Quantum heat engine using energy quantization and resources of ignorance (original) (raw)

Quantum heat engine using energy quantization and resources of ignorance

George Thomas, 1,∗{ }^{1, *} Debmalya Das, 2,†{ }^{2, \dagger} and Sibasish Ghosh 1,‡{ }^{1, \ddagger}
1{ }^{1} Optics and Quantum Information Group, The Institute of Mathematical Sciences,
HBNI, CIT Campus, Taramani, Chennai 600113, India
2{ }^{2} Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Allahabad 211 019, India

Abstract

We study a quantum Stirling cycle which extracts work using quantized energy levels. The work and the efficiency of the engine depend on the length of the potential well, and the Carnot efficiency is achieved in a low temperature limiting case. We show that the lack of information about the position of the particle inside the potential well can be converted into useful work without resorting to any measurement. In the low temperature limit, we calculate the amount of work extractable from distinguishable particles, fermions and bosons.

I. INTRODUCTION

The idea of the Maxwell’s demon occupies a central position in the understanding of thermodynamics and information. It was introduced in a thought experiment that envisaged a situation in which there could be a possible violation of the second law of thermodynamics [1, 2]. A classical analysis of the Maxwell demon was first developed in [3], in the form of the Szilard engine. The Szilard engine consists of an enclosed chamber containing a gas molecule. A thin and massless partition is inserted in the middle of the chamber [4]. The demon measures the position of the molecule to the right or to the left of the partition and records it. Based on the measurement, the demon then connects a mass to the partition on the same side as the molecule. Now by absorbing heat from a hot bath, the gas can expand isothermally to occupy the original full volume of the chamber. The partition, consequently, in pulling the mass, performs work of magnitude kBTln⁡2k_{B} T \ln 2 where TT is the temperature of the bath and kBk_{B} is the Boltzmann constant. Superficially, it seems that the involvement of the demon enables a Szilard engine, with a single gas molecule, to perform kBTln⁡2k_{B} T \ln 2 amount of work, leading to a decrease of entropy of the heat bath, measuring kBln⁡2k_{B} \ln 2. This is impossible, according to the second law of thermodynamics, as a minimum of an equivalent increase of entropy is required in some part of the global system. In [3] it was suggested that an equivalent amount of work is required in the measurement of the position of the gas molecule which saves the second law. However, it was not until [5] that the work done in the erasure of information in the demon’s memory was taken into consideration and the role played by measurement was refuted [5,6][5,6]. Landauer’s erasure principle showed that minimum amount of increase in the entropy has to be kBln⁡2k_{B} \ln 2 for erasing one-bit memory stored by the demon, establishing an intriguing connection between information and thermodynamics [5,6][5,6]. Further, Landauer’s erasure principle has been experimentally established us-

[1]ing a single colloidal particle [7].
In the quantum version of Szilard engine, the insertion and the removal of the barrier constitutes certain amount of work and heat exchange, unlike in the classical case [8]. Compared to the compression of the particle to the left (or right) side of the box, the insertion of the of the barrier needs less amount of work. This is because in an insertion scenario, the position of the particle is unknown. One has to perform a measurement after the insertion to find the position of the particle. Therefore, the state of the system after compression is equivalent to the state of the system, after insertion followed by the projective measurement. Similarly, during the removal process, the particle is delocalized due to tunneling, a factor that does not come into play during expansion. Hence the extractable work during the removal process is less compared to that obtained during expansion. There is an element of lack of information due to degeneracy and tunneling which causes a difference in the amount of work.

The modelling of a quantum Szilard engine begins with the conversion of a single infinite potential well to an infinite double well potential by introducing a barrier in the middle in the form of a delta potential. Consider a box of length 2a2 a with rigid walls containing a single molecule [9-11]. When a delta function barrier is introduced in the middle completely, the system is converted to an infinite double well potential, the even energy levels which have a node at the origin remain same while the odd energy levels shift upwards and overlap with the next even energy levels. The same methodology can be employed to investigate the cases where more than one barrier are inserted inside the potential box. Suppose the case of two barriers, inserted at distances 2a3\frac{2 a}{3} from the two walls of the box. Note that the third energy eigenstate, in the case of without having any barrier, has nodes at the above points. This implies that upon insertion of the two barriers, the third energy level remains unchanged while the first and the second energy levels shift to the third level. Similarly, all energy levels in the multiples of three remain unchanged on inserting two barriers and the others shift accordingly. Thus an infinite triple potential well has energy levels that are triply degenerate. Hence, for the case of N−1N-1 barriers, the original energy levels

1. • georget@imsc.res.in
     †{ }^{\dagger} debmalyadas@hri.res.in
     ‡\ddagger sibasish@imsc.res.in ↩︎

that are multiples of NN remain unchanged while the others shift and become degenerate with the former. This makes the energy levels of a potential well with N−1N-1 barriers degenerate, with degeneracy NN.

The main motivation for this work is to device a quantum heat engine which will work exclusively on quantum features [12]. The Szilard engine converts information into useful work and hence measurement is needed to extract work [8, 13-19]. Our analysis shows an effective way of converting lack of information to useful work without any measurement but using two different reservoirs. The amount of extractable work depends on the nature of the particle such as distinguishable particles, bosons or fermions [8, 15, 16, 18, 19]. The cycle we use is quite similar to the Stirling cycle. Quantum versions of Stirling engines have been studied in the recent past [20-23]. A classical Stirling cycle can operate at Carnot efficiency if a regenerator system is used in the isochoric branches. In this paper, we provide a model of Stirling engine that operates at Carnot efficiency at low temperature limit without using a regenerator and measurement. In Szilard engine model, Carnot efficiency can be achieved by erasing the information (obtained from the measurement) using a heat bath of lower temperature compared to the one attached to the engine [16]. We also discuss the case where more than one partitions are inserted with more number of particles.

The paper is organized as follows: In the Section II, we give a brief description on quantum model of Stirling-like engine. Section III is a brief discussion that points out the distinguishing features of our engine with the more conventional one that is based on expansion and compression. Section IV is devoted to discuss the limiting cases which includes a low temperature (reversible) limit and a classical limit (free particle). In Section V, we discuss the amount of work extractable from distinguishable particles and indistinguishable particles (fermions and bosons). We conclude the paper with some discussions in Section VI.

II. STIRLING-LIKE CYCLE

A Stirling cycle [20-22, 24] consists of four stages, two isothermal processes and two isochoric processes. In the first stage, a barrier is inserted isothermally in the middle of the well such that the working medium is in equilibrium with a hot bath at a temperature T1T_{1} during quasistatic insertion process. In the second stage, the working medium undergoes isochoric heat extraction by connecting it with a heat bath at a lower temperature T2T_{2}. Next, an isothermal removal of the barrier is effected by keeping the engine in equilibrium at the above temperature. In the final stage, the engine is once again connected to the hot bath at the temperature T1T_{1} and an isochoric heat absorption is carried out. The process is pictorially represented in Fig. 1

FIG. 1. The four stages of Stirling cycle. Stage 1 is isothermal insertion, Stage 3 is isothermal removal and Stages 2 and 4 are isochoric processes.

Let us consider a particle of mass mm inside a one dimensional potential box of length 2a2 a at equilibrium with a bath at temperature T1T_{1}. The nth energy level of the one dimensional potential well is given by

En=n2π2ℏ22m(2a)2(n=1,2,3⋯ )E_{n}=\frac{n^{2} \pi^{2} \hbar^{2}}{2 m(2 a)^{2}} \quad(n=1,2,3 \cdots)

This can be used to calculate the corresponding partition function of the system, given by

ZA=∑n=1∞e−EnkBT1=∑ne−n2π2ℏ22m(2a)2kBT1Z_{A}=\sum_{n=1}^{\infty} e^{-\frac{E_{n}}{k_{B} T_{1}}}=\sum_{n} e^{-\frac{n^{2} \pi^{2} \hbar^{2}}{2 m\left(2 a\right)^{2} k_{B} T_{1}}}

where kBk_{B} is the Boltzmann constant. The energy levels in Eq. (1) are evidently non-degenerate.

Suppose that a wall is inserted slowly in the middle of the box isothermally at this point. For all subsequent analyses and discussions we consider this point as the origin of coordinates. Immediately the problem is converted into an infinite double well potential. The energy levels get reoriented as a result of this action. The energy level corresponding to even values of nn remain unchanged while each energy level with odd value of nn shifts upwards and overlap with the nearest neighboring energy level. This leads to a degeneracy in the energy levels of this new setup. We can express an arbitrary energy level of the partitioned one dimensional potential box as

E2n=(2n)2π2ℏ22m(2a)2E_{2 n}=\frac{(2 n)^{2} \pi^{2} \hbar^{2}}{2 m(2 a)^{2}}

Accordingly, the new partition function changes to

ZB=∑n=1∞2e−(2n)2π2ℏ22m(2a)2kBT1=2Za,T1Z_{B}=\sum_{n=1}^{\infty} 2 e^{-\frac{(2 n)^{2} \pi^{2} \hbar^{2}}{2 m\left(2 a\right)^{2} k_{B} T_{1}}}=2 Z_{a, T_{1}}

where Za,T1Z_{a, T_{1}} is the canonical partition function for a particle in box with length aa in thermal equilibrium with temperature T1T_{1}. The internal energies UAU_{A} and UBU_{B} of the system can be calculated by employing the respective partition functions ZAZ_{A} and ZBZ_{B} from equations (2)

and (4) as UA/B=−∂ln⁡ZA/B/∂β1U_{A / B}=-\partial \ln Z_{A / B} / \partial \beta_{1}, where β1=1kBT1\beta_{1}=\frac{1}{k_{B} T_{1}}. The heat exchanged in the isothermal process (Stage (1) in Fig. 1) of introducing the wall is thus,

QAB=UB−UA+kBT1ln⁡ZB−kBT1ln⁡ZAQ_{A B}=U_{B}-U_{A}+k_{B} T_{1} \ln Z_{B}-k_{B} T_{1} \ln Z_{A}

In the next step (Stage (2) in Fig. 1), the system is connected to a heat bath at a lower temperature T2T_{2}. The energy levels remain the same, while the new partition function is given by

ZC=∑n=1∞2e−E2nkBT2=2Za,T2Z_{C}=\sum_{n=1}^{\infty} 2 e^{-\frac{E_{2 n}}{k_{B} T_{2}}}=2 Z_{a, T_{2}}

where Za,T2Z_{a, T_{2}} is the canonical partition function for a particle in box with length aa in equilibrium with a bath at temperature T2T_{2}. The heat exchanged is now the difference of the average energies of the initial and the final states.

QBC=UC−UBQ_{B C}=U_{C}-U_{B}

with UC=−∂ln⁡ZC/∂β2U_{C}=-\partial \ln Z_{C} / \partial \beta_{2} as the internal energy in the state C where β2=1kBT2\beta_{2}=\frac{1}{k_{B} T_{2}}. The wall is now removed slowly and isothermally (see Stage (3) in Fig. 1), with the system connected to the heat bath at temperature T2T_{2}. The energy levels are once again restored to initial values given in Eq. (1). while the partition function is now given by

ZD=∑n=1∞e−EnkBT2=∑n=1∞e−n2π2ℏ22m(2a)2kBT2Z_{D}=\sum_{n=1}^{\infty} e^{-\frac{E_{n}}{k_{B} T_{2}}}=\sum_{n=1}^{\infty} e^{-\frac{n^{2} \pi^{2} \hbar^{2}}{2 m\left(2 a\right)^{2} k_{B} T_{2}}}

If UD=−∂ln⁡ZD/∂β2U_{D}=-\partial \ln Z_{D} / \partial \beta_{2} is the internal energy in the state D , the heat exchanged in the process is

QCD=UD−UC+kBT2ln⁡ZD−kBT2ln⁡ZCQ_{C D}=U_{D}-U_{C}+k_{B} T_{2} \ln Z_{D}-k_{B} T_{2} \ln Z_{C}

In the final step (Stage (4) in Fig. 1), the system is connected to the higher temperature bath at T1T_{1} once again. The energy levels remain unchanged but the partition function changes to ZAZ_{A}. The corresponding heat exchanged is given by

QDA=UA−UDQ_{D A}=U_{A}-U_{D}

In our process, the total work done is

W=QAB+QBC+QCD+QDA=kBT1ln⁡ZBZA−kBT2ln⁡ZCZD\begin{aligned} W & =Q_{A B}+Q_{B C}+Q_{C D}+Q_{D A} \\ & =k_{B} T_{1} \ln \frac{Z_{B}}{Z_{A}}-k_{B} T_{2} \ln \frac{Z_{C}}{Z_{D}} \end{aligned}

Hence, the efficiency of the cycle is given by

η=1+QBC+QCDQDA+QAB\eta=1+\frac{Q_{B C}+Q_{C D}}{Q_{D A}+Q_{A B}}

It is to be noted that our engine represents an idealized case where the isothermal processes are done slow enough (compared to the tunneling time scales) to keep the system in equilibrium throughout the processes. We also consider the energy needed to couple and decouple the system with the baths are negligible.

III. COMPARISON WITH THE CONVENTIONAL SCENARIO

In a conventional Stirling engine, the isothermal expansion is carried out by keeping the engine in equilibrium with a hot bath while the compression is carried out using cold bath in contact, as given in the Fig. 2. On the other hand, in our cycle, the insertion is done when the system is in contact with the hot bath and isothermal removal of the barrier is assisted with cold bath. Consider the stage 1, discussed in Section II, a particle in a box of length 2a2 a and in equilibrium with a bath of temperature T1T_{1}. The canonical partition function is ZAZ_{A}. Now, consider a process in which isothermally inserting barrier in the middle of the box. The partition function at the end of the process is ZB=2Za,T1Z_{B}=2 Z_{a, T_{1}} where Za,T1Z_{a, T_{1}} is partition function for particle trapped in box of length aa. The factor 2 appears because of the degeneracy or in other words, due to the ignorance about the particle being in the left or right side of the box. Therefore, the work done by the engine is the difference in free energies. ΔF=kBT[ln⁡2+ln⁡Za,T1−ln⁡ZA]\Delta F=k_{B} T\left[\ln 2+\ln Z_{a, T_{1}}-\ln Z_{A}\right]. This work is lesser than that is needed to compress the box from 2a2 a to aa. In that case work needed is ΔF=kBT[ln⁡Za,T1−ln⁡ZA]\Delta F=k_{B} T\left[\ln Z_{a, T_{1}}-\ln Z_{A}\right]. This is due to the fact that in the compression scenario, the position of the particle is known whereas in the insertion scenario, there is a lack of knowledge about the position of the particle. Similarly, the work extracted in the removal process is kBT[ln⁡ZD−ln⁡2−ln⁡Za,T2]k_{B} T\left[\ln Z_{D}-\ln 2-\ln Z_{a, T_{2}}\right]. In this paper, we consider a cycle in which, we insert the barrier when the system is attached to a hot bath of temperature T1T_{1} and remove when the system is attached to a cold bath of T2T_{2} temperature. In certain limiting case, we show that the work done by the engine is kB(T1−T2)ln⁡2k_{B}\left(T_{1}-T_{2}\right) \ln 2. The appearance of ln⁡2\ln 2 is due to the ignorance about the position of the particle.

IV. LIMITING CASES

In this section, we discuss the extractable work from two limiting cases. In low temperature limit [8,15,16][8,15,16], the system works in a reversible manner at Carnot efficiency. The classical limit is obtained from a large width of the potential, where the particle behaves like a free particle and no work can be extracted.

A. Low temperature limit

Let us consider a box with length 2a2 a such that π2ℏ2/2m(2a)2>>kBT1\pi^{2} \hbar^{2} / 2 m(2 a)^{2}>>k_{B} T_{1}, where T1>T2T_{1}>T_{2}. It can be seen that the above condition holds good for low values of temperatures T1T_{1} as well as small values of aa. We will refer to this case as the low temperature limit in all subsequent discussions. In the low temperature limit, the occupational probability in the ground state is close to unity and the entropy approaches zero. When the partition

FIG. 2. Pressure-Volume (P-V) diagram for a classical Stirling cycle: AB and CD are the isothermal processes. BC and DA are isochoric (constant volume) processes. System is in contact with hot bath during DA and AB. The system is in contact with cold bath during BC and CD. In classical Stirling cycle, the working medium is ideal gas and it can work at Carnot efficiency using a regenerator during the stages BC and DA. The work is done only during isothermal branches.

FIG. 3. Plot of work /kBT2/ \mathrm{k}_{\mathrm{B}} \mathrm{T}_{2} (Eq. 11) vs the aa (in nanometers) which is width of each well of a double well potential. The horizontal line represents the low temperature limiting case. Inset shows the plot of efficiency (Eq.12) vs aa. The horizontal line represents the Carnot efficiency (1−T2T1)\left(1-\frac{T_{2}}{T_{1}}\right) obtained from the low temperature limit. Here, we have taken m=9.11×10−31m=9.11 \times 10^{-31} kg,T1=2K\mathrm{kg}, T_{1}=2 K and T2=1KT_{2}=1 K.
is inserted, the ground state of the double well becomes doubly degenerate with occupational probability 1/21 / 2 for each state and hence the entropy becomes ln⁡2\ln 2. Therefore, the total heat absorbed by the system from the hot bath becomes kBT1ln⁡2k_{B} T_{1} \ln 2. Similarly when the wall is removed, the heat exchanged between the system and the cold bath is −kBT2ln⁡2-k_{B} T_{2} \ln 2. Therefore, the work done and the efficiency in this case become respectively

W=kB(T1−T2)ln⁡2,η=1−T2T1W=k_{B}\left(T_{1}-T_{2}\right) \ln 2, \quad \eta=1-\frac{T_{2}}{T_{1}}

The system can attain Carnot efficiency and hence it is reversible. The dimensionless work (W/kBT2)\left(W / k_{B} T_{2}\right) and the efficiency are plotted with the length aa of the box in Fig. 3 and the corresponding values of these two quantities for
the low temperature limit, are also depicted. It is to be noted that, for a two-level system with energy-level spacing ω\omega at temperature TT, the canonical heat capacity can be written as ∂U∂T∣ω=(ω/T)2exp⁡(ω/T)/[1+exp⁡(ω/T)]2\frac{\partial U}{\partial T} \mid \omega=(\omega / T)^{2} \exp (\omega / T) /[1+\exp (\omega / T)]^{2}, where UU is the mean energy of the system [25, 26]. Therefore, in the limit ω>>T\omega>>T, the heat capacity goes to zero. Analogously, one can see that in the low temperature limit of the particle in a box, the heat capacity vanishes and hence the heat exchanged to lower or raise the temperature during Stage 2(QBC2\left(Q_{B C}\right. given in Eq. (7)) as well as Stage 4(QDA4\left(Q_{D A}\right. given in Eq. (10)) respectively also vanish.

B. Classical limit

The energy difference between the two adjacent levels ( nth and (n+1)(\mathrm{n}+1) th) are (2n+1)π2ℏ2/2m(2a)2(2 n+1) \pi^{2} \hbar^{2} / 2 m(2 a)^{2}. During the insertion of the barrier, the odd energy levels approaches the next even numbered level. When the barrier is fully inserted, each energy level will be doubly degenerate. Therefore the gap between the adjacent energy levels is responsible for the work. When a→∞a \rightarrow \infty, the energy gap goes to zero and there will be a continuum of energy levels. Therefore the work required to insert or remove the barrier goes to zero, as the particle becomes free particle in that limit. The classical limit has been explored earlier in the context of Szilard engine in [14].

V. ENGINE WITH DISTINGUISHABLE AND INDISTINGUISHABLE PARTICLES

We have looked at the dependencies of the work extracted from the Stirling engine at low temperature and classical limits. In this section, we explore the properties of the working fluid manifesting as the amount of extractable work in the low temperature limit. Let us consider two fermions and two bosons in the low temperature limiting case with a single partition. Before inserting the partition, the system is in the ground state (or, at least the system is highly probable to be in the ground state). The ground states of the system for bosons and fermions are given as

ΨB/F=12[ψn1(x1)ψn2(x2)±ψn2(x1)ψn1(x2)]\Psi_{B / F}=\frac{1}{\sqrt{2}}\left[\psi_{n_{1}}\left(x_{1}\right) \psi_{n_{2}}\left(x_{2}\right) \pm \psi_{n_{2}}\left(x_{1}\right) \psi_{n_{1}}\left(x_{2}\right)\right]

ψn1\psi_{n_{1}} and ψn2\psi_{n_{2}} represent the wavefunctions corresponding to the n1th n_{1}^{\text {th }} and n2th n_{2}^{\text {th }} energy eigenstates. The ground state for the case of fermion takes the values n1=1n_{1}=1 and n2=n_{2}= 2. On the other hand for bosons, both the particles can be in the same state and hence it can take n1=n2=1n_{1}=n_{2}=1. Upon the insertion of the wall, the ground state becomes doubly degenerate. Moreover, the number of ways for the arrangement of different classes of particles are also different.

It is interesting to study the quantity of work extracted from the engine in the case of different classes of particles and the effects of increasing the number of partitions

in the potential well. We would like to clarify here that the present analysis relates only to the limiting case (ie., the low temperature regime). Suppose after keeping our potential box with two distinguishable particles in the ground state, in equilibrium with a bath at temperature T1T_{1}, we insert a single partition in the middle. From the previous discussions, we know that the energy levels are doubly degenerate. Thus two particles can occupy the two states of the lowest energy level, one in each state, in two possible ways. Again, two particles can be in the same state in two different ways (see Fig. 4). Each of these possibilities comes with a probability 14\frac{1}{4}. Thus the entropy of the system is 2ln⁡22 \ln 2 and the heat absorbed from the hot reservoir upon isothermal insertion of the wall in the middle is 2kBT1ln⁡22 k_{B} T_{1} \ln 2. Now if the system is connected to a heat bath at a lower temperature T2T_{2} and the wall is removed isothermally, by the previous argument, the heat released is 2kBT2ln⁡22 k_{B} T_{2} \ln 2. Thus the work done by the system is 2kB(T1−T2)ln⁡22 k_{B}\left(T_{1}-T_{2}\right) \ln 2. The situation becomes even more exciting in the case of two fermions in the ground state. There is only one configuration in which two fermions can be arranged in the two states of same energy level after the barrier is inserted. Hence, the changes in entropy during insertion and removal processes are zero and consequently no work can be extracted from the engine in the case of two fermions in the ground state. This must be contrasted with the case in which the potential well contains two bosons in the ground state and the work extracted out of a Stirling-like cycle performed on it. In the ground state, two bosons can have three possible configurations, hence the change in entropy upon insertion or removal of the barrier is kBln⁡3k_{B} \ln 3. Thus the work that can be extracted out of the engine is kB(T1−T2)ln⁡3k_{B}\left(T_{1}-T_{2}\right) \ln 3.

Let us now examine the case in which we have a potential well with two distinguishable particles and insert two partitions, isothermally, at −a3-\frac{a}{3} and a3\frac{a}{3}. Upon insertion at a temperature T1T_{1}, the change in entropy is 2ln⁡32 \ln 3 and the heat exchanged is 2kBT1ln⁡32 k_{B} T_{1} \ln 3 as each energy level acquires a degeneracy 3 . Similarly the heat exchanged during the isothermal removal of the walls at a temperature T2T_{2} is 2kBT2ln⁡32 k_{B} T_{2} \ln 3. The amount of work that can now be extracted from the engine is 2kB(T1−T2)ln⁡32 k_{B}\left(T_{1}-T_{2}\right) \ln 3. Fermions, however, can occupy the three states of the ground level in three possible ways only and hence the work extracted can be only kB(T1−T2)ln⁡3k_{B}\left(T_{1}-T_{2}\right) \ln 3. Bosons, on the other hand can occupy these states in six possible ways and therefore the work done is kB(T1−T2)ln⁡6k_{B}\left(T_{1}-T_{2}\right) \ln 6. In general, upon inserting gg partitions in a box with nn particles, there are (g+1)n(g+1)^{n} possible ways to arrange distinguishable particles in the degenerate ground states, while bosons and fermions can be arranged in (n+g)!/(n!g!)(n+g)!/(n!g!) and (g+1)!/(n!(g+1−n)!)(g+1)!/(n!(g+1-n)!) number of different ways, respectively. The entire discussion is summarized in Table I. The magnitudes of work done by the engine for three particles of different classes is also summarized in Table II.

TABLE I. Comparison of work for the case of 2 particles.

TABLE II. Comparison for the case of 3 particles.

(a)

FIG. 4. Particle statistics after inserting the barrier: (a) Distinguishable particles, (b) Bosons and © fermions.

VI. CONCLUSION

We considered Stirling cycle which uses quantized energy levels to extract work. The lack of knowledge of the particle’s position can be effectively converted into useful work without involving measurement to locate the particle. Our engine operates exclusively using quantum features and do not work in classical limit where the width of the box is large. In the low temperature limit our engine attains the Carnot efficiency. The work obtained from the engine depends upon the number of partitions and the number particles as well as the spin-statistics nature of the particles. The extractable work from distinguishable particles, fermions and bosons are compared.

It is worth noting that we have discussed the effects of inserting one or more partitions, on the energy levels of a potential well, at particular points. To start off, we note that all the wavefunctions corresponds to even energy levels of an infinite single potential well have nodes at the origin. Hence, inserting a partition at the origin leaves them unchanged. Similarly, all energy levels with multiples of three have nodes at −a3-\frac{a}{3} and a3\frac{a}{3}. Thus to leave these energy levels unchanged, it is required to insert the

FIG. 5. (A) Plot of work /kBT2/ \mathrm{k}_{\mathrm{B}} \mathrm{T}_{2} vs the aa which is the half width of the total potential well, for different values of ϵ\epsilon in nanometers. The horizontal line represents the low temperature limiting case value (T1−T2)ln⁡2/T2\left(T_{1}-T_{2}\right) \ln 2 / T_{2}. (B) The plot shows the behavior of efficiency vs aa for different values ϵ\epsilon. The horizontal line represents the Carnot efficiency (1−T2T1)\left(1-\frac{T_{2}}{T_{1}}\right) obtained from the limiting case. Here, we have taken m=9.11×10−31m=9.11 \times 10^{-31} kg,T1=2K\mathrm{kg}, T_{1}=2 K and T2=1KT_{2}=1 K.
barrier at these precise points. The same argument holds for energy levels with multiples of NN, an arbitrary integer. Considering more practical situations, it is useful to explore the effects of inserting one or more partitions at some other points. All the energy levels would then shift resulting in different amount of work. Particularly, if a single partition is inserted ϵ\epsilon distance away from the
origin, say to the left, then the original first and second energy levels before insertion, do not completely merge but remain very close to each other. The width of the right and left wells are now a+ϵa+\epsilon and a−ϵa-\epsilon, respectively. Hence we get nearly degenerate levels for small ϵ\epsilon. An effect of degeneracy is the additional term ln⁡2\ln 2 in the work extracted. The near degeneracy ensures a value that is close to ln⁡2\ln 2. The efficiency of such an engine, in the low temperature limit, is thus close to the Carnot value as discussed earlier. However, for large ϵ\epsilon, the shifts do not bring the energy levels close enough, resulting in no such term. The work and efficiency of such an engine would be significantly lower. For different values of ϵ\epsilon, the work and efficiency are plotted versus half width of the total potential well in Fig. 5. It is to be noted that, in all our analysis, we restrict the length of the box to be much grater than the Compton wavelength and hence our analysis is completely non-relativistic [27].

Future direction includes modeling a heat engine with finite-time processes with finite barrier. One can also take different forms of potentials with interacting particles. Micrometer-sized Stirling engine is already realized with a single colloidal particle [24]. Realization of our model of Stirling engine in nanoscale with low temperatures is a future possibility.
[1] J. C. Maxwell, Life and Scientific Work of Peter Guthrie Tait, edited by C. G. Knott (Cambridge University Press, London, 1911).
[2] H. Leff and A. F. Rex, Maxwell’s Demon 2: Entropy, Classical and Quantum Information, Computing (Institute of Physics, Bristol, 2003).
[3] L. Szilard, Zeitschrift für Physik 53, 840 (1929).
[4] K. Maruyama, F. Nori, and V. Vedral, Rev. Mod. Phys. 81, 1 (2009).
[5] R. Landauer, IBM Journal of Research and Development 5, 183 (1961).
[6] C. H. Bennett, International Journal of Theoretical Physics 21, 905 (1982).
[7] A. Bérut, A. Arakelyan, A. Petrosyan, S. Ciliberto, R. Dillenschneider, and E. Lutz, Nature 483, 187 (2012).
[8] S. W. Kim, T. Sagawa, S. De Liberato, and M. Ueda, Phys. Rev. Lett. 106, 070401 (2011).
[9] D. J. Griffiths, Introduction to Quantum Mechanics (Pearson Prentice Hall, 2005).
[10] L. Schiff, Quantum Mechanics, International series in pure and applied physics (McGraw-Hill, 1955).
[11] E. Merzbacher, Quantum Mechanics (Wiley, 1998).
[12] D. Gelbwaser-Klimovsky, A. Bylinskii, D. Gangloff, R. Islam, A. Aspuru-Guzik, and V. Vuletic, ArXiv e-prints (2017), arXiv:1705.11180 [quant-ph].
[13] W. H. Zurek, eprint arXiv:quant-ph/0301076 (2003), quant-ph/0301076.
[14] H. Li, J. Zou, J.-G. Li, B. Shao, and L.-A. Wu, Annals of Physics 327, 2955 (2012).
[15] K.-H. Kim and S. W. Kim, Journal of the Korean Physical Society 61, 1187 (2012).
[16] C. Y. Cai, H. Dong, and C. P. Sun, Phys. Rev. E 85, 031114 (2012).
[17] Z. Zhuang and S.-D. Liang, Physics 12, 23915471 (2014).
[18] B. Paul, Phys. Rev. E 90, 052117 (2014).
[19] J. Bengtsson, M. Nilsson Tengstrand, A. Wacker, P. Samuelsson, M. Ueda, H. Linke, and S. M. Reimann, ArXiv e-prints (2017), arXiv:1701.08138 [quant-ph].
[20] H. Saygin and A. Şişman, Journal of Applied Physics 90, 3086 (2001).
[21] G. S. Agarwal and S. Chaturvedi, Phys. Rev. E 88, 012130 (2013).
[22] X.-L. Huang, X.-Y. Niu, X.-M. Xiu, and X.-X. Yi, The European Physical Journal D 68, 32 (2014).
[23] G. Thomas, M. Banik, and S. Ghosh, Entropy 19, 442 (2017).
[24] V. Blickle and C. Bechinger, Nat. Phys. 8, 143 (2011).
[25] A. E. Allahverdyan, R. S. Johal, and G. Mahler, Phys. Rev. E 77, 041118 (2008).
[26] R. S. Johal, Phys. Rev. E 80, 041119 (2009).
[27] P. Alberto, C. Fiolhais, and V. M. S. Gil, European Journal of Physics 17, 19 (1996).