Motivated by recent studies on models of particle and heat quantum pumps, we
study similar simple classical models and examine the possibility of heat
pumping. Unlike many of the usual ratchet models of molecular engines, the
models we study do not have particle transport. We consider a two-spin system
and a coupled oscillator system which exchange heat with multiple heat
reservoirs and which are acted upon by periodic forces. The simplicity of our
models allows accurate numerical and exact solutions and unambiguous
interpretation of results. We demonstrate that while both our models seem to be
built on similar principles, one is able to function as a heat pump (or engine)
while the other is not.Comment: 4 pages, 4 figure

Dhar, Abhishek

Jayannavar, A. M.

Marathe, Rahul

English

arXiv

Open AccessMotivated by recent studies of models of particle and heat quantum pumps, we study similar simple classical models and examine the possibility of heat pumping. Unlike many of the usual ratchet models of molecular engines, the models we study do not have particle transport. We consider a two-spin system and a coupled oscillator system which exchange heat with multiple heat reservoirs and which are acted upon by periodic forces. The simplicity of our models allows accurate numerical and exact solutions and unambiguous interpretation of results. We demonstrate that while both our models seem to be built on similar principles, one is able to function as a heat pump (or engine) while the other is not

Jayannavar, A.M.

Raman Research Institute Digital Repository

Physical Review E

Two simple models of classical heat pumps

Motivated by recent studies of models of particle and heat quantum pumps, we study similar simple classical models and examine the possibility of heat pumping. Unlike many of the usual ratchet models of molecular engines, the models we study do not have particle transport. We consider a two-spin system and a coupled oscillator system which exchange heat with multiple heat reservoirs and which are acted upon by periodic forces. The simplicity of our models allows accurate numerical and exact solutions and unambiguous interpretation of results. We demonstrate that while both our models seem to be built on similar principles, one is able to function as a heat pump (or engine) while the other is not

Indian Academy of Sciences

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Two simple models of classical heat pumps
Rahul Marathe1, Arun Jayannavar2, Abhishek Dhar1
1Raman Research Institute, Bangalore 560080, India and
2Institute of Physics, Bhubaneswar 751005,India
(Dated: February 3, 2008)
Motivated by recent studies on models of particle and heat quantum pumps, we study similar
simple classical models and examine the possibility of heat pumping. Unlike many of the usual
ratchet models of molecular engines, the models we study do not have particle transport. We
consider a two-spin system and a coupled oscillator system which exchange heat with multiple
heat reservoirs and which are acted upon by periodic forces. The simplicity of our models allows
accurate numerical and exact solutions and unambiguous interpretation of results. We demonstrate
that while both our models seem to be built on similar principles, one is able to function as a heat
pump (or engine) while the other is not.
PACS numbers: 05.70.Ln,05.40.-a,05.60.-k
The idea of constructing miniature versions of engines,
motors and pumps has been an interesting one. The ear-
liest theoretical construct of such a device is probably
Feynman’s pawl-and-ratchet model discussed in [1]. In
this article Feynman uses this simple microscopic model
to demonstrate why a Maxwell’s demon cannot work. In
the same article he also shows how this model can be used
to construct a microscopic heat engine and discusses its
efficiency. There have been a number of recent detailed
studies on the pawl-and-ratchet model and some subtle
flaws in Feynman’s original arguments have been pointed
out [2, 3]. A different class of ratchet models have also
been studied in [4]. In these models Brownian particles,
kept in an asymmetric periodic potential and acted upon
by periodic time-dependent forces, are found to exhibit
directed motion. A number of variations of this model
has been studied [5]. Among its applications it has been
proposed that this could provide a mechanism of trans-
port of motors in biological cells [6].
Ratchet models which work on somewhat different
principles are models of quantum pumps which are re-
cently being studied theoretically [7] and have also been
experimentally realized [8]. A typical and simple exam-
ple of such a device would be two coupled quantum dots
each separately in contact with particle reservoirs which
are at the same chemical potential. One applies AC gate
voltages v1 = v0 cos(Ωt) and v2 = v0 cos(Ωt + φ) to the
two dots respectively. This leads to a net flow of particle
current between the two reservoirs whose sign depends
on the phase φ. An essential ingredient seems to be a
periodic variation of at least two parameters such that
the area enclosed in parameter space is non-zero. Since
these pumps also work at zero temperature it appears
that noise is not an essential feature, which is unlike the
case for usual ratchet models. Motivated by the quantum
particle pump model, Segal and Nitzan have proposed a
model for a heat pump [9]. In this model a molecule with
two allowed energy levels interacts with two heat reser-
voirs kept at different temperatures. The energy levels
are modulated in a periodic way. Thus unlike the particle
pump model here only a single parameter is varied. How-
ever an asymmetry is incorporated by taking reservoirs
with different spectral properties and different couplings
to the molecule. This seems to lead to the desired pump-
ing of heat from the cold to the hot reservoir.
Motivated by the quantum pump model, in this paper,
we examine classical models which have the same basic
design as the quantum version. We consider two differ-
ent models: (i) a spin system consisting of two Ising spins
each driven by periodic magnetic fields with a phase dif-
ference and connected to two heat reservoirs and (ii) an
oscillator system of two interacting particles driven by
periodic forces with a phase difference and connected to
two reservoirs. In both cases we analyze the possibility
of the models to work either as pumps or as engines. Our
main result is that the spin system can work both as a
pump and as an engine. On the other hand the oscillator
model fails to perform either function.
Our first model consists of two Ising spins driven by
time-dependent magnetic fields hL(t) and hR(t) respec-
tively and each interacting with separate heat reservoirs.
The Hamiltonian of the system is given by
H = −Jσ1σ2 − hL(t)σ1 − hR(t)σ2 , σ1,2 = ±1 (1)
where J is the interaction energy between the spins.
The magnetic fields have the forms hL(t) = h0 cos(Ωt)
and hR(t) = h0 cos(Ωt + φ). The interaction of each
spin with the heat baths is modeled by a stochastic
dynamics. Here we assume that the time-evolution of
the spins is given by Glauber dynamics [10], general-
ized to the case of two heat baths, with temperatures
TL and TR. Thus the Glauber spin flip rates r
L and
rR, for the two spins respectively, are given by rLσ1σ2 =
r(1 − γLσ1σ2)(1 − νLσ1) and r
R
σ1σ2
= r(1 − γRσ1σ2)(1 −
νRσ2) where γL,R = tanh(J/kBTL,R), νL,R =
tanh(hL,R/kBTL,R) and r is a rate constant. The mas-
ter equation for evolution of the spin distribution func-
tion P̂ = [P (+, +, t), P (−, +, t), P (+,−, t), P (−,−, t)]T
is then given by:
∂P̂
∂t
= T P̂ , (2)
2
0 π/2 π 3π/2 2π
φ
-0.002
-0.001
0
0.001
0.002
0.003
q
L
q
R
w
0 π/2 π 3π/2 2π
φ
-0.002
-0.001
0
0.001
0.002
0.003
.
.
.
FIG. 1: (Color online) Plot of q̇L, q̇R, ẇ versus φ with both
baths at the same temperature. Inset shows the currents for
the case where the right bath is slightly colder.
where
T =




−rL++ − r
R
++ r
L
−+ r
R
+− 0
rL++ −r
L
−+ − r
R
−+ 0 r
R
−−
rR++ 0 −r
L
+− − r
R
+− r
L
−−
0 rR
−+ r
L
+− −r
L
−−
− rR
−−




.
We define Q̇L, Q̇R to be the rates (averaged over the
probability ensemble) at which heat is absorbed from the
left and right baths respectively while ẆL, ẆR are the
rates at which work is done on the left and right spins
by the external magnetic field. These can be readily ex-
pressed in terms of the spin distribution function and the
various transition rates. Thus we find:
Q̇L =
∑
σ1,σ2
P (σ1, σ2, t)r
L
σ1σ2
∆E1(σ1, σ2)
Q̇R =
∑
σ1,σ2
P (σ1, σ2, t)r
R
σ1σ2
∆E2(σ1, σ2)
ẆL = −〈σ1〉 ḣL = −ḣL
∑
σ1,σ2
σ1P (σ1, σ2, t)
ẆR = −〈σ2〉 ḣR = −ḣR
∑
σ1,σ2
σ2P (σ1, σ2, t) , (3)
where ∆E1 = 2 (Jσ1σ2 + hLσ1) and ∆E2 = 2 (Jσ1σ2 +
hRσ2) are the energy costs in flipping the first and second
spin respectively. The average energy of the system is
given by U = 〈H〉 =
∑
σ1,σ2
H(σ1, σ2)P (σ1, σ2, t). It is
easy to verify the energy conservation equation: U̇ =
Q̇L + Q̇R + ẆL + ẆR. Note that even at long times the
steady state distribution P̂ remains time-dependent. We
will also be interested in the following time averaged rates
of heat exchanges and work done, evaluated in the steady
state: q̇L,R =
1
τ
∫ τ
0
Q̇L,R dt , ẇL,R =
1
τ
∫ τ
0
ẆL,R dt,
where τ = 2π/Ω is the time period of the driving field.
We numerically solve the master equation Eq. (2) and
then evaluate the various steady-state energy exchange
0 π/2 π 3π/2 2π
φ
-0.04
-0.02
0
0.02
0.04
   q
L
   q
R
0 π/2 π 3π/2 2π
φ
-0.001
0
0.001
0.002
0.003
0.004
w
 
.
.
.
FIG. 2: (Color online) Plot of q̇L, q̇R, ẇ versus φ for parameter
values chosen such that the model performs as an engine.
rates q̇L,R and ẇL,R. In all our numerical calculations
we set r = 0.5 and J/kB = 1 and all other quantities
are measured in these units. In Fig. (1) we consider the
parameter values TL = TR = 0.5, h0 = 0.25, τ = 225 and
plot q̇L, q̇R and ẇ = ẇL + ẇR as functions of the phase φ.
It can be seen that, for certain values of the phase, both
q̇L and q̇R are negative while ẇ is positive. Following
our sign conventions, this means that all the work from
the external driving is getting dissipated into the two
baths. More interestingly we find that for certain values
of the phase we can get q̇L > 0 and q̇R < 0 which means
that there is heat flow from the left reservoir to the right
reservoir. The direction of heat flow can be reversed by
changing the phase. From continuity arguments it is clear
that this model can also sustain heat flow against a small
temperature gradient. Thus the inset of Fig. (1) shows
the currents when the right reservoir is kept at a slightly
lower temperature TR = 0.499. In the absence of any
driving we would get a steady current q̇L = −q̇R = 1.41×
10−4 from the left to right reservoir. In the presence of
driving and at a phase value φ = 2.2 we get q̇R = 3.674×
10−4, ˙qL = −1.025 × 10
−3 which means that heat flows
out of the cold reservoir. Thus we see that our model can
perform as a heat pump or a refrigerator. Similarly we
find that the model can also perform like an engine and
convert heat to work. This can be seen in Fig. (2) where
we consider the parameter values TL = 1.0, TR = 0.1,
h0 = 0.25, τ = 190. In this case we find that for certain
values of φ we can have ẇ < 0 which means that work
is being done on the external force. For typical values of
parameters that we have tried we find that the efficiency
of the engine is quite low. For example for Fig. (2) with
φ = 0.7π, we find η = |ẇ|/q̇L = 1.75 × 10
−2.
Finally in Fig.(3) we plot the time-dependent energy
transfer rates given by Eq. (3) for parameter values cor-
responding to the refrigerator and engine modes of oper-
ation. In both cases the initial configuration was chosen
with P (+, +, t = 0) = 1. At long times we see that all
quantities vary periodically with time with the same pe-
riod τ as the driving force. Fig. (3) corresponds to the
3
0 250 500 750
t
0
0.01
Q
L
Q
R
W
0 250 500 750
t
-0.1
0
0.1
.
.
.
FIG. 3: (Color online) Plot of Q̇L, Q̇R, Ẇ as a function of time
for parameters corresponding to pump and engine (inset).
parameter values TL = 0.5, TR = 0.499, h0 = 0.25, τ =
225 and φ = 2.2 while the inset corresponds to the engine
parameters TL = 1.0, TR = 0.1, h0 = 0.25, τ = 190 and
φ = 2.2.
The second model of our engine consists of two parti-
cles which separately interact with two reservoirs kept at
different temperatures. The particles interact with each
other and are also driven by two external periodic forces
with a phase difference. We consider the system to be
described by the Hamiltonian
H =
p21
2m
+
p22
2m
+
1
2
kx21 +
1
2
kx22 +
1
2
kc(x1 − x2)
2
− (fL(t) x1 + fR(t) x2) (4)
The two particles are acted on by external periodic forces
given by fL(t) = f0 cos(Ωt) and fR(t) = f0 cos(Ωt + φ)
respectively, where φ is a phase difference. The effect of
the heat baths at temperatures TL and TR is modeled by
Langevin equations. Thus the equations of motion are
mẍ1 = −(k + kc)x1 + kcx2 − γẋ1 + ηL + fL(t) ,
mẍ2 = −(k + kc)x2 + kcx1 − γẋ2 + ηR + fR(t) ,
where the two noise terms are Gaussian and uncorre-
lated and satisfy the usual fluctuation-dissipation rela-
tions 〈ηL,R(t)ηL,R(t
′)〉 = 2kBTL,Rγδ(t − t
′). Multiply-
ing the two equations above by ẋ1 and ẋ2 respectively
and adding them up we get: Ḣ = (−γẋ1 + ηL)ẋ1 +
(−γẋ2 + ηR)ẋ2 − ḟL(t)x1 − ḟR(t)x2, which has the ob-
vious interpretation of an energy conservation equation.
Averaging over noise we get U̇ = Q̇L + Q̇R + ẆL + ẆR.
where the various energy exchange rates have the same
interpretations as in the previous discussion and are
given by Q̇L = 〈(−γẋ1 + ηL)ẋ1〉 , Q̇R = 〈(−γẋ2 +
ηR)ẋ2〉 , ẆL = −〈ḟLx1〉 , ẆR = −〈ḟRx2〉. As before
we define the average energy transfer rates in the steady
state q̇L, q̇R, ẇL, ẇR. The present model being linear, it
is straightforward to exactly compute these as we now
show.
We first obtain the steady-state solutions of the equa-
tions of motion. We write the equations of motion in the
following matrix form:
MẊ = −ΦX − ΓẊ + η(t) + f(t) (5)
where X = [x1, x2]
T , η = [ηL, ηR]
T , f =
[f0 cos(Ωt), f0 cos(Ωt + φ)]
T , M and Γ are diagonal ma-
trices with diagonal elements m and γ respectively and
Φ is the force constant matrix. The steady state solution
of this equation is:
X(t) = XN (t) + XD(t) ,
where XN(t) =
∫
∞
−∞
dωe−iωtG(ω)η̃(ω) ,
XD(t) = Re[G(Ω)f̃ e
−iΩt] ,
with G(ω) = [Φ − ω2M + iωΓ]−1 , (6)
and η̃ =
∫
∞
−∞
dωe−iωt η(t), f̃ = {1, e−iφ}T . It is easy
to see that the matrix G(ω) has two independent ele-
ments and we denote them as A(ω) = G11 = G22 =
[k + kc − mω
2 − iγω]/[(k + kc − mω
2 − iγω)2 − k2c ] and
B(ω) = G12 = G21 = kc/[(k + kc − mω
2 − iγω)2 − k2c ].
Using the above solution in Eq. (6), and after some bit of
algebraic simplifications, we obtain the following results:
q̇L = −
f20Ω
2
[ AI(Ω) + BI(Ω) cos(φ) + D(Ω) sin(φ) ]
+
kBγk
2
c(TL − TR)
2(mk2c + (k + kc)γ
2)
,
q̇R = −
f20Ω
2
[ AI(Ω) + BI(Ω) cos(φ) − D(Ω) sin(φ) ]
+
kBγk
2
c(TR − TL)
2(mk2c + (k + kc)γ
2)
,
ẇL =
f20 Ω
2
[ AI(Ω) + BI(Ω) cos(φ) − BR(Ω) sin(φ) ] ,
ẇR =
f20 Ω
2
[ AI(Ω) + BI(Ω) cos(φ) + BR(Ω) sin(φ) ], (7)
where AR, AI , BR, BI are the real and imaginary parts
of A and B respectively and D(Ω) = 2γ2Ω2kc/Z(Ω)
where Z(Ω) = |(k + kc − mΩ
2 − iγΩ)2 − k2c |
2. From the
expressions in Eq. (7) it is clear that the heat transfer
rates can be separated into deterministic parts (depend-
ing on the driving strength f0) and noisy parts (depen-
dent on temperature of the two reservoirs). The work
terms are temperature independent. We now note that
the deterministic parts of q̇L and q̇R, are both negative.
This can be shown by using the facts that AI ≥ 0 and
A2I −B
2
I −D
2 = γ2Ω2[(k+kc−mΩ
2)2+γ2Ω2−k2c ]
2/Z2 ≥
0. This means that for TL > TR, we always get q̇R < 0
and hence we can never have heat transfer from the cold
to the hot reservoir. Thus this cannot work as a heat
pump. Also we note that while ẇL and ẇR can individ-
ually be negative, the total work done ẇL + ẇR is always
positive. This means that this model cannot work as an
engine either. These conclusions remain unchanged even
if we define work as ẆL = 〈fLẋ1〉, ẆR = 〈fRẋ2〉.
4
0 π/2 π 3π/2 2π
φ
-0.4
-0.2
0
0.2
0.4
   q
L
   q
R
   w
1
   w
2
   w 
.
.
.
.
.
FIG. 4: (Color online) Plots of heat transfer and work done
as a function of phase difference φ in the two particle model.
Here Ω = 2π/3.
In Fig. (4) we plot the dependence of the rates of
heat transfer and work done in the system on the phase
difference φ. The figures correspond to the parameter
values k = 2, kc = 3, m = 1, f0 = 1, γ = 1 and
TL = TR = T . The plots are independent of the tem-
perature T . Note that the only effect of the driving is to
pump in energy which is asymmetrically distributed be-
tween the two reservoirs. The asymmetric energy trans-
fer into the baths is an interesting effect considering that
there is no inbuilt directional asymmetry in the system.
In this model the heat baths and the external driving
seem to act independently on the system. It is clear that
the linearity of the model leads to this separability of the
effects of the driving and noise forces and this could be
the reason that the model is not able to function as a
heat pump. Hence it is important to consider the effect
of nonlinearity. We have numerically studied the effect of
including a nonlinear part of the form α[x41 + x
4
2 + (x1 −
x2)
4]/4 in the oscillator Hamiltonian. From simulations
with a large range of parameter values we find that the
basic conclusions remain unchanged and the model does
not work either as a pump or as an engine.
In conclusion, we have studied two models which have
the same ingredients as those on which recent models of
quantum pumps have been constructed. We find that
the first model performs as a heat pump to transfer heat
from a cold to a hot reservoir. Thus pumping is not
an essentially quantum-mechanical phenomena. Also our
model performs as an engine to do work on the driving
force. It is useful to compare our model with the other
well-studied microscopic model of a engine, namely the
Feynman pawl-and-ratchet. Recent detailed studies have
shown that this model can function both as an engine and
as a refrigerator [11]. One difference of this model from
ours is that there is no periodic external driving. However
this also means that in order for the model to work in a
cyclic way, at least one of the degrees of freedom has to
be a periodic (or angular) variable. This may not always
be a desirable feature in realistic models. Surprisingly
our second model, though apparently built on the same
principles, fails to perform either as a pump or as an
engine.
The important difference between microscopic models
of heat engines, such as those studied here, and usual
thermodynamic heat engines is that here the effects of
thermal fluctuations are important. A second difference
is that here the system is simultaneously in contact with
both the cold and hot baths. The understanding of these
microscopic models requires the use of nonequilibrium
statistical mechanics and there are currently no general
principles as in classical thermodynamics. It is clear that
further studies are necessary to understand the pumping
mechanism in simple models of molecular pumps and this
can perhaps lead to more realistic and practical models
of molecular pumps and engines.
[1] R. Feynman, R. Leighton, and M. Sands, The Feyn-
man Lectures on Physics (Adison-Wesley, Reading Mas-
sachussets, vol.1, chap. 46, 1966).
[2] J. M. R. Parrondo and P. Español, Am. J. Phys. 64, 1125
(1996); M.O.Magnasco and G.Stolovitzky, J. Stat. Phys.
93, 615 (1998).
[3] K. Sekimoto, Jr. Phy. Soc. Jpn. 66, 1234 (1997);
T. Hondou and F. Takagi, J. Phy. Soc. Jpn. 67, 2974
(1998); H. Sakaguchi, J. Phy. Soc. Jpn. 67, 709 (1998);
C. Jarzynski and O. Mazonka, Phys. Rev. E 59, 6448
(1999); Y. Lee et al., Phys. Rev. Lett. 91, 220601 (2003).
[4] M. O. Magnasco, Phys. Rev. Lett. 71, 1477 (1993); I.
Derényi and T. Vicsek, Phys. Rev. Lett. 75, 374 (1995);
F. Jülicher, A. Ajdari, and J. Prost, Rev. Mod. Phys.
69, 1269 (1997); P. Reimann, Physics Reports 361, 57
(2002); R. D. Astumian, Science 276, 917 (1997).
[5] T. Feldmann, E. Geva, R. Kosloff, and P. Salamon,
Am. J. Phys. 64, 485 (1996); I. Derényi and R. D. Astu-
mian, Phys. Rev. E 59, 6219 (1999); A. Gomez-Marin
and J. M. Sancho, Phys. Rev. E 71, 021101 (2005); M. C.
Mahato, T. P. Pareek, and A. M. Jayannavar, Int. J.
Mod. Phys. B 10, 3857 (1996).
[6] R. D. Astumian and I. Dernyi, European Biophysics
Journal 27, 474 (1997).
[7] M. Buttiker, H. Thomas, and A. Pretre, Z. Phys. B.
94, 133 (1994); P. W. Brouwer, Phys. Rev. B 58,
10135 (1998); L. Arrachea, Phys. Rev. B 72, 125349
(2005); L. Arrachea, Phys. Rev. B 72, 121306(R) (2005);
M. Strass, P. Hänggi, and S. Kohler, Phys. Rev. Lett. 95,
130601 (2005); S. Kohler, J. Lehmann, and P. Hänggi,
Phy. Rep. 406, 379 (2005); A. Agarwal and D. Sen, J.
Phys. Condens. Matter 19, 046205 (2007).
[8] M. Switkes et al., Science 283, 1905 (1999); P. Leek et
al., Phys. Rev. Lett. 95, 256802 (2005).
[9] D. Segal and A. Nitzan, Phys. Rev. E 73, 026109 (2006).
[10] R. Glauber, Jr. Math. Phys. 4, 294 (1963).
[11] C. van den Broeck, R. Kawai, and P. Meurs, Phys.
Rev. Lett. 93, 090601 (2004); C. Van den Broeck and
5
R. Kawai, Phys. Rev. Lett. 96, 210601 (2006).


Two simple models of classical heat pumps

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