# Stochastic thermodynamics of a finite quantum system coupled to two heat
baths

Heinz-Jürgen Schmidt1 and Jochen Gemmer1 1 Universität Osnabrück, Fachbereich
Physik, D - 49069 Osnabrück, Germany

###### Abstract

We consider a situation where an $N$-level system (NLS) is coupled
successively to two heat baths with different temperatures without being
necessarily thermalized and approaches a steady state. For this situation we
apply a general Jarzinski-type equation and conclude that heat and entropy is
flowing from the hot bath to the cold one. The Clausius relation between
increase of entropy and transfer of heat divided by a suitable temperature
assumes the form of two inequalities. Our approach is illustrated by an
analytical example. For the linear regime, i. e., for small temperature
differences between the two heat baths we derive an expression for the heat
conduction coefficient.

##  I Introduction

The study of the non-equilibrium thermodynamics of heat conduction has a long
history. As an example consider the sequence of papers of Lebowitz and Spohn
LS78 ; SL87 ; LS99  dealing with both classical and quantum aspects of this
problem. A considerable number of papers concerned with thermal conduction for
quantum systems consider a kind of chain of small quantum systems connected to
the two heat baths at their ends, and focus on the questions whether a local
thermal equilibrium is reached in the small quantum systems, whether a
constant temperature gradient exists and whether the heat transfer rate is
proportional to this gradient (“Fourier’s law"), see, e. g., STM96 ; MHGM03 ;
S03 ; MGM04 ; MMG05 ; JHR06 ; WHBGM07 ; P11 .

The present paper is not concerned with the validity of Fourier’s law for
special systems, but rather assumes a general $N$-level system (NLS) with
alternating contact with two heat baths and studies the heat and entropy
transfer in the steady state assumed after some time, without further
analyzing the internal structure of the NLS. For this purpose we use new
methods that have been developed during the last decades, in particular, the
approach via fluctuation theorems, see, e. g. LS99 . The famous Jarzynski
equation represents one of the rare exact results in nonequilibrium
statistical mechanics. It is originally a statement of the expectation value
of the exponential of the work $\left\langle e^{-\beta\,w}\right\rangle$
performed on a system initially in thermal equilibrium with inverse
temperature $\beta$, but may be far from equilibrium after the work process.
This equation was first formulated for classical systems BK77 ; BK81 ; J97
and subsequently proved for quantum systems P00 ; K00 ; T00 ; M03 . Extensions
for systems initially in local thermal equilibrium T00 , micro-canonical
ensembles TMYH13 , and large-canonical ensembles SS07 ; SU08 ; AGMT09 ; YTC11
; CHT11 ; E12 ; YKT12  have been published.

The most common interpretation of the quantum Jarzynski equation is to
consider sequential measurements, see P00 . This approach is also followed in
the present work. The general framework for such an approach was outlined in
SG20a  and SG20b . It is neither quantum mechanical nor classical per se and
will be referred to as “stochastic thermodynamics” in the present work. We
have recently applied this method to the problem of the interaction of an NLS
with a single heat bath SSG21 . Here we will extend this work to the
interaction with two heat baths of possibly different temperature. Again, it
is possible to describe the influence of the heat baths by an $N\times N$
stochastic matrix $T$ that gives the conditional probability of transitions
between the $N$ levels of the NLS. Moreover, it is sensible to identify the
steady state regime with fixed points of $T$.

However, it is not readily possible to infer the heat flow between the NLS and
heat baths from the change in energy of the NLS, analogously for the entropy
flow. At steady state the corresponding changes of energy and entropy would
vanish although the flow may be non-zero. To cope with this problem we
consider a step-wise interaction with the heat baths, see Figure 1: We start
with a preparatory energy measurement and a contact only with the first heat
bath, followed by a second energy measurement, together described by a
stochastic matrix $T^{(0)}$. This is followed by a separate contact with the
second heat bath and another measurement, together described by $T^{(1)}$. The
total interaction plus measurements is hence given by the product matrix
$T=T^{(1)}\,T^{(0)}$. Moreover, we will assume, similarly as in SSG21 , that
the single stochastic matrices $T^{(i)}$ have invariant probability
distributions of Gibbs type thereby introducing the two inverse temperatures
$\beta_{i},\;i=0,1$ of the heat baths. This enables us to investigate the
dependance between direction of heat and entropy flow and the temperature
difference, at least for the steady state. The main message is that, although
it would be very difficult to calculate the stochastic matrices $T^{(0)}$ and
$T^{(1)}$ for “real" systems, the validity of the second law-like statement
about the correct direction of heat and entropy flow only depends on some
general properties of these matrices and not of their particular entries.

The paper is organized as follows. In the following Section II we present the
pertinent definitions and assumptions that characterize the present approach
to the heat conduction problem. The main result is obtained in Section III
where we formulate the Jarzynski-type equation suited for the present problem
and apply Jensen’s inequality to derive two Clausius inequalities. These
inequalities immediately imply that, in the steady state and in the
statistical mean, heat and entropy always flow from the hotter to the colder
bath. Section IV is devoted to an analytical example where a three-level
system is alternately coupled to two harmonic oscillators with different
temperatures and the heat and entropy flows mentioned above can be explicitly
calculated by computer-algebraic means. Moreover, in this example the steady
state $p^{(s)}$ is characterized by two parameters $\beta^{(s)}$ and
$\gamma^{(s)}$, $\beta^{(s)}$ being some sort of inverse temperature and
$\gamma^{(s)}$ characterizing the deviation of the steady state from a Gibbs
state, and the dependence of $\beta^{(s)}$ and $\gamma^{(s)}$ on the inverse
temperatures of the two heat baths can be explicitly determined. In the linear
regime $\beta_{0}\approx\beta_{1}$, considered in Section V, the steady state
$p^{(s)}$ will only infinitesimally deviate from the Gibbs state of inverse
temperature $\beta_{0}$. This deviation can be calculated by means of first
order perturbation theory and used to determine the common slope $a$ of the
functions of entropy transfer and (reduced) heat transfer at
$\beta_{1}=\beta_{0}$. The result, of course, depends of the eigenvalues and
eigenvectors of $T^{(0)}$ which can be obtained for the analytical example of
Section IV but probably not, or only approximately, for “real" systems. We
close with a summary and outlook in Section VI.

Figure 1: Schematic sketch of one cycle of the alternate interaction of the
NLS (black squares) with two heat baths of inverse temperature
$\beta_{0}<\beta_{1}$, resp., interspersed with energy measurements (yellow
${\mathcal{M}}$). At steady state the heat $\Delta Q$ absorbed by the NLS
during contact with the first heat bath equals in amount the heat emitted
during contact with the second heat bath.

##  II Basic definitions

We consider an $N$-level system (NLS) described by a finite index set
${\mathcal{N}}$, energies $E_{n}$ and degeneracies $d_{n}$ for
$n\in{\mathcal{N}}$. The Hamiltonian for the NLS without interaction with the
heat baths will hence be of the form

$H=\sum_{n}E_{n}\,{\sf P}_{n}\;,$ (1)

with a complete family of mutually orthogonal projections ${\sf
P}_{n},\;n\in{\mathcal{N}},$ and $d_{n}=\mbox{Tr }{\sf P}_{n}$.

The Gibbs states for the NLS with inverse temperature $\beta>0$ are
characterized by probabilities

$p_{n}^{(\beta)}=\frac{d_{n}}{Z^{(\beta)}}\,\exp\left(-\beta\,E_{n}\right)\;,$
(2)

where the partition function $Z^{(\beta)}$ is given by

$Z^{(\beta)}=\sum_{n}d_{n}\,\exp\left(-\beta\,E_{n}\right)\;.$ (3)

We assume a rather general initial state $\rho^{(i)}$ of the NLS of the form

$\rho^{(i)}=\sum_{n}p_{n}^{(i)}\frac{{\sf P}_{n}}{d_{n}}\;,$ (4)

that is not necessarily a Gibbs state. After the interaction with the first
heat bath we perform a Lüders measurement of the energy that, without
selection according to the outcome, is assumed to yield the intermediate mixed
state

$\rho^{\prime}=\sum_{n}q_{n}^{(i)}\frac{{\sf P}_{n}}{d_{n}}\;,$ (5)

such that the transition between the probability distributions $p^{(i)}\mapsto
q^{(i)}$ is given by a left stochastic matrix $T^{(0)}$ characterizing the
first heat bath according to

$q^{(i)}_{n}=\sum_{m}T^{(0)}_{nm}\,p^{(i)}_{m},\quad\mbox{ for all
}n\in{\mathcal{N}}\;.$ (6)

Thereafter, an interaction with a second heat bath is assumed that, after a
second Lüders measurement of the energy, results in the final state

$\rho^{(f)}=\sum_{n}p_{n}^{(f)}\frac{{\sf P}_{n}}{d_{n}}\;,$ (7)

such that, analogously to the first step,

$p^{(f)}_{n}=\sum_{m}T^{(1)}_{nm}\,q^{(i)}_{m},\quad\mbox{ for all
}n\in{\mathcal{N}}\;,$ (8)

where $T^{(1)}$ is another (left) stochastic matrix characterizing the second
heat bath.

If we model the heat baths by quantum systems and their successive interaction
with the NLS by some unitary time evolution the form (5, 7) of the
intermediate state and the final state of the NLS does not follow
automatically but represents a crucial assumption of our approach. Only if the
NLS is non-degenerate, i. e., all $d_{n}=1$, then (5) and (7) will follow from
general principles. In other cases, these assumptions may fail or be only
approximately satisfied.

The stochastic matrices $T^{(0)}$ and $T^{(1)}$ generally possess invariant
probability distributions or fixed points, i. e., eigenvectors corresponding
to the eigenvalue $1$ with non-negative entries. At the moment we need not
assume that these probability distributions are unique, but we will rather
postulate that there exist invariant probability distributions of Gibbs type,
i. e.,

$\sum_{m}T^{(i)}_{nm}\,p_{m}^{(\beta_{i})}=p_{n}^{(\beta_{i})},\quad\mbox{ for
all }n\in{\mathcal{N}}\mbox{ and }i=0,1\;.$ (9)

It will be plausible to identify the inverse temperatures $\beta_{0}$ and
$\beta_{1}$ with the respective inverse temperatures of the heat baths:
According to the zeroth law of thermodynamics the interaction between a heat
bath and an NLS of the same temperature should not change the occupation
probabilities of the energy levels.

The total transition $p^{(i)}\mapsto q^{(i)}\mapsto p^{(f)}$ will be achieved
by the product matrix

$T=T^{(1)}\,T^{(0)}\;,$ (10)

which is again a stochastic matrix. In the case of $\beta_{0}=\beta_{1}$ the
Gibbs state probabilities $p^{(\beta_{0})}_{n}$ would form an invariant
probability distribution of $T$, but in the general case of
$\beta_{0}\neq\beta_{1}$ the fixed point of $T$ (not necessarily unique) will
be different from $p^{(\beta_{i})},\;i=0,1$. Starting with an arbitrary
initial distribution $p^{(i)}$ and iterating the above-described protocol of
successive interactions with the heat baths and energy measurements one would
obtain a sequence $T^{\nu}p^{(i)}$ of probability distributions that converge
towards a fixed point of $T$:

$p^{(s)}=\lim_{\nu\to\infty}T^{\nu}p^{(i)}\;.$ (11)

If the fixed point is not unique, $p^{(s)}$ will depend on the initial
distribution $p^{(i)}$. Physically, $p^{(s)}$ represents a steady state that
will be asymptotically assumed. The NLS transports heat and entropy between
the two heat baths, that are approximately unchanged (otherwise the stochastic
matrix $T$ could not be kept constant), see Figure 1. This is similar as for
the Carnot cycle of classical thermodynamics. The difference, however, is that
we do not consider additional extraction of work and allow for irreversible
processes, whereas the Carnot process is reversible.

The steady state distribution $p^{(s)}$ can be used to calculate the steady
flow of heat and entropy between the two heat baths due to the process
described above. A natural question is whether this flow is always directed
from the hotter bath to the colder one, as one would expect in accordance with
the second law of thermodynamics. This question will be addressed in the next
Section III.

##  III Direction of heat and entropy flow

We recall the general Jarzynski-type equation that holds for transitions
between $N$ states described by a stochastic matrix ${\sf T}$, and
interspersed with sequential measurements SSG21 :

$\left\langle\frac{{\sf p}^{(0)}_{n}\,{\sf q}_{m}}{{\sf p}^{(0)}_{m}\,{\sf
p}_{n}}\right\rangle=1\;.$ (12)

Here the terms in the bracket represent four random variables

$Y_{\nu}:{\mathcal{N}}\times{\mathcal{N}}\rightarrow{\mathbbm{R}}\;,$ (13)

defined on an “elementary event space" ${\mathcal{N}}\times{\mathcal{N}}$ with
probability function

$\displaystyle P$ $\displaystyle:$
$\displaystyle{\mathcal{N}}\times{\mathcal{N}}\rightarrow[0,1],$ (14)
$\displaystyle P(m,n)$ $\displaystyle:=$ $\displaystyle{\sf T}_{mn}{\sf
p}_{n}\;,$ (15)

where ${\sf p}$ is any positive probability distribution and
$\left\langle\ldots\right\rangle$ denotes the expectation value calculated by
means of the $P(m,n)$. The four random variables occurring in the Jarzynski-
type equation (12) are

$\displaystyle Y_{1}(m,n)$ $\displaystyle=$ $\displaystyle{\sf p}_{n},$ (16)
$\displaystyle Y_{2}(m,n)$ $\displaystyle=$ $\displaystyle{\sf
q}_{m}=\sum_{n}{\sf T}_{mn}\,{\sf p}_{n},$ (17) $\displaystyle Y_{3}(m,n)$
$\displaystyle=$ $\displaystyle{\sf p}_{n}^{(0)},$ (18) $\displaystyle
Y_{4}(m,n)$ $\displaystyle=$ $\displaystyle{\sf p}_{m}^{(0)},$ (19)

and ${\sf p}^{(0)}$ is a fixed point of ${\sf T}$, i. e.,

$\sum_{n}{\sf T}_{mn}\,{\sf p}^{(0)}_{n}={\sf p}^{(0)}_{m}\;,$ (20)

with positive components.

We will apply (12) in two ways: The first one is given by

${\sf T}=T^{(0)},\;{\sf p}_{n}=p_{n}^{(s)},\;\mbox{ and }{\sf
p}_{n}^{(0)}=p_{n}^{(\beta_{0})}\;,$ (21)

such that (20) holds according to (9). Using Jensens’s inequality (JI) and the
fact that $x\mapsto-\log x$ is a convex function, we conclude, analogously to
(SSG21, , (63 - 66)),

$\displaystyle 0$ $\displaystyle=$ $\displaystyle-\log
1\stackrel{{\scriptstyle(\ref{Jar})}}{{=}}-\log\left\langle\frac{p_{n}^{(\beta_{0})}\,q_{m}^{(s)}}{p_{m}^{(\beta_{0})}\,p_{n}^{(s)}}\right\rangle$
(22) $\displaystyle\stackrel{{\scriptstyle(JI)}}{{\leq}}$
$\displaystyle\left\langle-\log\frac{p_{n}^{(\beta_{0})}}{d_{n}}+\log\frac{p_{m}^{(\beta_{0})}}{d_{m}}-\log\frac{q_{m}^{(s)}}{d_{m}}+\log\frac{p_{n}^{(s)}}{d_{n}}\right\rangle$
(23) $\displaystyle=$ $\displaystyle\left\langle\beta_{0}E_{n}+\log
Z^{(\beta_{0})}-\beta_{0}E_{m}-\log
Z^{(\beta_{0})}\right\rangle-\sum_{m}q_{m}^{(s)}\,\log\frac{q_{m}^{(s)}}{d_{m}}+\sum_{n}p_{n}^{(s)}\,\log\frac{p_{n}^{(s)}}{d_{n}}$
(24) $\displaystyle=$ $\displaystyle\left\langle\Delta
S\right\rangle-\beta_{0}\left\langle\Delta Q\right\rangle\;.$ (25)

Here we have used the obvious definitions of the random variables “heat
transfer"

$\Delta Q(m,n):=E_{m}-E_{n}\;,$ (26)

and “entropy transfer"

$\Delta S(m,n):=\log\frac{{\sf p}_{n}}{d_{n}}-\log\frac{{\sf
q}_{m}}{d_{m}}\;.$ (27)

Hence we have shown the following Clausius inequality:

$\left\langle\Delta S\right\rangle\geq\beta_{0}\left\langle\Delta
Q\right\rangle\;.$ (28)

Note that the choice ${\sf p}_{n}=p_{n}^{(s)}$ is not necessary to derive
(28); similarly as in SSG21  this Clausius inequality would hold for arbitrary
initial probability distributions. However, the choice of a steady state
distribution will be crucial to derive the second Clausius inequality, see
(35) below.

The second way of applying (12) will be given by the choice

${\sf T}=T^{(1)},\;{\sf p}_{n}=\sum_{m}T^{(0)}_{nm}\,p_{m}^{(s)},\;\mbox{ and
}{\sf p}_{n}^{(0)}=p_{n}^{(\beta_{1})}\;,$ (29)

such that, again, (20) holds according to (9). Due to the fixed point property
of $p^{(s)}$ with respect to $T=T^{(1)}\,T^{(0)}$ the choice (29) implies that
the role of ${\sf p}_{n}$ and ${\sf q}_{m}$ is interchanged. This means that
at steady state the NLS absorbs the same amount of heat during contact with
the first heat bath as it emits during the contact with the second one,
analogously for the entropy. It follows that (28) again holds, but with the
replacements

$\displaystyle\left\langle\Delta S\right\rangle$ $\displaystyle\mapsto$
$\displaystyle-\left\langle\Delta S\right\rangle,$ (30)
$\displaystyle\left\langle\Delta Q\right\rangle$ $\displaystyle\mapsto$
$\displaystyle-\left\langle\Delta Q\right\rangle,\quad\mbox{ and}$ (31)
$\displaystyle\beta_{0}$ $\displaystyle\mapsto$ $\displaystyle\beta_{1}\;.$
(32)

This gives

$\displaystyle-\left\langle\Delta S\right\rangle$ $\displaystyle\geq$
$\displaystyle-\beta_{1}\left\langle\Delta Q\right\rangle\;,$ (33)
$\displaystyle\left\langle\Delta S\right\rangle$ $\displaystyle\leq$
$\displaystyle\beta_{1}\left\langle\Delta Q\right\rangle\;,$ (34)

and thus, together with (28), we obtain the Clausius inequalities

$\beta_{0}\left\langle\Delta Q\right\rangle\leq\left\langle\Delta
S\right\rangle\leq\beta_{1}\left\langle\Delta Q\right\rangle\;.$ (35)

These inequalities have the same form as the corresponding inequalities
derived in SSG21  and, under different assumptions, in JW04 ; JR10 ; Sa20 ;
S78 ; AMH21 ; SW21 , but generally a different meaning: For example, in SSG21
the inverse temperature $\beta_{1}$ refers to the NLS before the interaction
with the heat bath of inverse temperature $\beta_{0}$, but in (35) the inverse
temperature $\beta_{1}$ refers to the second heat bath, whereas the NLS is in
the steady state and, in general, has no temperature at all.

Immediate consequences of (35) are

$\left(\beta_{1}-\beta_{0}\right)\left\langle\Delta Q\right\rangle\geq 0\;,$
(36)

and

$\displaystyle\left(\beta_{1}-\beta_{0}\right)\left\langle\Delta
S\right\rangle$ $\displaystyle=$ $\displaystyle\beta_{1}\,\left\langle\Delta
S\right\rangle-\beta_{0}\,\left\langle\Delta S\right\rangle$ (37)
$\displaystyle\stackrel{{\scriptstyle(\ref{Clausius})}}{{\geq}}$
$\displaystyle\beta_{1}\left(\beta_{0}\left\langle\Delta
Q\right\rangle\right)-\beta_{0}\left(\beta_{1}\left\langle\Delta
Q\right\rangle\right)$ (38) $\displaystyle=$ $\displaystyle 0\;.$ (39)

These equations imply that, in the steady state regime and in the statistical
mean, heat and entropy always flow from the hotter bath to the colder one,

Figure 2: Plot of the transferred entropy $\left\langle\Delta S\right\rangle$
per cycle (red curve) and reduced heat transfer $\beta_{0}\,\left\langle\Delta
Q\right\rangle$ (blue curve) and $\beta_{1}\,\left\langle\Delta
Q\right\rangle$ (dark yellow curve) as a function of the inverse temperature
$\beta_{1}$ of the second heat bath and fixed inverse temperature
$\beta_{0}=1$ of the first one. The common tangent at $\beta_{1}=\beta_{0}=1$
is indicated by the dashed black line with slope $a=0.173895$ that will be
calculated in Section V. The calculations have been made for the analytical
example of Section IV and steady state regime. Note that the Clausius
inequalities (35) are satisfied and that the signs of $\left\langle\Delta
S\right\rangle$ and $\left\langle\Delta Q\right\rangle$ coincide with the sign
of $\beta_{1}-\beta_{0}$ in accordance with the second law of thermodynamics.

##  IV Analytical example

As an example where the time average of the transition matrix ${\sf T}(\beta)$
corresponding to a heat bath of inverse temperature $\beta$ can be exactly
calculated we have considered in SSG21  a single spin with spin quantum number
$s=1$ coupled to a harmonic oscillator that serves as a heat bath. Hence we
have a $N=3$-level system with ${\mathcal{N}}=\\{1,0,-1\\}$. The total
Hamiltonian and further details are given in SSG21  and need not be repeated
here. The Hamiltonian strongly resembles the Jaynes-Cummings model JC63 ; GE90
originally describing the interaction of a $2$-level system with a quantized
radiation field but also considered for $3$-level systems AB95 ; TLV15 . We
reproduce the analytical result SSG21  for the entries of ${\sf T}(\beta)$:

$\displaystyle{\sf T}_{1,1}(\beta)$ $\displaystyle=$
$\displaystyle\frac{1}{32}\left(e^{\beta}-1\right)\left(\frac{12}{e^{\beta}-1}+8e^{\frac{\beta}{2}}\coth^{-1}\left(e^{\frac{\beta}{2}}\right)+3e^{-\beta}\Phi\left(e^{-\beta},2,\frac{3}{2}\right)-8\right),$
(40) $\displaystyle{\sf T}_{1,0}(\beta)$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left(1-2\sinh\left(\frac{\beta}{2}\right)\tanh^{-1}\left(e^{-\frac{\beta}{2}}\right)\right),$
(41) $\displaystyle{\sf T}_{1,-1}(\beta)$ $\displaystyle=$
$\displaystyle\frac{3}{32}e^{-3\beta
0}\left(4e^{\beta}-\left(e^{\beta}-1\right)\Phi\left(e^{-\beta},2,\frac{3}{2}\right)\right),$
(42) $\displaystyle{\sf T}_{0,1}(\beta)$ $\displaystyle=$
$\displaystyle\frac{1}{4}e^{\beta}\left(1-2\sinh\left(\frac{\beta}{2}\right)\tanh^{-1}\left(e^{-\frac{\beta}{2}}\right)\right),$
(43) $\displaystyle{\sf T}_{0,0}(\beta)$ $\displaystyle=$
$\displaystyle\frac{1}{2},$ (44) $\displaystyle{\sf T}_{0,-1}(\beta)$
$\displaystyle=$
$\displaystyle\frac{1}{4}e^{-\frac{3\beta}{2}}\left(e^{\frac{\beta}{2}}+\left(e^{\beta}-1\right)\tanh^{-1}\left(e^{-\frac{\beta}{2}}\right)\right),$
(45) $\displaystyle{\sf T}_{-1,1}(\beta)$ $\displaystyle=$
$\displaystyle\frac{3}{32}e^{-\beta}\left(4e^{\beta}-\left(e^{\beta}-1\right)\Phi\left(e^{-\beta},2,\frac{3}{2}\right)\right),$
(46) $\displaystyle{\sf T}_{-1,0}(\beta)$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left(2\sinh\left(\frac{\beta}{2}\right)\tanh^{-1}\left(e^{-\frac{\beta}{2}}\right)+1\right),$
(47) $\displaystyle{\sf T}_{-1,-1}(\beta)$ $\displaystyle=$
$\displaystyle\frac{1}{32}e^{-3\beta}\left(4e^{2\beta}\left(11\sinh\left(\beta_{0}\right)+5\cosh\left(\beta\right)-4\sinh\left(\frac{\beta}{2}\right)\coth^{-1}\left(e^{\frac{\beta}{2}}\right)-2\right)+3\left(e^{\beta}-1\right)\Phi\left(e^{-\beta},2,\frac{3}{2}\right)\right)\;.$

Here $\Phi(z,s,a):=\sum_{k\in{\mathbbm{N}}}\frac{z^{k}}{(k+a)^{s}}$ denotes
Lerch’s transcendent, see (NIST, , $\S\,25.14$). It can be shown that ${\sf
T}(\beta)$ is a left stochastic matrix and leaves the probability distribution

$p^{(\beta)}=\frac{1}{e^{-\beta}+1+e^{\beta}}\left(e^{-\beta},1,e^{\beta}\right)$
(49)

invariant that corresponds to a Gibbs state with inverse temperature $\beta$.

Figure 3: Plot of the first parameter $\beta^{(s)}$ characterizing the steady
state distribution $p^{(s)}$ of the analytical example as a function of
$\beta_{1}$ for fixed $\beta_{0}=1$ (blue curve). The slope
$a_{1}=0.694846\ldots$ of the tangent at $\beta_{1}=\beta_{0}$ (dark yellow
line) and the asymptotic value of
$\lim_{\beta_{1}\to\infty}\beta^{(s)}=1.66098\ldots$ (green line) have been
calculated analytically.  Figure 4: Plot of the second parameter
$\gamma^{(s)}$ characterizing the steady state distribution $p^{(s)}$ of the
analytical example as a function of $\beta_{1}$ for fixed $\beta_{0}=1$ (blue
curve). The value of $\gamma^{(s)}=\gamma_{0}=0.0658087\ldots$ for
$\beta_{1}=0$ and the asymptotic value of
$\lim_{\beta_{1}\to\infty}\gamma^{(s)}=0.405262\ldots$ (green line) have been
calculated analytically. For $\beta_{1}=\beta_{0}=1$ the parameter
$\gamma^{(s)}$ vanishes since the corresponding Gibbs-type probability
distribution is a fixed point of $T$. Interestingly, there exists a second
zero of $\gamma^{(s)}$ at $\beta_{1}=\beta^{\prime}=0.578258\ldots$. In the
interval $\beta^{\prime}<\beta_{1}<1$ the parameter $\gamma^{(s)}$ has small
negative values and hence the function $E_{n}\mapsto\log p_{n}^{(s)}$ is
slightly concave.

Using this analytical result we set $T^{(0)}={\sf T}(\beta_{0})$ and
$T^{(1)}={\sf T}(\beta_{1})$ for the model with two heat baths and can
explicitly calculate the heat and entropy transfer $\left\langle\Delta
S\right\rangle(\beta_{1})$ and $\beta_{i}\,\left\langle\Delta
Q\right\rangle(\beta_{1})$, considered in Section III, as functions of
$\beta_{1}$. Although the result is too complex to be shown here these
transfer functions can be plotted for fixed values of $\beta_{0}$, say,
$\beta_{0}=1$, see Figure 2. Thus one can graphically confirm the Clausius
inequalities (35) and the correct sign of heat and entropy flow. The three
functions have a common tangent at $\beta_{0}=\beta_{1}$ the slope of which
will be calculated in Section V.

In the limit $\beta\to 0$ we obtain the bi-stochastic matrix

$\lim_{\beta\to 0}{\sf T}(\beta)=\frac{1}{8}\left(\begin{array}[]{ccc}3&2&3\\\
2&4&2\\\ 3&2&3\\\ \end{array}\right)\;,$ (50)

whereas for $\beta\to\infty$

$\lim_{\beta\to\infty}{\sf
T}(\beta)=\frac{1}{6}\left(\begin{array}[]{ccc}3&0&0\\\ 1&3&0\\\ 2&3&6\\\
\end{array}\right)\;.$ (51)

Both limits facilitate the analytical calculation of corresponding asymptotic
values of the pertinent functions. By the way, the latter one (51) is an
example of a stochastic matrix where the algebraic and geometric degeneracy of
the eigenvalues no longer coincide: the eigenvalue $t_{2}=1/2$ is doubly
degenerate but possesses only a one-dimensional eigenspace
${\mathbbm{R}}\,(0,1,-1)^{\top}$.

Next we will investigate the structure of the steady state distribution
$p^{(s)}$ as a function of $\beta_{0}$ and $\beta_{1}$ for the present
example. A general probability distribution $p$ of a $3$-level system with
energies $E_{1},E_{0},E_{-1}$ can be characterized by two parameters
$\beta,\gamma$ such that

$p_{n}=\frac{1}{Z}\exp\left(-\beta E_{n}+\gamma E_{n}^{2}\right),\quad\mbox{
for }n=1,0,-1\;,$ (52)

where $Z$ is chosen such that $p_{1}+p_{0}+p_{-1}=1$. A Gibbs-type probability
distribution would thus be characterized by $\gamma=0$ and $\beta$ being the
inverse temperature of the distribution. It is then a straightforward task to
calculate the parameters $\beta^{(s)},\gamma^{(s)}$ corresponding to the
steady state distribution $p^{(s)}$ as a function of $\beta_{0}$ and
$\beta_{1}$. Again, the result is too complex to be shown, but it can be
plotted for fixed $\beta_{0}$, say, $\beta_{0}=1$, see Figures 3 and 4.
Surprisingly, we find that for $\beta_{0}=1$ and
$\beta_{1}=\beta^{\prime}=0.578258\ldots$ the second parameter $\gamma^{(s)}$
vanishes and hence the corresponding steady state distribution is of Gibbs
type.

##  V The linear regime

So far, we have not made any assumptions about the relationship of the
stochastic matrices $T^{(0)}$ and $T^{(1)}$. They can be completely
independent except that they have Gibbs-type fixed points. If $T^{(0)}$ and
$T^{(1)}$ were derived from a physical model of the interaction of the NLS
with a heat bath of inverse temperature $\beta$, as in the example of Section
IV, there would probably be further relations between $T^{(0)}$ and $T^{(1)}$,
but we have not needed to consider them so far. However, if we turn to the
linear regime where the value of $\delta:=\beta_{1}-\beta_{0}$ is
infinitesimally small, we will need the following mild assumption, that is
fortunately satisfied for the analytical example of Section IV:

###### Assumption 1

There exists a smooth family $\delta\mapsto\mathbf{T}(\delta)$ of (left)
stochastic matrices defined for all $\left|\delta\right|<r$ and some $r>0$
such that

$T^{(0)}=\mathbf{T}(0),\quad\mbox{ and}\quad
T^{(1)}=\mathbf{T}(\delta)=\mathbf{T}(0)+\delta\dot{\mathbf{T}}(0)+O(\delta^{2})\;.$
(53)

Here the overdot $\dot{}$ means
$\frac{\partial}{\partial\delta}\left(\ldots\right)\left.\right|_{\delta=0}$.
We will henceforward omit the argument $\delta=0$ for sake of simplicity. For
example, for the total cycle we have

$T=T^{(1)}\,T^{(0)}=\mathbf{T}(\delta)\,\mathbf{T}=\mathbf{T}^{2}+\delta\,\dot{\mathbf{T}}\,{\mathbf{T}}+O(\delta^{2})\;.$
(54)

The steady state distribution $p^{(s)}$ will also be chosen as a smooth family
denoted by ${\mathbf{p}}(\delta)$ such that $T\,p^{(s)}=p^{(s)}$ now reads

$\mathbf{T}(\delta)\,\mathbf{T}\,{\mathbf{p}}(\delta)={\mathbf{p}}(\delta)\;,$
(55)

and

${\mathbf{p}}(\delta)={\mathbf{p}}+\delta\,\dot{{\mathbf{p}}}+O(\delta^{2})\;,$
(56)

where ${\mathbf{p}}=p^{(\beta_{0})}$.

We will calculate $\dot{{\mathbf{p}}}$ by means of first order perturbation
theory. This is similar in concept to the derivation of a Kubo formula in
Liouville space in MGM04 , where the temperature gradient introduced by the
two heat baths is treated as an external perturbation. Since the unperturbed
matrix will, in general, not be symmetric, some slight modifications of the
Rayleigh-Schrödinger perturbation theory known from quantum mechanics are
required that will be derived in what follows. To simplify these calculations
we will restrict ourselves to the “generic case" of

###### Assumption 2

The eigenvalue $t_{1}=1$ of ${\mathbf{T}}$ is algebraically non-degenerate and
for the other eigenvalues $t_{2},\ldots$ the algebraic and geometric
degeneracy coincides.

Hence there exist a complete basis of ${\mathbbm{R}}^{N}$ of (right)
eigenvectors $\left|{\mathbf{e}}_{n}\right\rangle$ of ${\mathbf{T}}$
satisfying

${\mathbf{T}}\,\left|{\mathbf{e}}_{n}\right\rangle=t_{n}\,\left|{\mathbf{e}}_{n}\right\rangle\;,$
(57)

for $n=1,\ldots,N$, where
$\left|{\mathbf{e}}_{1}\right\rangle={\mathbf{p}}=p^{(\beta_{0})}$ and
$t_{1}=1$. Here we have switched to Dirac’s bra-ket style which makes the
notation somewhat easier. The dual basis of left eigenvectors of
${\mathbf{T}}$ will be denoted by
$\left\langle{\mathbf{e}}^{n}\right|,\,n=1,\ldots,N$, such that the following
holds:

$\left\langle{\mathbf{e}}^{n}\left|\right.{\mathbf{e}}_{m}\right\rangle=\delta^{n}_{m}\quad\mbox{(Kronecker
delta)}\;,$ (58)

and

${\mathbbm{1}}=\sum_{n=1}^{N}\left|{\mathbf{e}}_{n}\right\rangle\left\langle{\mathbf{e}}^{n}\right|\;.$
(59)

The latter implies

${\mathbf{T}}={\mathbf{T}}\,{\mathbbm{1}}=\sum_{n=1}^{N}{\mathbf{T}}\,\left|{\mathbf{e}}_{n}\right\rangle\left\langle{\mathbf{e}}^{n}\right|\stackrel{{\scriptstyle(\ref{eigenT})}}{{=}}\sum_{n=1}^{N}t_{n}\,\left|{\mathbf{e}}_{n}\right\rangle\left\langle{\mathbf{e}}^{n}\right|\;.$
(60)

Obviously, $\left\langle{\mathbf{e}}^{1}\right|=(1,1,\ldots,1)$. Let
$\left\langle{\mathbf{e}}^{1}\right|^{\perp}$ denote the subspace

$\left\langle{\mathbf{e}}^{1}\right|^{\perp}:=\\{{\mathbf{x}}\in{\mathbbm{R}}^{N}\left|\left\langle{\mathbf{e}}^{1}|{\mathbf{x}}\right\rangle=\sum_{n=1}^{N}x_{n}=0\right.\\}\;.$
(61)

Upon differentiating (55) we obtain

$\dot{\mathbf{T}}\,{\mathbf{T}}\,{\mathbf{p}}+{\mathbf{T}}^{2}\,\dot{\mathbf{p}}=\dot{\mathbf{p}}\;,$
(62)

and hence, using ${\mathbf{T}}\,{\mathbf{p}}={\mathbf{p}}$,

$\left({\mathbbm{1}}-{\mathbf{T}}^{2}\right)\,\dot{\mathbf{p}}=\dot{\mathbf{T}}\,{\mathbf{p}}\;.$
(63)

The two vectors $\dot{\mathbf{p}}$ and $\dot{\mathbf{T}}\,{\mathbf{p}}$ lie in
the subspace $\left\langle{\mathbf{e}}^{1}\right|^{\perp}$ and the latter is
left invariant under the linear map
$\left({\mathbbm{1}}-{\mathbf{T}}^{2}\right)$. Hence (63) can be solved for
$\dot{\mathbf{p}}$ by inverting the restriction of
$\left({\mathbbm{1}}-{\mathbf{T}}^{2}\right)$ to the subspace
$\left\langle{\mathbf{e}}^{1}\right|^{\perp}$. By means of (60) the result can
be written as

$\dot{\mathbf{p}}=\sum_{n=2}^{N}\frac{1}{1-t_{n}^{2}}\,\left|{\mathbf{e}}_{n}\right\rangle\langle{\mathbf{e}}^{n}|\dot{\mathbf{T}}|{\mathbf{p}}\rangle\;.$
(64)

This completes the calculation of the steady state distribution in linear
order in $\delta=\beta_{1}-\beta_{0}$. Next we consider the corresponding
energy transfer $\left\langle\Delta Q\right\rangle$. Let
${\mathbf{E}}\in{\mathbbm{R}}^{N}$ be the “energy" vector with components
$\left\langle n\left|{\mathbf{E}}\right.\right\rangle=E_{n}$. The expectation
value $Q_{0}$ of the energy of the NLS before the contact with the first heat
bath is

$Q_{0}=\sum_{n}p_{n}(\delta)\,E_{n}=\sum_{n}p_{n}\,E_{n}+\delta\,\sum_{n}\dot{p}_{n}\,E_{n}+O(\delta^{2})\;.$
(65)

After the contact with the first heat bath and a subsequent energy measurement
we obtain the energy expectation value

$\displaystyle Q_{1}$ $\displaystyle=$
$\displaystyle\sum_{n}q_{n}\,E_{n}=\sum_{nm}T^{(0)}_{nm}\,p_{m}(\delta)\,E_{n}$
(66) $\displaystyle=$
$\displaystyle\sum_{nm}{\mathbf{T}}_{nm}\,p_{m}\,E_{n}+\delta\,\sum_{nm}{\mathbf{T}}_{nm}\,\dot{p}_{m}\,E_{n}+O(\delta^{2})$
(67) $\displaystyle=$
$\displaystyle\sum_{n}p_{n}\,E_{n}+\delta\,\sum_{nm}{\mathbf{T}}_{nm}\,\dot{p}_{m}\,E_{n}+O(\delta^{2})\;.$
(68)

Hence

$\left\langle\Delta Q\right\rangle=Q_{1}-Q_{0}=:\delta\,a+O(\delta^{2})$ (69)

where

$\displaystyle a$ $\displaystyle=$
$\displaystyle\sum_{nm}{\mathbf{T}}_{nm}\,\dot{p}_{m}\,E_{n}-\sum_{n}\dot{p}_{n}\,E_{n}$
(70) $\displaystyle=$
$\displaystyle\left\langle{\mathbf{E}}\left|{\mathbf{T}}-{\mathbbm{1}}\right|\dot{\mathbf{p}}\right\rangle$
(71) $\displaystyle\stackrel{{\scriptstyle(\ref{pdotexpl})}}{{=}}$
$\displaystyle\langle{\mathbf{E}}\,|\sum_{n=2}^{N}\frac{t_{n}-1}{1-t_{n}^{2}}\,{\mathbf{e}}_{n}\,\rangle\langle{\mathbf{e}}^{n}|\dot{\mathbf{T}}|{\mathbf{p}}\rangle$
(72) $\displaystyle=$
$\displaystyle-\sum_{n=2}^{N}\frac{1}{1+t_{n}}\,\langle{\mathbf{E}}\left|\right.{\mathbf{e}}_{n}\,\rangle\langle{\mathbf{e}}^{n}|\dot{\mathbf{T}}|{\mathbf{p}}\rangle\;.$
(73)

This completes the calculation of the slope $a$ of the tangent of
$\beta_{1}\mapsto\left\langle\Delta Q\right\rangle$ at $\beta_{1}=\beta_{0}$
which can be considered as a sort of heat conduction coefficient. The fact
that $a\geq 0$ is a consequence of (36).

This calculation establishes an “external Fourier law", similarly as in MGM04
, that holds independent of a possibly constant temperature gradient inside
the NLS.

In the example of Section IV and for $\beta_{0}=1$ the right and left
eigenvectors of ${\mathbf{T}}$ as well as $\dot{\mathbf{T}}$ can be explicitly
calculated although not be shown here due to their overwhelming complexity.
Inserting these terms into (73) gives $a=0.173895\ldots$, the slope of the
common tangent of the three function plots in Figure 2.

##  VI Summary and outlook

The second law of thermodynamics has several formulations of varying
generality; one of them is that heat always flows spontaneously from the
hotter to the colder body. In the field of quantum mechanics, the formulation,
let alone the proof, of a corresponding second law is highly obscure and
controversial, despite (or even because of) the large body of literature on
this subject. In this situation, it seems advisable to gather a few facts that
capture what we know for sure. We know, that the von Neumann entropy increases
in the course of a Lüders measurement vN32 . Further, again in the case of
sequential Lüders measurements, there hold various Jarzynski-type equations
and for each of them Jensen’s inequality produces second law-like statements,
see SG20a ; SG20b . In this paper, we have singled out a particular Jarzynski-
type equation so that the corresponding second law-like statement essentially
coincides with the above formulation, namely, that heat flows from the hotter
to the colder heat bath.

However, although this result is very general and relatively easy to prove, we
need to mention some restricting assumptions that were required for the proof.
The quantum system is an arbitrary $N$-level system (NLS) and we implicitly
assume that its states are completely characterized by the occupation
probabilities $p_{n}$ of the $n$-th energy level, not only at the beginning
but also during the whole heat conduction process. The two heat baths are not
analyzed in detail; they almost appear as “black boxes" only described by
stochastic matrices that contain the conditional transition probabilities
between the energy levels of the NLS and leave invariant Gibbs states of
inverse temperature $\beta_{i},i=0,1$. For technical reasons, we had to
alternate contacts of the NLS with the two heat baths, since we would lose the
overview of the heat and entropy transfer with a total contact of all three
systems. Hopefully, this special protocol will be a reasonable approximation
for real heat conduction processes. A a benefit of our approach we mention
that no thermalization assumptions are needed; the heat conduction process can
be arbitrarily irreversible. Our calculations are essentially confined to the
steady state regime. By the way, the approach to a steady state after a large
number of cycles is not an extra assumption but follows from the property that
all eigenvalues of a stochastic matrix except the eigenvalue $1$ are $<1$ in
absolute value.

It looks promising to apply the present approach to similar problems. For
example, a general thermodynamic process could be split into pure heat and
work parts that can be represented by (bi-)stochastic matrices analogously as
in this paper and the Jarzynski-Jensen method could be used to obtain second
law-like statements.

## Acknowledgment

This work was funded by the Deutsche Forschungsgemeinschaft (DFG) Grant No. GE
1657/3-1. We sincerely thank the members of the DFG Research Unit FOR2692 for
fruitful discussions. In addition, we are grateful to Philipp Strasberg for
expert comments and literature references.

## References

  * (1) J. L. Lebowitz, and H. Spohn, Transport properties of the Lorentz gas: Fourier’s law, J. Stat. Phy. 19, 633 - 654 (1978)
  * (2) H. Spohn, and J. L. Lebowitz, Irreversible Thermodynamics for Quantum Systems Weakly Coupled to Thermal Reservoirs., Advances in Chemical Physics: For Ilya Prigogine, Vol. 38, S. A. Rice (Ed.), John Wiley & Sons, Hoboken, New Jersey, (1987)
  * (3) J. L. Lebowitz, and H. Spohn, A Gallavotti-Cohen-Type Symmetry in the Large Deviation Functional for Stochastic Dynamics, J. Stat. Phy. 95 (1/2), 333 - 365 (1999)
  * (4) K. Saito, S. Takesue, S. Miyashita, Thermal conduction in a quantum system, Phys. Rev. E 54, 2404 - 2407, (1996)
  * (5) M. Michel, M. Hartmann, J. Gemmer, and G. Mahler, Fourier’s Law confirmed for a class of small quantum systems, Eur. Phys. J. B 34, 325 - 330 (2003)
  * (6) K. Saito, Strong evidence of normal heat conduction in a one-dimensional quantum system, Europhys. Lett. 61 (1), 34 - 40, (2003)
  * (7) M. Michel, J. Gemmer, and G. Mahler, Heat conductivity in small quantum systems: Kubo formula in Liouville space, Eur. Phys. J. B 42, 555 - 559, (2004)
  * (8) M. Michel, G. Mahler, and J. Gemmer, Fourier’s Law from Schrödinger Dynamics, Phys. Rev. Lett. 95, 180602, (2005)
  * (9) P. Jung, R. W. Helmes, and A. Rosch, Transport in Almost Integrable Models: Perturbed Heisenberg Chains, Phys. Rev. Lett. 96, 067202, (2006)
  * (10) H. Wichterich, M. J. Henrich, H.-P. Breuer, J. Gemmer, and M. Michel, Modeling heat transport through completely positive maps, Phys. Rev. E 76, 031115, (2007)
  * (11) T. Prosen, Exact Nonequilibrium Steady State of a Strongly Driven Open XXZ Chain, Phys. Rev. Lett. 107, 137201, (2011)
  * (12) G. N. Bochkov and I. E. Kusovlev, General theory of thermal fluctuations in nonlinear systems, Sov. Phys. JETP 45, 125 - 130, (1977) (Zh. Eksp. Theor. Fiz. 72, 238 - 247, 1977)In[199]:= rt = Transpose[r1]; rt // TableForm
  * (13) G. N. Bochkov and I. E. Kusovlev, Nonlinear Fluctuation-dissipation relations and stochastic models in nonequilibrium themrodynamics I, Physica A 106, 443 - 479, (1981)
  * (14) C. Jarzynski, Nonequilibrium equality for free energy differences,  Phys. Rev. Lett. 78 (17), 2690 (1997)
  * (15) B. Piechocinska, Information erasure, Phys. Rev. A 61, 062314, (2000)
  * (16) J. Kurchan, A quantum fluctuation theorem, arXiv:0007360v2 [cond-mat.stat-mech] (2000)
  * (17) H. Tasaki, Jarzynski Relations for Quantum Systems and Some Applications, arXiv:0000244v2 [cond-mat.stat-mech] (2000)
  * (18) S. Mukamel, Quantum Extension of the Jarzynski Relation: Analogy with Stochastic Dephasing.  Phys. Rev. Lett. 90 (17), 170604 (2003)
  * (19) P. Talkner, M. Morillo, J. Yi, and P. Hänggi, Statistics of work and fluctuation theorems for microcanonical initial states, New J. Phys. 15, 095001 (2013)
  * (20) T. Schmiedl and U. Seifert, Stochastic thermodynamics of chemical reaction networks, J. Chem. Phys. 126, 044101 (2007)
  * (21) K. Saito and Y. Utsumi, Symmetry in full counting statistics, fluctuation theorem, and relations among nonlinear transport coefficients in the presence of a magnetic field, Phys. Rev. B 78, 115429 (2008)
  * (22) D. Andrieux, P. Gaspard, T. Monnai, and S. Tasaki, The fluctuation theorem for currents in open quantum systems, New J. Phys. 11, 043014 (2009), Erratum in: New J. Phys. 11, 109802 (2009)
  * (23) J. Yi, P. Talkner, and M. Campisi, Nonequilibrium work statistics of an Aharonov-Bohm flux, Phys. Rev. E 84, 011138 (2011)
  * (24) M. Esposito, Stochastic thermodynamics under coarse graining, Phys. Rev. E 85, 041125 (2012)
  * (25) J. Yi, Y. W. Kim, and P. Talkner, Work fluctuations for Bose particles in grand canonical initial states, Phys. Rev. E 85, 051107 (2012)
  * (26) M. Campisi, P. Hänggi, and P. Talkner, Colloquium: Quantum fluctuation relations: Foundations and applications, Rev. Mod. Phys. 83, 771-791 (2011) ; Erratum: Rev. Mod. Phys. 83, 1653 (2011)
  * (27) H.-J. Schmidt and J. Gemmer, A Framework for Sequential Measurements and General Jarzynski Equations, Z. Naturforsch. A 75 3, 265 – 284, (2020)
  * (28) H.-J. Schmidt and J. Gemmer, Sequential measurements and entropy, J. Phys.: Conf. Ser. 1638, 012007 (2020).
  * (29) H.-J. Schmidt, J. Schnack and J. Gemmer, Stochastic thermodynamics of a finite quantum system coupled to a heat bath, Z. Naturforsch. A, 76 (8), 731 - 745 (2021).
  * (30) C. Jarzynski and D. K. Wójcik, Classical and Quantum Fluctuation Theorems for Heat Exchange, Phys.Rev.Lett. 92 (23), 230602 (2004).
  * (31) D. Jennings and T. Rudolph, Entanglement and the thermodynamic arrow of time, Phys.Rev.E 81, 061130 (2010).
  * (32) T. Sagawa, Entropy, Divergence, and Majorization in Classical and Quantum Thermodynamics, arXiv:2007.09974v3 [quant-ph] (2020)
  * (33) H. Spohn, Entropy production for quantum dynamical semigroups, J.Math.Phys. 19 (5), 1227 - 1230 (1978).
  * (34) T. Aoki, Y. Matsuzaki, and H. Hakoshima, Total thermodynamic entropy production rate of an isolated quantum system can be negative for the GKSL-type Markovian dynamics of its subsystem, 2103.05308v1 [quant-ph] (2021)
  * (35) P. Strasberg1 and A. Winter, First and Second Law of Quantum Thermodynamics: A Consistent Derivation Based on a Microscopic Definition of Entropy, arXiv:2002.08817v5 [quant-ph] (2021)
  * (36) E. T. Jaynes and F. W. Cummings, Comparison of quantum and semiclassical radiation theories with application to the beam maser, Proc. IEEE 51 (1), 89 - 109, (1963)
  * (37) C. C. Gerry and J. H. Eberly, Dynamics of a Raman coupled model interacting with two quantized cavity fields Phys. Rev. A 42 (11), 6805 - 6815, (1990)
  * (38) M. Alexanian and S. K. Bose, Unitary transformation and the dynamics of a three-level atom interacting with two quantized field modes, Phys. Rev. A 52 (3), 2218 - 2224, (1995)
  * (39) B. T. Torosov, S. Longhia, and G. Della Vallea, Mixed Rabi Jaynes-Cummings model of a three-level atom interacting with two quantized fields, Opt. Commun. 346, 110 - 114, (2015)
  * (40) NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, Release 1.1.1 of 2021-03-15. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds.
  * (41) J. von Neumann, Mathematische Grundlagen der Quantenmechanik, Springer-Verlag, Berlin, 1932, English translation: Mathematical Foundations of Quantum Mechanics, Princeton University Press, Princeton, 1955.


11institutetext: A. Saha 22institutetext: Physics Dept., Jalpaiguri Govt.
Engg. College, Japlaiguri 735102
22email<EMAIL_ADDRESS>33institutetext: A. Chanda 44institutetext:
Physics Dept., University of North Bengal, Rajarammohunpur 734013
44email<EMAIL_ADDRESS>55institutetext: S. Dey 66institutetext:
Physics Dept., University of North Bengal, Rajarammohunpur 734013
66email<EMAIL_ADDRESS>77institutetext: B. C. Paul 88institutetext:
Physics Dept., University of North Bengal, Rajarammohunpur 734013
88email<EMAIL_ADDRESS>99institutetext: S. Ghose 1010institutetext: ICARD,
Physics Dept., University of North Bengal, Rajarammohunpur 734013
1010email<EMAIL_ADDRESS>
# Rényi Holographic Dark Energy Models In Multidimensional Universe

A. Saha  A. Chanda  S. Dey  B. C. Paul  S. Ghose

###### Abstract

Rényi Holographic Dark Energy (RHDE) Models has been studied in different
dimensions considering a flat FRW metric, subsequently exploring the nature of
evolution of different cosmological parameters. It is seen that the number of
dimensions affect the evolution of the cosmological parameters and the
transition of the universe from decelerating to accelerating phase. The
statefinder diagnosis has been performed and it is seen that the model
corresponds to the $\Lambda CDM$ model for the present and future universe.

###### Keywords:

Rényi Holographic Dark Energy Higher Dimensions Stability

##  1 Introduction:

It has been two decades since supernovae observation suggested that the
present universe is passing through a phase of accelerating expansion.
Subsequent observations only strengthened this belief riess ; perlm ; perlm2 ;
perlm3 . The present phase of acceleration can be successfully modeled by
modifying the Einstein-Hilbert action in the first place nojde1 ; cappode .
Alternatively, some dynamical form of energy with negative pressure, termed as
dark energy, is thought to dominate the present epoch, leading to the
accelerated rate of expansion freese ; overduin ; fritzsch ; basilakos ; sola
; ratra1988 ; gibb ; armendariz2001 ; wetterich2017 ; escamilla2013 .
Dynamical dark energy or modification of gravity is not demanded by the late
phase of acceleration as it can be explained with a simple cosmological
constant($\Lambda$) also carroll2001cosmological . The motivation for
considering other approaches lies in the fact that the tiny value of the
$\Lambda$ as suggested by observation can not be explained from any
fundamental theory and there is a huge mismatch of about $~{}10^{60}$. Another
alternative approach, vastly studied in literature, is to explore dimensions
more than four chodos ; shafi ; wetterich ; ACCETTA ; kitaura ; bcp . Higher-
dimensional models are considered to serve the dual purpose of solving
cosmological riddles such as late-time acceleration and at the same time
providing us with a unified model of the fundamental forces of nature. The
idea behind these models is that, although undetectable at the present epoch,
the extra dimensions play a crucial role in driving the inflation in the early
universe or the acceleration at a later time. One of the earliest examples of
higher-dimensional models is the five-dimensional Kaluza-Klein theories kaluza
; klein ; leekk ; appelkk . Since then a plethora of higher dimensional
theories like the string theory cline2006 , superstring theory lidsey1999 ,
braneworld models brax2004  and others have been explored. Although none of
these models stands out as satisfactory, several of them show great prospect
both in the cosmological and the fundamental physics concepts. In the present
work we compare general FRW universes in different dimensions without diving
into any specific higher dimensional model.

Recently, the holographic principle has emerged as an essential proposal of
quantum gravity cohen ; susskind . The principle states that, _"the number of
degrees of freedom of any physical system should scale with the boundary and
not with the volume"_. The holographic principle leads to the Holographic dark
energy (HDE) models in the cosmological context with several interesting
features. Cosmology itself does not offer any unique choice of the long-
distance cutoff, also known as the infrared cut off (IR cut off) for any of
the HDE models granda2008 . One of the most natural choices of IR cut off is
the Hubble horizon (for cosmology in generaliged HDE and with arbitrarty cut
off see nojiriarb ; nojirisym ). However, in the original model of HDE (OHDE)
if one considers the Hubble radius as IR cut off and the Bekenstein entropy,
then both dark matter (DM) and OHDE are scaled with the same function of the
scale factor. Hsu subsequently noted that HDE cannot explain the present
accelerated phase of expansion hsu2004 . Later Zimdahl and Pavon showed that
an interaction between the dark sectors (dark energy and dark matter) would
lead to an accelerating universe in the late time zimdahl2007 . Very recently
Nojiri _et. al._ nojiriunify  have advanced an interesting idea of unifying an
early inflationary epoch and the late time accelerating phase of the universe
using the holographic approach. branches of HDE emerged as other non-extensive
entropy conditions were explored (see miao2011  for a detailed review). Recent
works suggest one such model, the Rényi Holographic Dark Energy (RHDE
hereafter), yields accelerated expansion in Kaluza-Klein framework even
without any interaction SAHA .
The motivation of the paper is to compare non-interacting Rényi Holographic
dark energy (RHDE) in different higher dimensional contexts. But, presently,
no specific higher dimensional model is dealt with. As suggested by numerous
observations, the universe is assumed to be completely homogenous and
isotropic in large scale. Evolution of different cosmological parameters are
studied and compared for different dimensions. A four dimensional universe is
emphasized in a sense that any higher dimensional theory must admit an
effective four dimensional scenario where diagnostics are physically
meaningful. However, it is worth exploring in this context if the presence of
extra dimension would alter the evolution of different parameters. As the
effect of dark energy is more prevalent in late time, the present work is more
focused on the late time evolution of the universe.

The paper is organised as follows: In sec. ( 2), a general idea of Rényi
entropy and Rényi dark energy density in higher dimensions is presented. Sec.
( 3) advances the relevant field Equation in higher dimensions. The
cosmological evolution is discussed in the sec. ( 4). In sec. ( 5), stability
and diagnostics of the model are studied, and finally, the findings are
discussed in the sec. ( 6).

##  2 General idea of Rényi entropy and Rényi dark energy density in higher
dimension:

It has been argued that the systems including long-range interactions are in
accord with the generalized entropy formalisms based on the power law
distribution of probabilities. Recently, such entropies have been employed to
build new holographic dark energy models that can describe the cosmic
evolution from matter dominates era to the current accelerating universe.

In non-extensive thermodynamics the Tsallis entropy ($S_{T}$) tsalis ;
biro2013q ; czinner2016renyi ; belin2013holographic  for a set of $W$ states
is defined as

$S_{T}=k_{B}\frac{1-\sum_{i=1}^{W}p_{i}^{q}}{q-1}\;\;{\left(\sum_{i=1}^{W}p_{i}=1;\;q\in\rm
I\\!R\right)},$ (1)

where $p_{i}$ is the probability associated to the $i^{th}$ microstate with
$\sum_{i}p_{i}=1$ and $q$ is any real number. Tsallis holographic dark energy
has been extensively studied in the context of cosmology sharma2020diagnosing
; varshney2019statefinder ; thdemain  The original Rényi entropy
($S_{R}^{Org}$), on the other hand, is defined as tsalis :

$S_{R}^{Org}=k_{B}\frac{ln\sum_{i=1}^{W}p_{i}^{q}}{q-1}=\frac{1}{1-q}ln\left[1+(1-q)S_{T}\right].$
(2)

It is interesting that both the eqs.(1) and (2) lead to Boltzmann-Gibbs
entropy for $q=1$. Recently, it has been proposed that the Benkenstein-Hawking
($S_{BH}$) entropy too is a kind of non-extensive entropy which leads to a
novel type of Rényi entropy czinner2016renyi  which is given by:

$S_{R}=\frac{1}{\delta}ln\left(1+\delta S_{BH}\right),$ (3)

where $\delta=1-q$ and for $\delta=0$, $S_{R}=S_{BH}$. In cosmology, Rényi
holographic dark energy is extensively consideredsharma2020statefinder ;
dubey2020diagnosing . It is shown in moradpour2017accelerated  that when Rényi
entropy is employed at the horizon it results in an accelerating universe in
the Rastall framework. In the present work, the Rényi entropy is considered
for describing the entropy on the Hubble horizon. As $S_{BH}=\frac{A}{4}$
(where $A$ is the area of the horizon), one obtains

$S_{R}=\frac{1}{\delta}ln\left(1+\frac{A}{4}\;\delta\right).$ (4)

In cosmological setup the relation $dE_{T}=TdS$ can be considered as the first
law of thermodynamics for the cosmological horizon cai . Here, $E_{T}$ denotes
the total energy of the universe, $T=\frac{1}{2\pi L}$, the Cai-Kim
temperature for a system with IR cutoff $L$, corresponds to the temperature of
the de-Sitter horizon and apparent horizon in flat FRW cosmology, $S$ is the
horizon entropy. Also in DE dominant present universe, one can use the
assumption $dE_{D}=\rho_{D}dV\propto dE_{T}=TdS$, where $E_{D}$ denotes the
energy content for the DE representative.
Note that the apparent horizon is a proper casual boundary for universe in
agreement with the thermodynamical laws and thus the energy-momentum
conservation law. Hence considering the Hubble radius as the IR cutoff
($L=H^{-1}$), the energy density for DE candidate is obtained as:

$\rho_{D}=\frac{C^{\prime}H}{2\pi}\frac{dV}{dS},$ (5)

where $C^{\prime}$ is the proportionality constant.

Again $n$ dimensional volume $(V_{n})$ and surface area $(S_{n-1})$ of radius
$R$ is given by

$V_{n}=\frac{\pi^{\frac{n}{2}}}{\Gamma\left(\frac{n}{2}+1\right)}R^{n},$ (6)

and

$S_{n-1}=\frac{2\pi^{\frac{n}{2}}}{\Gamma\left(\frac{n}{2}\right)}R^{n-1},$
(7)

Using the equation (5), (6), and (7) the energy density for RHDE can be
expressed as:

$\rho_{D}=\frac{B^{2}H^{2}}{1+\frac{\delta K}{H^{(n-1)}}},$ (8)

where $B^{2}=\frac{C^{2}}{8\pi}$,
$C^{2}=\frac{C^{\prime}(n-1)K^{\prime}K_{1}^{\frac{1}{n-1}}}{n}$,

$K^{\prime}=\frac{2\pi^{\frac{n}{2}}}{\Gamma\left(\frac{n}{2}\right)}\left[\frac{\Gamma\left(\frac{n}{2}+1\right)}{\pi^{\frac{n}{2}}}\right]^{\frac{n-1}{n}}$,
$K_{1}=\frac{\Gamma\left(\frac{n}{2}+1\right)}{\pi^{\frac{n}{2}}}$
and $K=\frac{K^{\prime}}{4K_{1}^{\frac{n}{n-1}}}$.

##  3 Field Equation for cosmology in Higher Dimension

The FRW metric in $(1+1+d)$ dimension is given by

$dS^{2}=dt^{2}-a^{2}(t)\left[\frac{dr^{2}}{1-kr^{2}}+r^{2}(dX_{d})^{2}\right],$
(9)

where $a(t)$ denotes the scale factor and $k=0,\pm 1$ represents the curvature
parameter for flat and closed (open) universe.
We consider higher dimensional flat universe filled by two types of cosmic
fluids. The total energy density is then $\rho=\rho_{D}+\rho_{m}$, where
$\rho_{D}$ corresponds to dark energy and $\rho_{m}$ is for matter including
cold dark matter with $\omega_{m}=0$.
The Einstein field equations for flat FRW universe in $(1+1+d)$ dimension are
as follows:

$H^{2}=\frac{16\pi}{d(d+1)}(\rho_{D}+\rho_{m})$ (10)

and

$H^{2}+\frac{2\dot{H}}{d+1}=-\frac{16\pi}{d(d+1)}P_{d}$ (11)

Here, we also assume that there is no energy exchange between the cosmic
fluids. For non-interacting fluids, the energy conservation equations are as
follows:

$\dot{\rho}_{m}+H(d+1)\rho_{m}=0,$ (12)
$\dot{\rho}_{D}+H(d+1)\rho_{D}(1+\omega_{D})=0.$ (13)

The dimensionless density parameters are defined as:

$\Omega_{m}=\frac{\rho_{m}}{\rho_{cr}},\;\;\Omega_{D}=\frac{\rho_{D}}{\rho_{cr}},$
(14)

where $\rho_{cr}=\frac{d(d+1)H^{2}}{16\pi}$. Eq. (10) can be written in terms
of density parameters as:

$\Omega_{m}+\Omega_{D}=1.$ (15)

The ratio of the energy densities is given by:

$r^{\prime}=\frac{\rho_{m}}{\rho_{D}}=\frac{\Omega_{m}}{\Omega_{D}}=\frac{1-\Omega_{D}}{\Omega_{D}}.$
(16)

###  3.1 Generalized Holographic Dark Energy

In the holographic principle, the holographic dark energy density is
proportional to the inverse squared infrared (IR) cut off $L_{IR}$, which is
related to the holographic DE density as,

$\rho_{HDE}=\frac{3C^{2}}{\kappa^{2}L_{IR}^{2}},$ (17)

where $\kappa^{2}=8\pi G$ and $C$ is a numerical constant that acts as a free
parameter. The IR cut off is assumed to be the particle horizon $L_{P}$ or the
future event horizon $L_{F}$, which are defined respectively as,

$L_{P}\equiv a\int_{0}^{t}\frac{dt}{a},\;\;\;\;\;\;\;\;\;\;L_{F}\equiv
a\int_{t}^{\infty}\frac{dt}{a}.$ (18)

The Hubble parameter can now be expressed in terms of $L_{P}$, $L_{F}$ and
their time derivatives. Taking time derivative of the above equation we get,

$H(L_{P},\dot{L_{P}})=\frac{\dot{L_{P}}}{L_{P}}-\frac{1}{L_{P}},\;\;\;\;\;\;\;\;H(L_{F},\dot{L_{F}})=\frac{\dot{L_{F}}}{L_{F}}+\frac{1}{L_{F}}.$
(19)

The general form of the cut off is given by (Ref.nojiriuorg ),

$L_{IR}=L_{IR}(L_{P},\dot{L_{P}},\ddot{L_{P}},......,L_{F},\dot{L_{F}},\ddot{L_{F}},.....a).$
(20)

The IR cutoff depends on other factors as well such as the Hubble parameter,
the Ricci Scalar and their derivatives. However, they can be transformed to
either $L_{P}$ and its derivatives or $L_{F}$ and its derivatives. The above-
mentioned cutoff can be chosen to be equivalent to a general covariant gravity
model as

$S=\int
d^{4}\sqrt{-g}F(R,R_{\mu\nu}R^{\mu\nu},R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma},\Box
R,\Box^{-1}R,\nabla_{\mu}R\nabla^{\mu}R,.......).$ (21)

These expressions can be used to derive a generalized IR cutoff for the RHDE.
For a higher dimensional spacetime one can write equation (17) as,

$\rho_{HDE}=\frac{d(d+1)C^{2}}{2\kappa^{2}L_{IR}^{2}}$ (22)

where $d=2$, gives the usual $4$ dimensional results. If one uses the
Bekenstein-Hawking (BH) entropy and the first law of thermodynamics ($TdS=dQ$,
where $S$ is the entropy and $Q$ is the increase of heat in the region inside
the cosmological horizon) then the Friedmann equation for a $(1+1+d)$
dimensional spacetime can be obtained. However, instead of the BH entropy one
may consider the Rényi entropy (eqn. 4) and apply the first law of
thermodynamics to obtain the following relation,

$\frac{\dot{\rho}}{H(d+1)}=\frac{d}{8\pi}\frac{1}{(1+\frac{\delta}{4}k_{A}H^{-d})}\dot{H}$
(23)

where, $k_{A}=\frac{2\pi^{\frac{n}{2}}}{\Gamma\left(\frac{n}{2}\right)}$, with
$n=(d+1)$ and we have considered the radius of the cosmological horizon to be
equal to the inverse of the Hubble rate ($R=\frac{1}{H}$).
The integration of the above equation yields an expression for the DE density
as,

$\rho=\frac{d(d+1)}{8\pi}\int{\frac{HdH}{(1+\frac{\delta}{4}k_{A}H^{-d})}}+C_{0}$
(24)

where $C_{0}$ is an integration constant. One can compare equations (22) and
(24) and argue that RHDE belongs to the generalized HDE family even in higher
dimensions. The corresponding infrared cutoff $L_{R}$ in this case depends on
the dimension and is given by,

$\frac{d(d+1)C^{2}}{2\kappa^{2}L_{R}^{2}}=\frac{d(d+1)}{8\pi}\int{\frac{HdH}{(1+\frac{\delta}{4}k_{A}H^{-d})}}+C_{0}.$
(25)

The above equation is too complicated to solve for a general $d$ value.
However, solutions for a particular dimension and the corresponding value for
the generalized cutoff can be computed. The EoS parameter can be obtained from
the conservation of equation of the HDE density $\rho_{HDE}$ :

$\Omega_{HDE}^{R}=-1-\frac{\rho_{D}}{(d+1)H\rho_{D}}=-1+\frac{2}{(d+1)HL_{R}}\frac{dL_{R}}{dt},$
(26)

where $L_{R}$ is given by eqn. (25). Thus we conclude that in higher
dimensions also, $\Omega_{HDE}^{R}$ is equivalent to $w_{D}$ as obtained in
nojirisym  for four dimensions.

##  4 Evolution of Universe With RHDE

Taking the time derivative of eq. (12) and eq. (13) we can reach

$\frac{\dot{H}}{H^{2}}=-\left(\frac{d+1}{2}\right)\left(1+\omega_{D}\Omega_{D}\right)$
(27)

In $(1+1+d)$ dimension one can easily obtain the expression of density
parameter as:

$\Omega_{D}=\frac{2C^{2}}{d(d+1)\left(1+\frac{\delta K}{H^{d}}\right)},$ (28)

Now, taking the time derivative of eq. (28) we have:

$\frac{\dot{\rho_{D}}}{\rho_{D}}=\frac{\dot{H}}{H}\Big{[}\frac{2C^{2}(d+2)-d^{2}(d+1)\Omega_{D}}{2C^{2}}\Big{]}.$
(29)

Combined with relation $\Omega_{D}^{{}^{\prime}}=\frac{\dot{\Omega_{D}}}{H}$
eq. (29) gives the evolution equation of RHDE density parameter as:

$\Omega_{D}^{{}^{\prime}}=-\frac{d(d+1)\Omega_{D}}{4C^{2}}\left[2C^{2}-d(d+1)\Omega_{D}\right]\left(1+\omega_{D}\omega_{D}\right),$
(30)

where $(^{\prime})$ denotes derivative with respect to $\ln a$ i.e,
$\Omega_{D}^{{}^{\prime}}=\frac{d\Omega_{D}}{d\ln a}$.

Figure 1: Evolution of cosmological parameters for different $\delta$ values
in four dimensions with $\delta=-0.0001$ (Green, Dashed), $\delta=-0.0002$
(Blue, Solid) and $\delta=-0.0005$ (Red, Dot-dashed).

EoS parameter is obtained from eqs. (13), (27), and (29):

$\omega_{D}=\frac{d\Big{[}2C^{2}-d(d+1)\Omega_{D}\Big{]}}{-2C^{2}(d+2)\Omega_{D}+4C^{2}+(d+1)d^{2}.\Omega_{D}}$
(31)

Using eqs. (27) and (31) the deceleration parameter can be written as:

$q=\Big{(}\frac{d-1}{2}\Big{)}+\Big{(}\frac{d+1}{2}\Big{)}\Big{[}\frac{d(2C^{2}-d(d+1)\Omega_{D}}{4C^{2}-2C^{2}(d+2)\Omega_{D}+d^{2}(d+1)\Omega_{D}^{2}}\Big{]}$
(32)

Figure 2: Evolution of cosmological parameters for different $\delta$ values
in five dimensions with $\delta=-0.0001$ (Green, Dashed), $\delta=-0.0002$
(Blue, Solid) and $\delta=-0.0005$ (Red, Dot-dashed). Figure 3: Evolution of
cosmological parameters for different $\delta$ values in ten dimensions with
$\delta=-0.0001$ (Green, Dashed), $\delta=-0.0002$ (Blue, Solid) and
$\delta=-0.0005$ (Red, Dot-dashed).

In the figs. (1)-(3) the cosmological parameters $\Omega_{d}$, $\omega_{d}$
and $q$ have been plotted respectively for different values of $\delta$ and
three different dimensions $4$, $5$ and $10$ of the universe, for the initial
condition $H(z=0)=0.07Gy^{-1}$ and $\Omega_{m0}=0.27$. These figures clearly
show that, depending on the values of $\delta$ the acceptable behaviour Of
$\Omega_{d}$, $\omega_{d}$ and $q$ can be obtained. In addition, comparing
figs. (1)-(3) with each other, we observe that the changes in the density
parameter become sharper as one goes for higher dimension and with the
increase in dimension of universe the behaviour of $\Omega_{D}$ with redshift
$z$ becomes almost independent of model parameter $\delta$.

From the evolution of the EoS parameter $w_{d}$ (figs. (1), fig. (2), and
(3)), it is evident that the dark energy begins in a quintessence era in early
universe and gradually progressing towards the phantom domain $i.e.$ $w<-1$.
However in the late universe the EoS parameter attains a value $w=-1$ and
never crosses the phantom divide irrespective of the dimensions. These
transitions are faster as in an higher dimensional universe.

Also, the deceleration parameter $(q)$ starts from positive value at the
earlier time and goes to negative in recent past, suggesting a decelerating
phase, sutable for structure formation as well as the late-time acceleration
of universe. For different value of the model parameter $\delta$, present
value of $q$ is seen to vary from $-0.4$ to $-0.1$ which is very much
acceptable in light of supernovae data. Moreover, eq. (32) suggests that the
transition redshift $z_{t}$ (at which the universe expansion phase is changed
from the decelerated matter dominated phase to an accelerated expansion)
occurs at the redshift for which $\Omega_{D}=\sqrt{\frac{2C^{2}}{d(d+1)}}$.
Hence the values of model parameter $\delta$ and diemension of the universe
$d$ also affect the value of transition redshift. It is also evident from the
figures that the universe remains ever accelerating in the future and this
behaviour is independent of both the dimensions and $\delta$

Figure 4: Evolution of the density parameter ($\Omega_{D}$), deceleration
parameter ($q$) and the EoS parameter ($w$) in higher dimensions. Four
dimensional (Blue), five dimensional (Red) and ten dimensional (Black) with
$\Omega_{D0}=-0.73$ and $H_{0}=0.07Gy^{-1}$.

##  5 Classical Stability and Diagonestics

###  5.1 Classical Stability of RHDE

In this section we study the classical stability of our model against
perturbations. For this the squared speed of sound $v^{2}$ is considered as an
indicator of stability. $v^{2}>0$ indicates that the given perturbation
propagates in the environment and hence shows the stability a model. The
squared speed of sound of in the present model is given by

$\begin{split}v^{2}&=\frac{d\rho_{D}}{dP_{D}}\\\ &=\omega_{d}\\\
&+\frac{d^{2}\Omega_{D}(2C^{2}-d(d+1)\Omega_{D})(-4C^{2}(d^{2}-2)}{(2C^{2}(d+2)-d^{2}(d+1)\Omega_{D})(4C^{2}-2C^{2}(d+2)\Omega_{D}+d^{2}(d+1)\Omega_{D}^{2})^{2}}\\\
&-\frac{4C^{2}d^{2}(d+1)\Omega_{D}+d^{3}(d+1)^{2}\Omega_{D}^{2}}{(2C^{2}(d+2)-d^{2}(d+1)\Omega_{D})(4C^{2}-2C^{2}(d+2)\Omega_{D}+d^{2}(d+1)\Omega_{D}^{2})^{2}}.\end{split}$
(33)

The above equation indicates that the stability of the model depends on model
parameter $\delta$. We have ploted $v^{2}$ against red-shift parameter $z$ for
three different dimensions in the fig. (5). It is interesting to note that for
some value of model parameter $\delta$ the model shows stability at present
time as well as from very recent past and continue to depict the same in
future also. We note that as the dimension increases the squared speed of
sound shows a discontinuity at a particular redshift however remaining stable
in other redshifts depending on $\delta$ values.

Figure 5: Evolution of the square speed of sound $v^{2}$ with redshift $z$ in
four (left) and five (right) dimensions for $\delta=-0.0001$ (Green, Dashed),
$\delta=-0.0002$ (Blue, Solid) and $\delta=-0.0005$ (Red, Dot-dashed).

###  5.2 Statefinder Diagnostic

Higher dimensions, as they might play crucial role in unifying the natural
forces or demystifying cosmic riddles, should be observationally equivalent to
a four dimensional universe at present. For example the compact Kaluza-Klein
theories achieve this feat by suggesting compactification of the extra
dimensions. So, the state finder diagnostics of RHDE is studied for the four
dimensional universe. The state finder pair $(r,s)$ is defined as follows
sahni2003statefinder ; alam2003exploring :

$r=\frac{\dddot{a}}{aH^{3}}=\frac{\ddot{H}}{H^{3}}+\frac{3\dot{H}}{H^{2}}+1$

and

$s=\frac{r-1}{3(q-\frac{1}{2})}$

Using Eqs. (10) and (11) the state finder parameters can be expressed as

$r=1+\frac{(d+1)(d-2)}{2}+\frac{(d+1)}{2}\left[(2d-1)+(d+1)\omega_{D}\right]\omega_{D}\Omega_{D}-\frac{(d+1)}{2}\Omega_{D}\omega_{D}^{{}^{\prime}}$
(34)

where,

$\omega_{D}^{\prime}=\frac{\dot{\omega}_{D}}{H}=\frac{d(-4C^{2}(d^{2}-2)-4C^{2}d^{2}(d+1)\Omega_{D}+d^{3}(d+1)^{2}\Omega_{D}^{2})}{(4C^{2}-2C^{2}(d+2)\Omega_{D}+d^{2}(d+1)\Omega_{D}^{2})^{2}}$
(35)

The variation of the statefinder pair $(r,s)$ with redshift $z$ are plotted in
case of four dimensions as shown in fig. (6). It is evident from the figure
that in four dimension the statefinder pair approach their corresponding
$\Lambda CDM$ values in the future. However, the present universe shows
significant departure from a $\Lambda CDM$ model. One interesting thing to
note here is that for higher dimensions the statefinder pair show a
significant departure from their $\Lambda CDM$ value at a particular positive
redshift. However RHDE model corresponds to $\Lambda CDM$ model for $z\leq 0$.

Figure 6: Evolution of the statefinder pair $r$ (left) and $s$ (right) with
redshift $z$ in four dimensions for $\delta=-0.0001$ (Green, Dashed),
$\delta=-0.0002$ (Blue, Solid) and $\delta=-0.0005$ (Red, Dot-dashed). Figure
7: $\mu(z)$ vs $z$ plot for different values of $\delta$ along with
observational data from Union 2 compilation, here $\delta=-0.0001$ (black) and
$\delta=-0.0002$ (red).

##  6 Discussion

We have made a comparative study of RHDE models in different dimensions. A few
points are worth noting. The density parameter is plotted in figs. (1), fig.
(2), and (3) for four, five and ten dimensions respectively taking three
different delta values. It is seen that density parameter exhibits a sharp
transition towards a value close to unity as dimension is increased. For
positive $z$, the variation in $\Omega_{D}$ with delta is seen from the
figures, but in the present universe and in near future the behaviour becomes
almost independent of model parameter $\delta$. This trend is same for all the
dimensions considered. Another thing to note here is that as the dimension is
increased $\Omega_{D}$ becomes almost independent of $\delta$ throughout the
evolution of the universe. From the plot of the eos parameter it is seen that
dark energy begins in a quintessence era in the early universe and the value
decreases reaching a value of $\omega_{D}=-1$, in the present universe. It
remains constant at $\omega_{D}=-1$ for negative $z$ never crossing the
phantom divide ($\omega_{D}<-1$). This behaviour is dimension independent. We
note that for $\delta=-0.0005$, the universe transitioned from a dust filled
phase to quintessence era. The universe transits from a decelerated phase to
accelerated phase as shown in figs. (1), fig. (2), and (3). For
$\delta=-0.0005$ the universe transits from decelerated to accelerated phase,
however for higher delta universe remains in the ever-accelerating phase. As
the dimension of the universe is increased higher $\delta$ values correspond
to ever accelerating universe as is evident from the figures (1)-(3). For
different value of the model parameter $\delta$, present value of $q$ is seen
to vary from $-0.4$ to $-0.1$ which is very much acceptable in light of Sn-Ia
data. Fig. (7) depicts a plot of distance modulus ($\mu(z)$) with redshift for
different values of the model parameter $\delta$ along with Union Compilation
data union . Clearly, some values of $\delta$ (e.g. $\delta=-0.0001$) is
observationally favoured. A full scale analysis of observational constraints,
however, is beyond the scope of the present work. The focus of this study has
been the late time evolution of the universe only and the effect of higher
dimension on this phase. However, it would be interesting to investigate some
unified scenarios based on HDE nojiriunify ; nojiriuorg  which also includes
an early inflationary phase.

## References

  * (1) A.G. Riess, A.V. Filippenko, P. Challis, A. Clocchiatti, A. Diercks, P.M. Garnavich, R.L. Gilliland, C.J. Hogan, S. Jha, R.P. Kirshner, et al., The Astronomical Journal 116(3), 1009 (1998)
  * (2) S. Perlmutter, S.C. Project, et al., AJ 116, 1009 (1998)
  * (3) S. Perlmutter, S.C.P. Collaboration, et al., Astron. J 116, 1009 (1999)
  * (4) R.A. Knop, G. Aldering, R. Amanullah, P. Astier, G. Blanc, M. Burns, A. Conley, S. Deustua, M. Doi, R. Ellis, et al., The Astrophysical Journal 598(1), 102 (2003)
  * (5) S. Nojiri, S.D. Odintsov, Physics Reports 505(2-4), 59 (2011)
  * (6) S. Capozziello, M. De Laurentis, Physics Reports 509(4-5), 167 (2011)
  * (7) K. Freese, F.C. Adams, J.A. Frieman, E. Mottola, Nuclear Physics B 287, 797 (1987)
  * (8) J. Overduin, F. Cooperstock, Physical Review D 58(4), 043506 (1998)
  * (9) H. Fritzsch, J. Sola, Classical and Quantum Gravity 29(21), 215002 (2012)
  * (10) S. Basilakos, M. Plionis, J. Sola, Physical Review D 80(8), 083511 (2009)
  * (11) J. Sola, A. Gomez-Valent, J. de Cruz Pérez, The Astrophysical Journal Letters 811(1), L14 (2015)
  * (12) B. Ratra, P.J. Peebles, Physical Review D 37(12), 3406 (1988)
  * (13) G. Gibbons, Physics Letters B 537(1-2), 1 (2002)
  * (14) C. Armendariz-Picon, V. Mukhanov, P.J. Steinhardt, Physical Review D 63(10), 103510 (2001)
  * (15) C. Wetterich, arXiv preprint arXiv:1711.03844 (2017)
  * (16) C. Escamilla-Rivera, G. García-Jiménez, O. Loaiza-Brito, O. Obregón, Classical and Quantum Gravity 30(3), 035005 (2013)
  * (17) S.M. Carroll, Living reviews in relativity 4(1), 1 (2001)
  * (18) A. Chodos and S. Detweiler, Physical Review D 21(8), 2167-2170 (1980)
  * (19) Q. Shafi and C. Wetterich, Nuclear Physics B 289, 787-809 (1987)
  * (20) C. Wetterich, Physics Letters B 113, (5) 377-381 (1982)
  * (21) S. Accetta Frank and M. Gleiser and R. Holman and E. W. Kolb, Nuclear Physics B 276, (3) 501-516 (1986)
  * (22) T. Kitaura and J. T. Wheeler, Physical Review D 48, 667-672 (1993)
  * (23) B. C. Paul and S. Mukherjee, Physical Review D 42, (8) 2595-2600 (1990)
  * (24) T. Kaluza, Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.) (arXiv: 1803.08616), 966 (1921)
  * (25) O. Klein, Zeitschrift für Physik 37(12), 895 (1926)
  * (26) _An Introduction to Kaluza-Klein Theories: Workshop on Kaluza-Klein Theories_ (World Scientific Pub. Co., 1984)
  * (27) _Modern Kaluza-Klein Theories_ (Addison-Wesley, 1987)
  * (28) J.M. Cline, in _Les Houches Summer School - Session 86: Particle Physics and Cosmology: The Fabric of Spacetime_ (2006)
  * (29) J.E. Lidsey, D. Wands, E.J. Copeland, Phys. Rept. 337, 343 (2000).  DOI 10.1016/S0370-1573(00)00064-8
  * (30) P. Brax, C. van de Bruck, A.C. Davis, Rept. Prog. Phys. 67, 2183 (2004).  DOI 10.1088/0034-4885/67/12/R02
  * (31) A.G. Cohen, D.B. Kaplan, A.E. Nelson, Physical Review Letters 82(25), 4971 (1999)
  * (32) L. Susskind, Journal of Mathematical Physics 36(11), 6377 (1995)
  * (33) L. Granda, A. Oliveros, Physics Letters B 669(5), 275 (2008)
  * (34) S. Nojiri and S. D. Odintsov The Europian Physical Jornal C 77, 528 (2017).
  * (35) S. Nojiri, S. D. Odintsov and T. Paul, Symmetry 13, 928 (2021)
  * (36) S.D. Hsu, Physics Letters B 594(1-2), 13 (2004)
  * (37) W. Zimdahl, D. Pavón, Classical and Quantum Gravity 24(22), 5461 (2007)
  * (38) S. Nojiri, S. D. Odintsov, V. K. Oikonomou, and T. Paul Physical Review D 102(19), 023540 (2020)
  * (39) L. Miao, L. Xiao-Dong, W. Shuang, W. Yi, Communications in theoretical physics 56(3), 525 (2011)
  * (40) A. Saha, S. Ghose, A. Chanda and B. C. Paul, Annals of Physics 426, 168403 (2021)
  * (41) C. Tsallis, L.J. Cirto, The European Physical Journal C 73(7), 2487 (2013)
  * (42) T.S. Biró, V.G. Czinner, Physics Letters B 726(4-5), 861 (2013)
  * (43) V.G. Czinner, H. Iguchi, Physics Letters B 752, 306 (2016)
  * (44) A. Belin, A. Maloney, S. Matsuura, Journal of High Energy Physics 2013(12), 50 (2013)
  * (45) U.K. Sharma, V.C. Dubey, A. Pradhan, IJGMM 17(2), 2050032 (2020)
  * (46) G. Varshney, U.K. Sharma, A. Pradhan, New Astronomy 70, 36 (2019)
  * (47) M. Tavayef, A. Sheykhi, K. Bamba, H. Moradpour, Physics Letters B 781, 195 (2018)
  * (48) U.K. Sharma, V.C. Dubey, New Astronomy p. 101419 (2020)
  * (49) V.C. Dubey, A.K. Mishra, U.K. Sharma, arXiv preprint arXiv:2003.07883 (2020)
  * (50) H. Moradpour, A. Bonilla, E.M. Abreu, J.A. Neto, Physical Review D 96(12), 123504 (2017)
  * (51) R. G Cai and S. P. Kim, Journal of High Energy Physics 2005(02), 050 (2005)
  * (52) V. Sahni, T. D. Saini, A. A. Starobinsky and U. Alam, Journal of Experimental and Theoretical Physics Letters 77(5), 201-206 (2003)
  * (53) U. Alam, V. Sahni, T. D. Saini and A. A. Starobinsky, Monthly Notices of the Royal Astronomical Society 344(4), 1057-1074 (2003)
  * (54) S. Nojiri, S. D. Odintsov, General Relativity and Gravitation 38, 1285–1304 (2006)
  * (55) N. Suzuki et. al., APJ 746, 85 (2012).