# Weighted Fractional Generalized Cumulative Past Entropy

Suchandan Kayal
Department of Mathematics, National Institute of Technology Rourkela,
Rourkela-769008, India<EMAIL_ADDRESS><EMAIL_ADDRESS>
Abstract

In this paper, we introduce weighted fractional generalized cumulative past
entropy of a nonnegative absolutely continuous random variable with bounded
support. Various properties of the proposed weighted fractional measure are
studied. Bounds and stochastic orderings are derived. A connection between the
proposed measure and the left-sided Riemann-Liouville fractional integral is
established. Further, the proposed measure is studied for the proportional
reversed hazard rate models. Next, a nonparametric estimator of the weighted
fractional generalized cumulative past entropy is suggested based on the
empirical distribution function. Various examples with a real life data set
are considered for the illustration purposes. Finally, large sample properties
of the proposed empirical estimator are studied.

Keywords: Weighted generalized cumulative past entropy; Fractional calculus;
Stochastic ordering; Reversed hazard rate model; Empirical cumulative
distribution function; Central limit theorem.

2020 Mathematics Subject Classifications: $94A17$, $60E15$, $26A33$

##  1 Introduction

Entropy plays an important role in several areas of statistical mechanics and
information theory. In statistical mechanics, the most widely applied form of
entropy was proposed by Boltzmann and Gibbs, and in information theory, that
was introduced by Shannon. Due to the growing applicability of the entropy
measures, various generalizations were proposed and their information
theoretic properties were studied. See, for instance, Rényi (1961) and Tsallis
(1988). We recall that most of the generalized entropies were developed based
on the concept of deformed logarithm. But, two generalized concept of
entropies: fractal (see Wang (2003)) and fractional (Ubriaco (2009)) entropies
were proposed based on the natural logarithm. Let $P=(p_{1},\ldots,p_{n})$ be
the probability mass function of a discrete random variable $X.$ The
Boltzmann-Gibbs-Shannon entropy of $X$ can be defined through an equation
involving the ordinary derivative as

$\displaystyle H_{BGS}(X)=-\frac{d}{du}\sum_{i=1}^{n}p_{i}^{u}\Big{|}_{u=1}.$
(1.1)

Ubriaco (2009) proposed a new entropy measure known as the fractional entropy
after replacing the ordinary derivative by the Weyl fractional derivative (see
Ferrari (2018)) in (1.1). It is given by

$\displaystyle H_{\alpha}(X)=\sum_{i=1}^{n}p_{i}(-\ln
p_{i})^{\alpha},~{}~{}0\leq\alpha\leq 1.$ (1.2)

The fractional entropy in (1.2) is positive, concave and non-additive.
Further, one can recover the Shannon entropy (see Shannon (1948)) from (1.2)
under $\alpha=1.$ From (1.2), we notice that the measure of information is
mainly a function of probabilities of occurrence of various events. However,
we often face with many situations (see Guiaşu (1971)) in different fields,
where the probabilities and qualitative characteristics of events need to be
taken into account for better uncertainty analysis. As a result, the concept
of weighted entropy was introduced by Guiaşu (1971), which is given by

$\displaystyle H^{w}(X)=-\sum_{i=1}^{n}w_{i}p_{i}\ln p_{i},$ (1.3)

where $w_{i}$ is a nonnegative number (known as weight) directly proportional
to the importance of the $i$th elementary event. Note that the weights
$w_{i}$’s can be equal. Following the same line as in (1.3), the weighted
fractional entropy can be defined as

$\displaystyle H_{\alpha}^{w}(X)=\sum_{i=1}^{n}w_{i}p_{i}(-\ln
p_{i})^{\alpha},~{}~{}0\leq\alpha\leq 1.$ (1.4)

Note that for $w_{i}=1,~{}i=1,\ldots,n$, (1.4) reduces to the fractional
entropy given by (1.2). Further, (1.4) equals to the weighted entropy given by
(1.3) when $\alpha=1.$

Recently, motivated by the aspects of the cumulative residual entropy due to
Rao et al. (2004) and the fractional entropy given by (1.2), Xiong et al.
(2019) introduced a new information measure, known as the fractional
cumulative residual entropy. The concept of multiscale fractional cumulative
residual entropy was described by Dong and Zhang (2020). Very recently,
inspired by the cumulative entropy (see Di Crescenzo and Longobardi (2009))
and (1.2), Di Crescenzo et al. (2021) proposed fractional generalized
cumulative entropy of a random variable $X$ with bounded support $(0,s)$,
which is given by

$\displaystyle CPE_{\gamma}(X)=\frac{1}{\Gamma(\gamma+1)}\int_{0}^{s}K(x)[-\ln
K(x)]^{\gamma}dx,~{}~{}\gamma>0,$ (1.5)

where $K$ is the cumulative distribution function (CDF) of $X.$ The fractional
generalized cumulative entropy is a generalization of the cumulative entropy
and generalized cumulative entropy proposed by Di Crescenzo and Longobardi
(2009) and Kayal (2016), respectively. We remark that the cumulative entropy
and generalized cumulative entropy are independent of the location. This
property appears as a drawback when quantifying information of an electronics
device or a neuron in different intervals having equal widths. Thus, to cope
with these situations, various authors proposed length-biased (weighted)
information measures. The weighted measures are also called shift-dependent
measures by some researchers. Readers may refer to Di Crescenzo and Longobardi
(2007), Misagh et al. (2011), Misagh (2016), Das (2017), Kayal and Moharana
(2017a), Kayal and Moharana (2017b), Mirali et al. (2017), Nourbakhsh and Yari
(2017), Kayal (2018) and Kayal and Moharana (2019) for some weighted versions
of various information measures. The existing weighted information measures
and the fractional generalized cumulative entropy in (1.5) inspire us to
consider the weighted fractional generalized cumulative past entropy (WFGCPE),
which has been studied in the subsequent sections of this paper. The following
definitions will be useful in order to obtain some ordering results for the
WFGCPE.

######  Definition 1.1.

Let $X_{1}$ and $X_{2}$ be two nonnegative absolutely continuous random
variables with probability density functions (PDFs) $k_{1},~{}k_{2}$ and CDFs
$K_{1},~{}K_{2}$, respectively. Then, $X_{1}$ is said to be smaller than
$X_{2}$ in the sense of the

  * (i)

usual stochastic order, denoted by $X_{1}\leq_{st}X_{2}$, if $K_{2}(x)\leq
K_{1}(x)$, for all $x\in\mathbb{R}$;

  * (ii)

hazard rate order, denoted by $X_{1}\leq_{hr}X_{2}$, if
$\bar{K}_{2}(x)/\bar{K}_{1}(x)$ is nondecreasing in $x>0,$ where
$\bar{K}_{1}=1-K_{1}$ and $\bar{K}_{2}=1-K_{2};$

  * (ii)

dispersive order, denoted by $X_{1}\leq_{disp}X_{2}$, if
$K_{1}^{-1}(u)-K_{1}^{-1}(v)\geq K_{2}^{-1}(u)-K_{2}^{-1}(v)$, for all
$0<u<v<1$, where $K_{1}^{-1}$ and $K_{2}^{-1}$ are the right continuous
inverses of $K_{1}$ and $K_{2}$, respectively;

  * (iii)

decreasing convex order, denoted by $X_{1}\leq_{dcx}X_{2},$ if and only if
$E(\tau(X_{1}))\leq E(\tau(X_{2}))$ holds for all nonincreasing convex real
valued functions for which the expectations are defined.

The rest of the paper is organized as follows. In Section $2$, we introduce
WFGCPE and study its various properties. Some ordering results are obtained.
It is shown that a less dispersed distribution produces smaller uncertainty in
terms of the WFGCPE. Some bounds are obtained. Further, a connection of the
proposed measure with the fractional calculus is discovered. The proportional
reversed hazard model is considered and the WFGCPE is studied under this set
up. Section $3$ deals with the estimation of the introduced measure. An
empirical WFGCPE estimator is proposed based on the empirical distribution
function. Further, large sample properties of the proposed estimator have been
studied. Finally, Section $4$ concludes the paper with some discussions.

Throughout the paper, the random variables are considered as nonnegative
random variables. The terms increasing and decreasing are used in wide sense.
The differentiation and integration exist whenever they are used. The notation
$\mathbb{N}$ denotes the set of natural numbers. Further, throughout the
paper, a standard argument $0=0.\ln 0=0.\ln\infty$ is adopted. The prime
$\prime$ denotes the first order ordinary derivatve.

##  2 Weighted fractional generalized cumulative past entropy

In this section, we propose WFGCPE and study its various properties. Consider
a nonnegative absolutely continuous random variable $X$ with support $(0,s)$
and CDF $K$ and PDF $k$. Then, the WFGCPE of $X$ with a general nonnegative
weight function $\psi(x)~{}(\geq 0)$ is defined as

$\displaystyle
CPE_{\gamma}^{\psi}(X)=\frac{1}{\Gamma(\gamma+1)}\int_{0}^{s}\psi(x)K(x)[-\ln
K(x)]^{\gamma}dx,~{}~{}\gamma>0,$ (2.1)

provided the right-hand-side integral is finite, where $\Gamma$ is a gamma
function. From (2.1), one can easily notice that the information measure
$CPE_{\gamma}^{\psi}(X)$ is always nonnegative. It is equal to zero when $X$
is degenerate. Note that the WFGCPE is nonadditive. We recall that an
information measure $H$ is additive if

$\displaystyle H(A+B)=H(A)+H(B),$ (2.2)

for any two probabilitically independent systems $A$ and $B$. If (2.2) is not
satisfied, then the information measure is said to be nonadditive. Several
information measures have been proposed in the literature since the
introduction of the Shannon entropy. Among those, probably Shannon’s entropy
and Renyi’s entropy (see Rényi (1961)) are additive and all other
generalizations (see, for example, Tsallis (1988)) are nonadditive. For
$\gamma\in\mathbb{N}$ and $s\rightarrow+\infty$, $CPE_{\gamma}^{\psi}(X)$
reduces to the weighted generalized cumulative entropy proposed by Tahmasebi
et al. (2020). Further, when $\psi(x)=1$, we get the fractional generalized
cumulative entropy due to Di Crescenzo et al. (2021). Let
$\Psi^{\prime}(x)=\frac{d}{dx}\Psi(x)=\psi(x).$ Then, when $\gamma\rightarrow
0^{+}$ and $0<s<+\infty$, we have from (2.1),

$\displaystyle CPE_{\gamma}^{\psi}(X)$ $\displaystyle=$
$\displaystyle\int_{0}^{s}\psi(x)dx-\int_{0}^{s}\psi(x)\bar{K}(x)dx$ (2.3)
$\displaystyle=$
$\displaystyle\Psi(s)-\Psi(0)-\int_{0}^{s}\psi(x)\left(\int_{x}^{s}k(y)dy\right)dx$
$\displaystyle=$
$\displaystyle\Psi(s)-\Psi(0)-\int_{0}^{s}\int_{0}^{y}\psi(x)k(y)dxdy$
$\displaystyle=$ $\displaystyle\Psi(s)-E[\Psi(X)].$

Thus,

$\displaystyle
CPE_{\gamma}^{\psi}(X)=\left\\{\begin{array}[]{ll}\Psi(s)-E[\Psi(X)],&\mbox{if}~{}\gamma\rightarrow
0^{+}\mbox{ and}~{}0<s<+\infty\\\ +\infty,&\mbox{if}~{}\gamma\rightarrow
0^{+}\mbox{ and}~{}s=+\infty.\end{array}\right.$ (2.6)

Moreover, in particular, for $\psi(x)=x$, we have

$\displaystyle
CPE_{\gamma}^{\psi}(X)=\left\\{\begin{array}[]{ll}\frac{1}{2}\left[s^{2}-E(X^{2})\right],&\mbox{if}~{}\gamma\rightarrow
0^{+}\mbox{ and}~{}0<s<+\infty\\\
CPE_{n}^{\psi(x)=x}(X),&\mbox{if}~{}\gamma=n\in\mathbb{N}~{}\mbox{and}~{}s=+\infty\\\
CPE^{\psi(x)=x}(X),&\mbox{if}~{}\gamma=1~{}\mbox{and}~{}s=+\infty\\\
+\infty,&\mbox{if}~{}\gamma\rightarrow 0^{+}\mbox{
and}~{}s=+\infty,\end{array}\right.$ (2.11)

where $CPE_{n}^{\psi(x)=x}(X)$ and $CPE^{\psi(x)=x}(X)$ are the shift-
dependent generalized cumulative past entropy of order $n$ (see Eq. (1.4) of
Kayal and Moharana (2019)) and weighted cumulative past entropy (see Eq. (10)
of Misagh (2016)), respectively.

Next, we consider an example to show that even though the fractional
generalized cumulative past entropy of two distributions are same, but the
WFGCPEs are not same.

######  Example 2.1.

Consider two random variables $X_{1}$ and $X_{2}$ with respective CDFs
$K_{1}(x)=x-a,~{}0<a<x<a+1$ and $K_{2}(x)=x-(a+1),~{}a+1<x<a+2<+\infty.$ Then,
the fractional generalized cumulative past entropy of $X_{1}$ and $X_{2}$ can
be obtained respectively as

$CPE_{\gamma}(X_{1})=\frac{1}{2^{\gamma+1}}=CPE_{\gamma}(X_{2}),~{}\gamma>0.$

That is, the fractional generalized cumulative past entropy of $X_{1}$ and
$X_{2}$ are same. Indeed, it is expected since the fractional generalized
cumulative past entropy is shift-independent (see Propositon $2.2$ of Di
Crescenzo et al. (2021)). In order to reach to the goal, let us consider
$\psi(x)=x.$ Then,

$CPE_{\gamma}^{\psi(x)=x}(X_{1})=\frac{1}{3^{\gamma+1}}+\frac{a}{2^{\gamma+1}}~{}\mbox{and}~{}CPE_{\gamma}^{\psi(x)=x}(X_{2})=\frac{1}{3^{\gamma+1}}+\frac{a+1}{2^{\gamma+1}},$

which show that the WFGCPEs of $X_{1}$ and $X_{2}$ are not same. Here,
$CPE_{\gamma}^{\psi}(X_{1})<CPE_{\gamma}^{\psi}(X_{2})$. Further, let
$\psi(x)=x^{2}.$ Thus, we have

$CPE_{\gamma}^{\psi(x)=x}(X_{1})=\frac{1}{4^{\gamma+1}}+\frac{2a}{3^{\gamma+1}}+\frac{a^{2}}{2^{\gamma+1}}~{}\mbox{and}~{}CPE_{\gamma}^{\psi(x)=x}(X_{2})=\frac{1}{4^{\gamma+1}}+\frac{2(a+1)}{3^{\gamma+1}}+\frac{(a+1)^{2}}{2^{\gamma+1}},$

which also reveal that the WFGCPEs of $X_{1}$ and $X_{2}$ are different from
each other.

From Example 2.1, we notice that when ignoring qualitative characteristic of a
given data set, the fractional generalized cumulative past entropy of two
distributions are same, as treated from the quantitative point of view.
However, when we do not ignore it, they are not same. In Table $1$, we provide
closed form expressions of the WFGCPE of various distributions for two choices
of $\psi(x)$. Let $K$ and $\bar{K}=1-K$ be the distribution and survival
functions of a symmetrically distributed random variable with bounded support
$(0,s)$. Di Crescenzo et al. (2021) showed that for this symmetric random
variable the fractional generalized cumulative residual entropy and the
fractional generalized cumulative entropy are same. However, this property
does not hold for the weighted versions of the fractional generalized
cumulative residual entropy and fractional generalized cumulative entropy.
Indeed,

$\displaystyle
CPE_{\gamma}^{\psi}(X)=\frac{1}{\Gamma(\gamma+1)}\int_{0}^{s}\psi(s-x)\bar{K}(x)[-\ln\bar{K}(x)]^{\gamma}dx,~{}~{}\gamma>0.$
(2.12)

Particularly, for a symmetric random variable $X$ with $\psi(x)=x$, we have

$\displaystyle CPE_{\gamma}^{\psi}(X)$ $\displaystyle=$
$\displaystyle\frac{s}{\Gamma(\gamma+1)}\int_{0}^{s}\bar{K}(x)[-\ln\bar{K}(x)]^{\gamma}dx-\frac{1}{\Gamma(\gamma+1)}\int_{0}^{s}x\bar{K}(x)[-\ln\bar{K}(x)]^{\gamma}dx$
(2.13) $\displaystyle=$ $\displaystyle
sCRE_{\gamma}(X)-CRE_{\gamma}^{\psi(x)=x}(X),~{}\mbox{say},$

where $CRE_{\gamma}(X)$ and $CRE_{\gamma}^{\psi(x)=x}(X)$ are respectively
known as the fractional generalized cumulative residual entropy and weighted
fractional generalized cumulative residual entropy. Di Crescenzo et al. (2021)
showed that the fractional generalized cumulative entropy of a nonnegative
random variable is shift-independent. In the following proposition, we
establish that the WFGCPE is shift-dependent. This makes the proposed weighted
fractional measure useful in context-dependent situations.

######  Proposition 2.1.

Let $Y=aX+b,$ where $a>0$ and $b\geq 0.$ Then,

$\displaystyle
CPE_{\gamma}^{\psi}(Y)=\frac{a}{\Gamma(\gamma+1)}\int_{0}^{s}\psi(ax+b)K(x)[-\ln
K(x)]^{\gamma}dx,~{}\gamma>0.$ (2.14)

###### Proof.

The proof follows from $K_{Y}(x)=K(\frac{x-b}{a})$. Thus, it is omitted. ∎

Table 1:  The closed form expressions of the WFGCPE of different distributions. For the case of Frèchet distribution, we assume that $\gamma>3/c.$ Model | Cumulative distribution function | $\psi(x)=x$ | $\psi(x)=x^{2}$
---|---|---|---
Power distribution |  $K(x)=\left(\displaystyle\frac{x}{b}\right)^{c},~{}0<x<b,~{}c>0$, | $\displaystyle\frac{b^{2}}{c(1+\frac{2}{c})^{\gamma+1}}$ | $\displaystyle\frac{b^{3}}{c(1+\frac{3}{c})^{\gamma+1}}$
Frèchet distribution | $K(x)=e^{-\displaystyle bx^{-c}},~{}x>0,~{}b,~{}c>0$ | $\displaystyle\frac{b^{\frac{2}{c}}\Gamma(\gamma-\frac{2}{c})}{c\Gamma(\gamma+1)}$ | $\displaystyle\frac{b^{\frac{3}{c}}\Gamma(\gamma-\frac{3}{c})}{c\Gamma(\gamma+1)}$

In particular, let us consider $\psi(x)=x.$ Then, after some simplification,
form (2.14) we get

$\displaystyle
CPE_{\gamma}^{\psi}(Y)=a^{2}CPE_{\gamma}^{\psi(x)=x}(X)+ab~{}CPE_{\gamma}(X),$
(2.15)

where

$\displaystyle
CPE_{\gamma}^{\psi(x)=x}(X)=\frac{1}{\Gamma(\gamma+1)}\int_{0}^{s}xK(x)[-\ln
K(x)]^{\gamma}dx$ (2.16)

and $CPE_{\gamma}(X)$ is given by (1.5). It is always of interest to express
various information measures in terms of the expectation of a function of
random variable of interest. Define

$\displaystyle\mu(t)=\int_{0}^{t}\frac{K(x)}{K(t)}dx,$ (2.17)

which is known as the mean inactivity time of $X$. Di Crescenzo and Longobardi
(2009) expressed cumulative entropy in terms of the expectation of the mean
inactivity time of $X$. Recently, Di Crescenzo et al. (2021) showed that the
fractional generalized cumulative entropy can be written as the expectation of
a decreasing function of $X.$ Below, we get similar findings for the case of
the WFGCPE.

######  Proposition 2.2.

Let $X$ be nonnegative absolutely continuous random variable with distribution
function $K(.)$ and density function $k(.)$ such that
$CPE_{\gamma}^{\psi}(X)<+\infty.$ Then,

$\displaystyle CPE_{\gamma}^{\psi}(X)=E[\tau_{\gamma}^{\psi}(X)],$ (2.18)

where

$\displaystyle\tau_{\gamma}^{\psi}(u)=\frac{1}{\Gamma(\gamma+1)}\int_{u}^{s}\psi(x)[-\ln
K(x)]^{\gamma}dx,~{}~{}\gamma>0.$ (2.19)

###### Proof.

Noting $K(x)=\int_{0}^{x}k(u)du$ and applying Fubini’s theorem, we have from
(2.1) that

$\displaystyle CPE_{\gamma}^{\psi}(X)$ $\displaystyle=$
$\displaystyle\frac{1}{\Gamma(\gamma+1)}\int_{0}^{s}\psi(x)[-\ln
K(x)]^{\gamma}\left(\int_{0}^{x}k(u)du\right)dx$ $\displaystyle=$
$\displaystyle\frac{1}{\Gamma(\gamma+1)}\int_{0}^{s}f(u)\left(\int_{u}^{s}\psi(x)[-\ln
K(x)]^{\gamma}dx\right)du$ $\displaystyle=$ $\displaystyle
E[\tau_{\gamma}^{\psi}(X)],$

where $\tau_{\gamma}^{\psi}(.)$ is given by (2.19). This complets the proof. ∎

Note that (2.18) reduces to Eq. (20) of Tahmasebi et al. (2020), for
$\gamma=n\in\mathbb{N}$ and $s=+\infty.$ For $\psi(x)=1,$ Proposition 2.2
turns out as Proposition $2.1$ of Di Crescenzo et al. (2021).

Similar to the normalized cumulative entropy, Di Crescenzo et al. (2021)
propoosed a normalized fractional generalized cumulative entropy of a random
variable $X$ with nonnegative bounded support. Here, we define a normalized
WFGCPE. It is assumed that the weighted cumulative past entropy with a general
nonnegative weight function is nonzero and finite, which is given by (see
Suhov and Sekeh (2015))

$\displaystyle CPE^{\psi}(X)=-\int_{0}^{s}\psi(x)K(x)\ln
K(x)dx,~{}~{}\psi(x)\geq 0.$ (2.20)

The normalized WFGCPE of $X$ can be defined as

$\displaystyle
NCPE_{\gamma}^{\psi}(X)=\frac{CPE_{\gamma}^{\psi}(X)}{(CPE^{\psi})^{\gamma}}=\frac{1}{\Gamma(\gamma+1)}\frac{\int_{0}^{s}\psi(x)K(x)[-\ln
K(x)]^{\gamma}dx}{\left(\int_{0}^{s}\psi(x)K(x)[-\ln
K(x)]dx\right)^{\gamma}},~{}~{}\gamma>0.$ (2.21)

Note that

$\lim_{\gamma\rightarrow
0^{+}}NCPE_{\gamma}^{\psi}(X)=\int_{0}^{s}\psi(x)K(x)dx~{}~{}\mbox{and}~{}~{}\lim_{\gamma\rightarrow
1}NCPE_{\gamma}^{\psi}(X)=1.$

The closed form expressions of the normalized WFGCPE of power and Frèchet
distributions are presented in Table 2 for two choices of the weight
functions.

Table 2:  The closed form expressions of the normalized WFGCPE of different distributions as Table $1$. For the case of the Frèchet distribution, $c>\max\\{3/\gamma,3\\},~{}\gamma>0.$ Model | $\psi(x)=x$ | $\psi(x)=x^{2}$
---|---|---
Power distribution | $\displaystyle\frac{(c+2)^{\gamma-1}}{\Gamma(\gamma+1)b^{2(\gamma-1)}}$ | $\displaystyle\frac{(c+3)^{\gamma-1}}{\Gamma(\gamma+1)b^{3(\gamma-1)}}$
Frèchet distribution | $\displaystyle\frac{1}{(\Gamma(\gamma+1))^{2}}\frac{c^{\gamma-1}}{b^{\frac{2}{c}(\gamma-1)}}\frac{\Gamma(\gamma-\frac{2}{c})}{(\Gamma(1-\frac{2}{c}))^{\gamma}}$ | $\displaystyle\frac{1}{(\Gamma(\gamma+1))^{2}}\frac{c^{\gamma-1}}{b^{\frac{3}{c}(\gamma-1)}}\frac{\Gamma(\gamma-\frac{3}{c})}{(\Gamma(1-\frac{3}{c}))^{\gamma}}$

Figure 1: Graphs of the normalized WFGCPE with $(a)$ $\psi(x)=x$ and $(b)$
$\psi(x)=x^{2}$ for the power distribution as in Table 2.

###  2.1 Some ordering results

In this subsection, we obtain some ordering properties for the WFGCPE. It can
be shown that the function $\tau_{\gamma}^{\psi}(u)$ given by (2.19) is
decreasing and convex when $\psi(x)$ is decreasing in $x$. Thus, for
decreasing $\psi,$ we have

$\displaystyle X_{1}\leq_{dcx}X_{2}\Rightarrow
E[\tau_{\gamma}^{\psi}(X_{1})]\leq E[\tau_{\gamma}^{\psi}(X_{2})]\Rightarrow
CPE_{\gamma}^{\psi}(X_{1})\leq CPE_{\gamma}^{\psi}(X_{2}).$ (2.22)

Di Crescenzo and Toomaj (2017) showed that more dispersed distributions
produce larger generalized cumulative entropy. Note that the generalized
cumulative entropy was introduced and studied by Kayal (2016). Recently, Di
Crescenzo et al. (2021) established similar property for the fractional
generalized cumulative entropy. In the following proposition, we notice that
analogous result holds for the proposed measure given by (2.1). We recall that
the dispersive order $X_{1}\leq_{disp}X_{2}$ can be equivalently characterized
by (see P. $160$, Oja (1981))

$\displaystyle k_{1}(K_{1}^{-1}(u))\geq k_{2}(K_{2}^{-1}(u)),~{}~{}u\in(0,1).$
(2.23)

######  Proposition 2.3.

Consider two nonnegative absolutely continuous random variables $X_{1}$ and
$X_{2}$ with CDFs $K_{1}$ and $K_{2}$, respectively. Then,

$\displaystyle X_{1}\leq_{disp}X_{2}\Rightarrow CPE_{\gamma}^{\psi}(X_{1})\leq
CPE_{\gamma}^{\psi}(X_{2}),$ (2.24)

provided $\psi$ is increasing.

###### Proof.

Using the transformation $K_{1}(x)=u$, we have

$\displaystyle
CPE_{\gamma}^{\psi}(X_{1})=\frac{1}{\Gamma(\gamma+1)}\int_{0}^{1}\psi(K_{1}^{-1}(u))u(-\ln
u)^{\gamma}\frac{du}{k_{1}(K_{1}^{-1}(u))}.$

Thus,

$\displaystyle
CPE_{\gamma}^{\psi}(X_{1})-CPE_{\gamma}^{\psi}(X_{2})=\frac{1}{\Gamma(\gamma+1)}\int_{0}^{1}u(-\ln
u)^{\gamma}\left[\frac{\psi(K_{1}^{-1}(u))}{k_{1}(K_{1}^{-1}(u))}-\frac{\psi(K_{2}^{-1}(u))}{k_{2}(K_{2}^{-1}(u))}\right]du.$
(2.25)

We know that $X_{1}\leq_{disp}X_{2}\Rightarrow X_{1}\leq_{st}X_{2}$, and as a
result, $K_{1}^{-1}(u)\leq K_{2}^{-1}(u)$, $u\in(0,1)$. Moreover, $\psi$ is
increasing. Thus, $\psi(K_{1}^{-1}(u))\leq\psi(K_{2}^{-1}(u)).$ Using this
inequality and (2.23) in (2.25), the hypothesis in (2.24) holds. This
completes the proof. ∎

Proposition 2.3 reduces to Proposition $1$ of Tahmasebi (2020) when
$s=+\infty$ and $\gamma=n\in\mathbb{N}.$ In the following, we obtain different
sufficient conditions involving the hazard rate order for the similar outcome
in (2.24). We recall that $X_{1}$ has decreasing failure rate (DFR) if the
hazard rate of $X_{1}$ is decreasing, equivalently,
$\bar{K}_{1}(x)=1-K_{1}(x)$ is log-convex.

######  Proposition 2.4.

For the random variables $X_{1}$ and $X_{2}$ as in Proposition 2.3, let
$X_{1}\leq_{hr}X_{2}$ hold. Further, let $X_{1}$ or $X_{2}$ be DFR. Then, for
$\gamma>0,$ one has $CPE_{\gamma}^{\psi}(X_{1})\leq
CPE_{\gamma}^{\psi}(X_{2})$.

###### Proof.

Making use of the result in Proposition 2.3, the proof follows from Theorem
$2.1(b)$ of Bagai and Kochar (1986). ∎

Next, we will study whether the usual stochastic order implies the ordering of
the WFGCPE. In doing so, we consider two random variables $X_{1}$ and $X_{2}$
with respective distribution functions $K_{1}(x)=x^{c_{1}}$ and
$K_{2}(x)=x^{c_{2}}$, $0<x<1$, $c_{1},~{}c_{2}>0.$ For $c_{1}\leq c_{2},$
clearly $K_{1}(x)\geq K_{2}(x)$ implies $X_{1}\leq_{st}X_{2}.$ Now, we plot
graphs of the difference of the WFGCPEs of $X_{1}$ and $X_{2}$ in Figure $2$,
for some values of $\gamma,$ which reveal that in general, the ordering
between the WFGCPEs may not hold.

Figure 2: The plots of $CPE_{\gamma}^{\psi}(X_{1})-CPE_{\gamma}^{\psi}(X_{2})$
for $(a)$ $\gamma=0.5$, $(b)$ $\gamma=0.75$, $(c)$ $\gamma=1.5$ and $(d)$
$\gamma=2.5$. Here, $\psi(x)=x$ is considered.

We end this subsection with a result which compares the WFGCPE measures when
two random variables are ordered in the sense of the usual stochastic order.
Here, prime denotes the ordinary derivative.

######  Proposition 2.5.

Consider two nonnegative absolutely continuous random variables $X_{1}$ and
$X_{2}$ with CDFs $K_{1}$ and $K_{2}$, respectively, such that
$X_{1}\leq_{st}X_{2}$. Further, assume that the means of $X_{1}$ and $X_{2}$
are finite but unequal. Then, for $\tau_{\gamma}^{\psi}(x)<+\infty$ and
$E[\tau_{\gamma}^{\psi}(X)]<\infty$, we have

$\displaystyle
CPE_{\gamma}^{\psi}(X_{1})=E[\tau_{\gamma}^{\psi}(X_{2})]+E[{\tau_{\gamma}^{\psi}}^{\prime}(V)][E(X_{1})-E(X_{2})],$
(2.26)

where $V$ is nonegative absolutely continuous random variable with density
function given by

$\displaystyle
k_{V}(x)=\frac{\bar{K}_{2}(x)-\bar{K}_{1}(x)}{E(X_{2})-E(X_{1})},~{}~{}x>0$
(2.27)

###### Proof.

We note that $CPE_{\gamma}^{\psi}(X_{1})=E[\tau_{\gamma}^{\psi}(X_{1})]$ (see
Proposition 2.2). Now, the rest of the proof follows from Theorem $4.1$ of Di
Crescenzo (1999). ∎

It can be easily seen that when $\psi(x)=1$, Proposition 2.5 reduces to
Proposition $3.4$ of Di Crescenzo et al. (2021). Further, when
$\gamma=n\in\mathbb{N}$ and $\psi(x)=1$, then Proposition 2.5 coincides with
Proposition $3.8$ of Di Crescenzo and Toomaj (2017). Here,
${\tau_{\gamma}^{\psi}}^{\prime}(v)\leq 0,$ for $v>0.$ Thus, under the
assumptions made in Proposition 2.5, a lower bound of
$CPE_{\gamma}^{\psi}(X_{1})$ can be obtained, which is given by

$CPE_{\gamma}^{\psi}(X_{1})\geq E[\tau_{\gamma}^{\psi}(X_{2})].$

In the next subsection, we discuss various bounds of the WFGCPE given by
(2.1).

###  2.2 Bounds

Di Crescenzo and Longobardi (2009) established that the cumulative entropy of
the sum of two independent nonnegative random variables is larger than the
maximum of the cumulative entropies of the individual random variables.
Similar result was obtained by Di Crescenzo et al. (2021) for the fractonal
generalized cumulative entropy. Below, we establish analogous result for the
WFGCPE under the assumption that the weight function $\psi(x)$ is increasing
in $x$ and the PDFs of the random variables are log-concave.

######  Proposition 2.6.

Let $X_{1}$ and $X_{2}$ be a pair of independent nonnegative absolutely
continuous random variables having log-concave PDFs. Then, for all increasing
function $\psi$, we have

$\displaystyle
CPE_{\gamma}^{\psi}(X_{1}+X_{2})\geq\max\\{CPE_{\gamma}^{\psi}(X_{1}),CPE_{\gamma}^{\psi}(X_{2})\\},~{}~{}\gamma>0.$
(2.28)

###### Proof.

Under the assumptions made, the proof follows from Theorem $3.B.7$ of Shaked
and Shanthikumar (2007) and Proposition 2.3. Thus, it is omitted. ∎

When $\gamma=n\in\mathbb{N}$, $s=+\infty$ and $\psi(x)=[K(x)]^{n}$, the result
in Proposition 2.6 yields Proposition $2$ of Tahmasebi et al. (2020). Further,
if we consider $\psi(x)=1$, then one can easily obtain the result stated in
Proposition $3.1$ of Di Crescenzo et al. (2021). Next, we obtain a bound of
the WFGCPE given by (2.1).

######  Proposition 2.7.

For a nonnegative random variable with support $(0,s)$ and
$\psi(x)=[\xi(x)]^{\gamma}$, $\xi(x)\geq 0,$ we have

$\displaystyle
CPE_{\gamma}^{\psi}(X)\left\\{\begin{array}[]{ll}\geq\displaystyle\frac{s^{1-\gamma}}{\Gamma(\gamma+1)}\left[CPE^{\xi}(X)\right]^{\gamma},&\mbox{if}~{}\gamma\geq
1\\\
\leq\displaystyle\frac{s^{1-\gamma}}{\Gamma(\gamma+1)}\left[CPE^{\xi}(X)\right]^{\gamma},&\mbox{if}~{}0<\gamma\leq
1,\end{array}\right.$ (2.31)

where $CPE^{\xi}(X)=-\int_{0}^{s}\xi(x)K(x)\ln K(x)dx$ is known as the
weighted cumulative past entropy with weight function $\xi(x).$

###### Proof.

Let $\gamma\geq 1.$ Then, for $0<x<s,$ $K(x)\geq[K(x)]^{\gamma}$. Thus, under
the assumptions made, from (2.1), we obtain

$\displaystyle
CPE_{\gamma}^{\psi}(X)\geq\frac{1}{\Gamma(\gamma+1)}\int_{0}^{s}\alpha_{\gamma}(\beta(x))dx,$
(2.32)

where $\beta(x)=-\xi(x)K(x)\ln K(x)\geq 0$ and
$\alpha_{\gamma}(t)=t^{\gamma}.$ It can be shown that $t^{\gamma}$ is convex
in $t\geq 0$, for $\gamma\geq 1.$ Thus, from Jensen’s integral inequality, the
rest of the proof follows. The case for $0<\gamma\leq 1$ can be proved
similarly. ∎

######  Proposition 2.8.

For a nonnegative random variable with support $(0,s)$ and $\gamma>0,$ we have

$\displaystyle
CPE_{\gamma}^{\psi}(X)\left\\{\begin{array}[]{ll}\geq\psi(s)CPE_{\gamma}(X),&\mbox{if}~{}\psi~{}\mbox{is
~{}decreasing}\\\ \leq\psi(s)CPE_{\gamma}(X),&\mbox{if}~{}\psi~{}\mbox{is
~{}increasing},\end{array}\right.$ (2.35)

where $CPE_{\gamma}(X)=\frac{1}{\Gamma(\gamma+1)}\int_{0}^{s}K(x)[-\ln
K(x)]^{\gamma}dx$ is known as the fractional generalized cumulative past
entropy.

###### Proof.

The proof is straightforward, and thus it is omitted. ∎

######  Proposition 2.9.

Let $X$ be an absolutely continuous random variable with support $(0,s)$ with
mean $E(X)=\mu<+\infty$. Then,

  * (i)

$CPE_{\gamma}^{\psi}(X)\geq\frac{1}{\Gamma(\gamma+1)}\int_{0}^{s}\psi(x)K(x)[1-K(x)]^{\gamma}dx$;

  * (ii)

$CPE_{\gamma}^{\psi}(X)\geq D(\gamma)e^{H(X)},$ where
$D(\gamma)=e^{\int_{0}^{1}\ln[\psi(K^{-1}(u))u(-\ln u)^{\gamma}]du}$ and
$H(X)$ is the differential entropy of $X$;

  * (iii)

$CPE_{\gamma}^{\psi}(X)\geq\tau_{\gamma}^{\psi}(\mu),$ provided $\psi$ is
decreasing.

###### Proof.

The first part of this proposition follows from the relation $\ln u\leq u-1$,
for $0<u<1$. To prove the second part, from the log-sum inequality, we have

$\displaystyle\int_{0}^{s}k(x)\ln\frac{k(x)}{\psi(x)K(x)[-\ln
K(x)]^{\gamma}}dx$ $\displaystyle\geq$
$\displaystyle\ln\frac{1}{\int_{0}^{s}\psi(x)K(x)[-\ln K(x)]^{\gamma}dx}$
(2.36) $\displaystyle=$ $\displaystyle-CPE_{\gamma}^{\psi}(X).$

Now, the rest of the proof follows using the arguments as in the proof of
Theorem $2$ of Xiong et al. (2019). Third part follows from Jensen’s
inequality. ∎

We end this subsection with the following result, which provides bounds of the
WFGCPE of $X_{2}$, where the CDF of $X_{2}$ is given by (2.42).

######  Proposition 2.10.

Let $X_{1}$ and $X_{2}$ be two random variables with CDFs $K_{1}$ and $K_{2}$,
respectively. Further, assume that the random variables satisfy proportional
reversed hazard model described in (2.42). Then,

$\displaystyle
CPE_{\gamma}^{\psi}(X_{2})\left\\{\begin{array}[]{ll}\leq\eta^{\gamma}CPE_{\gamma}^{\psi}(X_{1}),&\mbox{for}~{}\eta\geq
1\\\ \geq\eta^{\gamma}CPE_{\gamma}^{\psi}(X_{1}),&\mbox{for}~{}0<\eta\leq
1.\end{array}\right.$

###### Proof.

The proof is simple, and thus it is omitted. ∎

###  2.3 Connection with fractional calculus

Fractional calculus and its widely application have recently been paid more
and more attentions. We refer to Miller and Ross (1993) and Gorenflo and
Mainardi (2008) for more recent development on fractional calculus. Several
known forms of the fractional integrals have been proposed in the literature.
Among these, the Riemann-Liouville fractional integral of order $\gamma>0$ has
been studied extensively for its applications. See, for instance Dahmani et
al. (2010), Romero et al. (2013) and Tunc (2013). Let $\gamma>0$ and $f\in
L^{1}(a,b),~{}a\geq 0$. Then, the left-sided Riemann-Liouville fractional
integral in the interval $[a,b]$ is defined as follows

$\displaystyle
J_{a^{+}}^{\gamma}f(t)=\frac{1}{\Gamma(\gamma)}\int_{a}^{t}\frac{f(\tau)}{(t-\tau)^{1-\gamma}}d\tau,~{}t\in[a,b],$
(2.38)

where $f$ is a real-valued continuous function. We recall that the notion of
the left-sided Riemann-Liouville fractional integral given by (2.38) can be
elongated with respect to a strictly increasing function $h(.)$. In addition
to this strictly increasing property, we further assume that the first order
derivative $h^{\prime}(.)$ is continuous in the interval $(a,b)$. Then, for
$\gamma>0,$ the left-sided Riemann-Liouville fractional integral of $f$ with
respect to $h$ is given by

$\displaystyle
J_{a^{+};h}^{\gamma}f(t)=\frac{1}{\Gamma(\gamma)}\int_{a}^{t}\frac{h^{\prime}(\tau)f(\tau)}{(h(t)-h(\tau))^{1-\gamma}}d\tau,~{}t\in[a,b].$
(2.39)

One may refer to Samko et al. (1993) (Section $18.2$) for the representation
given in (2.39). Now, we will notice that the WFGCPE can be expressed in terms
of the limits of the integral (2.39) after suitable choices of the functions
$f(x)$ and $h(x)$, that is, the fractional nature of the proposed measure is
justified. Let us take

$h(x)=\ln K(x)~{}~{}\mbox{and}~{}~{}f(x)=\frac{\psi(x)(K(x))^{2}}{k(x)}.$

Then,

$\displaystyle\lim_{a\rightarrow 0,~{}t\rightarrow
s}J_{a^{+};h}^{\gamma+1}f(t)$ $\displaystyle=$
$\displaystyle\frac{1}{\Gamma(\gamma)}\int_{0}^{s}\psi(x)K(x)[-\ln
K(x)]^{\gamma}dx$ (2.40) $\displaystyle=$ $\displaystyle
CPE_{\gamma}^{\psi}(X).$

###  2.4 Proportional reversed hazards model

Let $X$ be a nonnegative absolutely continuous random variable with
distribution function $K$ and density function $k.$ Here, $X$ may be treated
as the lifetime of a unit. If $\lambda(t)=\frac{d}{dt}\ln K(t)$ denotes the
reversed hazard rate of $X$, then $\lambda(t)dt$ represents the conditional
probability the unit stopped working in an infinitesimal interval of width
$dt$ preceding $t$, given that the unit failed before $t$. In otherwords,
$\lambda(t)dt$ is the probability of failing in the interval $(t-dt,t)$ given
that the unit is found failed at time $t$. Let $X_{1}$ and $X_{2}$ be two
random variables with PDFs $k_{1}$ and $k_{2}$, CDFs $K_{1}$ and $K_{2}$ and
reversed hazard rate functions $\lambda_{1}$ and $\lambda_{2}$, respectively.
It is well known that $X_{1}$ and $X_{2}$ have proportional reversed hazard
rate model if

$\displaystyle\lambda_{2}(x)=\eta\lambda_{1}(x)=\eta\frac{k_{1}(x)}{K_{1}(x)},$
(2.41)

where $\eta>0$ is known as the proportionality constant. Note that (2.41) is
equivalent to the model

$\displaystyle K_{2}(x)=[K_{1}(x)]^{\eta},~{}~{}x\in\mathbb{R},~{}~{}\eta>0,$
(2.42)

where $K_{1}$ is the baseline distribution function (see Gupta et al. (1998),
Di Crescenzo (2000) and Gupta and Gupta (2007)). The PDF of $X_{2}$ is

$\displaystyle k_{2}(x)=\eta(K_{1}(x))^{\eta-1}k_{1}(x),~{}~{}x>0,~{}\eta>0.$
(2.43)

Next, we evaluate the WFGCPE of $X_{2}$. Making use of (2.43), from (2.1) and
(2.42), we have after some standard calculations that

$\displaystyle CPE_{\gamma}^{\psi}(X_{2})$ $\displaystyle=$
$\displaystyle\frac{1}{\Gamma(\gamma+1)}\int_{0}^{s}\psi(x)(K_{1}(x))^{\eta}[-\ln(K_{1}(x))^{\eta}]^{\gamma}dx$
(2.44) $\displaystyle=$
$\displaystyle-\frac{1}{\Gamma(\gamma+1)}\Bigg{[}\int_{0}^{s}x\psi(x)[-\ln(K_{1}(x))^{\eta}]^{\gamma}\eta(K_{1}(x))^{\eta-1}k_{1}(x)dx$
$\displaystyle-\gamma\int_{0}^{s}x\psi(x)[-\ln(K_{1}(x))^{\eta}]^{\gamma-1}\eta(K_{1}(x))^{\eta-1}k_{1}(x)dx$
$\displaystyle+\int_{0}^{s}\frac{x\psi^{\prime}(x)}{\eta\lambda_{1}(x)}[-\ln(K_{1}(x))^{\eta}]^{\gamma}\eta(K_{1}(x))^{\eta-1}k_{1}(x)dx\Bigg{]}$
$\displaystyle=$
$\displaystyle-\frac{1}{\Gamma(\gamma+1)}\Bigg{[}\int_{0}^{s}x\psi(x)[-\ln
K_{2}(x)]^{\gamma}k_{2}(x)dx$ $\displaystyle-\gamma\int_{0}^{s}x\psi(x)[-\ln
K_{2}(x)]^{\gamma-1}k_{2}(x)dx$
$\displaystyle+\int_{0}^{s}\frac{x\psi^{\prime}(x)}{\eta\lambda_{1}(x)}[-\ln
K_{2}(x)]^{\gamma}k_{2}(x)dx\Bigg{]}.$

Now, denote

$\displaystyle\mathcal{E}_{2}^{\eta}(\gamma)=\frac{1}{\Gamma(\gamma)}E\left[X_{2}\psi(X_{2})[-\ln
K_{2}(X_{2})]^{\gamma-1}\right]$ (2.45)

and

$\displaystyle\tilde{\mathcal{E}}_{2}^{\eta}(\gamma)=\frac{1}{\Gamma(\gamma)}E\left[\frac{X_{2}\psi^{\prime}(X_{2})}{\lambda_{1}(X_{2})}[-\ln
K_{2}(X_{2})]^{\gamma-1}\right].$ (2.46)

Thus, using (2.45) and (2.46) in (2.44), the following proposition can be
obtained.

######  Proposition 2.11.

Let (2.42) hold. Then, the WFGCPE of $X_{2}$ can be expressed as

$\displaystyle
CPE_{\gamma}^{\psi}(X_{2})=\mathcal{E}_{2}^{\eta}(\gamma)-\mathcal{E}_{2}^{\eta}(\gamma+1)-\eta^{-1}\tilde{\mathcal{E}}_{2}^{\eta}(\gamma+1),~{}~{}\gamma>0,$
(2.47)

provided the associated expectations are finite.

We note that when $\psi(x)=1$, (2.47) reduces to Eq. $(19)$ of Di Crescenzo et
al. (2021). An illustration of the result in Proposition 2.11 is provided in
the following example when $\psi(x)=x.$

######  Example 2.2.

Let $K_{1}(x)=x,~{}0<x<1$ be the baseline distribution function. We will find
the WFGCPE of a random variable $X_{2}$ with distribution function
$K_{2}(x)=[K_{1}(x)]^{c}=x^{c},~{}0<x<1,~{}c>0$. Under this set up, from
(2.45), for $\psi(x)=x$, we obtain

$\displaystyle\mathcal{E}_{2}^{\eta}(\gamma)=\frac{c^{\gamma}}{(2+c)^{\gamma}}=\tilde{\mathcal{E}}_{2}^{\eta}(\gamma).$
(2.48)

Now, using (2.48) in (2.47), we get

$\displaystyle CPE_{\gamma}^{\psi(x)=x}(X_{2})$ $\displaystyle=$
$\displaystyle\frac{c^{\gamma}}{(2+c)^{\gamma}}-\frac{c^{\gamma+1}}{(2+c)^{\gamma+1}}-\frac{c^{\gamma}}{(2+c)^{\gamma+1}}$
$\displaystyle=$ $\displaystyle\frac{c^{\gamma}}{(2+c)^{\gamma+1}},$

which coincides with the case of the Power distribution as in Table 1 for
$b=1$.

The WFGCPE of $X_{2}$ can be represented in terms of the WFGCPE with different
weight functions as follows

$\displaystyle CPE_{\gamma}^{\psi}(X_{2})=-CPE_{\gamma}^{\psi_{1}}(X_{2})-\eta
CPE_{\gamma}^{\psi_{2}}(X_{2})+\eta\gamma^{-1}CPE_{\gamma+1}^{\psi_{2}}(X_{2}),$
(2.49)

where $\psi$ is increasing, $\psi_{1}(x)=x\psi^{\prime}(x)$ and
$\psi_{2}(x)=x\psi(x)\lambda_{1}(x)$. Next, we show that a recurrence relation
can be constructed for the WFGCPE of $X_{2}$. It is shown that the WFGCPE of
$X_{2}$ of order $(\gamma+1)$ can be expressed in terms of that of order
$\gamma$. From (2.47),

$\displaystyle CPE_{\gamma+1}^{\psi}(X_{2})$ $\displaystyle=$
$\displaystyle\mathcal{E}_{2}^{\eta}(\gamma+1)-\mathcal{E}_{2}^{\eta}(\gamma+2)-\eta^{-1}\tilde{\mathcal{E}}_{2}^{\eta}(\gamma+2)$
(2.50) $\displaystyle=$
$\displaystyle\mathcal{E}_{2}^{\eta}(\gamma)-\mathcal{E}_{2}^{\eta}(\gamma+2)-\eta^{-1}[\tilde{\mathcal{E}}_{2}^{\eta}(\gamma+1)+\tilde{\mathcal{E}}_{2}^{\eta}(\gamma+2)]-CPE_{\gamma}^{\psi}(X_{2}).$

Further, when $\psi(x)=1$, (2.50) reduces to Eq. (22) of Di Crescenzo et al.
(2021). We note that the recurrence relation in (2.50) can be generalized for
any integer $n\geq 1,$ which is presented in the following proposition.

######  Proposition 2.12.

Let $n$ be a positive integer. Then, under the model in (2.42), for $\eta>0$
and $\gamma>0$, we obtain

$\displaystyle CPE_{\gamma+n}^{\psi}(X_{2})$ $\displaystyle=$
$\displaystyle\mathcal{E}_{2}^{\eta}(\gamma+n)-\mathcal{E}_{2}^{\eta}(\gamma+n+1)+(-1)^{n-1}[\mathcal{E}_{2}^{\eta}(\gamma)-\mathcal{E}_{2}^{\eta}(\gamma+1)]$
(2.51)
$\displaystyle+\eta^{-1}[(-1)^{n}\tilde{\mathcal{E}}_{2}^{\eta}(\gamma+1)-\tilde{\mathcal{E}}_{2}^{\eta}(\gamma+n+1)]+(-1)^{n}CPE_{\gamma}^{\psi}(X_{2}).$

###### Proof.

The proof follows using similar arguments as in the proof of Proposition $2.4$
of Di Crescenzo et al. (2021). Thus, it is omitted. ∎

We note that for the weight function $\psi(x)=1$, Proposition 2.12 coincides
with Proposition $2.4$ of Di Crescenzo et al. (2021). In this case, the terms
$\tilde{\mathcal{E}}_{2}^{\eta}(\gamma+1)$ and
$\tilde{\mathcal{E}}_{2}^{\eta}(\gamma+n+1)$ become zero.

##  3 Empirical WFGCPE

Let $T=(T_{1},\ldots,T_{n})$ be a random sample of size $n$ drawn from a
population with CDF $K.$ The order statistics of the sample $T$ are the
ordered sample values, denoted by $T_{1:n}\leq\ldots\leq T_{n:n}.$ Denote the
indicator function of the set $A$ by $I_{A},$ where

$I_{A}=\left\\{\begin{array}[]{ll}1,&\mbox{if }~{}A~{}\mbox{is~{} true}\\\
0,&\mbox{otherwise}.\end{array}\right.$

The empirical CDF on the basis of the random sample $T$ is given by

$\displaystyle\widetilde{K}_{n}(x)=\frac{1}{n}\sum_{i=1}^{n}I_{\\{T_{i}\leq
x\\}}=\left\\{\begin{array}[]{ll}0,&\mbox{if}~{}~{}x<T_{1:n}\\\
\frac{l}{n},&\mbox{if}~{}~{}T_{l:n}\leq x<T_{l+1:n}\\\ 1,&\mbox{if}~{}~{}x\geq
T_{n:n},\end{array}\right.$ (3.4)

where $l=1,\ldots,n-1.$ Using (3.4), for $\gamma>0$ and $\psi(x)\geq 0$, the
WFGCPE given by (2.1) can be expressed as

$\displaystyle CPE_{\gamma}^{\psi}(\widehat{K}_{n})$ $\displaystyle=$
$\displaystyle\frac{1}{\Gamma(\gamma+1)}\int_{0}^{s}\psi(x)\widehat{K}_{n}(x)[-\ln\widehat{K}_{n}(x)]^{\gamma}dx$
(3.5) $\displaystyle=$
$\displaystyle\frac{1}{\Gamma(\gamma+1)}\sum_{l=1}^{n-1}\int_{T_{l:n}}^{T_{l+1:n}}\psi(x)\widehat{K}_{n}(x)[-\ln\widehat{K}_{n}(x)]^{\gamma}dx$
$\displaystyle=$
$\displaystyle\frac{1}{\Gamma(\gamma+1)}\sum_{l=1}^{n-1}Z_{l}\left(\frac{l}{n}\right)\left(-\ln\frac{l}{n}\right)^{\gamma},$

where $Z_{l}=\Psi(T_{l+1:n})-\Psi(T_{l:n})$ and
$\Psi(x)=\int_{0}^{x}\psi(x)dx$. Note that when $\psi(x)=1$, we get the
empirical fractional generalized cumulative entropy (see Di Crescenzo et al.
(2021)) from (3.5). For $\psi(x)=x$ and $\gamma=1$, (3.5) coincides with the
empirical weighted cumulative entropy proposed by Misagh et al. (2011).
Further, let $\gamma$ be a natural number. Then, for $\psi(x)=x,$ (3.5)
reduces to the empirical shift-dependent generalized cumulative entropy due to
Kayal and Moharana (2019). Thus, we observe that the proposed empirical
estimate in (3.5) is a generalization of several empirical estimates developed
so far. In the following theorem, we show that the empirical WFGCPE converges
to the WFGCPE almost surely.

######  Theorem 3.1.

Consider a nonnegative absolutely continuous random variable $X$ with CDF $K.$
Then, for $X\in L^{p},~{}p>2$, we have

$CPE_{\gamma}^{\psi}(\widehat{K}_{n})\rightarrow CPE_{\gamma}^{\psi}(X),$

almost surely.

###### Proof.

We have

$\displaystyle\frac{\Gamma(\gamma+1)}{(-1)^{\gamma}}CPE_{\gamma}^{\psi}(\widehat{K}_{n})$
$\displaystyle=$
$\displaystyle\int_{0}^{1}\psi(x)\widehat{K}_{n}(x)\left[\ln\widehat{K}_{n}(x)\right]^{\gamma}dx+\int_{1}^{s}\psi(x)\widehat{K}_{n}(x)\left[\ln\widehat{K}_{n}(x)\right]^{\gamma}dx$
(3.6) $\displaystyle=$ $\displaystyle I_{1}+I_{2},~{}\mbox{say}.$

Now, using dominated convergence theorem and Glivenko-Cantelli theorem, the
rest of the proof follows as in Theorem $14$ of Tahmasebi et al. (2020). ∎

Next, we consider a data set, which was studied by Abouammoh et al. (1994). It
represents the ordered lifetimes (in days) of $43$ blood cancer patients, due
to one of the Ministry of Health Hospitals in Saudi Arabia.
———————————————————————————————————————
$115,181,255,418,441,461,516,739,743,789,807,865,924,983,1024,1062,1063,1165,1191,\\\
1222,1222,1251,1277,1290,1357,1369,1408,1455,1478,1549,1578,1578,15999,1603,1605,\\\
1696,1735,1799,1815,1852,1899,1925,1965.$
———————————————————————————————————————
Based on this dats set, tet us now compute the values of the WFGCPE with
weight functions $\psi(x)=\sqrt{x}$, $\psi(x)=x$ and $\psi(x)=x^{2}$ for
various values of $\gamma,$ which are presented in Table 3. Indeed, one can
compute the values of the WFGCPE with any positive valued weight functions.

Table 3:  Values of the WFGCPE based on the ordered lifetimes (in days) of $43$ blood cancer patients.  $\gamma$ | $\psi(x)=\sqrt{x}$ | $\psi(x)=x$ | $\psi(x)=x^{2}$
---|---|---|---
0.25 | 24004.3 | 881460 | $1.27542\times 10^{9}$
0.5 | 20065.8 | 707724 | $9.59358\times 10^{8}$
0.75 | 16858.4 | 570814 | $7.23578\times 10^{8}$
1.5 | 10279.3 | 309581 | $3.22149\times 10^{8}$
2.75 | 4489.63 | 114320 | $8.89639\times 10^{7}$

Figure 3: Graphs of the empirical WFGCPE with $(a)$ $\psi(x)=x$ and $(b)$
$\psi(x)=x^{2}$ based on the ordered lifetimes (in days) of $43$ blood cancer
patients, for $\gamma=0.25,0.5,0.75,1.5$ and $2.75$ (from above)

Next, we consider examples to illustrate the proposed empirical measure.

######  Example 3.1.

Let $T=(T_{1},\ldots,T_{n})$ be a random sample drawn from a population with
CDF $K(x)=x^{2},~{}0<x<1.$ Consider $\psi(x)=x.$ It can be shown that
$T_{l}^{2}$, $l=1,\ldots,n-1$ follow uniform distribution in the interval
$(0,1)$. Further, the sample spacings $T_{l+1:n}^{2}-T_{l:n}^{2}$,
$l=1,\ldots,n-1$ are independently beta distributed with parameters $1$ and
$n$. For details, please refer to Pyke (1965). Thus, from (3.5), for
$\gamma>0,$ we get

$\displaystyle
E[CPE_{\gamma}^{\psi}(\widehat{K}_{n})]=\frac{1}{\Gamma(\gamma+1)}\sum_{l=1}^{n-1}\frac{1}{2(1+n)}\left(\frac{l}{n}\right)\left[-\ln\frac{l}{n}\right]^{\gamma}$
(3.7)

and

$\displaystyle
Var[CPE_{\gamma}^{\psi}(\widehat{K}_{n})]=\frac{1}{(\Gamma(\gamma+1))^{2}}\sum_{l=1}^{n-1}\frac{n}{4(1+n)^{2}(2+n)}\left(\frac{l}{n}\right)^{2}\left[-\ln\frac{l}{n}\right]^{2\gamma}.$
(3.8)

We present the computed values of the means and variances of the empirical
estimator of WFGCPE under the set up explained in Example 3.1 in Table 4. From
Table 4, we observe that for fixed sample sizes, as $\gamma$ increases, the
mean and variance of the proposed estimator decrease. Further, for fixed
$\gamma,$ the mean and variance respectively increase and decrease, as the
sample size increases.

Table 4:  Numerical values of $E(CPE^{\psi}_{\gamma}(\widehat{\bar{K}}_{n}))$
and $Var(CPE^{\psi}_{\gamma}(\widehat{\bar{K}}_{n}))$ for the distribution as
in Example $3.1.$

$\gamma$ | $n$ | $E(CPE^{\psi}_{\gamma}(\widehat{\bar{K}}_{n}))$ | $Var(CPE^{\psi}_{\gamma}(\widehat{\bar{K}}_{n}))$ | $\gamma$ | $n$ | $E(CPE^{\psi}_{\gamma}(\widehat{\bar{K}}_{n}))$ | $Var(CPE^{\psi}_{\gamma}(\widehat{\bar{K}}_{n}))$
---|---|---|---|---|---|---|---
0.25 | 5 | 0.153878 | 0.004609 | 0.5 | 5 | 0.135721 | 0.003395
| 10 | 0.181591 | 0.003434 |  | 10 | 0.156472 | 0.002416
| 15 | 0.191238 | 0.002627 |  | 15 | 0.163420 | 0.001822
| 30 | 0.200941 | 0.001507 |  | 30 | 0.170268 | 0.001034
| 50 | 0.204774 | 0.000956 |  | 50 | 0.172941 | 0.000653
0.75 | 5 | 0.116302 | 0.002472 | 1.5 | 5 | 0.066611 | 0.000968
| 10 | 0.132732 | 0.001734 |  | 10 | 0.077849 | 0.000712
| 15 | 0.138160 | 0.001304 |  | 15 | 0.081549 | 0.000538
| 30 | 0.143500 | 0.000738 |  | 30 | 0.085119 | 0.000306
| 50 | 0.145593 | 0.000466 |  | 50 | 0.086481 | 0.000194

######  Example 3.2.

Consider a random sample $T$ from a Weibull population with CDF
$K(x)=1-e^{-\theta x^{2}},~{}x>0,~{}\theta>0.$ Using simple transformation
theory, it can be established that $T_{i}^{2},~{}i=1,\ldots,n$ follow
exponential distribution with mean $1/\theta$. Further, let $\psi(x)=x.$ Under
the present set up, the sample spacings $T_{l+1:n}^{2}-T_{l:n}^{2}$,
$l=1,\ldots,n-1$ are independent and exponentialy distributed with mean
$1/(\theta(n-l))$ (see Pyke (1965)). Thus, from (3.5), we obtain

$\displaystyle
E[CPE_{\gamma}^{\psi}(\widehat{K}_{n})]=\frac{1}{\Gamma(\gamma+1)}\sum_{l=1}^{n-1}\frac{1}{2\theta(n-l)}\left(\frac{l}{n}\right)\left[-\ln\frac{l}{n}\right]^{\gamma}$
(3.9)

and

$\displaystyle
Var[CPE_{\gamma}^{\psi}(\widehat{K}_{n})]=\frac{1}{(\Gamma(\gamma+1))^{2}}\sum_{l=1}^{n-1}\frac{1}{4\theta^{2}(n-l)^{2}}\left(\frac{l}{n}\right)^{2}\left[-\ln\frac{l}{n}\right]^{2\gamma}.$
(3.10)

######  Example 3.3.

Let $T$ be a random sample from a population with absolutely continuous CDF
$K$ and PDF $k$. Let $\psi(x)=k(x).$ Then,
$Z_{l}=K(T_{l+1:n})-K(T_{l:n}),~{}l=1,\ldots,n-1$ are independent and beta
distributed random variables with parameters $1$ and $n$. We refer to Pyke
(1965) for details. Thus, similar to (3.7) and (3.8), we have

$\displaystyle
E[CPE_{\gamma}^{\psi}(\widehat{K}_{n})]=\frac{1}{\Gamma(\gamma+1)}\sum_{l=1}^{n-1}\frac{1}{(1+n)}\left(\frac{l}{n}\right)\left[-\ln\frac{l}{n}\right]^{\gamma}$
(3.11)

and

$\displaystyle
Var[CPE_{\gamma}^{\psi}(\widehat{K}_{n})]=\frac{1}{(\Gamma(\gamma+1))^{2}}\sum_{l=1}^{n-1}\frac{n}{(1+n)^{2}(2+n)}\left(\frac{l}{n}\right)^{2}\left[-\ln\frac{l}{n}\right]^{2\gamma}.$
(3.12)

Hereafter, we provide central limit theorems for the empirical WFGCPE when the
random samples are drawn from $(i)$ a Weibull distribution with $\psi(x)=x$
and $(ii)$ a general CDF $K(x)$ with $\psi(x)=k(x)=\frac{d}{dx}K(x).$

######  Theorem 3.2.

Consider a random sample $T$ from a population with PDF $k(x)=2\lambda
xe^{-\lambda x^{2}},~{}x>0,~{}\lambda>0.$ Then, for $\gamma>0$ and
$\psi(x)=x,$

$\frac{CPE_{\gamma}^{\psi}(\widehat{K}_{n})-E(CPE_{\gamma}^{\psi}(\widehat{K}_{n}))}{\sqrt{Var(CPE_{\gamma}^{\psi}(\widehat{K}_{n}))}}\rightarrow
N(0,1)$

in distribution as $n\rightarrow\infty.$

###### Proof.

The proof is similar to that of Theorem $5.1$ of Kayal and Moharana (2019).
Thus, it is omitted. ∎

######  Theorem 3.3.

Consider a random sample $T$ from a population with CDF $K(x).$ Then, for
$\gamma>0$ and $\psi(x)=k(x),$

$\frac{CPE_{\gamma}^{\psi}(\widehat{K}_{n})-E(CPE_{\gamma}^{\psi}(\widehat{K}_{n}))}{\sqrt{Var(CPE_{\gamma}^{\psi}(\widehat{K}_{n}))}}\rightarrow
N(0,1)$

in distribution as $n\rightarrow\infty.$

###### Proof.

The proof is similar to that of Theorem $15$ of Tahmasebi et al. (2020). Thus,
it is omitted. ∎

##  4 Concluding remarks and some discussions

In this paper, we have proposed a weighted fractional generalized cumulative
past entropy of a nonnegative random variable having bounded support. A number
of results for the proposed weighted fractional measure have been obtained
when the weight is a general nonnegative function. It is noticed that WFGCPE
is shift-dependent and can be written as the expectation of a decreasing
function of the random variable. Some ordering results and bounds are
established. Based on the properties, it can be seen that the proposed measure
is a variability measure. Further, a connection between the proposed weighted
fractional measure and the fractional calculus is provded. The weighted
fractional generalized cumulative past entropy measure is studied for the
proportional reversed hazards model. A nonparametric estimator of the weighted
fractional generalized cumulative past entropy is introduced based on the
empirical cumulative distribution function. Few examples are considered to
compute mean and variance of the estimator. Finally, a large sample property
of the estimator is studied.

The proposed measure is not appropriate when uncertainty is associated with
past. Suppose a system has started working at time $t=0$. At a pre-specified
inspection time say $t\in(0,s)$, the system is found to be down. Then, the
random variable $X_{(t)}=X|X\leq t$, where $t\in(0,s)$ is known as the past
lifetime. The dynamic weighted fractional generalized cumulative past entropy
of $X_{(t)}$ is defined as

$\displaystyle
CPE_{\gamma}^{\psi}(X;t)=\frac{1}{\Gamma(\gamma+1)}\int_{0}^{t}\psi(x)\frac{K(x)}{K(t)}\left(-\ln\frac{K(x)}{K(t)}\right)^{\gamma}dx,~{}~{}\gamma>0,~{}\psi(x)\geq
0.$ (4.13)

One can prove most of the similar properties for $CPE_{\gamma}^{\psi}(X;t)$ as
established for the proposed measure given by (2.1).

Acknowledgements: The author Suchandan Kayal acknowledges the partial
financial support for this work under a grant MTR/2018/000350, SERB, India.

Conflict of interest statement: The author declares that he does not have any
conflict of interests.

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# Dicke Phase Transition and Collapse of Superradiant Phase in Optomechanical
Cavity with Arbitrary Number of Atoms

Xiuqin Zhao  Institute of Theoretical Physics, Shanxi University, Taiyuan,
Shanxi 030006, China  Department of Physics, Taiyuan Normal University,
Taiyuan, Shanxi 030001, China  Ni Liu<EMAIL_ADDRESS>Institute of
Theoretical Physics, Shanxi University, Taiyuan, Shanxi 030006, China  Xuemin
Bai  Institute of Theoretical Physics, Shanxi University, Taiyuan, Shanxi
030006, China  J.-Q. Liang<EMAIL_ADDRESS>Institute of Theoretical
Physics, Shanxi University, Taiyuan, Shanxi 030006, China

###### Abstract

We in this paper derive the analytical expressions of ground-state energy,
average photon-number, and the atomic population by means of the spin-
coherent-state variational method for arbitrary number of atoms in an
optomechanical cavity. It is found that the existence of mechanical oscillator
does not affect the phase boundary between the normal and superradiant phases.
However, the superradiant phase collapses by the resonant damping of the
oscillator when the atom-field coupling increases to a so-called turning
point. As a consequence the system undergoes at this point an additional phase
transition from the superradiant phase to a new normal phase of the atomic
population-inversion state. The region of superradiant phase decreases with
the increase of photon-phonon coupling. It shrinks to zero at a critical value
of the coupling and a direct atomic population transfer appears between two
atom-levels. Moreover we find an unstable nonzero-photon state, which is the
counterpart of the superradiant state. In the absence of oscillator our result
reduces exactly to that of Dicke model. Particularly the ground-state energy
for $N=1$ (i.e. the Rabi model) is in perfect agreement with the numerical
diagonalization in a wide region of coupling constant for both red and blue
detuning. The Dicke phase transition remains for the Rabi model in agreement
with the recent observation.

Dicke Phase Transition; Optomechanical Cavity; Spin-Coherent-State Variational
Method

###### pacs:

03.75.Mn, 71.15.Mb, 67.85.Pq

##  I Introduction

The Dicke model describing $N$ two-level atoms interacting with a single-mode
bosonic field is a paradigm to study fascinating collective-quantum-phenomena
of the light-matter system Dic54 ; JLJ06 ; LEB04 ; DVi04 ; EBr03 ; LJo04 ;
HLi73 , the ground state of which undergoes a quantum phase transition (QPT)
by the variation of atom-field coupling LEB04 ; EBr03  in the infinite-$N$
limit. It is believed that this QPT is possible only when the critical atom-
field coupling reaches the same order of magnitude as the atomic level
splitting Dic54 ; EBr03 . The Dicke model has opened an exciting avenue of
research in a variety of context from quantum optics to condensed matter
physics since it is a striking example for the macroscopic many-particle
quantum state, which can be solved rigorously. The experimental realization
BGB10 ; KMB11  of QPT from normal phase (NP) to superradiant phase (SP) is a
milestone in this field. This is achieved with a Bose-Einstein condensate
(BEC) in an optical cavity by detecting the photon numbers BGB10 ; KMB11 . The
QPT of Dicke model has been well studied Dic54 ; EBr03 ; CLL06  theoretically
based on the variational method with the help of Holstein-Primakoff
transformation ETB03  to convert the pseudospin operators into a one-mode
bosonic operator in the thermodynamic limit. The ground-state properties were
also revealed in terms of the catastrophe formalism GNa78 , the coherent state
theory LZL12 ; CNL11 , the dynamic method DEP07 ; HRi01 , and the boson
expansion ETB03  as well. Collapse and revival of oscillations are
demonstrated in a parametrically excited BEC in combined harmonic and optical
lattice trap VermaP .

The optical cavity coupled with a mechanical oscillator was originally used to
explore the boundaries between classical and quantum mechanics BVT80 . This
hybrid system receives renewed interest due to the experimental progress with
the laser cooling of the mechanical mode SDN06 ; CCI07 , which is a
substantial step toward the quantum regime MMT97 ; MSP03 . The study of such
system now becomes a new research branch known as cavity optomechanics, which
is a major resource for implementing high-precision measurement and quantum-
information processing AKM . The micro-engineered mechanical oscillator is
coupled with the cavity mode by the radiation pressure, which generates a
nonlinear interaction between the photon of cavity mode and the phonon of
nano-mechanical oscillation. The influence of mirror motion on the QPT for an
optomechanical Dicke model is investigated by the semi-classical steady-state
analysis including also the dissipation damping AggarwalN . In a recent review
paper the cavity quantum electrodynamics is presented for ultracold atoms in
optical and optomechanical cavities DebnathK . Recently the variational
ground-state and related QPT for a BEC trapped in an optomechanical cavity
were investigated LLL13  with the Holstein-Primakoff transformation under the
large-$N$ limit. It was also suggested that this QPT could be observed by
measuring the dynamics of nano-mechanical oscillator SSF10 .

We in this paper study the ground-state properties of two-level atoms in an
optomechanical cavity by means of the recently developed spin coherent-state
Rad71 ; MVe03 ; AHa12  (SCS) variational method LZL12 ; ZLL14 . This method is
not only valid for arbitrary atom number $N$ but also has advantages to
include the inverted pseudospin state ($\Uparrow$), which was firstly
considered BMS12  in the nonequilibrium dynamics of Dicke model. The SCS
variational method is universal from the $N$-atom Dicke model to one-atom Rabi
model. We report the mechanical-oscillator induced collapse of the SP and the
related multiple Dicke phase transitions for the finite $N$ different from the
QPT with $N$ tending to infinity..

##  II Spin coherent-state variational method

The optomechanical system consists of a high-finesse single-mode optical
cavity of frequency $\omega$ with a fixed mirror and a movable mirror, which
is coupled to a mechanical oscillator PHH99 . We assume that $N$ two-level
87Rb atoms with transition frequency $\omega_{a}$ are trapped in the quantized
cavity shown schematically in Fig. 1. Although the QPT has been realized
experimentally with an external pump laser BGB10 ; DEP07 ; CSD07  we in this
paper consider only the simple Dicke-model cavity in order to demonstrate the
effect of oscillator in a clear manner. The optomechanical cavity with
$N$-atom can be described by the following Hamiltonian BGB10 ; DEP07 ; ABh09
(with the convention $\hbar$ $=$ $1$):

$H=H_{DM}+\omega_{b}b^{{\dagger}}b-\frac{\zeta}{\sqrt{N}}\left(b^{{\dagger}}+b\right)a^{{\dagger}}a\text{,}$
(1)

in which

$H_{DM}=\omega
a^{{\dagger}}a+\omega_{a}J_{z}+\frac{g}{2\sqrt{N}}\left(a^{{\dagger}}+a\right)\left(J_{+}+J_{-}\right)$
(2)

is the standard Dicke model Hamiltonian WHi73 ; Hio73 . Where $a^{{\dagger}}$
($a$) is the photon creation (annihilation) operator and $b$ ($b^{{\dagger}}$)
is the phonon annihilation (creation) operator of the single vibrational mode
of the nano-oscillator. The ensemble of $N$ atoms is represented by collective
spin operators $J_{z}$, $J_{\pm}$, which satisfy the angular momentum
commutation relations $\left[J_{+},J_{-}\right]=2J_{z}$ and $[J_{z},J_{\pm}]$
$=$ $\pm J_{\pm}$ with eigenvalue $j=N/2$. $g$ denotes the collective atom-
field coupling strength. $\zeta$ is the coupling constant between the nano-
oscillator and cavity mode via radiation pressure KV08 ; MG09 ; FK09 ; AGHK10
; HWAH11 . A three-body interaction term AKM ; SimonG ; SantosJP ; ChangY
denoted by $\chi(b^{{\dagger}}+b)(a^{{\dagger}}+a)J_{x}$ in Hamiltonian (1) is
removed by specific choice of the coupling parameters SantosJP  $\zeta$,
$\chi$, since only the two-body (photon, phonon) radiation-pressure
interaction is realized experimentally SimonG .

Figure 1: Schematic diagram for $N$ atoms in a single-mode optical cavity of
frequency $\omega$. The cavity consists of a fixed mirror and a movable
mirror, which is coupled to a mechanical oscillator.

We begin with the average of Hamiltonian Eq. (1) in boson-operator states only

$\overset{\\_}{H}\left(\alpha,\beta\right)=\langle u|H|u\rangle\text{,}$ (3)

where the trial wave-function

$|u\rangle=\left|\alpha\right\rangle\left|\beta\right\rangle\text{,}$

is considered as the direct product of photon and phonon coherent states
$a\left|\alpha\right\rangle=\alpha\left|\alpha\right\rangle$,
$b\left|\beta\right\rangle=\beta\left|\beta\right\rangle$. In this average
both photon and phonon operators are replaced by the complex boson-operator
eigenvalues such that $\langle\alpha|a\left|\alpha\right\rangle=\alpha$,
$\langle\alpha|a^{{\dagger}}\left|\alpha\right\rangle=\alpha^{\ast}$;
$\langle\beta|b\left|\beta\right\rangle=\beta$,
$\langle\beta|b^{{\dagger}}\left|\beta\right\rangle=\beta^{\ast}$. Moreover we
parameterize the complex eigenvalues $\alpha$, $\beta$ as

$\alpha=\gamma e^{i\eta}\text{,\qquad}\beta=\rho e^{i\xi}\text{.}$

After the average in the boson coherent-state the Hamiltonian becomes

$\overset{\\_}{H}\left(\alpha,\beta\right)=(\omega-\frac{2\zeta\rho\cos\xi}{\sqrt{N}})\gamma^{2}+\omega_{b}\rho^{2}+H_{sp}\text{,}$
(4)

in which

$H_{sp}=\omega_{a}J_{z}+\frac{g\gamma\cos\eta\left(J_{+}+J_{-}\right)}{\sqrt{N}}$

is an effective spin-Hamiltonian. Different from the variational method with
the Holstein-Primakoff transformation, we in the following are going to
diagonalize the spin Hamiltonian $H_{sp}\left(\alpha,\beta\right)$ in terms of
the SCSs $|\mp\mathbf{n}\rangle$ such that

$H_{sp}\left(\alpha,\beta\right)|\mp\mathbf{n}\rangle=E_{sp}^{\mp}|\mp\mathbf{n}\rangle\text{.}$
(5)

We can visualize $|\mp\mathbf{n}\rangle$ as two eigenstates of a projection
angular momentum operator, $\mathbf{J}\cdot\mathbf{n|\mp n}\rangle=\mp j$
$|\mp\mathbf{n}\rangle$, with
$\mathbf{n}=\left(\sin\theta\cos\varphi,\sin\theta\sin\varphi,\cos\theta\right)$
being the unit vector. The directional angles $\theta$ and $\varphi$ as
unknown parameters are to be determined from the eigenstate equation Eq. (5).
The SCSs $|\mp\mathbf{n}\rangle$ called respectively the normal ($\Downarrow$)
and inverted ($\Uparrow$) pseudospin states BMS12  can be generated from the
extreme Dicke state $\left|j,-j\right\rangle$ ( $J_{z}$
$\left|j,-j\right\rangle$ $=-j\left|j,-j\right\rangle$) with the SCS
transformation, $\left|\mp\mathbf{n}\right\rangle=U(\mathbf{n})\left|j,\mp
j\right\rangle$. The unitary operator is given by ACG72

$U(\mathbf{n})=e^{\frac{\theta}{2}(J_{+}e^{-i\varphi}-J_{-}e^{i\varphi})}\text{.}$
(6)

In the SCS, spin operators satisfy the minimum uncertainty relation, for
example, $\Delta J_{+}\Delta J_{-}=\left\langle J_{z}\right\rangle/2$ and
therefore it is called the macroscopic quantum state. Applying the unitary
transformation Eq. (6) to the eigenvalue equation Eq. (5) and using the
unitary transformation relations

$\left\\{\begin{array}[]{c}U^{\dagger}J_{z}U=J_{z}\cos\theta+\frac{1}{2}e^{i\varphi}J_{-}\sin\theta+\frac{1}{2}e^{-i\varphi}J_{+}\sin\theta\text{,}\\\
U^{\dagger}J_{+}U=J_{+}\cos^{2}\frac{\theta}{2}-e^{i\varphi}J_{z}\sin\theta-\frac{1}{2}J_{-}e^{2i\varphi}\sin^{2}\frac{\theta}{2}\text{,}\\\
U^{\dagger}J_{-}U=J_{-}\cos^{2}\frac{\theta}{2}-e^{-i\varphi}J_{z}\sin\theta-\frac{1}{2}J_{+}e^{-2i\varphi}\sin^{2}\frac{\theta}{2}\text{,}\end{array}\right.$

it is easy to realize that the SCSs $\left|\mp\mathbf{n}\right\rangle$ indeed
are the eigenfunctions of effective spin-Hamiltonian
$H_{sp}\left(\alpha,\beta\right)$ if the following conditions

$\left\\{\begin{array}[]{c}\frac{\omega_{a}}{2}e^{-i\varphi}\sin\theta+\frac{g\gamma}{\sqrt{N}}\cos\eta\left(\cos^{2}\frac{\theta}{2}-e^{-2i\varphi}\sin^{2}\frac{\theta}{2}\right)=0\text{,}\\\
\frac{\omega_{a}}{2}e^{i\varphi}\sin\theta+\frac{g\gamma}{\sqrt{N}}\cos\eta\left(\cos^{2}\frac{\theta}{2}-e^{2i\varphi}\sin^{2}\frac{\theta}{2}\right)=0\text{,
}\end{array}\right.$ (7)

are satisfied. The energy eigenvalues are found as

$E_{sp}^{\mp}=\mp\frac{N}{2}A\left(\alpha,\theta,\varphi\right)\text{,}$

in which the parameter function is defined by

$A\left(\alpha,\theta,\varphi\right)=\omega_{a}\cos\theta-\frac{2g}{\sqrt{N}}\gamma\cos\eta\cos\varphi\sin\theta\text{.}$

The angle parameters $\theta$ and $\varphi$ can be determined from Eq. (7). We
obtain after a tedious algebra

$A\left(\gamma\right)=\omega_{a}\sqrt{1+f^{2}(\gamma)}\text{,}$ (8)

which becomes a one parameter function only with

$f(\gamma)=\frac{2g}{\omega_{a}\sqrt{N}}\gamma\text{.}$

The total trial wave-function CNL11 ; LLM96 ; CLS04  is a direct product of
the SCSs $\left|\mp\mathbf{n}\right\rangle$ and the boson coherent-states
$|u\rangle$

$\left|\psi_{\mp}\right\rangle=|u\rangle\left|\mp\mathbf{n}\right\rangle\text{.}$
(9)

The variational energy-function in the trial state is evaluated as

$\displaystyle E_{\mp}\left(\gamma,\rho,\xi\right)$ $\displaystyle=$
$\displaystyle\langle\psi_{\mp}|H\left|\psi_{\mp}\right\rangle$ (10)
$\displaystyle=$
$\displaystyle\omega\gamma^{2}-\frac{2\zeta\rho\cos\xi}{\sqrt{N}}\gamma^{2}+\omega_{b}\rho^{2}$
$\displaystyle\mp\frac{N}{2}A\left(\gamma\right)\text{,}$

which depends on three variational parameters $\gamma$, $\rho$, and $\xi$. The
ground state can be determined by the variation of energy function
$E_{\mp}\left(\gamma,\rho,\xi\right)$ with respect to the three variational
parameters. The energy functions Eq. (10) are valid for any atom number $N$
unlike the variational method with the Holstein-Primakoff transformation, in
which large-$N$ limit is required.

##  III Ground state and phase diagram

The ground state is considered as the variational minimum of energy function
$E\left(\gamma,\rho,\xi\right)$ with respect to the variation parameters
$\gamma$, $\rho$, $\xi$. From the usual extremum condition of the energy
function $\partial E/\partial\xi=0$, and $\partial E/\partial\rho=0$ we find
the relation

$\text{ \ }\rho=\frac{\zeta\gamma^{2}}{\sqrt{N}}\text{,}$ (11)

and the isolated parameter $\xi$ is determined as $\cos\xi=1$. Replacing the
parameter $\rho$ in the energy function Eq. (10) by the relation Eq. (11) the
variation-energy becomes a one-parameter function only

$E_{\mp}(\gamma)=\omega\gamma^{2}-\frac{\zeta^{2}}{N\omega_{b}}\gamma^{4}\mp\frac{N}{2}A\left(\gamma\right)\text{.}$
(12)

The extremum condition of energy function $\partial E_{\mp}/\partial\gamma=0$
possesses always a zero photon-number solution ($\gamma=0$), which is called
the NP, if it is stable with a positive second-order derivative

$\frac{\partial^{2}E_{\mp}(\gamma=0)}{\partial\gamma^{2}}=2\left(\omega\mp\frac{g^{2}}{\omega_{a}}\right)>0\text{.}$
(13)

From the stability condition Eq. (13) it is easy to find that the NP state for
the normal spin ($\Downarrow$) denoted by $N_{-}$ exists only when

$g<g_{c}=\sqrt{\omega\omega_{a}}\text{,}$

where $g_{c}$ is the well known critical point EBr03 ; ETB03  of the phase
transition between NP and SP in the Dicke model. The zero-photon solution for
the inverted spin ($\Uparrow$) denoted by $N_{+}$ exists, however, in the
whole region of $g$. The mechanical oscillator does not affect the NP and the
critical point $g_{c}$ at all. The extremum condition of energy function for
the nonzero photon solution

$\frac{\partial E_{\mp}(\gamma)}{\partial\gamma}=0$

becomes effectively a cubic power equation of the variable
$\overset{\\_}{\gamma}^{2}=\gamma^{2}/N$ seen to be

$p_{\mp}(\overset{\\_}{\gamma})=\omega-\frac{2\zeta^{2}\overset{\\_}{\gamma}^{2}}{\omega_{b}}\mp\frac{g^{2}}{A(\overset{\\_}{\gamma})}=0\text{,}$
(14)

where
$A(\overset{\\_}{\gamma})=\omega_{a}\sqrt{1+f^{2}(\overset{\\_}{\gamma})}$
with $f(\overset{\\_}{\gamma})=2g\overset{\\_}{\gamma}/\omega_{a}$. The
Eq.(14) can be solved graphically. According to the experimental parameters
BGB10 ; KMB11 , we set the atom resonant frequency $\omega_{a}=$ $1$ $MHz$ and
the collective atom-field coupling strength $g$ should be in the same order
BGB10 ; KMB11  to realize the Dicke phase transition. The frequency of nano-
oscillator $\omega_{b}$ ranged from megahertz to gigahertz AAU09  and the
photon-phonon coupling constant $\zeta$ is of the order of megahertz AKM . In
the numerical evaluation the coupling constants, field frequency, and energy
are measured in the unit of atom frequency $\omega_{a}$ throughout the paper.
We assume $\omega_{b}=10$ in this paper. Fig. 2 displays the plots of
polynomial $p_{\mp}(\overset{\\_}{\gamma})$ and scaled energy

$\varepsilon_{\mp}(\overset{\\_}{\gamma})=\frac{E_{\mp}(\gamma)}{N}$ (15)

as functions of the variable $\overset{\\_}{\gamma}$ for a given photon-phonon
coupling $\zeta=1$ and various atom-field coupling values to show the
$g$-dependence of the solutions. The second-order derivative of the energy
function

$\frac{\mathbf{\partial}^{2}\mathbf{\varepsilon}_{\mp}\left(\overset{\\_}{\gamma}\right)}{\mathbf{\partial}\overset{\\_}{\gamma}^{2}}=2\left(\omega\mathbf{-}\frac{6\zeta^{2}\overset{\\_}{\gamma}^{2}}{\omega_{b}}\mathbf{\mp}\frac{g^{2}\omega_{a}^{\frac{1}{2}}}{A^{\frac{3}{2}}(\overset{\\_}{\gamma})}\right)$
(16)

serves as the stability condition of the nonzero photon state. The solution of
energy extremum condition Eq. (14) denoted by $\gamma_{s}^{-}$ is a stable
state, which is the root of the polynomial $p_{-}(\overset{\\_}{\gamma})$ with
a positive slope of the curve [Fig. 2(a2)] and, therefore, the positive
second-order derivative of the energy function
$\partial^{2}\mathbf{\varepsilon}_{-}\left(\overset{\\_}{\gamma}\right)/\partial\overset{\\_}{\gamma}^{2}>0$.
The state $\gamma_{s}^{-}$, which is a local minimum of the energy function
[Fig. 2(b2)], is called the SP in the phase diagram. The higher value solution
denoted by $\gamma_{us}^{-}$ [Fig. 2(a2)] with a negative slope (namely the
negative second-order derivative of the energy function) is a unstable state
corresponding to the local maximum of the energy function (b2). For the fixed
photon-phonon coupling $\zeta=1$, two solutions move close to each other with
the increase of the atom-field coupling $g$ and finally coincide at a critical
value $g_{t}=1.763$ (a3) called the turning point, where the energy function
becomes a flexing point (b3). Above this point the SP no longer exists. The
unstable state $\gamma_{us}^{-}$ extends also to the NP region below the
critical point $g_{c}$ seen from Fig. 2(a1), (b1). The variation of the SP-
state $\gamma_{s}^{-}$ with the photon-phonon coupling $\zeta$ is shown in the
Fig. 3 for a fixed atom-field coupling $g=1.5$. The two solutions
$\gamma_{s}^{-}$ and $\gamma_{us}^{-}$ also move close to each other with the
increase of $\zeta$. Particularly when the photon-phonon coupling reaches a
critical value $\zeta=1.203$ the two states $\gamma_{s}^{-}$,
$\gamma_{us}^{-}$ coincide and the energy curve becomes a flexing point
corresponding to the turning point $g_{t}=1.5$ seen from Fig. 4.

Figure 2: (Color online) Graphical solution of the energy extremum equations
$p_{-}(\overset{-}{\gamma})=0$ (upper panel), $p_{+}(\overset{-}{\gamma})=0$
(lower panel) for given photon-phonon coupling $\zeta=1.0$ and different atom-
field couplings $g=0.8$ (a1)(c1), $1.5$ (a2)(c2), $1.763$ (a3)(c3). The
corresponding average-energy curves are plotted in (b1)(d1), (b2)(d2),
(b3)(d3). $\gamma_{s}^{-}$ denotes stable nonzero photon solution of normal
spin ($\Downarrow$) with a local energy minimum and $\gamma_{us}^{-}$ is the
unstable one (energy maximum). There is no any stable solution for the
inverted spin ($\Uparrow$) but the unstable one denoted by $\gamma_{us}^{+}$.
Figure 3: (Color online) $\zeta$-dependence of the nonzero photon solution of
the energy extremum equation $p_{-}(\overset{\\_}{\gamma})=0$ (a) and the
corresponding energy curve $\varepsilon_{-}$ (b) with $\zeta=1.0$ (1), $1.203$
(2), for the fixed atom-field coupling $g=1.5$. With the increase of $\zeta$
the stable and unstable solutions move close to each other and finally
coincide at the turning point seen from (a1) (a2).

For the inverted spin ($\Uparrow$) we only find the unstable nonzero photon
state denoted by $\gamma_{us}^{+}$ in the system considered [Fig. 2(c)]. Fig.
4 is the phase diagram of $g$-$\zeta$ plane obtained from the extremum Eq.
(14) and stability condition Eq. (16) with $\omega=1$. The mechanical
oscillator does not affect the NP and the critical point $g_{c}$ between NP
and SP. The SP is, however, bounded by the turning-point line $g_{t}$, beyond
which the SP collapses by the resonant damping of the nano-oscillator. In the
region above the turning-point line $g_{t}$ the zero photon solution $N_{+}$
of Eq. (14) still exists for the inverted spin ($\Uparrow$) and becomes the
ground state. This NP with the state $N_{+}$ is denoted by the phase notation
$NP(N_{+})$ in the phase diagram (Fig. 4). Thus the system undergoes an
additional phase transition at the turning point $g_{t}$ from the SP to the
$NP(N_{+})$.

Figure 4: Phase diagram in $g$-$\zeta$ plane. The SP is restricted by two
boundary lines of $g_{c}$ and $g_{t}$. When $\zeta=0$ the phase transition
reduces exactly to that of Dicke model from the $NP(N_{-})$ to the SP at the
critical point $g_{c}$. The region of SP extends from $g_{c}$ to infinity. On
the other hand the region of SP decreases with the increase of $\zeta$ and
disappears completely when $\zeta=3.0$ resulting in the direct transfer
between two NP-states $N_{-}$ and $N_{+}$.

The turning point shifts back to the lower value direction of atom-field
coupling $g$ with the increase of photon-phonon coupling $\zeta$ and the SP
disappears completely at the value $\zeta=3$.

##  IV  Collapse of the superradiant phase and atomic population transfer

The nano-oscillator induces collapse of the SP due to resonant damping. To see
detail of the effect we study the variations of average photon number, energy
and atomic population with respect to the coupling constants $g$, $\zeta$. The
mean photon number in the SP is obviously

$n_{p}=\frac{\left\langle\alpha|a^{{\dagger}}a|\alpha\right\rangle}{N}=\left(\gamma_{s}^{-}\right)^{2}\text{,}$
(17)

here $\gamma_{s}^{-}$ means the corresponding $\overset{\\_}{\gamma}$-value in
the SP of normal spin ($\Downarrow$) as indicated in Figs. 2 and 3. The atomic
population difference is evaluated from average of the collective pseudospin
operator in the normal spin state $|-\mathbf{n\rangle}$

$\displaystyle\Delta n_{a}$ $\displaystyle=$
$\displaystyle\frac{\left\langle-\mathbf{n}|J_{z}|-\mathbf{n}\right\rangle}{N}=\frac{\left\langle
j,-j|U^{{\dagger}}(\mathbf{n})|J_{z}U(\mathbf{n})\mathbf{|}j,-j\right\rangle}{N}$
(18) $\displaystyle=$
$\displaystyle-\frac{1}{2\sqrt{1+f^{2}(\gamma_{s}^{-})}}\text{,}$

which becomes the known value in the NP, that $\Delta
n_{a}(\gamma_{-}=0)=-1/2$. While the atomic population difference in the state
$N_{+}$ is $\Delta n_{a}(\gamma_{+}=0)=1/2$ indicating the atomic population
inversion. The average of phonon number is proportional to the square of
photon number that

$n_{b}=\frac{\left\langle\beta|b^{{\dagger}}b|\beta\right\rangle}{N}=\frac{\zeta^{2}}{\omega_{b}^{2}}n_{p}^{2}\text{.}$
(19)

Figure 5: (Color online) The photon-phonon coupling dependence of average
photon number $n_{p}$ (a), atomic population difference $n_{a}$ (b) and
average energy $\varepsilon$ (c). The typical phase transition of Dicke model
($\zeta=0$) is shown in (1), where $g_{c}$ is the critical point between
$NP$($N_{-}$) (black solid line) and $SP$($\gamma_{s}^{-}$) (black solid
line). The SP collapses at the turning point $g_{t}$ (2), (3) and the zero
photon state $N_{+}$ becomes the ground state (red solid line). The system
undergoes the second phase transition from SP to the $NP(N_{+})$. When
$\mathbf{\zeta=}3.0$ (4) the SP disappears and we see the direct atomic
population transfer between two states $N_{-}$ and $N_{+}$ [or spin flip
between normal ($\Downarrow$) and inverted ($\Uparrow$) spins]. Beyond the
turning point $g_{t}$ the nonzero photon state turns back to an upper branch
$\gamma_{us}^{-}$ (blue dotted line), which is unstable. $\gamma_{us}^{+}$
(red dotted line) denotes unstable state of inverted spin.

The average photon number can be considered as an order parameter with
$n_{p}>0$ denoting the SP and $n_{p}=0$ for the NP. We plot in Fig. 5 the
$n_{p}$ curves for different phonon-photon couplings $\zeta$ compared with the
normal Dicke model ($\zeta=0$) shown in Fig. 5(1) to demonstrate the nano-
oscillator induced effect. Below the critical point $g_{c}$ we have the
bistable zero photon states $N_{\mp}$ , in which the $N_{-}$ (black solid
line) with lower energy [Fig. 5(c)] is the ground state and $N_{+}$ is the
excited state (red dash line). The phase transition from $NP(N_{-})$ to SP of
the nonzero photon state $\gamma_{s}^{-}$ takes place at the critical point
$g_{c}$. In this region the zero photon state $N_{+}$ (red dash lines) is
still excited state with higher energy than superradiant state
$\gamma_{s}^{-}$ seen from Fig. 5(c1-c3). The SP collapses at the turning
point $g_{t}$ and the zero photon state $N_{+}$ of inverted spin ($\Uparrow$)
(the atomic population inversion state) shown in Fig. 5(b) becomes the ground
state (red solid line). An additional phase transition from SP to the
$NP(N_{+})$ appears at the turning point $g_{t}$. This multiple phase
transitions generated by the mechanical oscillator does not exist at all in
the usual Dicke model [Fig. 5(1)]. With the increase of photon-phonon coupling
$\zeta$ the region of SP is suppressed and the turning point $g_{t}$ shifts
back to the lower value direction of $g$ [Fig. 5(2,3)]. The SP disappears
completely at a critical value $\zeta=3$ and the phase transition becomes the
transition from the $NP(N_{-})$ to $NP(N_{+})$ within the same phase of zero
order-parameter $\gamma=0$ [Fig. 5(a4)] but different average energy [Fig.
5(c4)] and atomic population [Fig. 5(b4)]. We observe a interesting phenomenon
of atomic population transfer (or spin flip) between two atom levels (b4).
Since the atomic population inversion plays a important role in the laser
physics, the controllable population transfer certainly has technical
applications. Beyond the turning point $g_{t}$ the nonzero photon state turns
back to a upper branch state $\gamma_{us}^{-}$ (blue dotted lines) so that we
have a dual state for one value of $g$. This is similar to the optical
bistability, however, the state $\gamma_{us}^{-}$ is unstable in the
considered system. The unstable state $\gamma_{us}^{+}$ (red dotted lines) of
inverted spin ($\Uparrow$) possesses more higher values of photon-number,
atomic population, and average energy as well.

It may be worthwhile to emphasize again that the SCS variational method LLL13
has advantage to avoid the thermodynamic limit and the Dicke phase transition
demonstrated in this paper is valid for arbitrary atom number $N$. We in the
following section compare our results with those in the literature.

##  V Dicke phase transition for Rabi model

The Rabi model describes a two-level atom in a single-mode cavity Rab36  and
has been widely used in atomic, optic, and condensed matter physics SZu97 .
The validity of the model has been experimentally verified in various systems
RBH01 ; LBM03 ; NPT01 ; FLM10 ; Lon11 ; CLO12 ; LWe11 . In the absence of the
mechanical oscillator ($\zeta=0$) the Hamiltonian Eq. (1) of optomechanical
cavity reduces exactly to that of the Rabi model for $N=1$. The energy
function has a simple form LZL12 ; LLM96 ; CLS04  (we in this section consider
only the normal spin ($\Downarrow$) and neglect the subscript ”-” for the
simplicity). The energy function becomes simply

$E\left(\gamma\right)=\omega\gamma^{2}-\frac{A(\gamma)}{2}\text{.}$ (20)

The extremum equation $\partial E(\gamma)/\partial\gamma=0$ gives rise to the
stable zero-photon solution $\gamma=0$, namely the NP, below the critical
point

$g_{c}=\sqrt{\omega\omega_{a}}.$

The photon number in the SP above $g_{c}$ is found as

$n_{p}=\langle\alpha|a^{{\dagger}}a|\alpha\rangle=\gamma_{s}^{2}=\frac{1}{4}\left(\frac{g^{2}}{\omega^{2}}-\frac{\omega_{a}^{2}}{g^{2}}\right),$
(21)

which is exactly the same as the average photon number $n_{p}$
=$\langle\alpha|a^{{\dagger}}a|\alpha\rangle/N$ in the normal Dicke model
ETB03 ; CLL06 ; LZL12  with $N$ atoms. The ground state energy

$E\left(g\right)=\left\\{\begin{array}[]{c}-\frac{\omega_{a}}{2}\text{, }g\leq
g_{c}\text{,}\\\
-\frac{\omega}{4}\left(\frac{g^{2}}{\omega^{2}}+\frac{\omega_{a}^{2}}{g^{2}}\right)\text{,
}g>g_{c}\text{.}\end{array}\right.$ (22)

equals also to the average energy of Dicke model ETB03 ; CLL06 ; LZL12 . We
present the energy curves obtained from the analytic formula of Eq. (22) and
the numerical diagonalization together in Fig. 6. We see the perfect
coincidence in a wide range of coupling value and for both the red and blue
detuning. The small deviations come out in the large detuning as shown by the
dashed lines. The deviation decreases with the increase of photon-number $n$
of Fock state in the numerical diagonalization. The phase transition indeed
exists in the Rabi model in agreement with the recent observation HPP15 .

Figure 6: (Color online) The energy curves obtained from both analytic formula
Eq.(22) and the numerical diagonalization with Fock state up to $300$-photon
for the red detuning (a) and blue detuning (b). The deviation of numerical-
diagonalization line from the analytic result is indicated by the dashed line.

##  VI Conclusion and discussion

The SCS variational method is a universal tool in the study of ground-states
for various atom-cavity systems with arbitrary atom number $N$. The Dicke
phase transition is a common phenomenon independent of the atom number $N$.
The only assumption is that the cavity field is in the coherent state or in
other words the macroscopic quantum state. The nano-oscillator coupled with
the cavity mode by radiation-pressure does not affect the phase transition
boundary from the NP to SP. However the SP state collapses at a turning point
due to the resonant damping of the oscillator. As a consequence of collapse
the zero-photon state-$N_{+}$ of inverted spin ($\Uparrow$) becomes a ground-
state and an additional phase transition from the SP to $NP(N_{+})$ takes
place at the turning point. Particularly a direct atomic population-transfer
between two atom-levels is realized by the manipulation of photon-phonon
coupling. This novel observation may have technical applications in laser
physics. In the absence of mechanical oscillator ($\zeta=0$) our result
recover the phase transition of Dicke model. The phase transition remains for
Rabi model with $N=1$ in agreement with the recent observation HPP15 . This
conclusion is also verified by the quantitative agreement of average energy
with the result of numerical diagonalization in a wide region of coupling
constant for both the red and blue detuning. The dual nonzero-photon states
similar to the optical bistability are also found, however, the upper branch
of state is unstable in the system considered.

##  VII Acknowledgments

This work was supported by the National Natural Science Foundation of China
(Grant Nos. 11275118, 11404198, 91430109), and the Scientific and
Technological Innovation Programs of Higher Education Institutions in Shanxi
Province (STIP) (Grant No. 2014102), and the Launch of the Scientific Research
of Shanxi University (Grant No. 011151801004), and the National Fundamental
Fund of Personnel Training (Grant No. J1103210). The natural science
foundation of Shanxi Province (Grant No. 2015011008).

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