Download Feedback!control and! fluctuation!theorems! in! classical systems!

Feedback!control and!
fluctuation! theorems!
in! classical systems!
Juan MR Parrondo!
GISC and Universidad Complutense de Madrid
Observation ➔ Operation
Jordan Horowitz
Feedback!control and!
fluctuation! theorems!
in! classical systems!
Juan MR Parrondo!
GISC and Universidad Complutense de Madrid
The Szilard engine.!
Fluctuation theorems.!
Feedback control.!
Optimal control protocols.!
Multiparticle Szilard engine.
Jordan Horowitz
The Szilard engine
A measurement is
necessary to implement a
reversible expansion
T
Wextract =
Z
Wextract
Vfin
P dV =
Vinit
Z
Vfin
Vinit
kT
Vfin
dV = kT log
V
Vinit
1
= kT log
= kT log 2 > 0
1/2
Thermal
bath
Q
W
“Soft” Szilard
1
T
Wextract
1
= kT log
>0
1/2
Thermal
bath
↵
↵
Q
W
Feedback vs. blind protocols
Feedback protocol: action depends on measurement.
Wextract
1
= kT log
1/2
Wextract = kT log
1
1/2
Blind protocol: no measurement.
L
Wextract
= kT log
?
?
R
Wextract
= kT log
Average work:
Wextract
h p
R
L
Wextract
+ Wextract
=
= kT log 2
(1
2
i
) 0
1
1/2
1/2
Fluctuation theorems
L
Wextract
= kT log
?
?
R
Wextract
= kT log
1
1/2
1/2
Kawai, JMRP, van den Broeck, PRL (2007)
Q+
E+
F =
m
hWextract
i
hWextract i =
X
m
Probability of m
T S tot
+
F 
m
pF (m)hWextract
i

pF (m)
kT log
pB (m)
kT
X
m
pF (m) log
Probability of m in
the reverse process
pF (m)
2nd law for cyclic
0
pB (m)
isothermal processes
Relative entropy
0
Fluctuation theorems and control
Wextract = kT log
Wextract
1
1/2
1
= kT log
1/2
Probability of m
m
hWextract
i
BUT now:
+
X
m
hWextract i =
X
m
F 
pF (m)
kT log m
pB (m)
pm
B (m) 6= 1
m
pF (m)hWextract
i

(
kT
X
)
The process depends on m
pm
B (m) = 1
m
X
m
0
Probability of m in
the reverse process
pF (m) log
pF (m)
pm
B (m)
Extracted work
can be positive
Not a relative entropy!
Optimal control protocol
hWextract i 
kT
X
m
pF (m)
pF (m) log m
pB (m)
m
p
The maximum work is obtained for: B (m) = 1
hWextract imax =
kT
X
for all m
pF (m) log pF (m) = kT Hshannon (m)
m
The reverse of this step
prepares the system in state m
Information gathered in
the measurement
T
Wextract = kT H = kT log 2
Optimal control protocol
The recipe:
pm
B (m) = 1
for all m
The protocol after measuring m must be
such that, when it is run backwards in time,
prepares the system in state m
The reverse of this step
prepares the system in state m
T
An example: multiparticle Szilard engines
PRL 106, 070401 (2011)
Selected for a Viewpoint in Physics
PHYSICAL REVIEW LETTERS
week ending
18 FEBRUARY 2011
Quantum Szilard Engine
Sang Wook Kim,1,2 Takahiro Sagawa,2 Simone De Liberato,2 and Masahito Ueda2
1
Department of Physics Education, Pusan National University, Busan 609-735, Korea
2
Department of Physics, University of Tokyo, Tokyo 113-0033, Japan
(Received 23 June 2010; revised manuscript received 28 November 2010; published 14 February 2011)
The Szilard engine (SZE) is the quintessence of Maxwell’s demon, which can extract the work from a
heat bath by utilizing information. We present the first complete quantum analysis of the SZE, and derive
an analytic expression of the quantum-mechanical work performed by a quantum SZE containing an
arbitrary number of molecules, where it is crucial to regard the process of insertion or removal of a wall as
a legitimate thermodynamic process. We find that more (less) work can be extracted from the bosonic
(fermionic) SZE due to the indistinguishability of identical particles.
DOI: 10.1103/PhysRevLett.106.070401
Maxwell’s demon is a hypothetical being of intelligence
that was conceived to illuminate possible limitations of the
second law of thermodynamics [1,2]. Szilard conducted a
classical analysis of the demon, considering an idealized
heat engine with a one-molecule gas, and directly associated the information acquired by measurement with a
physical entropy to save the second law [3]. The basic
working principle of the Szilard engine (SZE) is schematically illustrated in Fig. 1. If one acquires the information
concerning which side the molecule is in after dividing the
box, the information can be utilized to extract work, e.g.,
via an isothermal expansion. The crucial question here is
how this cyclic thermodynamic process is compatible with
the second law. Now it is widely accepted that the measurement process including erasure or reset of demon’s
memory requires the minimum energy cost of at least
kB ln2 (kB is the Boltzmann constant), associated with
the entropy decrease of the engine, and that this saves the
second law [4–7].
Although the SZE deals with a microscopic object,
namely, an engine with a single molecule, its fully quantum
analysis has not yet been conducted except for the measurement process [8,9]. In this Letter we present the first
complete quantum analysis of the SZE. The previous literature takes for granted that insertion or removal of the
wall costs no energy. This assumption is justified in classical mechanics but not so in quantum mechanics [10]
because the insertion or removal of the wall alters the
boundary condition that affects the eigenspectrum of the
system. As shown below, a careful analysis of this process
leads to a concise analytic expression of the total net work
H=
PACS numbers: 05.30.!d, 03.67.!a, 05.70.!a, 89.70.Cf
performed perfectly. The case of imperfect measurement
is discussed in terms of mutual information in Ref. [7].
To define the thermodynamic work in quantum mechanics, let us consider a closed system described as H c n ¼
En c n , where H, c n , and En are the Hamiltonian of the
system, its nth eigenstate, and eigenenergy, respectively.
The
P internal energy U of the system is given as U ¼
n En Pn , where Pn is the mean occupation number of
the nth eigenstate. In equilibrium Pn obeys the canonical
distribution. From the derivative of U, one obtains dU ¼
P
n ðEn dPn þ Pn dEn Þ. Analogous to the classical thermodynamic first law, TdS ¼ dU þ dW, where S and W are
the entropy and work done by the system, respectively, the
quantum P
thermodynamic work (QTW)Pcan be identified as
dW ¼ ! n Pn dEn [11,12]. Note that n En dPn should be
1
1
1
3
log 4 + log 4 + log 2 = log 2
4
4
2
2
Wextract = kT
✓
◆
2
2
log 2 + log 2 = kT log 2
4
4
We waste half a bit.Can we extract work from
?
L=2. One finds f0( ¼ f1( ¼ 1 since in these cases the wall
m
reaches the end of the box so that Zðlm
eq Þ ¼ Zm ðleq Þ
(
¼ 1 is always true
(m ¼ 0; 1) is satisfied. Note that fm
for m ¼ 0 and N. Together with f0 ¼ f1 ¼ 1=2, we obtain
Wtot ¼ kB T ln2, implying the work performed by the quanSelected for a Viewpoint in Physics
week ending
tum
SZE is equivalent
SZE.
P H Y S I C A LtoR Ethat
V I E W of
L E T Tthe
E R S classical 18 FEBRUARY
2011
PRL 106, 070401 (2011)
However, consideration of individual processes reveals an
Quantum Szilard Engine
important distinction between
the classical and quantum
Sang Wook Kim, Takahiro Sagawa, Simone De Liberato, and Masahito Ueda
SZEs. For Department
the quantum
SZE one obtains WinsKorea
¼ %! þ
of Physics Education, Pusan National University, Busan 609-735,
Department of Physics, University of Tokyo, Tokyo 113-0033, Japan
kB T ln2,
Wexp
¼2010;!,revised
and
Wrem
¼28 0November
for 2010;
each
process,
where
(Received
23 June
manuscript
received
published
14 February 2011)
P1 demon,%!E
The Szilard engine (SZE) is the quintessence of Maxwell’s
whichncan
ðlÞextract the work from a
! ¼ ln½zðLÞ=zðL=2Þ&,
zðlÞ
e analysis,of theand
Ederive
heat bath by utilizing information. We
present¼
the first complete
SZE, and
n ðlÞ ¼
n¼1 quantum
an analytic
expression of the quantum-mechanical work performed by a quantum SZE containing an
2
2
2
h n =ð8Ml
Þ with
hwherebeing
the
Planck
constant.
Inas the
arbitrary number
of molecules,
it is crucial to
regard the
process of insertion
or removal of a wall
a legitimate thermodynamic process. We find that more (less) work can be extracted from the bosonic
low-temperature
! isof simply
given as E1 ðL=2Þ %
(fermionic) SZE due to limit,
the indistinguishability
identical particles.
PACS numbers:
05.30.!d, 03.67.!a,
05.70.!a,
89.70.Cf
E1 ðLÞ. DOI:
If 10.1103/PhysRevLett.106.070401
the insertion process were
ignored
in the
classical
SZE, the second law would be violated because ! ) kB T
performed perfectly. The case of imperfect measurement
Maxwell’s demon is a hypothetical being of intelligence
in Ref. [7].
that was
to illuminate possible limitations
of the Inis discussed
in
theconceived
low-temperature
limit.
fact, in!terms
forof mutual
the information
expansion
To define the thermodynamic work in quantum mechansecond law of thermodynamics [1,2]. Szilard conducted a
ics, let us consider
a closed for
systeminserting
described as H c ¼
classical analysis
the demon, considering anby
idealized
process
isof compensated
the work
required
heat engine with a one-molecule gas, and directly associE c , where H, c , and E are the Hamiltonian of the
ated the
information
acquired
by measurement
with a
system, its
eigenstate,between
and eigenenergy,
respectively.
the
wall.
In the
end,
a tiny difference
ofnthwork
these
physical entropy to save the second law [3]. The basic
The
P internal energy U of the system is given as U ¼
working principle
of the Szilard engine
(SZE) is schematiP , where P classical
is the mean occupation
number of
two
processes
results
in the Eprecise
value,
cally illustrated in Fig. 1. If one acquires the information
the nth eigenstate. In equilibrium P obeys the canonical
whichAs
side the
molecule
is in after dividing the
distribution. From
derivative of U, one
obtains dU ¼
kconcerning
the
temperature
increases,
thetheclassical
results
B T ln2.
P
box, the information can be utilized to extract work, e.g.,
ðE dP þ P dE Þ. Analogous to the classical thermovia an individual
isothermal expansion. The
crucial question here
of
processes
areis recovered,
Wins
0,W are
dynamic first law, TdSi.e.,
¼ dU þ dW,
where!
S and
how this cyclic thermodynamic process is compatible with
the entropy and work done by the system, respectively, the
the exp
second!
law. k
Now
it isln2,
widely accepted
the rem
mea- ¼quantum
thermodynamic
(QTW)Pcan be identified as
W
and thatW
0 since
!work approaches
BT
P
surement process including erasure or reset of demon’s
dW ¼ ! P dE [11,12]. Note that
E dP should be
requires
thethis
minimum
energy cost of at least
T
ln2
in
limit.
kmemory
B
k ln2 (k is the Boltzmann constant), associated with
the entropy
the engine, and that
this saveswith
the
For decrease
the ofquantum
SZE
more than one particle,
second law [4–7].
Although the SZE
deals with a effects
microscopic object,
dramatic
quantum
come into play. Let us consider
namely, an engine with a single molecule, its fully quantum
has not yet been conducted
for the mea-in a symmetric potential well
aanalysis
two-particle
SZE except
confined
surement process [8,9]. In this Letter we present the first
( ¼ f( ¼ 1 for the
completeaquantum
analysis
of¼
the L=2.
SZE. The One
previous also
litand
wall
at
l
finds
f
0
2
erature takes for granted that insertion or removal of the
distinguishability for bosons (fermions) as the temperature
increases.
The inset of Fig. 3 clearly shows that one can extract
more work from the bosonic SZE but less work from the
An example: multiparticle Szilard engines
1,2
2
2
2
1
2
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
B
n
n
n
n
n
n
B
wall costs no energy. This assumption is justified in classical mechanics but not so in quantum mechanics [10]
because the insertion or removal of the wall alters the
boundary condition that affects the eigenspectrum of the
system. As shown below, a careful analysis of this process
leads to a concise analytic expression of the total net work
performed by the quantum SZE. If more than one particle
is present in the SZE, we encounter the issue of indistinguishability of quantum identical particles. Indeed, how
much work is extracted from the quantum SZE strongly
depends crucially on whether it consists of either bosons
or fermions. We also show that the crossover from
indistinguishability to distinguishability occurs as the tem-
FIG. 3 (color online). f0 as a function of T for bosons (solid
curve), fermions (dashed curve), and classical particles (dashdotted line) in the case of the infinite potential well. The
temperature is given in units of E1 ðLÞ=kB . (a) Three possible
ways in which two identical bosons are assigned over two states.
(b) Four possible ways in which two distinguishable particles are
allocated over two places. The inset shows Wtot =Wc as a function
of T.
070401-3
FIG. 1 (color online). Schematic diagram of the thermodynamic processes of the classical SZE. Initially a single molecule
is prepared in an isolated box. (A) A wall depicted as a vertical
gray bar is inserted to split the box into to two parts. The
molecule is represented by the dotted circles to indicate that at
this stage we do not know in which box the molecule is. (B) By
the measurement, we find where the molecule is. (C) A load is
attached to the wall to extract a work via an isothermal expan-
We waste half a bit. Can we extract work from
?
An example: multiparticle Szilard engines
Designing Optimal Feedback Engines
8
WB IB on the compressed box size. To simplify our analysis, we only consider boxes
such that lx = 2ly . In figure 3, we plot WB IB as a function of the box size parameter
= lx /d = 2ly /d. The smaller the smaller the box. Notice that WB IB < 0. We also
0.0
WB IB
0.1
0.2
0.3
3.0
3.5
4.0
4.5
5.0
⇥
Figure 3. Plot of the deviation from reversibility WB IB for the two-particle Szilard
engine protocol implemented in response to measuring each particle in a separate half
of the box (outcome B) as a function of the box size parameter = lx /d = 2ly /d.
observe that the process becomes reversible (WB IB = 0) when < 4 (lx < 4d and
ly < 2d); the box is so small when < 4 that both particles cannot fit into the same
No information is wasted
(W=kTH) if the box is small
and the particle have a finite
size.!
Many particles (Hal Tasaki)
Traps
This protocol is optimal if the number
of traps exactly matches the number
of particles in each side of the box.
A simple preparation protocol
measurement
Slow evolution
Conclusions
hWextract i 
kT
X
m
pF (m)
pF (m) log m
pB (m)
Not normalized
The protocol after measuring m must be
such that, when it is run backward in time,
prepare the system in state m
Thanks!