$ when varying $p_{D}$ with $p_{C}=1$, and as a
function of $N(1-p_{C})$ when varying $p_{C}$ with $p_{D}=1$.

In our numerical realizations with $p_{D}\neq p_{C}$, we have also recorded
the average time $T$ needed to reach the final absorbing state. Figure 5 shows
results for $N^{-2}T$ in the case where $p_{C}=1$ and $p_{D}$ changes (cf.
lower panel of Fig. 2). In contrast with the case with $p_{D}=p_{C}=1$,
rescaling of the time $T$ with $N^{2}$ leaves a remnant discrepancy between
results for different population sizes $N$. Specifically, for $p_{D}<1$, $T$
grows faster than $N^{2}$. Moreover, $T$ is nonmonotonic as a function of
$p_{D}$, exhibiting a minimum which shifts towards $p_{D}=1$ as $N$ grows. The
same dependence with $N$ and $p_{C}$ is observed when we fix $p_{D}=1$ and let
$p_{C}$ vary.

Figure 5: Total time $T$ needed to reach the final absorbing state, normalized
by the squared population size $N^{2}$, as a function of the probability
$p_{D}$ ($p_{C}=1$). Bézier curves have been plotted as a guide to the eye.

Summarizing our results for a one-dimensional population with two-agent
groups, we can say that the possibility that both consensus and disagreement
spread over the system makes it possible to find absorbing collective states
of either full consensus, with all the agents having the same opinion, or
maximal disagreement, where opposite opinions alternate between consecutive
neighbor agents. For large populations, the relative prevalence of collective
consensus and disagreement is controlled by how the probabilities $p_{D}$ and
$p_{C}$ compare with each other. Our results suggest that, in the limit
$N\to\infty$, the condition $p_{C}>p_{D}$ univocally leads to full consensus
and vice versa. For smaller sizes, however, the system can approach full
consensus even when $p_{D}>p_{C}$, and vice versa —presumably due to finite-
size fluctuations.

##  3 Larger groups on two-dimensional arrays

A two-dimensional version of the above model, where agents occupy the
$N=L\times L$ sites of a regular square lattice with periodic boundary
conditions, can be defined as follows. The reference group $G^{\prime}$ at
each interaction event is a randomly chosen $2\times 2$-agent block. The
corresponding active group $G$ is formed by the eight nearest neighbors to the
agents in $G^{\prime}$ which are not in turn members of the reference group.
The active group, thus, surrounds $G^{\prime}$. Of the sixteen possible
opinion configurations of the reference group, two correspond to full
consensus —with the four agents sharing the same opinion— and six correspond
to maximal disagreement —with two agents in each opinion. The remaining eight
configurations correspond to partial consensus, with only one agent
disagreeing with the other three. The dynamical rules are the following: (1)
if $G^{\prime}$ is in full consensus, all the agents in $G$ copy the common
opinion in $G^{\prime}$; (2) if $G^{\prime}$ is in maximal disagreement, each
agent in $G$ adopts the opinion opposite to that of the nearest neighbor in
$G^{\prime}$; (3) otherwise, nothing happens. Hence, both consensus and
disagreement spread outwards from the reference group. Probabilities $p_{D}$
and $p_{C}$ for the spreading of disagreement and consensus are introduced
exactly as above. The left part of Fig. 6 shows, up to rotations and opinion
inversions, the three possible outcomes of a single interaction event.

Figure 6: Left: The three possible outcomes of the interaction, up to $\pm
90^{\circ}$ rotations and opinion inversions, on the two-dimensional lattice.
The active and the reference groups, $G$ and $G^{\prime}$, are respectively
formed by the outermost and innermost agents. Right: Four snapshots of a
population with $L=120$, for $n_{+}(0)=0.35$ and $p_{D}=p_{C}=1$, including
the initial condition and two intermediate states. At time $t=3207$, an
absorbing state of maximal disagreement has been reached. Black and white dots
correspond, respectively, to opinions $+1$ and $-1$.

The states of full collective consensus —with all the agents in the population
having the same opinion— and of maximal collective disagreement —with the two
opinions alternating site by site along each direction over the lattice— are
absorbing states, in correspondence with the one-dimensional case. However,
for $p_{D}=p_{C}=1$, the system cannot be reduced anymore to a collection of
sublattices governed by the voter model. The definition of $G$ and
$G^{\prime}$ establish now correlations between the opinion changes in the
active group at each interaction event. Moreover, some opinion configurations
in the reference group induce evolution in the active group, while others do
not. Figure 6 shows, to its right, four snapshots of a $120\times 120$-agent
population, along a realization starting with $n_{+}(0)=0.35$ and
$p_{D}=p_{C}=1$. Note the formation of consensus clusters at rather early
stages, and the final prevalence of disagreement. The line boundaries between
disagreement regions are also worth noticing.

Following the same lines as for the one-dimensional array, we study first the
probability $P_{\rm cons}$ of reaching full collective consensus as a function
of the initial fraction of agents with opinion $+1$, $n_{+}(0)$, in the case
$p_{D}=p_{C}=1$. Opinions are homogeneously distributed all over the
population. For very small $n_{+}(0)$, as expected, we find $P_{\rm
cons}\approx 1$. However, in sharp contrast with the one-dimensional case (see
Fig. 3), $P_{\rm cons}$ remains close to its maximal value until
$n_{+}(0)\approx 0.35$, where it drops abruptly to $P_{\rm cons}\approx 0$.
The width of the transition zone decreases as a nontrivial power of the system
size, $\sim L^{0.83\pm 0.04}$, as illustrated in the insert of Fig. 7. Our
best estimate for the critical value of $n_{+}(0)$ at which $P_{\rm cons}$
drops is $n_{+}^{\rm crit}=0.353\pm 0.001$. The main plot in the figure shows
the collapse of numerical measurements of $P_{\rm cons}$ as a function of
$n_{+}(0)$ for different sizes $L$, averaged over $100$ realizations, when
plotted against the rescaled shifted variable $L^{0.83}[n_{+}(0)-0.353]$.

Figure 7: Numerical results for the probability of reaching full consensus,
$P_{\rm cons}$, on a two-dimensional lattice with $p_{D}=p_{C}=1$, obtained
from $100$ realizations for each parameter set. Collapse for several system
sizes $L$ is obtained plotting $P_{\rm cons}$ against
$L^{0.83}[n_{+}(0)-0.353]$. Insert: Scaling of the width of the transition
zone of $P_{\rm cons}$, determined from fitting a sigmoidal function, as a
function of the size $L$. The straight line has slope $-0.83$.

These results suggest that, for very large populations, the probability of
reaching full consensus jumps discontinuously from $P_{\rm cons}=1$ to $0$ at
$n_{+}(0)=n_{+}^{\rm crit}$. Compare this with the smooth, size-independent
behavior of the one-dimensional case. Note also that $n_{+}^{\rm crit}$ is
close to, but does not coincide with, $n_{+}(0)=1/3$. At this latter value, in
the initial condition with homogeneously distributed opinions, the probability
of finding a $2\times 2$-agent block in full consensus becomes lower than that
of maximal disagreement as $n_{+}(0)$ grows.

In the above simulations, we have also measured the average total time $T$
needed to reach the final absorbing state. Results are shown in Fig. 8. Again
in contrast with the one-dimensional case, $T$ exhibits a remarkable change in
its scaling with the system size as $n_{+}(0)$ overcomes the critical value
$n_{+}^{\rm crit}$.

Figure 8: Total time $T$ needed to reach the final absorbing state in a two-
dimensional lattice, as a function of $n_{+}(0)$, for different sizes $L$.

Going now to the dependence of $P_{\rm cons}$ on the probability of
disagreement spreading $p_{D}$ —with $p_{C}=1$ and $n_{+}(0)=0.5$— it
qualitatively mirrors that of the one-dimensional case, shown in Fig. 3.
Namely, as $p_{D}$ decreases from $1$, $P_{\rm cons}$ grows from $0$ to $1$ in
an interval whose width decreases with the population size. In the two-
dimensional system, however, the transition takes place at a critical
probability $p_{D}^{\rm crit}$ that can be clearly discerned from $p_{D}=1$.
Our estimate is $p_{D}^{\rm crit}=0.984\pm 0.002$. Moreover, the scaling of
the transition width with the population size exhibits a nontrivial exponent,
decreasing as $L^{-0.93\pm 0.05}$. Collapse of the rescaled numerical results
for various sizes, obtained from averages of $100$ realizations, are shown in
Fig. 9, where we plot $P_{\rm cons}$ as a function of $L^{0.93}(0.984-p_{D})$
(cf. Fig. 4). The insert displays the power-law dependence of the width on the
size $L$. Analogous results are obtained if the probability of consensus
spreading $p_{C}$ is varied, with $p_{D}=1$.

Figure 9: Collapse of numerical results for the probability of reaching full
consensus, $P_{\rm cons}$, on a two-dimensional lattice with $p_{C}=1$ and
$n_{+}(0)=0.5$, for several system sizes $L$ when plotted against
$L^{0.93}(0.984-p_{D})$. Insert: Scaling of the width of the transition zone
of $P_{\rm cons}$ as a function of the size $L$. The straight line has slope
$-0.93$.

Finally, we have found that the transition in $P_{\rm cons}$ as a function of
the disagreement probability $p_{D}$ shows a dependence on the initial
fraction of agents with opinion $+1$. To characterize this effect in a way
that highlights the relative prevalence of disagreement and consensus, we have
measured the value of $p_{D}$ at which the probability of getting full
collective consensus reaches $P_{\rm cons}=0.5$, as a function of $n_{+}(0)$.
The parameter plane $(n_{+}(0),p_{D})$, thus, becomes divided into regions
where a final state of full consensus is more probable than that of maximal
disagreement, and vice versa. Results for a $120\times 120$-agent population
are presented in Fig. 10.

Figure 10: Zones of relative prevalence of full consensus and maximal
disagreement in a two-dimensional lattice with $L=120$, plotted on the
parameter plane $(n_{+}(0),p_{D})$. Symbols stand for numerical results, and
the curve serves as a guide to the eye.

In summary, while spreading of consensus and disagreement on a two-dimensional
lattice bears superficial qualitative similarity with the one-dimensional
case, the probability that the population reaches full collective consensus in
two dimensions exhibits a quite different dependence on the system size, on
the initial conditions, and on the spreading probabilities. In particular, our
results reveal the existence of critical phenomena involving scaling laws with
nontrivial exponents.

##  4 Conclusion

In this paper, we have considered the emergence of collective opinion in a
population of interacting agents where, instead of imitation between
individual agents, opinions are transmitted through the spreading of local
consensus and disagreement toward their neighborhoods. The basic interacting
units in this mechanism are not individual agents but rather small groups of
agents which mutually compare their internal degrees of consensus and modify
their opinions accordingly. In this sense, it extends the basic mechanism
underlying such models as the majority-rule and Sznajd-like dynamics [1, 8,
13], where the opinion of each individual agent changes in response to the
collective state of a reference group. It is expected that in real social
systems the dissemination of individual opinions through agent-to-agent
imitation on one side, and the spreading of consensus and disagreement by
group interaction on the other, are complementary mechanisms simultaneously
shaping the overall opinion distribution. Here, in order to gain insight on
the specific effects of the second class, we have focused on models solely
driven by the spreading of consensus and disagreement. The combined effects of
the two mechanisms is a problem open to future work.

Our numerical simulations concentrated on two-opinion models evolving on one-
and two-dimensional arrays [14]. In both cases, absorbing states with all the
population bearing the same opinion (full consensus) and with half of the
population in each opinion (maximal disagreement) are possible final states
for the system. Maximal disagreement states are characterized by alternating
opinions between neighbor sites along the arrays.

A relevant quantity to characterize the behavior is the probability of
reaching full consensus, as a function of the initial condition —i.e., the
initial fraction of the population with each opinion— and of the relative
probabilities of consensus and disagreement spreading. The total time needed
to reach the final absorbing state, averaged over realizations, has also been
measured as a characterization of the dynamics. We have found that, in several
cases, these quantities display critical phenomena when the control parameters
are changed, with power-law scaling laws as functions of the system size,
pointing to the presence of discontinuities in the limit of infinitely large
populations. It is interesting to remark that the scaling laws are rather
simple for one-dimensional arrays, but involve nontrivial exponents and
critical points in the case of two-dimensional systems.

Within the same one- and two-dimensional models analyzed here, an aspect that
deserves further exploration is the dynamics and mutual interaction of the
opinion domains that develop since the first stages of evolution (Figs. 1 and
6). However, the most interesting extension of the present analysis should
progress along the direction of considering more complex social structures.
The interplay between the dynamical rules of consensus and disagreement
spreading and the topology of the interaction pattern underlying the
population might bring about the emergence of new kinds of collective self-
organization phenomena.

###### Acknowledgements.

We acknowledge enlightening discussions with Eduardo Jagla. Financial support
from ANPCyT (PICT2011-545) and SECTyP UNCuyo (Project 06/C403), Argentina, is
gratefully acknowledged.

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# Inverse Laplace transform based on Widder’s method for Tsallis exponential

S.S. Naina<EMAIL_ADDRESS>K. Jeevanandham1, A. Basherrudin
Mahmud Ahmed2, Md. Manirul Ali3<EMAIL_ADDRESS>and R.
Chandrashekar4<EMAIL_ADDRESS>1 Department of Physics,
Government Arts College, Udumalpet, Tiruppur District - 642126, Tamilnadu,
India  2 School of Physics, Madurai Kamaraj University, Madurai 625021, India
3 Centre for Quantum Science and Technology, Chennai Institute of Technology,
Chennai 600069, India  4 Centre for Quantum Information, Communication and
Computing, Indian Institute of Technology Madras, Chennai 600036, India

###### Abstract

A generalization of the Laplace transform based on the generalized Tsallis
$q$-exponential is given in the present work for a new type of kernel. We also
define the inverse transform for this generalized transform based on the
complex integration method. We prove identities corresponding to the Laplace
transform and inverse transform like the $q$-convolution theorem, the action
of generalized derivative and generalized integration on the Laplace
transform. We then derive a $q$-generalization of the inverse Laplace
transform based on the Post-Widder’s method which bypasses the necessity for a
complex contour integration. We demonstrate the usefulness of this in
computing the Laplace and inverse Laplace transform of some elementary
functions. Finally we use the Post-Widder’s method based inverse Laplace
transform to compute the density of states from the partition function for the
case of a generalized classical ideal gas and linear harmonic oscillator in
$D$-dimensions.

##  1 Introduction

A mathematical function ‘$f$’ in the ‘$x$’ domain can be transformed to a
function ‘$F$’ in the ‘$u$’ domain using the integral

$F(u)=\int_{a}^{b}f(x)\,K(u,x)dx.$ (1)

This process is known as an integral transform and the function $K(u,x)$ is
the kernel of the transformation. Here the integral transform maps the problem
in a given domain to another domain in which it is simpler to solve. Laplace
transform is one of the widely used integral transforms in physics. It is very
useful in solving convolution integral equations and differential equations.
The inverse Laplace transform is an equally important transform with several
applications. In general an inverse Laplace transform is carried out using a
Bromwich contour integral in the complex plane [1] . But this practice of
using a complex variable technique is done only for the sake of convenience.
In fact an inverse Laplace transform based only on real variables was
introduced by Post [2] and later it was refined by Widder [3]. This method has
been investigated in several works [4, 5, 6] considering different
applications.

The kernel of a Laplace transform is an exponential function of the form
$\exp(-st)$. Tsallis introduced [7] generalizations of the logarithm and the
exponential functions as follows:

$\ln_{q}(x)\equiv\frac{x^{1-q}-1}{1-q};\qquad\exp_{q}(x)\equiv[1+(1-q)x]^{\frac{1}{1-q}},$
(2)

where $q\in\mathbb{R}_{+}$ is the generalization parameter. These functions
are generally referred to as Tsallis q-logarithm and Tsallis q-exponential and
are inversely related. The $q$-logarithm and the $q$-exponential function
reduces to the usual logarithm and exponential functions in the $q\rightarrow
1$ limit. These generalized functions have been investigated in a wide variety
of fields like astrophysics [8, 9], high energy physics [10, 11] neutrino
physics [12], mathematical physics [13, 14] and nonequilibrium statistical
physics [15].

A generalization of the Laplace transform has been done in Ref. [16, 17, 18,
19] using the Tsallis q-exponential. Here again the inverse was based on using
a contour integral in the complex plane. In our present work we extend the
concept of Post and Widder’s method to the generalized Laplace transform based
on Tsallis q-exponential.

The work is organized as follows: In Section 2, we give a brief introduction
to the Laplace transform, Post-Widders technique and the Tsallis
$q$-exponential. The Laplace and inverse Laplace based on type - I kernel is
defined in the third section. The properties of Laplace transform are derived
in Section 4. In Section 5 we give the $q$-generalization of the Post-Widder’s
method for the inverse Laplace transform. The Laplace transform and the
inverse Laplace transform based on the Widder’s method for different functions
are computed in Section 6. As an application of this method we calculate the
density of states of a $D$-dimensional classical ideal gas and a linear
harmonic oscillator from the partition function using the Post-Widder’s method
based inverse Laplace transform.

##  2 A primer on Laplace transform, Post- Widder’s technique and Tsallis
$q$-exponential

In this section we briefly review the concept of Laplace transform and its
inverse. Then we describe the Post-Widder’s method to calculate the inverse
Laplace transform. The Laplace transform of a function $f(t)$ denoted by
$\mathcal{L}[f(t)]=F(s)$ is defined as

$F(s)=\int_{0}^{\infty}\exp(-st)f(t)dt.$ (3)

Here $f\in S(\mathbb{R})$ where $S(\mathbb{R})$ represents the Schwartz space
of functions $f:\mathbb{R}\rightarrow\mathbb{C}$ with $f\in
C^{\infty}(\mathbb{R})$, i.e., $f$ is infinitely differentiable on
$\mathbb{R}$. In general $s=\sigma+i\tau$ with $\sigma$ and $\tau$ being real
numbers. The integral converges when ${\rm Re}[s]=\sigma>0$ and for
$\sigma<0$, $F(s)=0$. For the Laplace transform of $F(s)$, the inverse
transform is defined as

$f(t)=\frac{1}{2\pi
i}\lim_{\tau\rightarrow\infty}\int_{\sigma-i\tau}^{\sigma+i\tau}\exp(st)F(s)ds.$
(4)

The direct Laplace transform Eq. (3) and the inverse Laplace transform Eq. (4)
are inverses of each other for the functions $f\in S(\mathbb{R})$. In order to
calculate the inverse Laplace transform we need to perform a Bromwich contour
integration over the complex plane. Below we explain an alternative method
introduced by Post and Widder which does a Laplace inverse using only real
variables.

Let us consider the $n^{th}$ derivative of the Laplace transform $F(s)$ with
respect to the variable ‘$s$’,

$\frac{{\rm d}^{n}}{{\rm d}s^{n}}F(s)\equiv
F^{(n)}(s)=(-1)^{n}\int_{0}^{\infty}t^{n}\exp(-st)f(t)dt.$ (5)

We use the following three steps, (i) First we use the variable transformation
$s=n/x$, (ii) We follow it by multiplying the numerator and the denominator by
$x^{n}$ and (iii) Finally we use the variable change $y=t/x$, and recast the
integral in Eq. (5) to

$F^{(n)}\left(\frac{n}{x}\right)=(-1)^{n}x^{n+1}\int_{0}^{\infty}(ye^{-y})^{n}f(xy)dy.$
(6)

Here we would like to point out that the function $ye^{-y}$ has a single
maximum at $y=1$ and for the function $(ye^{-y})^{n}$ this maximum is sharply
peaked at $y=1$ and hence Eq. (6) can be rewritten as

$F^{(n)}\left(\frac{n}{x}\right)\approx(-1)^{n}x^{n+1}f(x)\int_{0}^{\infty}(ye^{-y})^{n}dy.$
(7)

Evaluating the integral we get

$F^{(n)}\left(\frac{n}{x}\right)\approx(-1)^{n}n!\left(\frac{x}{n}\right)^{n+1}f(x).$
(8)

In the limit $n\rightarrow\infty$, we can use the transformation
$y=\frac{t}{x}\big{|}_{y=1}\Rightarrow x=t$ and so $s=\frac{n}{x}\Rightarrow
s=\frac{n}{t}$ we can rewrite Eq. (8) as

$f(t)=\lim_{n\to\infty}\frac{(-1)^{n}}{n!}\,s^{n+1}F^{n}(s)|_{s=n/t}.$ (9)

Thus we observe that the Laplace transform and the inverse Laplace transform
can be expressed as functions of real variables alone. This sequence converges
very fast since the rate of convergence is at least $1/n$.

The Laplace transform exists for a piecewise continuous function of
exponential order. Similarly, the $q$-Laplace transform has been defined for a
piecewise continuous function of q-exponential order. The $q$-exponential
based Laplace transform can be defined using three different types of kernels
as noted in Ref. [16] and these kernels are

$\displaystyle K_{{\rm I}}(q;s,t)$ $\displaystyle=$
$\displaystyle\exp_{q}(-st),$ (10) $\displaystyle K_{{\rm II}}(q;s,t)$
$\displaystyle=$ $\displaystyle[\exp_{q}(-t)]^{s},$ (11) $\displaystyle
K_{{\rm III}}(q;s,t)$ $\displaystyle=$ $\displaystyle[\exp_{q}(t)]^{-s}.$ (12)

Of these three kernels, Laplace transform has been defined using the first and
the second kernels in Ref . [16, 18, 19]. In the extensive limit
($q\rightarrow 1$), both these generalized Laplace transforms reduce to the
ordinary Laplace transform. For the generalized Laplace transform based on the
first kernel [19], the inverse Laplace transform has not been defined. But for
the case where the Laplace transform was defined using the second kernel, the
inverse transform was defined using the complex integration method.

##  3 Laplace and Inverse transform based on type - I kernel

The Laplace transform due to type-I kernel which was introduced in [19] is
examined in this section. For this generalization, the inverse Laplace
transform has not been introduced so far and we restrict and present only the
$q<1$ case here. Below we define the inverse Laplace transform and prove its
inverse nature. A function ‘$f$’ is said to be of $q$-exponential order ‘$c$’,
if there exists $c$, $M>0$, $T>0$, such that $|f(t)|\leq M\exp_{q}(ct)$
$\forall$ $t>T$. If a function is piecewise continuous and is of
$q$-exponential order $c$, then $F_{q}(s)=L_{q}[f(t)]$ exists for $s>c$ and
$\lim_{s\rightarrow\infty}F(s)=0$. Under these conditions, the $q$-Laplace
transform is defined as

$L_{q}[f(t)](s)=F_{q}(s)=\int_{0}^{\infty}f(t)\exp_{q}(-st)dt.$ (13)

The corresponding inverse Laplace transform is defined as

$L_{q}^{-1}[F_{q}(s)](t)=f(t)=\frac{2-q}{2\pi
i}\int_{c-i\infty}^{c+i\infty}F_{q}(s)[\exp_{q}(-st)]^{2q-3}ds,$ (14)

where in the limit $q\rightarrow 1$, we have
$[\exp_{q}(-st)]^{2q-3}\rightarrow\exp(st)$. Here $c$ is a real constant that
exceeds real part of all the singularities of $F_{q}(s)$. To prove the inverse
relationship between $L_{q}$ and $L_{q}^{-1}$, we verify the following two
identities:

$\displaystyle f(t)$ $\displaystyle=$ $\displaystyle L_{q}^{-1}[L_{q}[f(t)]],$
(15) $\displaystyle F_{q}(s)$ $\displaystyle=$ $\displaystyle
L_{q}[L_{q}^{-1}[F_{q}(s)]].$ (16)

Using the inverse Laplace transform we can write (16),

$\displaystyle L_{q}[L_{q}^{-1}[F_{q}(s)]]$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\exp_{q}(-st)\;L_{q}^{-1}[F_{q}(s)]$ (17)
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}dt\exp_{q}(-st)\;\frac{(2-q)}{2\pi
i}\;\int_{c-i\infty}^{c+i\infty}ds^{\prime}\;F_{q}(s^{\prime})[\exp_{q}(-s^{\prime}t)]^{2q-3}.$

We can rewrite the above expression as

$L_{q}[L_{q}^{-1}[F_{q}(s)]]=\frac{(2-q)}{2\pi
i}\;\int_{c-i\infty}^{c+i\infty}ds^{\prime}\;F_{q}(s^{\prime})\int_{0}^{\infty}dt\exp_{q}(-st)\;[\exp_{q}(-s^{\prime}t)]^{2q-3}.$
(18)

The second integral converges when ${\rm Re}[s^{\prime}]=c<{\rm Re}[s]$ and
the resulting solution is

$\mathcal{I}_{q}(s,s^{\prime})=\int_{0}^{\infty}dt\exp_{q}(-st)\;[\exp_{q}(-s^{\prime}t)]^{2q-3}=\frac{1}{(2-q)}\;\frac{1}{(s-s^{\prime})}.$
(19)

Substituting this in Eq. (18) we get

$L_{q}[L_{q}^{-1}[F_{q}(s)]]=\frac{1}{2\pi
i}\;\int_{c-i\infty}^{c+i\infty}ds^{\prime}\frac{F_{q}(s^{\prime})}{(s-s^{\prime})}.$
(20)

The function $F_{q}(s)$ is an entire function and does not have any poles in
the complex plane. The integrand $F_{q}(s^{\prime})/(s-s^{\prime})$ has a
simple pole at $s=s^{\prime}$. To evaluate the integral we draw a straight
line at $s^{\prime}$ and an arc enclosing the pole of the integrand. The
solution of the integral is

$\int_{c-i\infty}^{c+i\infty}ds^{\prime}\frac{F_{q}(s^{\prime})}{(s-s^{\prime})}=2\pi
iF_{q}(s).$ (21)

Substituting this in Eq. (20) we can observe that
$L_{q}[L_{q}^{-1}[F_{q}(s)]]=F_{q}(s)$. Next we verify the first identity Eq.
(15) as follows:

$L_{q}^{-1}[L_{q}[f(t)]]\equiv
L_{q}^{-1}[F_{q}(s)]=L_{q}^{-1}\left[\int_{0}^{\infty}dt\exp_{q}(-st)f(t)\right].$
(22)

Using the definition of the inverse Laplace transform

$\displaystyle L_{q}^{-1}[L_{q}[f(t)]]=\frac{2-q}{2\pi
i}\int_{c-i\infty}^{c+\infty}[\exp_{q}(-st)]^{2q-3}\int_{0}^{\infty}dt^{\prime}f(t^{\prime})\exp_{q}(-st^{\prime})\;ds.$
(23)

Rewriting the above equation (23) we arrive at

$L_{q}^{-1}[L_{q}[f(t)]]=\frac{2-q}{2\pi
i}\;\int_{0}^{\infty}dt^{\prime}f(t^{\prime})\int_{c-i\infty}^{c+\infty}[\exp_{q}(-st)]^{2q-3}\exp_{q}(-st^{\prime})\;ds.$
(24)

Using the definition of the Dirac delta function based on the $q$-exponential
described in Ref. [20, 21] we get

$L_{q}^{-1}[L_{q}[f(t)]]=\int_{0}^{\infty}dt^{\prime}f(t^{\prime})\delta(t-t^{\prime})\equiv
f(t).$ (25)

Thus in the present section, we have introduced the inverse for the Laplace
transform defined using the type-I kernel using the complex integration
technique.

##  4 Properties of $q$-Laplace transform

In this section, we list some of the properties of the $q$-Laplace transform:

  1. 1.

${\rm I}^{st}$ Identity on Limits:

$\lim_{s\rightarrow\infty}sL_{q}[f(t)]=\lim_{t\rightarrow
0}\frac{f(t)}{1+(1-q)},$ (26)

Proof: Let us consider a general convergent function which can be expressed in
terms of a power series $f(t)=\sum_{n=0}^{\infty}a_{n}t^{n}$. The $q$-Laplace
transform of this general function is

$L_{q}[f(t)]=\int_{0}^{\infty}\left(\sum_{n=0}^{\infty}a_{n}t^{n}\right)[1-(1-q)st]^{\frac{1}{1-q}}\,dt.$
(27)

Since the function $f(t)$ is a convergent function we can rewrite the above
Equation (27) as

$L_{q}[f(t)]=\sum_{n=0}^{\infty}a_{n}\int_{0}^{\infty}t^{n}[1-(1-q)st]^{\frac{1}{1-q}}dt.$
(28)

To solve Eq. (28) we use the integration by parts method choosing $u=t^{n}$
and $dV=[1-(1-q)st]^{\frac{1}{1-q}}$. The resulting solution reads:

$\displaystyle sL_{q}[f(t)]$ $\displaystyle=$
$\displaystyle-\frac{1}{1+(1-q)}\bigg{(}f(t)[1-(1-q)st]^{\frac{1}{1-q}+1}\big{|}_{0}^{\infty}$
(29)
$\displaystyle-\int_{0}^{\infty}[1-(1-q)st]\left[\frac{d}{dt}f(t)\right]\exp_{q}(-st)dt\bigg{)}.$

On applying the limits corresponding to the integration in Eq. (29) we get

$\displaystyle sL_{q}[f(t)]$ $\displaystyle=$
$\displaystyle\frac{f(0)}{1+(1-q)}+\frac{1}{1+(1-q)}$ (30)
$\displaystyle\int_{0}^{\infty}[1-(1-q)st]\;\frac{df(t)}{dt}[1-(1-q)st]^{\frac{1}{1-q}}dt.$

Under the limiting condition $s\rightarrow\infty$, Eq. (30) gives

$\lim_{s\rightarrow\infty}sL_{q}[f(t)]=\lim_{s\rightarrow\infty}\frac{f(t)}{1+(1-q)}.$
(31)

We can observe that the RHS is independent of $s$ and can be expressed as a
limiting value of the parameter $t$ and this gives us

$\lim_{s\rightarrow\infty}sL_{q}[f(t)]=\lim_{t\rightarrow
0}\frac{f(t)}{1+(1-q)}.$ (32)

Thus we prove the first identity on the limits of a Laplace transform.

  2. 2.

${\rm II}^{nd}$ Identity on Limits:

$\lim_{s\rightarrow
0}sL_{q}[f(t)]=\lim_{t\rightarrow\infty}\frac{f(t)}{1+(1-q)}.$ (33)

Proof: To prove this limit let us consider the Eq. (29) and evaluate the
limit, $s\rightarrow 0$ and the resulting expression is

$\displaystyle\lim_{s\rightarrow 0}sL_{q}[f(t)]$ $\displaystyle=$
$\displaystyle\lim_{s\rightarrow
0}\frac{f(0)}{1+(1-q)}+\frac{1}{1+(1-q)}\int_{0}^{\infty}\frac{df}{dt}dt$ (34)
$\displaystyle=$
$\displaystyle\frac{f(0)}{1+(1-q)}+\frac{f(t)}{1+(1-q)}\bigg{|}_{0}^{\infty}.$

Applying the limits we get

$\lim_{s\rightarrow 0}sL_{q}[f(t)]=\frac{f(\infty)}{1+(1-q)}.$ (35)

The RHS of the equation can be rewritten as

$\lim_{s\rightarrow
0}sL_{q}[f(t)]=\lim_{t\rightarrow\infty}\frac{f(t)}{1+(1-q)}.$ (36)

Thus we prove the second identity on the Laplace transform.

  3. 3.

Scaling

$L_{q}[f(at)]=\frac{1}{a}F_{q}(s/a).$ (37)

Proof: Let us consider the Laplace transform of a function $f(at)$

$L_{q}[f(at)]=\int_{0}^{\infty}dt[1-(1-q)st]^{\frac{1}{1-q}}f(at).$ (38)

Substituting $at=x$, we get

$L_{q}[f(at)]=\frac{1}{a}\,\int_{0}^{\infty}dx\bigg{[}1-(1-q)\frac{sx}{a}\bigg{]}^{\frac{1}{1-q}}f(x)=\frac{1}{a}F_{q}(s/a).$
(39)

Hence the scaling relation is proved.

  4. 4.

Shifting

$F_{q}(s-s_{0})=L_{q}\left[f(t)\exp_{q}\left[\frac{s_{0}t}{1-(1-q)st}\right]\right].$
(40)

Proof: Let us consider the definition of the Laplace transform

$F_{q}(s)=\int_{0}^{\infty}\exp_{q}(-st)f(t)dt,$ (41)

and introduce a shift in ‘$s$’ as ‘$s\rightarrow s-s_{0}$’ and this yields

$F_{q}(s-s_{0})=\int_{0}^{\infty}\exp_{q}(-(s-s_{0})t)f(t)dt.$ (42)

This can be rewritten as

$F_{q}(s-s_{0})=\int_{0}^{\infty}\exp_{q}(-st)\exp_{q}\bigg{(}\frac{s_{0}t}{1-(1-q)st}\bigg{)}f(t)dt.$
(43)

Hence we have proved

$F_{q}(s-s_{0})=L_{q}\bigg{[}f(t)\exp_{q}\bigg{(}\frac{s_{0}t}{1-(1-q)st}\bigg{)}\bigg{]}.$
(44)

  5. 5.

$q$-translation

$L_{q}[f(t)]\big{[}\exp_{q}(st_{0})\big{]}^{1+(1-q)}=L_{q}\bigg{[}f\left(\frac{t-t_{0}}{1-(1-q)st_{0}}\right)\Theta\left(\frac{t-t_{0}}{1-(1-q)st_{0}}\right)\bigg{]}.$
(45)

Proof: The RHS of the above identity gives

$\displaystyle I$ $\displaystyle\equiv$ $\displaystyle
L_{q}\bigg{[}f\left(\frac{t^{\prime}-t_{0}}{1-(1-q)st_{0}}\right)\Theta\left(\frac{t^{\prime}-t_{0}}{1-(1-q)st_{0}}\right)\bigg{]}$
$\displaystyle=$ $\displaystyle\int
dt^{\prime}f\left(\frac{t^{\prime}-t_{0}}{1-(1-q)st_{0}}\right)\Theta\left(\frac{t^{\prime}-t_{0}}{1-(1-q)st_{0}}\right)\exp_{q}(-st^{\prime}),$

where $\Theta(x)$ is the Heaviside step function such that $\Theta(x)=0$ when
$x<0$ and $\Theta(x)=1$ for $x\geq 0$. Using the scaling
$t=\frac{t^{\prime}-t_{0}}{1-(1-q)st_{0}}$, we can rewrite the integral as

$I=\int_{-\alpha}^{\infty}\;dt\exp_{q}(-st)\;f(t)[\exp_{q}(-st_{0})]^{2-q}\;\Theta(t).$

where $\alpha=\frac{t_{0}}{1-(1-q)st_{0}}$. On applying the Heaviside step
function we get

$I=L_{q}[f(t)][\exp_{q}(-st_{0})]^{2-q}.$ (46)

Hence the $q$-translation identity has been proved.

The properties of linearity and $q$-convolution have been established in Ref.
[19] and here we are stating them just for the sake of completeness. The two
properties of linearity and $q$-convolution are

  1. 1.

Linearity:

$L_{q}[a_{1}f_{1}(t)+a_{2}f_{2}(t)]=a_{1}L_{q}[f_{1}(t)]+a_{2}L_{q}[f_{2}(t)].$
(47)

  2. 2.

$q$-convolution:
Let $f(t)$ and $g(t)$ be two positive scalar functions of ‘$t$’, and
$F_{q}(s)$ and $G_{q}(s)$ be their q-Laplace transforms, then

$L_{q}[f(t)\ast g(t)]=F_{q}(s)\ast G_{q}(s),$ (48)

where $f(t)\ast g(t)=\int_{0}^{t}f(\tau)\ast g(t-\tau)d\tau$.

Laplace transform of $q$-derivatives and $q$-integrals: The properties of the
$q$-Laplace transform based on derivatives and integrals have been derived in
Ref. [19]. For the sake of completeness, we give below the expression for the
generalized Laplace transform of a derivative function

$\displaystyle L_{q}\left[\frac{{\rm d}^{n}}{{\rm d}t^{n}}f(t)\right]$
$\displaystyle=$
$\displaystyle-\left(f^{(n-1)}(t)+(1-\delta_{1n})\sum_{\ell=1}^{n-1}Q_{\ell-1}(q)s^{\ell}f^{(n-\ell-1)}(t)\right)\Bigg{|}_{t=0}$
(49) $\displaystyle+Q_{n-1}(q)s^{n}L_{\frac{a_{n+1}}{a_{n}}}(f(t))(a_{n}s),$

where, $Q_{n}(q)=\prod_{j=0}^{n}a_{j}$ with $a_{j}=jq-(j-1)$ and $\delta_{1j}$
is the Kronecker delta function. The Laplace transform of the integral reads:

$L_{q}\left[\int_{0}^{\infty}f(x)dx\right]=\frac{2-q}{s}L_{\frac{1}{2-q}}\left[f(t)\right]\left(s(2-q)\right).$
(50)

In the present work, we derive the $q$-Laplace transform of $q$-calculus of
functions. The concept of $q$-calculus was first introduced in [22], where the
authors defined the derivative and integral based on $q$-deformation. Many
such derivatives and integrals were investigated in subsequent works [13, 16,
19]. For the present work we use the $q$-derivative defined in [23] and its
corresponding $q$-deformed integral. The relevant $q$-derivative and
$q$-integral operators are

$D_{q}(s)=\frac{1}{1-(1-q)\left(s\frac{d}{ds}\right)}\frac{d}{ds};\qquad\int
d_{q}x=\int dx\left[1-(1-q)x\frac{d}{dx}\right].$ (51)

  1. 1.

Derivative of Laplace transform:

$D_{q}^{n}(s)\\{L_{q}[f(t)]\\}(s)=L_{q}[(-t)^{n}f(t)].$ (52)

Proof: Let us consider the first derivative of the $q$-Laplace transform

$\displaystyle D_{q}\\{L_{q}[f(t)]\\}(s)$ $\displaystyle=$ $\displaystyle
D_{q}(s)\left\\{\int_{0}^{\infty}dt\exp_{q}(-st)f(t)\right\\}$ (53)
$\displaystyle=$ $\displaystyle\int_{0}^{\infty}dt\,D_{q}(s)\exp_{q}(-st)f(t)$
$\displaystyle=$ $\displaystyle L_{q}[(-t)f(t)].$

The second derivative of the $q$-Laplace transform is

$D_{q}^{2}\\{L_{q}[f(t)]\\}(s)=L_{q}[(-t)^{2}f(t)].$ (54)

For the $n^{th}$ $q$-derivative we get

$D_{q}^{n}\\{L_{q}[f(t)]\\}(s)=L_{q}[(-t)^{n}f(t)].$ (55)

  2. 2.

Integral of a Laplace transform:

$\int_{0}^{\infty}d_{q}s\,F_{q}(s)=L_{q}\left\\{\frac{f(t)}{t}\right\\}.$ (56)

Let us consider the definition of the generalized Laplace transform apply the
$q$-integral operator

$\int_{0}^{\infty}d_{q}s\,F_{q}(s)=\int_{s}^{\infty}d_{q}s\int_{0}^{\infty}\exp_{q}(-st)f(t)dt.$
(57)

Rearranging the order of the integrals and evaluating the $q$-integral with
respect to ‘$s$’ gives

$\int_{0}^{\infty}d_{q}s\,F_{q}(s)=\int_{0}^{\infty}\frac{1}{t}\exp_{q}(-st)f(t)dt=L_{q}\left[\frac{f(t)}{t}\right].$
(58)

Hence proved.

##  5 Post-Widders method of inverse Laplace transform

The inverse Laplace transform can be calculated using a Bromwich contour
integral over the complex plane. As an alternative we introduce a $q$-deformed
generalization of the Post-Widder’s method to compute the inverse Laplace
transform using only real variables. To derive the $q$-Widder’s formula we
will have to rewrite the type I kernel as
$K_{q}(s,t)=\exp\left(\frac{1}{1-q}\ln(1-(1-q)st)\right)$. We then have to
take the $k^{th}$ derivative of the Laplace transform and scale the function
$f(t)$ to $f(\xi_{m}t)$ and the resulting expression is

$F_{q}^{(k)}(s)=\int_{0}^{\infty}dt[-(1-q)t]^{k}\frac{\Gamma\left(\frac{2-q}{1-q}\right)}{\Gamma\left(\frac{2-q}{1-q}-k\right)}(1-(1-q)st)^{\frac{1}{1-q}-k}f(t\xi_{m}).$
(59)

Through a change of variables $t=xy$ and $s=k/x$ we get:

$F_{q}^{(k)}\left(\frac{k}{x}\right)=[-(1-q)]^{k}x^{k+1}\frac{\Gamma\left(\frac{2-q}{1-q}\right)}{\Gamma\left(\frac{2-q}{1-q}-k\right)}\int_{0}^{\infty}(1-(1-q)ky)^{\frac{1}{1-q}-k}y^{k}dyf(xy\xi_{m}).$
(60)

The function $y^{k}(1-(1-q)ky)^{\frac{1}{1-q}-k}$ has a single maximum which
is sharply peaked at $y=1$. Hence we can replace the function $f(\xi_{m}xy)$
by $f(\xi_{m}x)$ and get

$F_{q}^{(k)}\left(\frac{k}{x}\right)=f(\xi_{m}x)[-(1-q)]^{k}x^{k+1}\frac{\Gamma\left(\frac{2-q}{1-q}\right)}{\Gamma\left(\frac{2-q}{1-q}-k\right)}\int_{0}^{\infty}(1-(1-q)ky)^{\frac{1}{1-q}-k}y^{k}dy.$
(61)

The solution of the integral in the above equation is

$\int_{0}^{\infty}(1-(1-q)ky)^{\frac{1}{1-q}-k}y^{k}dy=\frac{\Gamma(k+1)\Gamma\left(\frac{2-q}{1-q}-k\right)}{\Gamma\left(\frac{3-2q}{1-q}\right)}.$
(62)

Replacing the value of the integral in Eq. (61) and simplifying we have

$F_{q}^{(k)}\left(\frac{k}{x}\right)=(-1)^{k}f(x\xi_{m})x^{k+1}\frac{\Gamma(k+1)}{k^{k+1}(2-q)}.$
(63)

where $\xi_{m}=\left[\frac{1+(1-q)}{Q_{m}(2-q)}\right]^{\frac{1}{m-1}}$ valid
for $m\geq 2$ and ill defined for $m=1$ and the polynomial
$Q_{m}(q)=\prod_{j=1}^{m}(1-(1-q)j)$ exists for only for $j\geq 1$.
Substituting $t=\xi_{m}x$, the $q$-deformed Widder’s formula for the inverse
Laplace transform is

$f(t)=\lim_{k\rightarrow\infty}\frac{(-1)^{k}}{\Gamma(k+1)}F_{q}^{(k)}(s)(2-q)s^{k+1}|_{s=\frac{k\xi_{m}}{t}}.$
(64)

##  6 Inverse Laplace transform for some elementary functions

In this section we evaluate the Laplace transform of some elementary functions
and then compute their inverse transform using Post-Widders method.

Algebraic function:
Let us consider an algebraic function $f(t)=t^{m-1}$, the $q$-Laplace
transform of this function is

$L_{q}\left[t^{m-1}\right]=\displaystyle\int_{0}^{\infty}dt\exp_{q}(-st)t^{m-1}.$
(65)

Evaluating the Laplace transform we get

$F_{q}(s)=\frac{\Gamma(m)}{Q_{m}(2-q)}\frac{1}{s^{m}}\;,\qquad\mbox{for}\;m\geq
2.$ (66)

The inverse of the $q$-Laplace transform can be computed using a $q$-version
of the Widder’s formula. For this we calculate the $k^{th}$-derivative of
$F_{q}(s)$

$F_{q}^{k}(s)=\frac{1}{Q_{m}(2-q)}(-1)^{k}\frac{\Gamma(m+k)}{s^{m+k}}.$ (67)

Substituting Eq. (67) in the $q$-Widder’s formula we get

$\displaystyle f(t)$ $\displaystyle=$
$\displaystyle\displaystyle\lim_{k\rightarrow\infty}\frac{(-1)^{k}}{\Gamma(k+1)}\left(\frac{1}{Q_{m}(2-q)}(-1)^{k}\frac{\Gamma(m+k)}{s^{m+k}}\right)Q_{1}(2-q)s^{k+1}\Biggr{|}_{s=\frac{k\xi_{m}}{t}},$
(68) $\displaystyle=$
$\displaystyle\displaystyle\lim_{k\rightarrow\infty}\frac{(-1)^{k}}{\Gamma(k+1)}\left(\frac{1}{Q_{m}(2-q)}(-1)^{k}\frac{\Gamma(m+k)}{s^{m-1}}\right)Q_{1}(2-q)\Biggr{|}_{s=\frac{k\xi_{m}}{t}}.$

where, $\xi_{m}=\left(\frac{Q_{1}(2-q)}{Q_{j}(2-q)}\right)^{\frac{1}{j-1}}$
and the subscript ‘$j$’ depends on the exponent of ‘$s$’, since the exponent
of ‘$s$’ is ‘$m-1$’ the value of ‘$j$’ is ‘$m$’. Substituting the value of
‘$s$’ we get:

$\displaystyle f(t)$ $\displaystyle=$
$\displaystyle\frac{Q_{1}(2-q)}{Q_{m}(2-q)}\frac{t^{m-1}}{\xi^{m-1}}\left(\displaystyle\lim_{k\rightarrow\infty}\frac{k^{1-m}\Gamma(m+k)}{\Gamma(k+1)}\right).$
(69)

On substitution of the limits and simplifying Eq. (69) we get the algebraic
limit $f(t)=t^{m-1}$. This validates the Post-Widder’s method of computing
inverse Laplace transform for any Algebraic function.

Exponential functions:
Regular exponential: Let us consider a generalized form of the exponential
function $f(t)=\exp\left(\pm\sigma_{\epsilon}\;t\right)$ where
$\sigma_{\epsilon}=i^{\epsilon}\alpha$ with $\alpha\in\mathcal{R}$ and
$\epsilon=0\;\mbox{or}\;1$. We can write this definition in the following way:

$\displaystyle
f(t)=\exp\left(\pm\sigma_{\epsilon}\;t\right)=\left\\{\begin{array}[]{ll}\exp(\pm\alpha
t)&\mbox{for}\;\epsilon=0\\\ \exp(\pm i\alpha t)&\mbox{for}\;\epsilon=1\\\
\end{array}\right.$ (72)

In terms of hypergeometric functions, the classical exponential given in Eq.
(72) can be expressed as:

$f(t)=\exp\left(\pm\sigma_{\epsilon}\;t\right)={}_{0}F_{0}\left(\;;\;;\pm\sigma_{\epsilon}\;t\right).$
(73)

The $q$-Laplace transform of $f(t)$ gives:

$\displaystyle
L_{q}\left[{}_{0}F_{0}\left(\;;\;;\pm\sigma_{\epsilon}\;t\right)\right]$
$\displaystyle=$
$\displaystyle\displaystyle\int_{0}^{\infty}dt\;\exp_{q}(-st)\;{}_{0}F_{0}\left(\;;\;;\pm\sigma_{\epsilon}t\right)$
(74) $\displaystyle=$
$\displaystyle\frac{1}{s\;Q_{1}(2-q)}\;{}_{1}F_{1}\left(1;\frac{3-2q}{1-q};\pm\frac{\sigma_{\epsilon}}{(1-q)s}\right).$

To find the $q$-inverse Laplace transform using Widder like formula for the
Tsallis $q$-case, we take the $k^{th}$-derivative of $F_{q}(s)$ given by Eq.
(74) which is

$F_{q}^{(k)}(s)=\frac{1}{Q_{1}(2-q)}\left[\displaystyle\sum_{n=0}^{\infty}\frac{1}{\left(\frac{3-2q}{1-q}\right)_{n}}\frac{\left(\pm\frac{\sigma_{\epsilon}}{1-q}\right)^{n}}{n!}(-1)^{k}\frac{\Gamma(n+k+1)}{s^{n+k+1}}\right].$
(75)

Substituting Eq. (75) in the $q$-Widder’s formula, Eq. (64) and simplifying it
we get:

$f(t)=\displaystyle\sum_{n=0}^{\infty}\frac{1}{\left(\frac{3-2q}{1-q}\right)_{n}}\frac{1}{n!}\left(\pm\frac{\sigma_{\epsilon}t}{(1-q)\xi_{m}}\right)^{n}\left[\lim_{k\rightarrow\infty}k^{1-n}\left(1+\frac{n}{k}\right)\frac{\Gamma(k+n)}{\Gamma(k+1)}\right].$
(76)

Applying the limits in Eq. (76) and simplifying the expression we get the
initial function $f(t)=\exp(\pm\sigma_{\epsilon}t)$ validating the Post-
Widder’s method.

Generalized exponential: In analogy with the exponential, we can write the
generalized exponential as
$f(t)=\exp_{q^{\prime}}\left(\pm\sigma_{\epsilon}\;t\right)$ where
$\sigma_{\epsilon}=i^{\epsilon}\alpha$ with $\alpha\in\mathcal{R}$ and
$\epsilon=0\;\mbox{or}\;1$. This definition of $f(t)$ can be written as

$\displaystyle
f(t)=\exp_{q^{\prime}}\left(\pm\sigma_{\epsilon}\;t\right)=\left\\{\begin{array}[]{ll}\exp_{q^{\prime}}(\pm\alpha
t)&\mbox{for}\;\epsilon=0\\\ \exp_{q^{\prime}}(\pm i\alpha
t)&\mbox{for}\;\epsilon=1\\\ \end{array}\right.$ (79)

The generalized exponential given in equation (79) can be written in terms of
hypergeometric functions

$f(t)=\exp_{q^{\prime}}\left(\pm\sigma_{\epsilon}\;t\right)={}_{1}F_{0}\left(-\frac{1}{1-q^{\prime}};\;;\mp(1-q^{\prime})\sigma_{\epsilon}\;t\right).$
(80)

Applying the $q$-Laplace transform on $f(t)$ we get:

$\displaystyle L_{q}[f(t)]$ $\displaystyle=$
$\displaystyle\displaystyle\int_{0}^{\infty}dt\;\exp_{q}(-st)\;{}_{1}F_{0}\left(-\frac{1}{1-q^{\prime}};\;;\mp(1-q^{\prime})\sigma_{\epsilon}\;t\right)$
$\displaystyle=$
$\displaystyle\frac{1}{s\;Q_{1}(2-q)}\;{}_{2}F_{1}\left(1,-\frac{1}{1-q^{\prime}};\frac{3-2q}{1-q};\mp\frac{(1-q^{\prime})\sigma_{\epsilon}}{(1-q)s}\right).$

To find the inverse $q$-Laplace transform, using the Widder’s method we
compute the $k^{th}$ derivative of $F_{q}(s)$ given in the previous
expression. The derivative so calculated is

$F_{q}^{(k)}(s,q^{\prime})=\displaystyle\sum_{n=0}^{\infty}\frac{\left(-\frac{1}{1-q^{\prime}}\right)_{n}\left(\mp(1-q^{\prime})\sigma_{\epsilon}\right)^{n}}{Q_{n+1}(2-q)\;\;n!}\frac{(-1)^{k}\Gamma(n+k+1)}{s^{n+k+1}}.$
(81)

Thus, to find the $q$-Widders formula we substitute Eq. (81) in Eq. (64) in
this case to get:

$f(t;q^{\prime})=Q_{1}(2-q)\left(\displaystyle\sum_{n=0}^{\infty}\frac{\left(-\frac{1}{1-q^{\prime}}\right)_{n}\left(\mp(1-q^{\prime})\sigma_{\epsilon}\;t\;\right)^{n}}{Q_{n+1}(2-q)\;\xi_{m}^{n}\;\;n!}\left[\displaystyle\lim_{k\rightarrow\infty}\frac{(n+k)(k)_{n}}{k^{2+n}}\right]\right).$
(82)

To get the inverse $q$-transform we apply the limits in the above expression
and simplify using the identities of Pochammer and this gives us the
generalized exponential function defined in Eq. (82). This result confirms the
use of Post-Widder’s method.

Gaussian function:
Regular Gaussian: The $q$-Laplace transform of a Gaussian function
$f(t)=\exp(-\alpha t^{2})$ is

$L_{q}\left[\exp\left(-\alpha
t^{2}\right)\right]=\displaystyle\int_{0}^{\infty}dt\;\exp_{q}\left(-st\right)\exp\left(-\alpha
t^{2}\right).$ (83)

and can be written in terms of the hypergeometric function as follows:

$F_{q}(s)=\frac{1}{s\;Q_{1}(2-q)}\;{}_{2}F_{2}\left(1,\frac{1}{2};\frac{3-2q}{2(1-q)},\frac{4-3q}{2(1-q)};-\frac{\alpha}{(1-q)^{2}s^{2}}\right).\qquad$
(84)

The $q$-inverse Laplace transform can be found using the Widder’s method and
for this purpose we evaluate the $k^{th}$-derivative of $F_{q}(s)$ which is

$F_{q}^{(k)}(s)=\frac{1}{Q_{1}(2-q)}\displaystyle\sum_{n=0}^{\infty}\frac{\left(\frac{1}{2}\right)_{n}\left(-\frac{\alpha}{(1-q)^{2}}\right)^{n}}{\left(\frac{3-2q}{2(1-q)}\right)_{n}\left(\frac{4-3q}{2(1-q)}\right)_{n}}\frac{(-1)^{k}(2n+k)!}{(2n)!}\frac{1}{s^{2n+k+1}}.\qquad$
(85)

Using Eq. (85) in Eq. (64) we can derive the Widder’s formula, for the
Gaussian function as

$f(t)=\displaystyle\left[\displaystyle\sum_{n=0}^{\infty}\frac{\left(\frac{1}{2}\right)_{n}\left(-\frac{\alpha}{(1-q)^{2}}\right)^{n}}{\left(\frac{3-2q}{2(1-q)}\right)_{n}\left(\frac{4-3q}{2(1-q)}\right)_{n}}\left(\lim_{k\rightarrow\infty}\frac{(-1)^{k}}{\Gamma(k+1)}\frac{(-1)^{k}(2n+k)!}{k^{2n}}\right)\frac{t^{2n}}{(2n)!\xi_{m}^{2n}}\right].$
(86)

Applying the limiting value for $k$ as

$\displaystyle\lim_{k\rightarrow\infty}\frac{(-1)^{k}}{\Gamma(k+1)}\frac{(-1)^{k}(2n+k)!}{k^{2n}}=\displaystyle\lim_{k\rightarrow\infty}\frac{\Gamma(2n+k+1)}{\Gamma(k+1)}k^{1-(2n+1)}\rightarrow
1.$

Using the identities of Pochammer symbol
$\left(\frac{3-2q}{2(1-q)}\right)_{n}\left(\frac{4-3q}{2(1-q)}\right)_{n}=\frac{1}{4^{n}}\frac{Q_{2n+1}(2-q)}{(1-q)^{2n}Q_{1}(2-q)}$,
and the identity of factorials $(2n)!=4^{n}\;n!\;\left(\frac{1}{2}\right)_{n}$
we can simplify Eq. (86). This yields the Gaussian function $f(t)=\exp(-\alpha
t^{2})$ confirming the usefulness of Post-Widder’s method.

Generalized Gaussian: The $q$-Laplace transform of a classical Gaussian
function $f(t)=\exp_{q^{\prime}}(-\alpha t^{2})$ is:

$L_{q}\left[\exp_{q^{\prime}}\left(-\alpha
t^{2}\right)\right]=\displaystyle\int_{0}^{\infty}dt\;\exp_{q}\left(-st\right)\exp_{q^{\prime}}\left(-\alpha
t^{2}\right),$ (87)

which on integration gives the hypergeometric function

$F_{q}(s;q^{\prime})=\frac{1}{sQ_{1}(2-q)}{}_{3}F_{2}\left(1,\frac{1}{2},-\frac{1}{1-q^{\prime}};\frac{3-2q}{2(1-q)},\frac{4-3q}{2(1-q)};\frac{(1-q^{\prime})\alpha}{(1-q)^{2}s^{2}}\right).$
(88)

The inverse of the $q$-Laplace transform can be computed using the Widder’s
method and to find it we take the $k^{th}$-derivative of the function
$F_{q}(s)$ as

$F_{q}^{(k)}(s;q^{\prime})=\displaystyle\sum_{n=0}^{\infty}\frac{\left(-\frac{1}{1-q^{\prime}}\right)_{n}}{Q_{2n+1}(2-q)}\frac{((1-q^{\prime})\alpha)^{n}}{n!}(-1)^{k}\frac{\Gamma(k+2n+1)}{s^{2n+k+1}}.$
(89)

Substituting Eq. (89) in the $q$-Widder’s formula in Eq. (64) we get:

$f(t,q^{\prime})=\displaystyle\sum_{n=0}^{\infty}\frac{Q_{1}(2-q)\left(-\frac{1}{1-q^{\prime}}\right)_{n}}{Q_{2n+1}(2-q)}\frac{((1-q^{\prime})\alpha
t^{2})^{n}}{n!\;\;\xi^{2n}}\frac{1}{s^{2n}}\left[\lim_{k\rightarrow\infty}\frac{\Gamma(k+2n+1)}{k^{2n}\Gamma(k+1)}\right].$
(90)

On applying the appropriate limits and using the Pochammer identities, we will
recover the generalized Gaussian function indicating that the Post-Widder’s
method works.

Circular functions:
The circular functions (both cosine and sine) can be expressed in terms of
hypergeometric function and written in a compact form as follows:

$f(t)=(\alpha
t)^{\delta}{}_{0}F_{1}\left(\;;\frac{1}{2}+\delta;-\frac{1}{4}\alpha^{2}t^{2}\right)=\left\\{\begin{array}[]{ll}\cos\alpha
t&\mbox{for}\;\delta=0\\\ \sin\alpha t&\mbox{for}\;\delta=1\\\
\end{array}\right.$ (91)

The $q$-Laplace transform of Eq. (91) is

$\displaystyle F_{q}(s)$ $\displaystyle=$
$\displaystyle\frac{1}{s}\left(\frac{\alpha}{s}\right)^{\delta}\frac{1}{Q_{\delta+1}(2-q)}$
(92)
$\displaystyle\times{}_{1}F_{2}\left(1;\frac{1}{2}\left(\frac{3-2q}{1-q}+\delta\right),\frac{1}{2}\left(\frac{4-3q}{1-q}+\delta\right);-\frac{\alpha^{2}}{4(1-q)^{2}s^{2}}\right)$
$\displaystyle=$ $\displaystyle\left\\{\begin{array}[]{ll}\cos\alpha
t&\mbox{for}\;\delta=0\\\ \sin\alpha t&\mbox{for}\;\delta=1\\\
\end{array}\right.$ (95)

For calculating the inverse $q$-Laplace transform, we find the $k^{th}$
derivative of Eq. (92) as follows:

$\displaystyle
F_{q}^{(k)}(s)=\frac{1}{Q_{\delta+1}(2-q)}\displaystyle\sum_{n=0}^{\infty}\left(\frac{(-1)^{n+k}\Gamma(2n+\delta+k+1)\alpha^{2n+\delta}}{(1-q)^{2n}\left(\frac{3-2q}{1-q}+\delta\right)_{2n}(2n+\delta)!}\right)\frac{1}{s^{2n+\delta+k+1}}.$
(96)

(i) $\cos\alpha t$: For $\delta=0$, the $k^{th}$ derivative of
$F_{q}^{(k)}(s)$ can be written from Eq. (96) as

$\displaystyle
F_{q}^{(k)}(s)=\frac{1}{Q_{1}(2-q)}\displaystyle\sum_{n=0}^{\infty}\left(\frac{(-1)^{n+k}\Gamma(2n+k+1)\alpha^{2n}}{(1-q)^{2n}\left(\frac{3-2q}{1-q}\right)_{2n}(2n)!}\right)\frac{1}{s^{2n+k+1}}.\qquad$
(97)

To calculate the inverse Laplace transform we use the $q$-generalization of
the Widder’s method and for this we substitute Eq. (97) in Eq. (64) to obtain

$f(t)=\displaystyle\sum_{n=0}^{\infty}\left(\frac{(-1)^{n}(\alpha
t)^{2n}}{(1-q)^{2n}\left(\frac{3-2q}{1-q}\right)_{2n}(2n)!\xi^{2n}}\right)\left[\displaystyle\lim_{k\rightarrow\infty}\frac{k^{1-(2n+1)}\Gamma(2n+k+1)}{\Gamma(k+1)}\right].$
(98)

Under the limiting conditions of $k$ and using the proper Pochammer identities
we get

$f(t)=\displaystyle\sum_{n=0}^{\infty}\frac{\left(-\;\alpha^{2}t^{2}\right)^{n}}{(2n)!}=\cos\alpha
t.$ (99)

(ii) $\sin\alpha t$: Considering $\delta=1$, the $k^{th}$ derivative of
$F_{q}^{(k)}(s)$ can be written as

$F_{q}^{(k)}(s)=\frac{1}{Q_{2}(2-q)}\displaystyle\sum_{n=0}^{\infty}\left(\frac{(-1)^{n+k}\Gamma(2n+k+2)\alpha^{2n+1}}{(1-q)^{2n}\left(\frac{4-3q}{1-q}\right)_{2n}(2n+1)!}\right)\frac{1}{s^{2n+k+2}}.$
(100)

Substituting Eq. (100) in $q$-Widder’s formula in Eq. (64) we get:

$f(t)=\displaystyle\sum_{n=0}^{\infty}\left(\frac{(-1)^{n}(\alpha
t)^{2n+1}}{(1-q)^{2n}\left(\frac{4-3q}{1-q}\right)_{2n}(2n+1)!\xi^{2n+1}}\right)\left[\displaystyle\lim_{k\rightarrow\infty}\frac{k^{1-(2n+2)}\Gamma(2n+k+2)}{\Gamma(k+1)}\right].$
(101)

On application of the limiting value of $k$ and using the Pochammer identities
we get:

$f(t)=\displaystyle\sum_{n=0}^{\infty}(-1)^{n}\frac{\left(\alpha
t\right)^{2n+1}}{(2n+1)!}=\sin\alpha t.$ (102)

Thus we prove that the inverse of the $q$-Laplace transform works.

Generalized circular functions:
Using the generalized exponential a set of circular functions have been
defined in Ref. [24]. In terms of the hypergeometric functions these
generalized circular functions can be written in a compact form as

$\displaystyle f(t,q^{\prime})$ $\displaystyle=$ $\displaystyle(\alpha
t)^{\delta}{}_{0}F_{1}\left(\frac{1}{2}\left(-\frac{1}{1-q^{\prime}}+\delta\right),\frac{1}{2}\left(-\frac{q^{\prime}}{1-q^{\prime}}+\delta\right);\frac{1}{2}+\delta;-(1-q^{\prime})^{2}\alpha^{2}t^{2}\right),$
(105) $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\cos_{q^{\prime}}\alpha
t&\mbox{for}\;\delta=0\\\ \sin_{q^{\prime}}\alpha t&\mbox{for}\;\delta=1\\\
\end{array}\right.$

The $q$-Laplace transform of the generalized circular function is

$\displaystyle F_{q}(s;q^{\prime})$ $\displaystyle=$
$\displaystyle\frac{1}{s}\left(\frac{\alpha}{s}\right)^{\delta}\frac{1}{Q_{\delta+1}(2-q)}\times$
(108)
$\displaystyle{}_{3}F_{2}\left(1,\frac{-a+\delta}{2},\frac{-a+1+\delta}{2};\frac{b+\delta}{2},\frac{b+1+\delta}{2};-\frac{(1-q^{\prime})^{2}\alpha^{2}}{(1-q)^{2}s^{2}}\right),$
$\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}L_{q}\left[\cos_{q^{\prime}}\left(\alpha
t\right)\right]&\mbox{for}\;\delta=0\\\
L_{q}\left[\sin_{q^{\prime}}\left(\alpha
t\right)\right]&\mbox{for}\;\delta=1\\\ \end{array}\right.$

where we have used the relations:

$a=\frac{1}{1-q^{\prime}};\qquad b=\frac{1}{1-q}+2.$ (109)

The $k^{th}$-derivative of equation (108) is calculated to find the inverse
$q$-Laplace transform using the Widder’s method. The $k^{th}$ derivative is

$\displaystyle F_{q}^{(k)}(s;q^{\prime})$ $\displaystyle=$
$\displaystyle\frac{1}{Q_{\delta+1}(2-q)}\displaystyle,$
$\displaystyle\sum_{n=0}^{\infty}\left(\frac{(-1)^{n+k}\Gamma(2n+\delta+k+1)\left(-\frac{1}{1-q^{\prime}}+\delta\right)_{2n}\frac{(1-q^{\prime})^{2n}}{(1-q)^{2n}}\alpha^{2n+\delta}}{Q_{\delta+1}(2-q)\left(\frac{3-2q}{1-q}+\delta\right)_{2n}(2n+\delta)!s^{2n+\delta+k+1}}\right).$

(i) $\cos_{q^{\prime}}\alpha t$: For $\delta=0$, the $k^{th}$-derivative of
$F_{q}^{k}(s;q^{\prime})$ can be written as

$\displaystyle
F_{q}^{(k)}(s;q^{\prime})=\frac{(-1)^{k}}{Q_{1}(2-q)}\displaystyle\sum_{n=0}^{\infty}\frac{(-1)^{n}\left(-\frac{1}{1-q^{\prime}}\right)_{2n}\Gamma(2n+k+1)\frac{(1-q^{\prime})^{2n}}{(1-q)^{2n}}\alpha^{2n}}{\left(\frac{3-2q}{1-q}\right)_{2n}(2n)!\;s^{2n+k+1}}.$
(111)

Substituting in the formula for the Post-Widder’s method in Eq. (64) we get:

$f(t;q^{\prime})=\displaystyle\sum_{n=0}^{\infty}\left(\frac{\left(-\frac{1}{1-q^{\prime}}\right)_{2n}\left(\frac{(1-q^{\prime})\,\alpha
t}{1-q}\right)^{2n}}{(-1)^{n}\left(\frac{3-2q}{1-q}\right)_{2n}\xi^{2n}\;(2n)!}\right)\left[\displaystyle\lim_{k\rightarrow\infty}\frac{k^{1-(2n+1)}\Gamma(2n+k+1)}{\Gamma(k+1)}\right].$
(112)

Applying the appropriate limits and also using Pochammer identities we get

$\displaystyle
f(t;q^{\prime})=\displaystyle\sum_{n=0}^{\infty}(-1)^{n}\frac{\left(-\frac{1}{1-q^{\prime}}\right)_{2n}\left((1-q^{\prime})\;\alpha
t\right)^{2n}}{(2n)!}=\cos_{q^{\prime}}\left(\alpha t\right).$ (113)

(ii) $\sin_{q^{\prime}}\alpha t$: The choice of $\delta=1$ gives the
generalized sine function. The $k^{th}$-derivative of
$F_{q}^{(k)}(s;q^{\prime})$ of this function is

$F_{q}^{(k)}(s;q^{\prime})=\frac{(-1)^{k}}{Q_{2}(2-q)}\displaystyle\sum_{n=0}^{\infty}\frac{(-1)^{n}\left(-\frac{q^{\prime}}{1-q^{\prime}}\right)_{2n}\Gamma(2n+k+2)\frac{(1-q^{\prime})^{2n}}{(1-q)^{2n}}\alpha^{2n+1}}{\left(\frac{4-3q}{1-q}\right)_{2n}(2n+1)!\;s^{2n+k+2}}.$
(114)

Under the appropriate limits and using the proper Pochammer identities we get

$f(t,q^{\prime})=\displaystyle\sum_{n=0}^{\infty}(-1)^{n+1}\left(-\frac{1}{1-q^{\prime}}\right)_{2n+1}\frac{\left((1-q^{\prime})\;\alpha
t\right)^{2n+1}}{(2n+1)!}=\sin_{q^{\prime}}\left(\alpha t\right).$ (115)

Thus the $q$-Laplace transform and the inverse $q$-Laplace transform computed
for the $\sin_{q^{\prime}}\alpha t$ give consistent results proving that the
Post-Widder’s method works.

Hyperbolic functions:
The hyperbolic functions $\cosh\alpha t$ and $\sinh\alpha t$ can be expressed
in terms of the hypergeometric function and can be written in compact form as:

$f(t)=(\alpha
t)^{\delta}{}_{0}F_{1}\left(\;;\frac{1}{2}+\delta;\frac{1}{4}\alpha^{2}t^{2}\right)=\left\\{\begin{array}[]{ll}\cosh\alpha
t&\mbox{for}\;\delta=0\\\ \sinh\alpha t&\mbox{for}\;\delta=1\\\
\end{array}\right.$ (116)

The $q$-Laplace transform of Eq. (116) is:

$\displaystyle F_{q}(s)$ $\displaystyle=$
$\displaystyle\frac{1}{s}\left(\frac{\alpha}{s}\right)^{\delta}\frac{1}{Q_{\delta+1}(2-q)}{}_{1}F_{2}\left(1;\frac{b+\delta}{2},\frac{b+1+\delta}{2}\frac{\alpha^{2}}{4(1-q)^{2}s^{2}}\right)$
(119) $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}L_{q}[\cosh\alpha
t]&\mbox{for}\;\delta=0\\\ L_{q}[\sinh\alpha t]&\mbox{for}\;\delta=1\\\
\end{array}\right.$

where the factor $b$ used in Eq. (119) is defined in Eq. (109). The inverse
$q$-Laplace transform can be calculated using the Post-Widder’s method. For
this we compute the $k^{th}$-derivative of Eq. (119):

$\displaystyle
F_{q}^{(k)}(s)=\frac{1}{Q_{\delta+1}(2-q)}\displaystyle\sum_{n=0}^{\infty}\left(\frac{(-1)^{k}\Gamma(2n+\delta+k+1)\alpha^{2n+\delta}}{(1-q)^{2n}\left(\frac{3-2q}{1-q}+\delta\right)_{2n}(2n+\delta)!}\right)\frac{1}{s^{2n+\delta+k+1}}.$
(120)

(i) $\cosh\alpha t$: For the $\cosh\alpha t$ ($\delta=0$) function, the
$k^{th}$-derivative is

$\displaystyle
F_{q}^{(k)}(s)=\frac{1}{Q_{1}(2-q)}\displaystyle\sum_{n=0}^{\infty}\left(\frac{(-1)^{k}\Gamma(2n+k+1)\alpha^{2n}}{(1-q)^{2n}\left(\frac{3-2q}{1-q}\right)_{2n}(2n)!}\right)\frac{1}{s^{2n+k+1}}.$
(121)

Substituting in the $q$-generalization of Widder’s formula, and applying the
appropriate limits and Pochammer identities we get

$f(t)=\displaystyle\sum_{n=0}^{\infty}\frac{\left(\;\alpha^{2}t^{2}\right)^{n}}{(2n)!}=\cosh\alpha
t.$ (122)

which confirms the $q$-Laplace transform as well as the generalized Widder’s
method defined in our work.

(ii) $\sinh\alpha t$: For the choice $\delta=1$ the $k^{th}$-derivative of
$F_{q}^{(k)}(s)$ is

$\displaystyle
F_{q}^{(k)}(s)=\frac{1}{Q_{2}(2-q)}\displaystyle\sum_{n=0}^{\infty}\left(\frac{(-1)^{k}\Gamma(2n+k+2)\alpha^{2n+1}}{(1-q)^{2n}\left(\frac{4-3q}{1-q}\right)_{2n}(2n+1)!}\right)\frac{1}{s^{2n+k+2}}.$
(123)

Applying the appropriate limits and using the Pochammer under the limit
$k\rightarrow\infty$ in the Equation above gives us

$\displaystyle f(t)=\displaystyle\sum_{n=0}^{\infty}\frac{\left(\alpha
t\right)^{2n+1}}{(2n+1)!}=\sinh\alpha t.$ (124)

These results confirms that the $q$-Laplace transform as well as the inverse
transform work as expected.

Generalized hyperbolic functions:
The generalized hyperbolic functions viz the $\cosh_{q^{\prime}}(\alpha t)$
and $\sinh_{q^{\prime}}(\alpha t)$ can be expressed in terms of the
hypergeometric function and can be written in a compact form as

$\displaystyle f(t,q^{\prime})$ $\displaystyle=$ $\displaystyle(\alpha
t)^{\delta}{}_{2}F_{1}\left(\frac{1}{2}\left(-\frac{1}{1-q^{\prime}}+\delta\right),\frac{1}{2}\left(-\frac{q^{\prime}}{1-q^{\prime}}+\delta\right);\frac{1}{2}+\delta;(1-q^{\prime})^{2}\alpha^{2}t^{2}\right)$
(127) $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\cosh_{q^{\prime}}\alpha
t&\mbox{for}\;\delta=0\\\ \sinh_{q^{\prime}}\alpha t&\mbox{for}\;\delta=1\\\
\end{array}\right.$

The $q$-Laplace transform of the $q$-hyperbolic function is

$\displaystyle F_{q}(s;q^{\prime})$ $\displaystyle=$
$\displaystyle\frac{1}{s}\left(\frac{\alpha}{s}\right)^{\delta}\frac{1}{Q_{\delta+1}(2-q)}\times$
(130)
$\displaystyle{}_{3}F_{2}\left(1,\frac{-a+\delta}{2},\frac{-(a-1)+\delta}{2};\frac{b+\delta}{2},\frac{b+1+\delta}{2};\frac{(1-q^{\prime})^{2}\alpha^{2}}{(1-q)^{2}s^{2}}\right)$
$\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}L_{q}\left[\cosh_{q^{\prime}}\left(\alpha
t\right)\right]&\mbox{for}\;\delta=0\\\
L_{q}\left[\sinh_{q^{\prime}}\left(\alpha
t\right)\right]&\mbox{for}\;\delta=1\\\ \end{array}\right.$

where the factors used in the hypergeometric function are defined in Eq.
(109). The $k^{th}$-derivative of $F_{q}^{(k)}(s)$ is

$\displaystyle
F_{q}^{(k)}(s;q^{\prime})=\displaystyle\sum_{n=0}^{\infty}\left(\frac{(-1)^{k}\Gamma(2n+\delta+k+1)\left(-\frac{1}{1-q^{\prime}}+\delta\right)_{2n}\frac{(1-q^{\prime})^{2n}}{(1-q)^{2n}}\alpha^{2n+\delta}}{Q_{\delta+1}(2-q)\left(\frac{3-2q}{1-q}+\delta\right)_{2n}(2n+\delta)!s^{2n+\delta+k+1}}\right).$
(131)

For the case (i) $\cosh_{q^{\prime}}(\alpha t)$ we can choose $\delta=0$ we
can calculate the $k^{th}$ derivative of $F_{q}(s,q^{\prime})$ by allowing
$\delta=1$ and substituting this in the $q$-Widder’s formula and choosing the
appropriate limits and using the Pochammer identities we get:

$f(t,q^{\prime})=\displaystyle\sum_{n=0}^{\infty}\frac{\left(-\frac{1}{1-q^{\prime}}\right)_{2n}\left((1-q^{\prime})\;\alpha
t\right)^{2n}}{(2n)!}=\cosh_{q^{\prime}}\left(\alpha t\right).$ (132)

By choosing $\delta=1$ we can get the results corresponding to the case (ii)
$\sinh_{q^{\prime}}(\alpha t)$ by repeating the procedure as being done in all
the earlier cases. The final result reads:

$f(t,q^{\prime})=\displaystyle\sum_{n=0}^{\infty}\left(-\frac{q^{\prime}}{1-q^{\prime}}\right)_{2n}(1-q^{\prime})^{2n}\frac{(\alpha
t)^{2n+1}}{(2n+1)!}=\sinh_{q^{\prime}}\left(\alpha t\right).$ (133)

Thus we compute the generalized Laplace transform and its inverse using the
Post-Widders method and we observe consistent results.

##  7 Applications in statistical mechanics

The Laplace transform has several applications in physics. Notable among these
applications is the use of Laplace transform to interrelate the partition
function and the density of states. The partition function is the fundamental
quantity using which all the thermodynamic quantities are computed in the
canonical ensemble which is used to study a closed system. The thermodynamic
quantities are calculated using the density of states in the microcanonical
ensemble which is used to describe an isolated system. In both the canonical
and the microcanonical ensemble, the number of particles remain fixed. The
density of states is the number of states of a given energy in a system and
the partition function is the average of the density of states weighted by a
Boltzmann distribution. Hence these two quantities viz partition function and
density of states can be connected by a Laplace transform [25].

To investigate the statistical mechanics of systems involving long-range
interactions a generalization of Boltzmann-Gibbs statistics was proposed by
Tsallis [7] using the deformed $q$-exponential. When the average of the
density of states is weighted by the Tsallis statistics i.e., using a deformed
$q$-exponential, we can connect the partition function to the density matrix
using a $q$-deformed Laplace transform. While both the partition function and
the density of states can be calculated using first principles, the density of
states which completely characterizes a isolated system is generally hard to
compute. To overcome this one can calculate the partition function in the
canonical ensemble and use the inverse Laplace transform to obtain the density
of states. In our case we use the partition function of the Tsallis statistics
and use the inverse of the $q$-Laplace transform and obtain the density of
states.

Classical Ideal gas:
As an illustration let us consider a system of classical ideal gas in $D$
dimensions with the Hamiltonian [25, 26, 27, 28]

$H=\displaystyle\sum_{i=1}^{DN}\frac{p_{i}^{2}}{2m},$ (134)

where $p$ and $m$ are the momentum and mass of the gas. The nonextensive
statistical mechanics begins with the introduction of a generalization of the
entropy. The probability distribution for the microstates can be obtained by
optimizing the entropy using physically appropriate constraints. In the course
of the development of this field, three different ways of optimizations
emerged. Of these three methods, the third constraint method [29, 30] known as
the escort probability distribution is the one presently accepted. Using this
constraint leads to implicit thermodynamics relations which are hard to solve.
To overcome this one can solve the problem in the second constraint approach
and the results obtained can be transformed to those in the third constraint
using a transformation rule based on the inverse temperature. Since we are
interested in demonstrating the usefulness of Laplace transform, we prefer to
avoid the complications arising with the third constraint and instead use the
second constraint formulation. But we would like to note that any result we
obtain in the second constraint can be transformed to third constraint. The
partition function of the system in the second constraint formulation is

$Z_{q}(\beta)=\frac{1}{h^{DN}N!}\displaystyle\int_{0}^{\infty}\;d\bar{x}\;d\bar{p}\;\exp_{q}\left(-\beta\displaystyle\sum_{i=1}^{DN}\frac{p_{i}^{2}}{2m}\right),$
(135)

where $\bar{x}$ and $\bar{p}$ are $\left(x_{1},x_{2},\ldots,x_{DN}\right)$,
$\left(p_{1},p_{2},\ldots,p_{DN}\right)$. On calculating the integral we get:

$Z_{q}(\beta)=\frac{V^{N}(2\pi
m)^{\frac{DN}{2}}}{h^{DN}N!}\left(\frac{\Gamma\left(\frac{1}{1-q}+1\right)}{(1-q)^{\frac{DN}{2}}\;\Gamma\left(\frac{1}{1-q}+\frac{DN}{2}+1\right)}\right)\frac{1}{\beta^{\frac{DN}{2}}}.$
(136)

Using the inverse of the $q$-Laplace transform one can calculate the density
of states. Here we use the $q$-generalization of the Widder’s method to get:

$g(E)=\displaystyle\lim_{k\rightarrow\infty}\frac{(-1)^{k}}{\Gamma(k+1)}Z_{q}^{(k)}(\beta)\beta^{k+1}Q_{1}(2-q)\Biggr{|}_{\beta=\frac{k\xi_{m}}{E}}.$
(137)

Substituting for the partition function Eq. (136) in the expression for the
density of states Eq. (137), we apply the limiting value for $\beta$ and
simplify the expression to get:

$g(E)=\frac{V^{N}(2\pi
m)^{\frac{DN}{2}}}{h^{DN}N!\left(\frac{DN}{2}-1\right)!}E^{\frac{DN}{2}-1},$
(138)

which is the density of states of the classical ideal gas [25].

A system is in thermodynamic equilibrium when it is in thermal, mechanical and
chemical equilibrium with its surroundings and consequently we need three
macroscopic variables to describe the system. The macroscopic variable can be
either an extensive variable dependent on the number of particle or an
intensive variable which is independent of the particle number. The energy
($E$) of a system is an example for an extensive variable and the temperature
($T$) is an example of an intensive variable. Thus from the thermal,
mechanical and chemical equilibrium we can choose from any of the two
variables and hence we have in total eight different ensembles. Of the eight
ensembles there are four ensembles in which the temperature is fixed and these
are referred to as the isothermal ensemble and four in which the heat function
is fixed and these ensembles are known as the adiabatic ensembles. The four
isothermal ensembles are characterized by the partition function, while the
four adiabatic ensembles are characterized by the density of states. The
density of states is the number of states with same value for the heat
function in a system and the partition function is the average of the density
of states weighted by a distribution and in general cases we use a Boltzmann
distribution. The eight ensembles can be grouped into four pairs with each
pair comprising of an isothermal ensemble and an adiabatic ensemble. The
partition of an isothermal ensemble can be related to the density of states of
the adiabatic ensemble through a Laplace transform. In the discussion above we
saw the correspondence between the microcanonical and the canonical ensemble.
Similarly, the constant pressure ensemble and the isoenthalpic-isobaric
ensemble are also related through a Laplace transform. A theory of the
thermodynamic formulation of all the eight ensembles are given in Ref. [31]
for the generalized statistical mechanics based on Tsallis $q$-exponential.

Linear harmonic oscillator:
Next let us consider a system of $N$ number of $D$-dimensional non-interacting
harmonic oscillators. The Hamiltonian of the oscillator is [25]

$H=\displaystyle\sum_{i=1}^{DN}\left(\frac{p_{i}^{2}}{2m}+\frac{1}{2}m\omega^{2}x_{i}^{2}\right),$
(139)

where $x$ and $p$ are the position and momentum co-ordinates of the
oscillator. There are three different approaches to calculate the probability
distribution in the current work we use the second constraint approach. The
partition function of the system in the second constraint formulation is

$Z_{q}(\beta)=\frac{1}{h^{DN}}\displaystyle\int_{0}^{\infty}\;d\bar{x}\;d\bar{p}\;\exp_{q}\left(-\beta\displaystyle\sum_{i=1}^{DN}\left(\frac{p_{i}^{2}}{2m}+\frac{1}{2}m\omega^{2}x_{i}^{2}\right)\right),$
(140)

where $\bar{x}$ and $\bar{p}$ are
$\left(x_{1},x_{2},\ldots,x_{DN}\right)$,$\left(p_{1},p_{2},\ldots,p_{DN}\right)$.
Evaluating the integral we get:

$Z_{q}(\beta)=\frac{1}{\left(\hbar\omega\right)^{DN}}\left(\frac{\Gamma\left(\frac{1}{1-q}+1\right)}{(1-q)^{DN}\;\Gamma\left(\frac{1}{1-q}+DN+1\right)}\right)\frac{1}{\beta^{DN}}.$
(141)

To calculate the thermodynamics of a set of isolated harmonic oscillator we
use the microcanonical ensemble and so we need to find the density of states.
The density of states can be computed from the partition function using the
inverse of the $q$-Laplace transform. Instead of using the regular complex
analysis based approach to calculate the inverse transform we use the
$q$-generalization of the Widder’s method introduced in the present work.
Using this method the density of states obtained is

$\displaystyle
g(E)=\displaystyle\lim_{k\rightarrow\infty}\frac{(-1)^{k}}{\Gamma(k+1)}Z_{q}^{(k)}(\beta)\beta^{k+1}Q_{1}(2-q)\Biggr{|}_{\beta=\frac{k\xi}{E}}.$
(142)

Substituting the partition function Eq. (141) in the expression for the
density of states Eq. (142) and substituting the value of $\beta$ and applying
the limiting value of $k$ we get

$g(E)=\frac{1}{\left(\hbar\omega\right)^{DN}\left(DN-1\right)!}E^{DN-1},$
(143)

which is the expression of the density of states of a system of $N$ non-
interacting $D$-dimensional oscillators in the microcanonical ensemble [25].
Thus we use the generalized Post-Widder’s method was based inverse Laplace
transform to transform the partition function of a non-relativistic classical
ideal gas and linear harmonic oscillator to their respective density of
states.

##  8 Conclusion

A generalization of the Laplace transform based on the Tsallis $q$-exponential
is defined in the present work based on the kernel
$K_{I}(q;s,t)=\exp_{q}(-st)$. While this definition was also already
introduced in Ref. [19], an inverse transformation was not defined. In our
work we define the inverse transform using the complex integration method. We
also provide an alternative method based on Post-Widder’s method to compute
the inverse of the transform using only real variables. Several properties of
the Laplace transform and the inverse transform are verified. Then we compute
the Laplace transform of some functions like algebraic functions, Gaussian
functions, $q$-Gaussian functions, circular functions, $q$-circular function,
hyperbolic function, and $q$-hyperbolic functions. On the resulting
expressions we use the inverse Laplace transform based on the Post-Widders
method to derive the original functions. Finally we show the applications of
these methods in statistical mechanics where we compute the density of states
of a $N$-Dimensional classical ideal gas and linear harmonic oscillator from
their partition function.

## Acknowledgements

Md. Manirul Ali was supported by the Centre for Quantum Science and
Technology, Chennai Institute of Technology, India, vide funding number
CIT/CQST/2021/RD-007. R. Chandrashekar was supported in part by a seed grant
from IIT Madras to the Centre for Quantum Information, Communication and
Computing.

##  9 References:

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