# Microscopic Work Distribution of Small System in Quantum Isothermal Process

H. T. Quan  Theoretical Division, MS B213, Los Alamos National Laboratory, Los
Alamos, NM, 87545, U.S.A.  S. Yang  Institute of Theoretical Physics, Chinese
Academy of Sciences, Beijing, 100080, China  C. P. Sun  Institute of
Theoretical Physics, Chinese Academy of Sciences, Beijing, 100080, China

###### Abstract

For a two-level quantum mechanical system, we derive microscopically the exact
expression for the fluctuation of microscopic work in a multi-step non-
equilibrium process, and we rigorously prove that in an isothermal process,
the fluctuation is vanishingly small, and the most probabilistic work just
equals to the difference of the free energy. Our study demonstrates that the
convergence of the microscopic work in the isothermal process is due to the
nature of isothermal process rather than usual thermodynamic limit condition.
Our investigation justifies the validity of “minimum work principle”
formulation of the second law even for a small system far from thermodynamic
limit.

###### pacs:

05.70.Ln, 05.40.-a

##  I INTRODUCTION

Thermodynamics usually deals with the systems of infinite number of degree of
freedoms, in which relative fluctuations of the observable, e.g., energy,
particle number, are inversely proportional to the square root of the numbers
of the particles of the system kersonhuang . Hence for a macroscopic system
consisting of infinite number of particles, the fluctuations are vanishingly
small and the ensemble average can describe thermodynamic phenomena
completely. However, concerning small systems, usually the fluctuations of the
microscopic values of thermodynamic observable will become appreciable, and
ensemble average alone can not longer give a complete description smallsystem
. In recent years, increasing interests are drawn to the study of
thermodynamics of small system, and the emphases are put on the fluctuations
of the microscopic value of the observable, instead of their ensemble average.
Some notable progresses have been made, examples including the Jarzynski
equality JE ; crooks  and the Fluctuation Theorem evans . The former connects
the free energy difference of two equilibrium states with ensemble average of
microscopic work in non-equilibrium process while the later illustrates the
probabilistic “entropy decrease” of a closed system within short time, or
transient “violation” of the second law. These studies shed new light on the
understanding of non-equilibrium thermodynamical processes of biological
motors in cells and promise important applications to the design of small-size
machines. In all these studies, for small systems, though fluctuations of most
observables are appreciable, there exists an exception – the work done during
a slowest reversible equilibrium process (we use isothermal processes to
replace slowest reversible processes hereafter). It has been pointed out that
the fluctuation of microscopic work done by or on a small system during a
slowest reversible process is vanishingly small JE ; kawai . Nevertheless,
though the fluctuations of microscopic work for small systems in finite-time
irreversible processes has been extensively studied microscopic work , and the
vanishing fluctuation of microscopic work of classical small systems specially
concerning thermodynamic isothermal process has been point out, to our best
knowledge, a rigorous proof of the above result from microscopic aspect is
still lacking, and its quantum mechanical generalization has not been studied
yet.

In this paper, we will investigate this problem by simulating a quantum
isothermal process with infinite number of infinitesimal quantum adiabatic
process (QAP) and quantum isochoric process (QIP) arnaud ; kieu06 ; quan5 . We
prove rigorously from microscopic aspect the above result that, for a two-
level system, the fluctuations of the microscopic work during an quantum
isothermal process quan5  is vanishingly small. We emphasize that, different
from most cases in conventional statistical mechanics, where fluctuations
vanishes in the thermodynamic limit, the vanishing work fluctuations for a
small system in an isothermal process is due to the intrinsic nature of
isothermal process. Our study also verify the universal validity of the
“minimum work principle” formulation of the second law: it holds even for a
small system!

Figure 1: Schematic illustration of a quantum isothermal process quan5
$\widehat{AB}$. Here the horizontal axis $P_{e}$ is the occupation probability
in the excited state of the two-level system, and the vertical axis indicates
the level spacing of the two-level system. The smooth curve $\widehat{AB}$
represents the isothermal process, whose “equation of state” can be expressed
as $\Delta(t)=-\beta^{-1}\ln(P_{e}^{-1}-1)$. The horizontal and vertical lines
represent QIC and QAP quan5 . We can use many small QAP and QIP to model the
quantum isothermal process. For example, we use a “five-step stair” path
(green) $A\longrightarrow C\longrightarrow D\cdots\longrightarrow B$ to
simulate the smooth curve $\widehat{AB}$. “One-step” path (blue) and “twenty-
step” path (orange) are also illustrated.

##  II The thermodynamic process in parameter space

We consider a two-level quantum mechanical system with excited (ground) states
$\left|e\right\rangle$ ($\left|g\right\rangle$) with instantaneous eigen-
energy $E_{e}(t)$ ($E_{g}(t)$) depending on time $t$. This two-level system
can be modeled as a spin-1/2 in an external magnetic field. It interacts with
a heat bath of inverse temperature $\beta$, which can be universely modeled as
a collection of many bosons with creation (annihilation) operators
$a_{q}^{{\dagger}}$ ( $a_{q})$ leggett . The model Hamiltonian reads spinboson
; berman .

$H=\Delta(t)\sigma_{z}+\sum_{q}\omega_{q}a_{q}^{{\dagger}}a_{q}+\sum_{q}(\lambda_{q}\sigma_{-}a_{q}^{{\dagger}}+h.c.),$
(1)

where $\sigma_{-}=\left|g\right\rangle\left\langle
e\right|=(\sigma_{x}-i\sigma_{y})/2$ and
$\sigma_{z}=(\left|e\right\rangle\left\langle
e\right|-\left|g\right\rangle\left\langle g\right|)/2$. Initially, let the
two-level system be thermalized to equilibrium. Then we alter the magnetic
field slowly so that the energy level spacing $\Delta(t)$ slowly changes from
$\Delta_{A}$ to $\Delta_{B}$. During the controlling process illustrated by
the smooth curve $\widehat{AB}$ in Fig. 1, the work is done on the system. In
the infinitely slow process, which can be alternatively regarded as a quantum
isothermal process quan5 , the two-level system is in the thermal equilibrium
at every instant, which is described by the diagonal reduced density matrix
$\rho_{S}(t)=P_{e}(t)\left|e\right\rangle\left\langle
e\right|+[1-P_{e}(t)]\left|g\right\rangle\left\langle g\right|$, where
$P_{e}(t)=\exp[{-\Delta(t)]/}(1+\exp[{-\beta\Delta(t)]})$ satisfies the Gibbs
distribution. It should be pointed out that, during the isothermal process,
there is a heat exchange between the two-level system and the heat bath.

For such an isothermal process, it is difficult to calculate the microscopic
work distribution directly. According to Ref. arnaud ; kieu06 ; quan5 ,
however, this process can be simulated by a series of QAP and QIP. In QAP
(QIC) processes, there is only work done (heat exchange). Hence, using the
changes of eigen-energies of microscopic state at instant $t=A,C,D$ , we can
indirectly calculate the microscopic work done (heat exchange) JE ; crooks
${\mathchar 22\relax\mkern-11.0mud}W=E_{\alpha}(C)-E_{\alpha}(A)$ (${\mathchar
22\relax\mkern-11.0mud}Q=E_{\alpha}(D)-E_{\beta}(C)$), for $\alpha,\beta=e,g$.
In the parameter space, these QAP and QIP series processes are represented by
the “stair” path ($A\longrightarrow C\longrightarrow
D\longrightarrow\cdots\longrightarrow B$) in Fig. 1. When every step of the
“stair” path becomes infinitesimal, the “stair” path becomes equivalent to the
isothermal process $\widehat{AB}$. In this way we simulate the quantum
isothermal process with $N$ equal-height steps (see Fig. 1) with the small
height $\Delta=(\Delta_{B}-\Delta_{A})/N$ where $\Delta_{A}$ and $\Delta_{B}$
are the level spacings at point $A$ and point $B$ respectively. The level
spacings of the two-level system after the $(j-1)$-th QIC is

$\Delta_{j}=\Delta_{A}+(j-1)\Delta,$ (2)

for $j=1,2,\cdots,N+1.$ The initial and final point $A$ and $B$ corresponds to
$j=1$ and $j=N+1$ respectively. When we fix the initial point $A$, and the
final point $B$, the jump $\Delta$ in every step decrease with the increase of
the step number $N$, and $\Delta$ approaches zero when $N$ becomes infinity.
Obviously, when $N\longrightarrow\infty$, the “stair” path approaches its
asymptotic behavior - the isothermal path (see Fig. 1). When the system
reaches thermal equilibriums, the occupation probabilities obeys the Gibbs
distribution defined by

$P_{e}^{j}=e^{-\beta\Delta_{j}}[1+e^{-\beta\Delta_{j}}]^{-1};P_{g}^{j}=P_{e}^{j}e^{\beta\Delta_{j}}$
(3)

We remark that there are three time scales in our process: $\tau_{a}$ for
quantum adiabatic approximation, $\tau_{c}$ the control time of changing the
magnetic field, hence the level spacing, and $\tau_{r}$ the relaxation of the
two-level system. According to Ref. berman  , $\tau_{r}$ is determined by the
coupling strength $\lambda_{q}$ (1). We consider the case that
$\tau_{a}\ll\tau_{c}\ll\tau_{r}$ for a quantum adiabatic process where we can
define the microscopic work in every realization of the process.

##  III Microscopic work distribution

Having defined the “path” in the parameter space ($\Delta-P_{e}$) space, we
can further introduce the microscopic work and its corresponding probabilities
for a given “path”. Actually, the definition of microscopic work is very
similar to that in Ref. crooks . In the above path divided into many “steps”,
the first step $A\longrightarrow C\longrightarrow D$ consists of a QAP
$A\longrightarrow C$, and a QIP $C\longrightarrow D$. At the beginning (the
point $A$ of Fig. 1), the system is initially in a thermal equilibrium state
$\rho_{S}(A)$, which implies that the system is either in its microscopic
state $\left|g\right\rangle$ or $\left|e\right\rangle$ with probabilities
$P_{g}^{1}$ and $P_{e}^{1}$ respectively. We choose the ground state in the
energy reference point so that the microscopic energy $E(A)$ of the system at
initial pint $A$ can take $E_{e}(A)=\Delta_{A}$ or $E_{g}(A)=0$, with
probability $P_{e}^{1}$ and $1-P_{e}^{1}$ respectively. In the first QAP
$A\longrightarrow C$, the system remains in its microscopic state
$\left|g\right\rangle$ ($\left|e\right\rangle$) if the system is initially in
its microscopic state $\left|g\right\rangle$ ($\left|e\right\rangle$). As
there is no heat exchange in the QAP, the work done by external controller is
just the change of the microscopic energy
$W_{\alpha}=E_{\alpha}(C)-E_{\alpha}(A)$ for $\alpha=e,g$. Correspondingly the
work done during $A\longrightarrow C$ can be either $\Delta_{C}-\Delta_{A}$ or
$0$ with probabilities $P_{e}^{1}$ or $1-P_{e}^{1}$ respectively. This also
agrees with the definition of work in quantum mechanical system: work is
associated with the change of the level spacing quan5 ; kieu06 ; quan1 .

After the QAP, a quantum isochroc process $C\longrightarrow D$ (see Fig. 1)
follows. Here, there is no work done according to the definition of work in
quantum mechanical system quan5 ; kieu06 ; quan1 , because there is no change
in the eigenergies. Nevertheless, there is heat exchange between the system
and the bath. The QIP last long enough ($\gg\tau_{r}$) so that the system can
reach thermal equilibrium with the heat bath. After a thermolization for long-
time, the two-level system reach thermal equilibrium with the heat bath again
(3) at instant $D$ indicated in Fig. 1. Then a second step $D\longrightarrow
E\longrightarrow F$ begins. Similarly, the microscopic work $0$ or
$\Delta_{3}-\Delta_{2}$ is done in this step with probabilities $1-P_{e}^{2}$
or $P_{e}^{2}$. The microscopic work done and their probabilities for the
remaining steps can be obtained through a similar analysis. Because in every
QIP, the system is independently thermalized by the heat bath, then there
should be no correlations of the probabilities distributions in every two
neighbor steps, or alternatively, this process is Markovian process. Hence,
the total microscopic work done after the $N$-step is a sum of microscopic
works done in all steps and the joint probabilities for the $N$-step as a
whole is the product of that of all steps.

For a special example that the microscopic work done during the whole process
is $W=N\Delta$ where $\Delta$ is that for each QIP step, the joint
probabilities for the system keeping in $\left|e\right\rangle$ in every QIP is
$P\left[N\Delta\right]=P_{e}^{1}P_{e}^{2}\cdots P_{e}^{N}$. The more general
case with microscopic work $W=\left(N-k\right)\Delta$ corresponds to a
microscopic process, in which $k$ out of $N$ QIPs ends with the system in its
microscopic state $\left|g\right\rangle$. The probability
$P(k):=P\left[\left(N-k\right)\Delta\right]$ with the microscopic work
$W=\left(N-k\right)\Delta$ in the $N$-step path is given by the following
eqution:

$P(k)=\left(\prod_{j=1}^{N}P_{e}^{j}\right)\left(\prod_{l=0}^{k-1}\frac{e^{\beta\Delta_{B}}-e^{\beta(\Delta_{A}+l\Delta)}}{e^{\beta(l+1)\Delta}-1}\right),$
(4)

To prove the above result, we first consider the case with $k=1$. For this
case, there is one and only one out of the $N$ QIPs, in which the system ends
up in the microscopic state $\left|g\right\rangle$. Then the corresponding
probability can be caculated as $P(1)=\left(1-P_{e}^{1}\right)P_{e}^{2}\cdots
P_{e}^{N}+P_{e}^{1}\left(1-P_{e}^{2}\right)\cdots
P_{e}^{N}+\cdots+P_{e}^{1}P_{e}^{2}\cdots\left(1-P_{e}^{N}\right)=\left(\prod_{j=1}^{N}P_{e}^{j}\right)e^{\beta\Delta_{B}}\sum_{x_{1}=1}^{N}e^{-\beta\Delta
x_{1}}$ or

$P(1)=\left(\prod_{j=1}^{N}P_{e}^{j}\right)\frac{\left(e^{\beta\Delta_{B}}-e^{\beta\Delta_{A}}\right)}{e^{\beta\Delta}-1},$
(5)

That means thre Eq. (4) holds for $k=1$. Similarly we can check the case with
$k=2$. For this case, there are two out of the $N$ QIP, in which the system
ends up in the microscopic state $\left|g\right\rangle$. Hence its probability
can be expressed as
$P(2)=\left(1-P_{e}^{1}\right)\left(1-P_{e}^{2}\right)P_{e}^{3}\cdots
P_{e}^{N}+\left(1-P_{e}^{1}\right)P_{e}^{2}\left(1-P_{e}^{3}\right)\cdots
P_{e}^{N}+\cdots+P_{e}^{1}P_{e}^{2}\cdots\left(1-P_{e}^{N-1}\right)\\\
\left(1-P_{e}^{N}\right)=\left(\prod_{j=1}^{N}P_{e}^{j}\right)e^{2\beta\Delta_{B}}\sum_{x_{1}=1}^{N}\sum_{x_{2}=1}^{x_{1}-1}e^{-\beta\Delta(x_{1}+x_{2})}$
or

$P(2)=\left(\prod_{j=1}^{N}P_{e}^{j}\right)\frac{\left[e^{\beta\Delta_{B}}-e^{\beta\Delta_{A}}\right]\left[e^{\beta\Delta_{B}}-e^{\beta(\Delta_{A}+\Delta)}\right]}{(e^{\beta\Delta}-1)(e^{2\beta\Delta}-1)},$
(6)

Hence Eq. (4) also holds for $k=2$ case. In general, for an arbitrary $k$, the
corresponding probability can be expressed as

$\displaystyle\begin{split}P(k)=\left(\prod_{j=1}^{N}P_{e}^{j}\right)e^{k\beta\Delta_{B}}\chi(k),\end{split}$
(7)

where
$\chi(k)=\sum_{x_{1}=1}^{N}\sum_{x_{2}=1}^{x_{1}-1}\cdots\sum_{x_{k}=1}^{x_{k-1}-1}e^{-\beta\Delta(x_{1}+\cdots+x_{k})}$.
As $\chi(k),(k=1,2,\cdots,N)$ satisfy the following relation

$\displaystyle\begin{split}\chi(k)&=\sum_{i=1}^{k-1}\left[\left(\prod_{j=1}^{i}\frac{-1}{1-e^{-j\Delta}}\right)\left(-e^{-i\Delta}\right)\chi(k-i)\right]\\\
&+\left(\prod_{j=1}^{k}\frac{-1}{1-e^{-j\Delta}}\right)\left[e^{-k(N+1)\Delta}-e^{-k\Delta}\right],\end{split}$
(8)

we can use the complete induction method to prove that the $\chi(k)$ can be
generally expressed as

$\displaystyle\begin{split}\chi(k)=\prod_{l=0}^{k-1}\frac{e^{-\beta
N\Delta}\left[e^{\beta N\Delta}-e^{\beta
l\Delta}\right]}{e^{\beta(l+1)\Delta}-1}\end{split}$ (9)

Substituting Eq. (9) into Eq. (7), we obtain Eq. (4). Hence, by now we prove
the general result given by Eq. (4).

##  IV Most probabilistic distribution and fluctuation

The above equation (4) can result in the main conclusion in this paper. From
the above microscopic work distribution function (4), we obtain the ratio
$R(k)=P(k+1)/P(k)$ of distributions for two close microscopic work, i.e.,

$R(k)=\frac{\left(e^{\beta\Delta_{B}}-e^{\beta\Delta_{A}+k\beta\Delta}\right)}{e^{\beta\left(k+1\right)\Delta}-1}.$
(10)

Let $\tilde{k}$ maximaze the probability distribution $P(k)$ for the
microscopic work $[N-(\tilde{k}+1)]\Delta$ . Then $P(\tilde{k})\geq
P(\tilde{k}\pm 1),$or $R(\tilde{k})\leq 1$ or $R(\tilde{k}-1)\geq 1$. For very
large $\tilde{k},R(\tilde{k})\simeq 1$ that

$\tilde{k}\Delta=\frac{1}{\beta}\ln\left[\frac{1+\exp[\beta\Delta_{B}]}{\exp[\beta(\Delta_{B}-\Delta_{A})/N]+\exp[\beta\Delta_{A}]}\right].$
(11)

In the large $N$ limit, the above equation determines the microscopic work
$\tilde{W}=(N-\tilde{k})\Delta$ with most probabilistic distribution

$\tilde{W}=\frac{1}{\beta}\ln\left(\frac{1+e^{\beta\Delta_{B}}}{1+e^{\beta\Delta_{A}}}\right)$
(12)

which is just the free energy difference $\Delta F_{AB}=F_{B}-F_{A}$ where
$F_{j}=\ln[1+\exp(\beta\Delta_{j})]/\beta$ for $j=A,B$

Figure 2: Microscopic work distribution of an $N$-step “stair” process. The
horizontal axis indicates the possible microscopic work ranging from $0$ to
$\Delta_{B}-\Delta_{A}$, and the vertical axis is their probabilities. Here,
$\exp(-\beta\Delta_{A})=1/2$, and $\exp(-\beta\Delta_{B})=1/3$. The steps are
chosen to be $N=1$, $5$, $20$, $100$, $1000$, and $10000$ respectively. “Path”
corresponding to $N=1$, $5$, $20$ are given in Fig. 1. From these figures it
can be inferred that when $N$ is small the process is irreversible, and the
fluctuation is appreciable. The relative fluctuation of the microscopic work
vanishes when $N\rightarrow\infty$, or the fluctuation of an isothermal
process approaches zero. Besides, the most probabilistic work from the
(numerical) figures $\tilde{W}=0.29(\ln 3-\ln 2)k_{B}T$ agrees well with the
(analtical) free energy difference $\Delta
F_{AB}=[\ln(1+1/2)-\ln(1+1/3)]k_{B}T$.

Next let us give a heuristic analysis of the dispersion of the work
distribution (4). Because all steps in the “stair” path are independent with
each other, thus the whole process can be regarded as Markovian. So the
variance of total microscopic work done during the whole process equals to the
sum of variance of local microscopic work in every step, i.e., $\left\langle
W_{AB}^{2}\right\rangle-\left\langle
W_{AB}\right\rangle^{2}=\sum_{j=1}^{N}(\left\langle
W_{j}^{2}\right\rangle-\left\langle W_{j}\right\rangle^{2})$, where $W_{j}$ is
the microscopic work done during the $j$th QAP, and the local fluctuations

$\left\langle W_{j}^{2}\right\rangle-\left\langle
W_{j}\right\rangle^{2}=\Delta^{2}[P_{e}^{j}-(P_{e}^{j})^{2}]$ (13)

for different $j$ are similar. Here $\Delta$ is inversely proportional to $N$,
and $\left\langle W_{AB}\right\rangle$ being independent of $N$, the relative
variance of $W_{AB}$ is inversely proportional to

$\sqrt{\frac{\left\langle W_{AB}^{2}\right\rangle-\left\langle
W_{AB}\right\rangle^{2}}{\left\langle
W_{AB}\right\rangle}}\propto\frac{1}{\sqrt{N}}$ (14)

We numerically plot the work distribution function (see Fig.2) based on the
above analytical result (4) to test the above analysis. Here we choose the
step number $N$ from $1$ to $10000$. For $N=1$, the “stair” path becomes a
“one-step” path consists of an QAP and an QIP (see Fig. 1). The microscopic
work corresponding to the “one-step” path is either $\Delta_{B}-\Delta_{A}$ or
$0$ with the probability $P(W=\Delta_{B}-\Delta_{A})=P_{e}^{1}$ or
$P(W=0)=1-P_{e}^{1}$. In the above figures, we choose
$\exp(-\beta\Delta_{A})=1/2$, ($P_{e}^{1}=1/3$), and the numerical result
agrees well with our analysis. For $N=5$ (see Fig. 1), the possible
microscopic work can be $W=i(\Delta_{B}-\Delta_{A})/5,i=0,1,2,\cdots,5$. The
numerical result indicates vanishing probability for
$W=\Delta_{B}-\Delta_{A}$. For $N=20$ (see Fig. 1), the numerical result show
even more vanishing probabilities of microscopic work. That is, the dispersion
(fluctuation) of microscopic work decrease with the increase of $N$. Actually,
from the above numerical figures, it is not difficult to find that the
dispersion of the microscopic work distribution is inversely proportional to
the square root of $N$. For example, the dispersion for $N=100$ is ten times
that for $N=10000$ case (see Fig. 2). Hence, numerical results agrees well
with our heuristic analysis and both they verified our main result, when
$N\longrightarrow\infty$, the fluctuations of microscopic work vanishes.

##  V Minimum work principle for a two-level system

As we have mentioned before, for small systems and within short time, the
formulation “entropy never decrease for a closed system” of the second law may
be transiently “violated” probabilistically due to appreciable fluctuations
evans . A straightforward question is: will the other formulations of the
second law, e.g., the minimum work principle kawai ; minimum , also be
transiently “violated” probabilistically for small systems? The “minimum work
principle” states that “when varying the speed of a given process for an
initially equilibrium system, the work is minimal for the slowest realization
of the process” kawai ; minimum . In the following we will test the validity
of “minimum work principle” for a two-level system by utilizing the formula
(4) we derived above. The average work over all possible realizations for a
given $N$-step path can be expressed as

$\left\langle
W\right\rangle_{N}=\sum_{k=0}^{N}\left(\prod_{j=1}^{N}P_{e}^{j}\right)\left(\prod_{l=0}^{k-1}\frac{e^{\beta\Delta_{B}}-e^{\beta(\Delta_{A}+l\Delta)}}{e^{\beta(l+1)\Delta}-1}\right)(N-k)\Delta,$
(15)

Figure 3: Averaged work $\left\langle W\right\rangle_{N}$ as a function of $N$
(15). The steps $N$ chosen here are
$5^{0}=1,5^{1}=5,5^{2}=25,5^{3}=125,5^{4}=625,5^{5}=3125$, and $5^{6}=15625$.
It can be seen that the averaged work is a monotonically deceasing function of
$N$. In the one-step path ($N=1$), the averaged work equals to $\left\langle
W\right\rangle_{1}=(\ln{3}-\ln{2})/3\approx 0.135155k_{B}T$; In the
$15625$-step path, the averaged work equals to $\left\langle
W\right\rangle_{5^{6}}=(\ln{3}-\ln{2})/3\approx 0.11784k_{B}T$, which is very
close to its asymptotic value $\Delta
F_{AB}=[\ln(1+1/2)-\ln(1+1/3)]k_{B}T\approx 0.117783k_{B}T$. Thus, it can be
inferred that the “minimum work principle” still holds for a two-level system.

In Fig. 3 we plot the averaged work $\left\langle W\right\rangle_{N}$ as a
function of $N$ (15). It can be seen that for the two-level system,
$\left\langle W\right\rangle_{N}$ is a monotically decreasing function of $N$
(time $t$), and when $N\rightarrow\infty,(t\rightarrow\infty)$, the averaged
work $\left\langle W\right\rangle_{N}$ approaches an asymptotic value, and its
minimum value – the difference of the free energy. Thus, from the numerical
result it can be inferred that the “minimum work principle” still holds for a
two-level system.

The above proof of minimum work principle can be alternatively understood in
the following way. From the above analytical and numerical result, we observed
that the fluctuation of microscopic work in an isothermal process vanishes,
and then the work of the most probabilistic distribution equals to the
difference of the free energy $\tilde{W}=\Delta F$. According to Ref. JE ,
$\left\langle W_{\mathrm{irre}}\right\rangle\geqslant\Delta F$, where
$\left\langle W_{\mathrm{irre}}\right\rangle$ is the average work done during
an irreversible process. Combining the two results, we have $\left\langle
W_{\mathrm{irre}}\right\rangle\geqslant\tilde{W}$. Thus we proved the minimum
work principle for small system.

##  VI Discussion and conclusion

Before concluding this paper, we would like to emphasize the following points:
First, the technique of simulating isothermal processes with adiabatic
processes and isochoric processes are important to our proof, which enables us
to establish the connection between large time limit and large N limit.
Second, the calculation of exact expression of microscopic work in our paper
is non-trivial because the work contributions in the different steps are not
identically distributed. Hence, it is different from the law of large numbers,
with time as the large number kawai . Third, we proved the “minimum work
principle” formulation of the second law stands for even small system, though
other formulations may be transiently “violated” probabilistically evans .
This is not surprising because “minimum work principle” concerns infinite-
long-time processes, which has no contradiction with the transient “violation”
of the second law for small systems predicted by the Fluctuation Theorem.
Actually, the Fluctuation Theorem does not constitute real violation of the
second law, which is a statistical law and holds when averaged over different
realization of the process. Fourth, the isothermal process is reversible, but
the finite $N$ “step path” is irreversible, due to the QIP (thermolization) is
irreversible. We can thus expect that the work dissipation kawai ; microscopic
work  for the finite $N$ step path will be finite and will decrease with the
increase of $N$, and finally vanishes when $N$ approaches infinity.

In summary, by simulating an quantum isothermal process with infinite many
infinitesimal QAP and QIP, we obtain the analytical expressions of microscopic
work distribution in an isothermal process. Through both analytical and
numerical analysis, we rigorously verify that the fluctuations of microscopic
work distribution vanishes even for a small system in an isothermal process.
This result is different from the usual fluctuations in statistical mechanics,
e.g., the energy fluctuation and particle number fluctuation in canonical
ensamble and grand canonical ensamble, where the fluctuations of energy and
particle nubmers approaches zero when the system approaches thermodynamic
limit (particle number approaches infinity $N_{P}\longrightarrow\infty$).
Here, however, even for single particle system, we microscopically demonstrate
the vanishing of microscopic work fluctuation. Because
$N\longrightarrow\infty$ is a must to simulate an isothermal process, we
conclude that the vanishing of microscopic work fluctuations is due to the
intrinsic nature of isothermal process, rather than the thermodynamic limit of
the system size. We also prove that for a small system, the “minimum work
principle” formulation of the second law holds though other formulations maybe
transiently “violated” probabilistically. Finally we would like to point it
out that our result is universal and does not depend on the specific model
used here, because the technique of simulating the isothermal process with the
isochoric process and the adiabatic process can be applied to any systems.
Generalizations of our current discussion to other models will be given in the
future.

##  VII acknowledgments

The authors thank a anonymous referee for pointing out a mistake in our
previous version of the manuscript. HTQ thanks Rishi Sharma for stimulating
discussions and gratefully acknowledges the support of the U.S. Department of
Energy through the LANL/LDRD Program for this work; CPS is supported by NSFC
with grant Nos. 90203018, 10474104, 60433050, 10704023 and NFRPC with grant
Nos. 2006CB921205, 2005CB724508.

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