Operational characteristics of single particle heat engines and refrigerators with time asymmetric protocol

0.91 MB·

by  P. S. Pal

Operational characteristics of single particle heat engines and refrigerators
with time asymmetric protocol.
P. S. Pala,∗, Arnab Sahaa, A. M. Jayannavara
aInstitute of Physics, Sachivalaya Marg, Bhubaneswar - 751005, India
6
1
0 Abstract
2
We have studied the single particle heat engine and refrigerator driven by time asymmetric protocol of
n
a finite duration. Our system consists of a particle in a harmonic trap with time-periodic strength that
J drives the particle cyclically between two baths. Each cycle consists of two isothermal steps at different

5 temperatures and two adiabatic steps connecting them. The system works in irreversible mode of operation
even in the quasistatic regime. This is indicated by finite entropy production even in the large cycle time

]
h limit. Consequently, Carnot efficiency for heat engine or Carnot Co-efficient of performance (COP) for
c refrigerators are not achievable. We further analysed the phase diagram of heat engines and refrigerators.
e They are sensitive to time-asymmetry of the protocol. Phase diagram shows several interesting features,
m
often counterintuitive. The distribution of stochastic efficiency and COP is broad and exhibits power law
- tails.
t
a
t
s
In recent years a lot of interest has been generated in the study of stochastic single particle heat engines
.
t
a andrefrigerators[1,2,3,4,5,6,7,8,9,10,11]. Enginesatnanoscaleareubiquitousinbiology[12,13,14]and
m becomeincreasinglypertinentsynthetically. Withtheprogressoftechnologymicrometersizedstochasticheat
- engines have been realised experimentally [15, 16, 17, 18]. At these length scales thermal fluctuation plays
d a pivotal role in determining the performance characteristics of the system. Typical energy transformations
n
(work and heat) in these systems are of the order of k T, where T is the temperature of the surrounding
o B
reservoir.Therefore taking account of thermal fluctuations is an absolute necessity to achieve engineering
c
[ capabilities in designing such small scale devices [19].

The apt theory for the thermodynamics of small scale devices comes under the frame work of stochastic
1
thermodynamics where the macroscopic thermodynamic variables (e.g. work, heat, total entropy, internal
v
4 energy etc.) are defined over a single trajectory and thereby differs stochastically from one measurement to
5 another [20, 21, 22, 23, 24, 25, 26, 27]. Besides validating macro thermodynamics after averaging over all
8 possible trajectories, the new frame work offers first law like equality defined over a single trajectory and
0
fluctuationtheorems[28,29,30,31,32,33],asetofequalitiesbetweenstochasticallyvaryingthermodynamic
0
variables that put rigorous constraints to their distributions.
.
1
Using stochastic thermodynamics microscopic heat engines and refrigerators have been explored. Ex-
0
tensive studies including both quasistatic and nonquasistatic regime have been done on systems consisting
6
1 of a harmonically trapped Brownian particle driven periodically (with period τ) by the time dependent
: strength of the confining potential within two thermal reservers having different temperatures T and T
v h l
where T >T [10, 11]. The protocol studied in [10, 11] consists of two isotherms having equal length along
i h l
X time axis (that is why the protocol can be termed as time-symmetric) and two adiabatic path connecting
r them by instantaneous jumps. We found that in this case the system operates in four thermodynamically
a
favourable modes: a.) Engine - heat from hot bath is converted partially into work and the rest is supplied
to the cold bath, b.) Heater I - work done on the system is divided into two parts that heat up both the
∗Correspondingauthor
Email addresses: [email protected](P.S.Pal),[email protected](ArnabSaha),[email protected](A.M.
Jayannavar)
Preprint submitted to Elsevier January 6, 2016
baths, c.) Heater II - the system takes heat from hot bath and with the help work done on it, the heat is
transferred from hot to the cold bath, d.) Refrigerator - the system takes heat from the cold bath and with
the help of work done transfers heat to the hot bath. Different operational mode of the system appears at
different values of τ and Th, which are described by the phase diagram in τ T plane for fixed T .
Tl − h l
In the following we extend our previous studies [10, 11] by driving the single particle heat engine and
refrigerator with time-asymmetric protocols. Here we will analyse the thermodynamics of the system in
quasistatic as well as nonquasistatic regime, driven by the protocol having fixed τ but with unequal lengths
of the isothermal steps along the time axis (i.e. the protocol is time-asymmetric) together with the equal
jumps of the protocol along the adiabatic steps. We find that in nonquasistatic limit, tuning the lengths
of the isothermal steps keeping τ fixed, the phase diagram can be modified. Secondly, we find that in
quasistatic regime with high friction, the heights of the adiabatic jumps of the protocol and the ratio of
the bath temperatures together will determine a generic condition for the system operating under reversible
mode of operation.
Inthispaper,firstwedescribethemodelandtheprotocolsforthedrive. Thenweanalysethequasistatic
behaviour of the system driven by all the protocols both in the underdamped and overdamped limit. Next,
afterbrieflydiscussingthebasicsofstochasticthermodynamicsforcompleteness,weprovidedetailedanalysis
of non-quasistatic behaviour of the system focussing on the effects of time asymmetry of the protocols used.
Finally we summarise our result and conclude.
1. Model
Our system consists of a Brownian particle of mass m having position x and velocity v, confined in a
harmonictrap. Thestiffnessofthetrapk(t)isvariedperiodicallyintimeusingatime-asymmetricprotocols.
Fortheunderdampedcase,theequationofmotionoftheparticleincontactwithaheatbathattemperature
T is given by [34, 35]
(cid:112)
mv˙ = γv k(t)x+ γk Tξ(t) (1)
B
− −
In overdamped limit, the equation of motion reduces to
(cid:112)
γx˙ = k(t)x+ γk Tξ(t). (2)
B
−
Here the fluctuation dissipation relation between noise strength, temperature of the bath (T) and friction
coefficient (γ) is maintained. In further analysis, the mass of the particle, friction coefficient and Boltzmann
constant k are set to unity. The noises from the bath ξ are Gaussian distributed with zero mean and are
B
delta correlated, i.e., ξ(t) =0 and ξ(t )ξ(t ) =2δ(t t ).
1 2 1 2
(cid:104) (cid:105) (cid:104) (cid:105) −
Two types of time-asymmetric periodic protocols of periodicity τ have been applied on our system viz.
engine protocol and refrigerating protocol(reverse of the previous one). Each of the protocol consists of
four steps: two isothermal and two adiabatic. The isothermal processes takes place in finite time whereas
the adiabatic processes occur instantaneously. During one isothermal process the system is connected to a
hot bath at temperature T and in the other isothermal process the system is connected to cold bath at
h
temperature T . Time-asymmetry in protocols refers to the fact that the contact time with two heat baths
l
during the isothermal processes is different. These protocols are described below.
Engine protocol: Inthefirststepthesystemundergoesanisothermalexpansionincontactwithhotbath
and the stiffness is changed from an initial value a to a/2. In second step the stiffness is changed from a/2
to a/4 instantaneously to perform the adiabatic expansion process. In this step the system is disconnected
from the hot bath and instantly connected to the cold bath. Next, another isothermal process takes place
in which the trapped is compressed and the stiffness is changed from a/4 to 3a/4. In the last step adiabatic
compression takes place and the stiffness is changed instantaneously from 3a/4 to a. In this step the system
is again connected back to the hot bath. The ratio of the contact times with two heat baths during the
(cid:16) (cid:17)
isothermal processes is r : s i.e., the duration of isothermal expansion is τ = r τ and the duration
1 r+s
(cid:16) (cid:17)
of the isothermal compression is τ = s τ. The time dependency of stiffness is given in the following
2 r+s
2
equations,
(cid:18) (cid:19)
r+s t
k(t) = a 1 0 t<τ
1
− 2r τ ≤
= a/4 t=τ
1
(cid:18) (cid:19)
1 r r+s t
= a + τ t<τ
1
4 − 2s 2s τ ≤
= a t=τ. (3)
k(t)
k(t)
a a
Th Th Th
3a/4 3a/4
Th Th Th
a/2 a/2 Tl Tl Tl
a/4 Tl Tl Tl
a/4
⌧ ⌧ ⌧ ⌧
1 2 1 2
t t
τ1 ⌧ τ τ1 ⌧ τ
A B
Figure 1: (Color online) A. Engine protocol consisting of two isothermal steps at two different temperatures T and T, with
h l
twoinstantaneousadSaituardbaya 2 tJaincuarsy t16epsconnectingthem. Threedifferentprotocolshadbeenshownforthreedifferentcontacttime
ratios(r:s)-1:1,1:3and3:1. τ1 denotesthetimeduringwhichthesystemundergoesanisothermalexpansionincontactwith
the hot bath and τ1 denotes the time during which the system undergoes an isothermal compression in contact with the cold
bath. B. Refrigerator protocol obtained by reversing the engine protocol. In this protocol the system undergoes isothermal
compressionincontactwiththehotbathforatimedurationofτ1 andisothermalexpansionincontactwiththecoldbathfor
atimedurationofτ2.
Refrigerator protocol: Therefrigeratorprotocolisthereverse(intime)oftheengineprotocol. Inthefirst
step the system undergoes an isothermal compression in contact with the hot bath for a time duration of
τ where the stiffness of the trap is increased from a/2 to a. Next there is adiabatic expansion process in
1
which the stiffness of the trap is decreased instantaneously from a to 3a/4. In the third step the system go
isothermal expansion process in contact with the cold bath for a duration τ . In this step the stiffness is
2
decreased from 3a/4 to a/4. In the last step the system undergoes adiabatic compression and the stiffness
is instantly changed back to a/2. The time dependency of stiffness is given in the following equations,
(cid:18) (cid:19)
1 r+s t
k(t) = a 1+ 0 t<τ
1
2 r τ ≤
= 3a/4 t=τ
1
(cid:18) (cid:19)
3 r r+s t
= a + τ t<τ
1
4 2s − 2s τ ≤
= a/2 t=τ. (4)
The protocols are shown in Fig. 1 .
2. Stochastic thermodynamics
Before we investigate further, for completeness here we describe the essentials of stochastic thermody-
namics for our system. For underdamped case, using equation of motion one can write the first law along a
3
trajectory of the particle as
∆u=w q, (5)
−
where u= 1mv2+ 1k(t)x2, w =(cid:82) ∂udt and q = (cid:82)( γv+√γTξ)vdt are change of internal energy of the
2 2 ∂t − −
particle,workdoneontheparticleandheatexchangebetweentheparticleandthethermalbathrespectively.
Note that, though w and q depends on the points along a single trajectory of the particle but ∆u depends
only on the initial and final points.
Using the definition of work along a single trajectory one can write the work along the isothermal steps
as
(cid:90) τ1 ∂u (cid:90) τ ∂u
w = dt + dt
isoth
∂t ∂t
0 τ1
(cid:90) τ1 1(cid:16) (cid:17) (cid:90) τ 1(cid:16) (cid:17)
= dt k˙x2 + dt k˙x2 . (6)
0 2 T=Th τ1 2 T=Tl
As the adiabatic steps are instantaneous, the work along those steps is
w =[u(τ+) u(τ−)]+[u(τ+) u(τ−)] (7)
ad 1 − 1 −
and therefore the total work along a single trajectory is given by
w =w +w . (8)
isoth ad
The heat transfer along the isothermal paths are given by (from the first law)
(cid:90) τ1 1(cid:16) (cid:17)
q = dt k˙x2 +[u(τ−) u(0)] (9)
1 − 0 2 T=Th 1 −
(cid:90) τ 1(cid:16) (cid:17)
q = dt k˙x2 +[u(τ−) u(τ+)]. (10)
2 − τ1 2 T=Tl − 1
Sincetheheattransferalongtheadiabaticstepsarezero,thetotalheattransferalongthetotalheattransfer
alongacycleisq =q +q . Usingthedefinitionofheatandworkforengineprotocolonecandefineefficiency
1 2
as
w
η = − (11)
q
1
−
and for refrigerator protocol COP as
q
2
(cid:15)= − . (12)
w
η and(cid:15)dependontheindividualtrajectoryoftheparticleandthereforetheyvarystochasticallyfordifferent
cycles of the engine/refrigerator. Here we study their distributions in the following sections.
Running the dynamics for large number of cycles (N), we define the average work, power and heats as
1 (cid:88) W 1 (cid:88) 1 (cid:88)
W = w; P = ; = q ; = q . (13)
1 1 2 2
N τ Q N Q N
allcycles allcycles allcycles
Note that using and , we can calculate the change of average bath entropy ∆S = Q1 + Q2. In
Q1 Q2 (cid:104) bath(cid:105) −Th Tl
timeperiodicsteadystateandlargeenoughN, ∆S = ∆S . Thereforefromnumericswecanexplore
bath tot
(cid:104) (cid:105) (cid:104) (cid:105)
non-quasistatic as well as quasistatic behaviour of thermodynamic quantities (W, P, , ) by varying τ
1 2
Q Q
from small to large values. Similar analysis can be done in the overdamped limit with u= 1k(t)x2.
2
4
3. Quasistatic results
3.1. Engine protocol
3.1.1. Underdamped dynamics
We calculate the thermodynamic quantities like average work and heat exchanges for different steps
of a cycle. During the isothermal processes the system instantaneously adjusts to the equilibrium state
corresponding to the value of the protocol at that instant. Hence the work done along any isothermal
process is the the free energy difference between the initial and the final state. In the first step, i.e.,
isothermal expansion the work done on the system is
T (cid:18)k(t=τ−)(cid:19) T (cid:18)a/2(cid:19) T (cid:18)1(cid:19)
W =∆F = h ln 1 = h ln = h ln (14)
1 h
2 k(0) 2 a 2 2
At t = τ , the system is in equilibrium with the bath at temperature T with stiffness constant a/2. The
1 h
second step being instantaneous, no heat will be dissipated and the phase space distribution given by
1 (cid:114)a (cid:18) ax2 v2 (cid:19)
P (x,v)= exp (15)
τ1 2πT 2 −4T − 2T
h h h
remains unaltered. Correspondingly, the average work done on the particle is the change in its internal
energy,
(cid:90) ∞ (cid:16)a a(cid:17)x2 T
h
W = dxdv P (x,v)= . (16)
2 4 − 2 2 τ1 − 4
−∞
Similarly in the isothermal compression step the work done on the particle in the quasistatic limit is
(cid:18) (cid:19) (cid:18) (cid:19)
T k(t=τ) T 3a/4 T
l l l
W =∆F = ln = ln = ln3. (17)
3 l
2 k(τ ) 2 a/4 2
1
At the end of the third step the system is in equilibrium with the cold bath with stiffness constant 3a/4 and
the corresponding distribution is
1 (cid:18) 3ax2 v2 (cid:19)
P (x,v)= √3aexp . (18)
τ
4πT − 8T − 2T
l l l
In the last step, i.e., adiabatic compression step the work done on the system is
(cid:90) ∞ (cid:18) 3a(cid:19)x2 T
l
W = dxdv a P (x,v)= . (19)
4 τ
− 4 2 6
−∞
Hence, the average total work done in a full cycle of the engine protocol in the quasistatic limit is given by
(cid:18) (cid:19)
T 1 T T T
h h l l
W =W +W +W +W = ln + ln3+ (20)
tot 1 2 3 4
2 2 − 4 2 6
Heatexchangedwiththehotbathintheisothermalexpansionprocessisobtainedbycalculatingtheinternal
energy change and using first law of thermodynamics. At t = 0−, the system was in contact with the cold
bath whereas at t = 0+ the system is connected to hot bath. Thus the system has to relax into a new
equilibrium after a sudden change of temperature. This relaxation leads to a heat exchange between the
system and the hot bath which accounts to change in internal energy. One can readily obtain the change in
internal energy as
7
∆U =U(τ−) U(0+)=T T . (21)
1 − h− 6 l
Now, using first law, the average heat absorption from the hot bath for the first step is
(cid:18) (cid:19)
7 T 1
h
Q =∆U W =T T ln . (22)
1 1 h l
− − − 6 − 2 2
5
Similarly we can obtain the heat transferred from the cold bath to the system,
3 T
l
Q =T T ln3. (23)
2 l h
− − 4 − 2
From Eq.22 and Eq. 23, Q is negative and Q is positive for all values of the temperature ratio T /T in
1 2 h l
the quasistatic limit. But, from Eq. 20, W is positive when (T /T ) < 1.2 and negative otherwise. This
h l
implies that in the quasistatic limit, the system will act as heater II when (T /T ) < 1.2 and as an engine
h l
when (T /T )>1.2.
h l
The average efficiency and the average entropy production in the quasistatic limit is given by
W Th ln2+ Th 1ln3+ 1
η = − tot = 2Tl 4Tl − 2 6, (24)
q −Q1 TThl − 67 + 2TThl ln2
3T 7T 1
h l
∆S = + ln6, (25)
q
4T − 6T 2
l h
From the above expression it can be shown that ∆S never vanishes for any y >1. Hence there will be
q
positive heat dissipation implying that the system always work in irreversible mode. Therefore the system
cannot reach Carnot efficiency in the quasistatic limit.
3.1.2. Overdamped dynamics
In the overdamped limit, the dynamics of the system is described by Eq. 2 where the inertial effects
are ignored. This approximation is valid when the time steps of observation is large compared to m/γ.
The internal energy is given only by the potential energy term. The analytical calculations for the average
thermodynamic quantities in the quasistatic limit are similar to the underdamped case. The work done in
the isothermal expansion and compression are exactly the same as Eq. 14 and 17. The work done along the
adiabatic steps will be same as given by Eq. 16 and 19, except the fact that the probability distribution will
depend only on the position of the particle and not on its velocity. The total work done on the system in a
whole cycle will be the same as that obtained in the underdamped case(Eq. 20). Using the same arguments
similar to the underdamped case and keeping in mind the fact that there is only one phase space variable,
namely position, the averaged internal energy change in the first step is given by
1 2
∆U =U(τ−) U(0+)= T T . (26)
1 − 2 h− 3 l
Using the first law, the average heat absorption from the hot bath in the first step is
(cid:18) (cid:19)
1 2 T 1
h
Q =∆U W = T T ln (27)
1 1 h l
− − 2 − 3 − 2 2
and the heat exchanged with the cold bath is given by,
T 1 1
l
Q = ln3+ T T . (28)
2 h l
2 4 − 2
From the above expressions, it is easy to show that Q is negative and Q is positive for all values of
1 2
the temperature ratio T /T in the quasistatic limit. Similar to the underdamped dynamics, the system
h l
acts in the same operational behaviour in the overdamped dynamics when the driving protocol is applied
quasistatically i.e., it works as an engine when (T /T )>1.2 and as heater II otherwise.
h l
The average efficiency and the average total entropy are given by
W T ln2+ 1T T ln3 1T
η = − tot = h 2 h− l − 3 l, (29)
q Q T +T ln2 4T
− 1 h h − 3 l
6
Q Q 1 1T 2 T
1 2 h l
∆S = − + = ln6+ . (30)
q
T T 2 4 T − 3T
h l l h
From the above expression of η , we conclude that for any Th >1, η =η =1 Tl. Hence the system
q Tl q (cid:54) c − Th
works irreversibly. It is also in compliance with the fact that ∆S 0, evident from Eq.30. In fig.2 we
q
numerically calculate the total entropy with varying τ at Th =2. Ap≥art from the nonquasistatic behaviour
Tl
oftotalentropyforsmallτ,itshowsthatforlargerτ,thetotalentropysaturatestoanonzeropositivevalue
(cid:12)
which is, ∆S (cid:12) .
q(cid:12)Th=2
Tl
1.1
1
0.9
0.8
0.7
�S
h toti 0.6
0.5 1:1
0.4 1:3
3:1
0.3
4:1
0.2
20 40 60 80 100 120
⌧
k(t)
k(t)
Figure2: Averagetotalentropyproductionasafunctionofcycletimeforfourdifferentcontacttimeratios: red-1:1,green-
1:3,blue-3:1,pink-4:1. TheahotbathtemperatureismaintainedatTh=0.2. a
3a/4 k(t) 3a/4
k(t)
a
aa a
a/2 a/2
r a
1
T
h
3a/4 3a/4
a/4
r a a/4
2
T
l
ar/2a a/2
3
t
t
τ ⌧ τ τ τ
1
1
a/4
Figure 3: (Color online)Carnot type engine protocol with arbitrary jump heights. The system first undergoaes/4an isothermal
expansion in presence of hot bath where the stiffness of the trap is changed from a to r2a. In the second step the system
is subjected to adiabatic expansion by changing the stiffness instantaneously from r2a to r3a. Next the stiffness is changed
isothermallychangedinpresenceofcoldbathfromr3ator1a. Inthefinalstep,thestiffnessoftheissuddenlychangedfrom
r1atoa. Oneshouldnotethatr3<r2<r1<1. Inordertogetareversiblemodeinthequasistaticlimitoneneedstomain
a certain temperature ratio (cid:16)TThl(cid:17) and r1,r2&r3 shouldτmaintain a definite ratio(derived in theτtextt). As this is aquasistatic τ τ t
result one should not consideTrhuarsbdaoyu 3t1 Dtehceembteirm 15e-asymmet1ry of the protocol. That is why the protocol shown in the figure is 1
time-symmetricbuthavedifferentjumpheights.
As opposed to the protocols in [10, 11], our protocol does not exhibit Carnot efficiency for any values
of T and T in the overdamped quasistatic limit. This is due to the equal adiabatic jump heights in our
h l
protocol. One can change the jump heights and obtain the Carnot value given the fact that the heights and
7
the temperature of the two baths maintain a certain relation. To demonstrate this, let’s consider a general
protocol as shown in Fig. 3 and calculate the efficiency in the quasistatic regime.
The work done in the first isothermal step when the system is attached to hot bath at temperature T is
h
given by
(cid:34) (cid:35)
T k(t= τ−) T (cid:104)r a(cid:105) 1
W = h ln 2 = h ln 2 = T ln(r ) (31)
1 h 2
2 k(0) 2 a 2
At the end of the first step the system is in thermal equilibrium with the hot bath with stiffness r a.
2
The average work done in the second step is given by average internal energy change,
(cid:90) ∞ 1 (cid:114) r a (cid:18) r a (cid:19) 1
W = dx (r a r a)x2 2 exp 2 x2 = (r r )T (32)
2 3 2 3 2 h
2 − 2πT −2T 2r −
−∞ h h 2
Work done in the third step i.e., when the system is undergoing an isothermal process in contact with
the cold bath is
T (cid:20)k(t=τ−)(cid:21) T (cid:20)r a(cid:21) 1 (cid:18)r (cid:19)
l l 1 1
W = ln = ln = T ln (33)
3 2 k(τ) 2 r a 2 l r
2 3 3
The last step being adiabatic compression step, the stiffness of the harmonic trap is changed instan-
taneously from r a to a while the system is still in equilibrium with the cold bath. As a result the heat
1
dissipation vanishes and the work done is readily given by the internal energy change
(cid:90) ∞ 1 (cid:114) r a (cid:18) r a (cid:19) 1
W = dx (a r a)x2 1 exp 1 x2 = (1 r )T . (34)
4 1 1 l
2 − 2πT −2T 2r −
−∞ l l 1
Hence the average total work in the quasistatic limit is given by
W = W +W +W +W
tot 1 2 3 4
(cid:18) (cid:19)
1 1 1 r 1
1
= T ln(r )+ (r r )T + T ln + (1 r )T (35)
h 2 3 2 h l 1 l
2 2r − 2 r 2r −
2 3 1
Using the similar arguments as before one can calculate the internal energy change in the isothermal expan-
(cid:16) (cid:17)
sion as ∆U = 1 T 1 T . The average heat absorbed from the hot bath in this step is given by
2 h− r1 l
(cid:18) (cid:19)
1 1 1
Q =∆U W = T T T ln(r ). (36)
1 1 h l h 2
− − 2 − r − 2
1
The average efficiency and the average entropy production in the quasistatic regime is given by
(cid:16) (cid:17)
Th ln(r )+ Th (r r )+ln r1 + 1 (1 r )
η = −Wtot = Tl 2 Tlr2 3− 2 r2 r1 − 1 , (37)
q −Q1 TThl ln(r2)− TThl + r11
(cid:18) (cid:19)
Q Q T 1T 1 T 1 r r
1 2 h l h 1 2
∆S = − + =1 (r r ) ln . (38)
q 3 2
T T − 2T − 2r T − 2r − T − 2 r
h l l 1 h 2 l 3
(cid:16) (cid:17)
From the above expression, it can be easily shown that η equals Carnot efficiency η =1 Tl when
q c − Th
boththeconditionsnamely Th = r2 andr = r3 aresatisfied. Undertheseconditionstheentropyproduction
Tl r3 1 r2
vanishes and hence the system works in reversible mode [36].
8
3.2. Refrigerator protocol
3.2.1. Underdamped dynamics
Here we calculate the average thermodynamical quantities in quasistatic limit under the application of
refrigerator protocol. In the first step, i.e., isothermal compression step the work done on the particle is the
free energy difference as given by,
T (cid:18)k(τ−)(cid:19) T (cid:18) a (cid:19) T
W(cid:48) =∆F = h ln 1 = h ln = h ln2 (39)
1 h 2 k(0) 2 a/2 2
At t=τ−, the system is in equilibrium with the hot bath and the corresponding probability distribution is
1
given by
(cid:20) (cid:18)ax2 v2 (cid:19)(cid:21)
P (x,v)=N(cid:48)exp + , (40)
τ1 1 − 2T 2T
h h
√
where N(cid:48) = a . The second is an adiabatic expansion process. This step is instantaneous and hence no
1 2πTh
heat is dissipated to the bath. So, the work done on the particle is the instantaneous change in its internal
energy, given by
(cid:90) ∞ 1(cid:18)3a (cid:19) 1
W(cid:48) = a x2P (x,v)dxdv = T (41)
2 2 4 − τ1 −8 h
−∞
Similar to the first step, the work done on the particle in the isothermal expansion(third step) is given by,
T (cid:18)k(τ−)(cid:19) T (cid:18) a/4 (cid:19) T (cid:18)1(cid:19)
W(cid:48) = l ln = l ln = l ln . (42)
3 2 k(τ+) 2 3a/4 2 3
1
At the end of the third step, the system is in equilibrium with the cold bath and probability distribution of
the state of the particle is,
(cid:20) (cid:18)ax2 v2 (cid:19)(cid:21)
P (x,v)=N(cid:48)exp + , (43)
τ 2 − 8T 2T
l l
√
where N(cid:48) = a . In the last step, the average work done on the particle is given by
2 4πTl
(cid:90) ∞ 1(cid:16)a a(cid:17) 1
W(cid:48) = x2P (x,v)dxdv = T . (44)
4 2 2 − 4 τ 2 l
−∞
The total average work is given by
(cid:18) (cid:19)
T 1 T 1 1
Wref =W(cid:48) +W(cid:48) +W(cid:48) +W(cid:48) = h ln2 T + l ln + T . (45)
tot 1 2 3 4 2 − 8 h 2 3 2 l
To obtain heat absorption from the cold bath (Qref) first we have to calculate internal energy change along
the third step. The average internal energy at t2= τ+ is U(cid:0)τ+(cid:1) = (cid:82)∞ (cid:16)3ax2 + v2(cid:17)P (x,v)dxdv = 7T .
1 2 −∞ 8 2 τ1 8 h
Since the system is in equilibrium with the cold bath at t=τ−, the average internal energy will be T . This
l
leads to the change in internal energy in the third step, (cid:0)T 7T (cid:1). Using first law, we obtain average heat
l− 8 h
dissipated to the cold bath,
(cid:18) (cid:19)
T 1 7
Qref = l ln T + T . (46)
2 2 3 − l 8 h
Similarly one can obtain the heat transferred to the hot bath,
T 3
Qref = h ln2 T + T . (47)
1 2 − h 2 l
It can be easily shown from Eq. 45, Wref is positive for all values of T /T i.e., work is always done on the
tot h l
system in the quasistatic limit. But Eq. 46 and Eq. 47 gives us three regimes where the system acts in
three different operational mode depending on (T /T ). When (T /T )<1.77, Qref is negative and Qref is
h l h l 2 1
positive and hence the system act as a refrigerator in the quasistatic limit. When 1.77 < (T /T ) < 2.29,
h l
both Qref and Qref are positive. Under this condition the system works as heater I. When 2.29<(T /T ),
2 1 h l
Qref is positive and Qref is negative and the system behaves as heater II.
2 1
9
3.2.2. Overdamped dynamics
Quasistatic calculations similar to the underdamped case can also be done for overdamped limit with
onlypotentialenergycontributingtothetotalinternalenergy. Thetotalaverageworkdoneinacycleinthe
overdampedlimitissameasthatobtainedfortheunderdampedcase. Att=τ+,theaverageinternalenergy
1
is given by U(τ+) = 3T . At t = τ−, the system is in equilibrium with cold bath and the corresponding
1 8 h
average internal energy is U(τ−) = 1T . Hence the change in internal energy in the isothermal expansion
process is (cid:0)1T 3T (cid:1). Hence, the av2erlage heat that is transferred to the cold bath in the quasistatic limit
2 l− 8 h
is given by
(cid:18) (cid:19)
T 1 1 3
Qref = l ln T + T . (48)
2 2 3 − 2 l 8 h
The heat exchanged between the hot bath and the system is given by
T 1
Qref = h ln2 T +T . (49)
1 2 − 2 h l
Similar to the underdamped case the work done on the system in overdamped dynamics is always positive.
Eq. 48andEq. 49givesusthreedomains,dependingontheratioofthehotandthecoldbathtemperatures,
where the system works in three different operational modes. When (T /T ) < 2.79, Qref is positive and
h l 1
Qref isnegativei.e.,heatiscarriedfromcoldbathtothehotbathusingtheworkdoneonthesystem. Under
2
this condition the system works as a refrigerator in the quasistatic limit. When 2.79<(T /T )<6.51, both
h l
Qref and Qref are positive i.e., work done on the system is being used to heat up both the baths and the
1 2
system acts as a heater I. When 6.51<(T /T ), Qref is negative and Qref is positive. In this temperature
h l 1 2
regime the system performs as heater II.
4. Numerical results
In this section we explore the non-quasistatic regime. We evolve the system using discretised Langevin
dynamicswithtimestepdt=0.001intheunderdampedaswellasoverdampedlimit[Eq. 1andEq. 2]. The
system is driven by time periodic protocols [Eq. 3 and Eq. 4]. We follow Heun’s method [37]. We have set
γ =1andm=1. Allthephysicalquantitiesareindimensionlessform. Throughoutthepaperwehavefixed
a = 5 and T = 0.1. We have considered four different values of the ratio r : s namely 1 : 1, 1 : 3, 3 : 1 and
l
4 : 1 and compared the results between them. We made sure that, after the initial transient regime( 103
∼
cycle time), the system settles to a TPSS i.e., P (x,v,t+τ)=P (x,v,t).
ss ss
4.1. Engine Protocol
4.1.1. Underdamped dynamics
Phase diagram: For each (τ,T ) pair, we calculate W, , and and thereby obtained the phase
h 1 2
Q Q
diagram [Fig. 4] for the operational modes of the system for four different asymmetric protocols. From
these phase diagrams it is clear that the area of different regions changes depending on the asymmetry of
the protocol.
10