Interacting Multi-particle Classical Szilard Engine

0.28 MB·

by  P. S. Pal

InteractingMulti-particleClassicalSzilardEngine
P. S. Pala,b and A. M. Jayannavara,b∗
aInstitute of Physics, Sachivalaya Marg, Bhubaneswar-751005, India
bHomiBhabhaNationalInstitute,TrainingSchoolComplex,AnushaktiNagar,Mumbai400085,India
Szilard engine(SZE) is one of the best example of how information can be used to extract work from a
system.Initially,theworkingsubstanceofSZEwasconsideredtobeasingleparticle.Lateron,researchershas
extendedthestudiesofSZEtomulti-particlesystemsandeventoquantumregime. Herewepresentadetailed
study of classical SZE consisting N particles with inter- particle interactions, i.e., the working substance is
alowdensitynon-idealgasandcomparetheworkextractionwithrespecttoSZEwithnon-interactingmulti
particle system as working substance. We have considered two cases of interactions namely: (i) hard core
interactionsand(ii)squarewellinteraction. Ourstudyrevealsthatworkextractionislesswhenmoreparticles
are interacting through hard core interactions. More work is extracted when the particles are interacting via
7 squarewellinteraction. Anotherimportantresultforthesecondcaseisthatasweincreasetheparticlenumber
1
the work extraction becomes independent of the initial position of the partition, as opposed to the first case.
0
Work extraction depends crucially on the initial position of the partition. More work can be extracted with
2
largernumberofparticleswhenpartitionisinsertedatpositionsneartheboundarywalls.

n
a PACSnumbers:05.70.-a
J Keywords:SzilardEngine,information,interactingparticles

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]
h I. INTRODUCTION
c
e
m In1929, LeoSzilardputforwardaclassicdemonstrationofMaxwell’sdemon[1,2]bydescribinganengine
thatusestheinformationacquiredbymeasurementofthestateoftheworkingsubstancetoperformsomework
-
t [3]. Szilard considered a single particle in a box and executed the following four steps to extract work from
a
the system:(a) insert a partition at the middle of the box, (b) measure the whether the particle is in the right or
t
s left half, (c) attach the box with a bath at inverse temperature β and allow the partition to move isothermally
.
t andquasistaticallytilltheendoftheboxand(d)removethepartitionsothatthesystemregainsitsinitialstate.
a
m Nowherearisesacrucialquestionregardingthecompatibilitybetweenthiscyclicthermodynamicprocessandthe
second law of thermodynamics. Apparently, at first stance one might think that work has been extracted out of
-
d nowhere. Butitisnotthecase. Itisnowawidelyknownfactthatthemeasurementprocessincludingerasurehas
n anentropiccostwhichsavesthesecondlaw[4,5]. Employmentofinformationtoextractworkoutofasystem
o hasledtosomeinsightfulphysicsintheareaofnon-equilibriumthermodynamics[6–10].
c AlthoughSzilardenginehasbeeninitiallypicturedinclassicalframeworkwithsingleparticle,recentlyithas
[
beentranslatedtoquantumregimealongwithmulti-particlescenarioandasymmetriceffects[11–14].Inthiswork

1 weconsideredamulti-particleclassicalSzilardengine,buttheseparticledoesnotbehavelikeanidealgas.Instead
v wetakeN particlenon-idealgasasourworkingsubstancei.e.,theparticleshaveinteractionsbetweenthem. Here
8
wehaveassumedthegastobeoflowdensitysothatwecansafelyusetheequationofstateofthegasuptosecond
8
virialco-efficient.Weworkedontwotypesofinteractions:(i)hardcoreinteractionand(ii)squarewellinteraction.
1
Ourstudyrevealsthatworkextractionislesswhenmoreparticlesareinteractingthroughhardcoreinteractions.
7
0 Moreworkisextractedwhentheparticlesareinteractingviasquarewellinteraction. Anotherimportantresultfor
. thesecondcaseisthatasweincreasetheparticlenumbertheworkextractionbecomesindependentoftheinitial
1
positionofthepartition,asopposedtothefirstcase. Workextractiondependscruciallyontheinitialpositionof
0
thepartition. Moreworkcanbeextractedwithlargernumberofparticleswhenpartitionisinsertedatpositions
7
1 neartheboundarywalls.
: The gas is filled in a box of length L and surface area A(total volume V = A.L). The partition is inserted
v
vetically at l = xL (0 < x < 1). Next the number of particle is measured on the left half, say n. The system
i
X undergoesanisothermalexpansionincontactwiththeheatbathandthefinalpositionofthepartitionbeln .Lastly
eq
r thepartitionisremoved. TheworkdonebytheSZEisgivenby[11,13]
a
N (cid:18) (cid:19)
W =−1 (cid:88)p ln pn . (1)
tot β n f
n
n=0
∗[email protected],[email protected]
β = (k T)−1,wherek denotestheBoltzmannconstantwhichistakentobeunityinsubsequentexpressions.
B B
Herep andf aregivenby
n n
p = Zn,N−n(xL) ;f = Zn,N−n(lenq) , (2)
n (cid:80) Z (xL) n (cid:80) Z (ln )
n(cid:48) n(cid:48),N−n(cid:48) n(cid:48) n(cid:48),N−n(cid:48) eq
whereZ (X) = Z (X)Z (L−X)isapartitionfunctionthatdescribesthesituationofnparticlesto
n,N−n n N−n
theleftofthepartitionandtheremainingN −ntotheright,inathermalequilibrium. Physically,p denotesthe
n
probabilitythattherearenparticlestotheleftafterpartitionandf representstheprobabilitytochoosethecase
n
ofnparticlesontheleftsideofthewallwhenthewallisinsertedatln inthetimebackwardprocess. Notethat
eq
onecanchooselfreelywhenthewallisinsertedwhileln isdeterminedfromtheforce(pressure)balanceonboth
eq
sidesofthewallFL+FR = 0. Thewholeproblemnowboilsdowntothecalculationofthepartitionfunction
fornon-idealgasinaconfinedvolumefortwointeractionpotentials, asmentionedabove. SectionIIdealswith
thepartitionfunctionfornon-idealgases. SectionsIIIAandIIIBdescribesworkextractionusingnon-idealgases
with hard-core potential and square well interactions respectively. A short discussion is added in Section IV.
FinallyweconcludeinSectionV.Eachsectionismadeselfconsistent.
II. PARTITIONFUNCTIONOFNON-IDEALGAS
Partition function of non-ideal gas consisting of N interacting particles confined in a box of length L and
volumeV atinversetemperatureβ isgivenby
(cid:20) N2 (cid:90) (cid:21)
Z =Z 1+ {e−βU((cid:126)r)−1}d3(cid:126)r . (3)
id 2V
HereZ isthepartitionfunctionforidealgasandisgivenby
id
cLN
Z = , (4)
id N!
wherecissomeconstant.U((cid:126)r)istheinteractionpotentialbetweentwoparticles.IfU((cid:126)r)istakentobespherically
symmetriconecanintegrateouttheangulardegreesoffreedomresulting
(cid:20) N2 (cid:90) (cid:21)
Z =Z 1+ 4π r2{e−βU(r)−1}dr ,
id 2V
cLN (cid:20) N2 (cid:21)
= 1− B (β) , (5)
N! V 2
whereB (β) = −2π(cid:82) r2{e−βU(r)−1}dr isthesecondvirialco-efficientthatarisesintheequationofstateof
2
thenon-idealgas.
A. HardCoreinteraction
Inthismodeltheinteractionpotentialisgivenby
U(r)=∞ r <σ,
=0 σ <r, (6)
whereσisthehardcoreradius.
2
FIG.1. Hardcorepotential.
TheexpressionforB (β)is
2
(cid:90) σ 2
B (β)=2π r2dr = πσ3, (7)
2 3
0
andthecorrepondingpartitionfunctionisgivenby
cLN (cid:20) 2 (cid:21)
Z = 1− πσ3N2 . (8)
N! 3V
Equationofstateforsuchanon-idealgaswithhardcorepotentialis
N
βP = , (9)
V − 2 πσ3N
3V
whereP isthepressureofthegas.
B. Squarewellinteraction
Inthiscasetheinteractionpotentialisgivenby
U(r)=∞ r <σ
=−ε σ <r <Rσ
=0 Rσ <r. (10)
3
FIG.2. Squarewellpotential.
TheexpressionforB (β)is
2
(cid:90) σ (cid:90) Rσ
B (β)=−2π{− r2dr+(eβε−1) r2dr}
2
0 σ
2
= πσ3{1−(R3−1)(eβε−1)}
3
=C −C eβε, (11)
1 2
whereC = 2πσ3R3andC = 2πσ3(R3−1). Thepartitionfunctionisgivenby
1 3 2 3
cLN (cid:26) N2 (cid:27)
Z = 1− (C −C eβε) . (12)
N! V 1 2
Correspondingequationofstateforthenon-idealgaswithsquarewellinteractionbetweenparticlesis
N (cid:18)N(cid:19)2 N (cid:18)N(cid:19)2
βP = +B (β) = +(C −C eβε) . (13)
V 2 V V 1 2 V
III. WORKEXTRACTION
In classical systems, insertion and removal of partition does not involve work done on the system. Work
extractionisonlypossibleduringtheisothermalprocessanditisgivenbyEq.1. Thepartitionisinitiallyinserted
atl = xLandduringtheisothermalprocessitshiftstol = ln dependingonthenumberofparticlesontheleft/
eq
rightside. Sincethepartitionfunctioncannowbecalculatedwecaneasilycomputep ,f andtheworkforthe
n n
abovementionedinteractions.
A. Hardcoreinteraction
Inthistypeofinteractionbetweenparticles,p iscalculatedusingEq.2and8
n
Z (xL)
p = n,N−n
n (cid:80) Z (xL)
n(cid:48) n(cid:48),N−n(cid:48)
Z (xL)Z ((1−x)L)
= n N−n
(cid:80)
Z (xL)Z ((1−x)L)
n(cid:48) n(cid:48) N−n(cid:48)
(cid:0)N(cid:1)xn(1−x)N−n(1− αn2)[1− α(N−n)2]
= n x 1−x (14)
(cid:80) (cid:0)N(cid:1)xn(cid:48)(1−x)N−n(cid:48)(1− αn(cid:48)2)[1− α(N−n(cid:48))2]
n(cid:48) n(cid:48) x 1−x
4
Hereα = 2 πσ3. Beforethecomputationoff ,weneedtocalculatetheequilibriumlengthln uptowhichthe
3V n eq
partitionmovesduetothepressuredifferenceontwosidesofthepartition. Forthiswerequirepressureexerted
onthepartitionbythegasonbothsidestobesame,i.e.,PL =PR,whichcanberewrittenusingtheequationof
state(Eq.9)as
n N −n
= ,
Aln − 2 πσ3n A(L−ln )− 2 πσ3(N −n)
eq 3V eq 3V
2 2
An(L−ln )− πσ3(N −n)n=(N −n)Aln − πσ3n(N −n),
eq 3V eq 3V
n
ln = L. (15)
eq N
Nowtheexpressionoff canbeeasilycalculatedandisgivenby
n
(cid:0)N(cid:1)(n)n(1− n)N−n(1−αnN)[1−α(N −n)N]
f = n N N (16)
n (cid:80) (cid:0)N(cid:1)(n)n(cid:48)(1− n)N−n(cid:48)(1−αn(cid:48)N)[1−α(N −n(cid:48))N]
n(cid:48) n(cid:48) N N
α=0.001
0.7
x=0.2
x=0.3
0.65 x=0.4
x=0.5
α=0, x=0.5
0.6
W 0.55
0.5
0.45
0.4
1 2 3 4 5 6 7 8 9 10 11 12
N
FIG.3. Averageworkextractionasfunctionofnumberofparticles.
Fig.3representstheaverageworkasthefunctionofnumberofparticleswithhardcoreinteractionsfordifferent
initial position of the partition. Similar to the non-interacting case, work extraction decreases when number of
particles increases. For comparison, we have also included in the figure, the work extraction plot with non-
interactingparticles(α=0)whenthepartitionisinitiallyplacedatthemid-pointx=0.5. Ascanbeseen,work
extractionlowerwhenwearedealingwithnon-idealgasesratherthanidealgases.
5
α=0.001, N=2
0.7 α=0.001, N=5
α=0, N=2
α=0, N=5
0.6
W 0.5
0.4
0.3
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
FIG.4. Averageworkextractionasfunctionofpositionofthepartition.
Fig.4depictsthebehaviorofaverageworkasafunctionofthepositionwherethepartitionisinitiallyinserted.
The plot clearly shows that interactions plays a major role in work extraction as the number of particles is in-
creased. Inter-particle interactions decreases work extraction. Average work is symmetric about x = 0.5 for
obvious reason. Another point to be noted is that work extraction is more for x (cid:54)= 0.5, as particle number is
increased.
B. Squarewellpotential
Inthistypeofinteractionbetweenparticles,p iscalculatedusingEq.2and12
n
Z (xL)
p = n,N−n ,
n (cid:80) Z (xL)
n(cid:48) n(cid:48),N−n(cid:48)
Z (xL)Z ((1−x)L)
= n N−n ,
(cid:80)
Z (xL)Z ((1−x)L)
n(cid:48) n(cid:48) N−n(cid:48)
(cid:0)N(cid:1)xn(1−x)N−n(1− γn2)[1− γ(N−n)2]
= n x 1−x , (17)
(cid:80) (cid:0)N(cid:1)xn(cid:48)(1−x)N−n(cid:48)(1− γn(cid:48)2)[1− γ(N−n(cid:48))2]
n(cid:48) n(cid:48) x 1−x
whereγ = C(cid:48) −C(cid:48)eβε withC(cid:48) = 1C = 2πσ3R3 = αR3 andC(cid:48) = 1C = 2πσ3(R3−1) = α(R3 −1). To
1 2 1 V 1 3V 2 V 2 3V
calculate the equilibrium position of the partition we need to equate the pressure (given by Eq.13) on the both
sidesofthepartitionandhenceweneedtosolvethefollowingequation:
n (cid:18) n (cid:19)2 N −n (cid:20) N −n (cid:21)2
+(C −C eβε) = +(C −C eβε) . (18)
Aln 1 2 Aln A(L−ln ) 1 2 A(L−ln )
eq eq eq eq
Thephysicalsolutionoftheaboveequationisln =(n/N)L. UsingEq.2,f canbecalculated
eq n
Z (ln )
f = n,N−n eq ,
n (cid:80) Z (ln )
n(cid:48) n(cid:48),N−n(cid:48) eq
Z (ln )Z (L−ln )
= n eq N−n eq ,
(cid:80) Z (ln )Z (L−ln )
n(cid:48) n(cid:48) eq N−n(cid:48) eq
(cid:0)N(cid:1)(n)n(1− n)N−n(1−γnN)[1−γ(N −n)N]
= n N N (19)
(cid:80) (cid:0)N(cid:1)(n)n(cid:48)(1− n)N−n(cid:48)(1−γn(cid:48)N)[1−γ(N −n(cid:48))N]
n(cid:48) n(cid:48) N N
6
α=0.001, R=20, ε=1
2.2
x=0.5 No Interaction
2 x=0.1
x=0.2
1.8 x=0.3
x=0.4
1.6
x=0.5
1.4
W 1.2
1
0.8
0.6
0.4
0.2
1 2 3 4 5 6 7 8 9 10 11 12
N
FIG.5. Averageworkextractionasfunctionofnumberofparticles.
Fig.5showsthebehaviorofaverageworkextractionasweincreasethenumberofparticle. Theplotshowsthat
for small number of particles work extraction is less than the non-interacting case. But with increasing particle
number the more work is extracted using interacting particle system. Another important thing to notice is that
workextractionisindependentoftheinitialpositionofthepartitionforhighervaluesofN. Thisfactissupported
by the plot given in Fig.6, where the amount of work is plotted against x - initial position of wall- for different
N values. Thisgraphdepictsthefactthatasoneincreasethenumberofparticles,themaximumworkextraction,
which was initially at x = 0.5, shifts to either sides and finally for N = 12, the effect of asymmetry on work
extractionbecomesverylessintherange0.1≤x≤0.9.
α=0.001, R=20, ε=1
2.2
N=2, No Interaction
2 N=2
1.8 N=5
N=10
1.6 N=12
1.4
1.2
W
1
0.8
0.6
0.4
0.2
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
FIG.6. Averageworkextractionasfunctionofpositionofthepartition.
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α=0.001, R=20, ε=1
100
10
W
1 β=1
β=0.1
β=0.05
β=0.02
β=0.01
β=0.01, Hard core
0.1
1 2 3 4 5 6 7 8 9 10 11 12
N
FIG.7. Averageworkextractionasfunctionofnumberofparticlesfordifferenttemperaturevalues.
Fig.7showsthebehaviorofaverageworkatdifferenttemperaturesinpresenceofsquarewellinteractionbe-
tween particles. Work extraction increases as the temperature is increased. At high enough temperatures the
particlesdonotfeelthenegativewelloftheinteractionandtheworkingsubstanceisexpectedtoactasanon-ideal
gaswithhardcoreinteraction. Intheplotacomparisonofworkextractionisshownbetweenparticleswithsquare
wellinteractionsandparticleswithhardcoreinteractionforβ =0.01.
α=0.001, N=5, ε=1
7
R=10
R=20
6
R=30
R=40
5 R=50
4
W
3
2
1
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
FIG.8. AverageworkextractionasfunctionofinitialpositionofthepartitionfordifferentRvalues.
Workextractioncanbeincreasedbyvaryingthewellwidthanddepth. Figs.8and9showthebehaviorofwork
extraction when well width and depth is varied respectively. More work is extracted as one increases the width
andthedepthofthewell.
8
α=0.001, N=5, R=20
ε=0.5
1.2 ε=1
ε=1.5
1
0.8
W
0.6
0.4
0.2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
FIG.9. Averageworkextractionasfunctionofinitialpositionofthepartitionfordifferentvaluesofwelldepth.
IV. DISCUSSIONS
We now discuss the role of interactions in work extraction by providing a comparative study between non-
interactingandinteractingparticlesystems. Figs. 10and11comparetheworkextractionforthreecases. Work
extractioninsystemswithsquarewellinteractionsbetweenparticlesismorewithrespecttonon-interactingsystem
andparticleswithhardcoreinteractions.Thisisduetotheformationofboundstates.Ifprobabilityofformationof
boundstatesislargeduetoincreasedwellwidthanddepth,therewillbehigherprobabilitytofindlargenumberof
particlesononeside. Asaresultthedistancebetweentheinitialandthefinalequilibriumpositionofthepartition
willbemore. Thiswillinturnincreasetheworkextraction. Thiseffectisnotseenincaseofparticlesinteracting
viahardcoreinteractionandhenceworkextractiondecreasesinthatcase.
α=0.001, R=20, ε=1, x=0.3
2.2
No Interaction
2 Hard core
Square well
1.8
1.6
1.4
W
1.2
1
0.8
0.6
0.4
1 2 3 4 5 6 7 8 9 10 11 12
N
FIG.10. Comparisonofaverageworkextractionasfunctionofnumberofparticles.
9
α=0.001, R=20, ε=1, N=5
2.2
No Interaction
2 Hard core
1.8 Square well
1.6
1.4
1.2
W
1
0.8
0.6
0.4
0.2
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
FIG.11. Comparisonofaverageworkextractionasfunctionofpositionofthepartition.
V. CONCLUSION
In conclusion, we have studied the interacting multi-particle classical Szilard engine. In our article, we dealt
withworkingsystemsconsistingofparticleswhichinteractwiththemselvesviahardcoreandsquarewellinter-
actions. Amajorassumptioninourworkisthatweconsideredtheworkingsystemstobenon-idealgaswithlow
density,sothatwecansafelyusetheequationofstateofthegasuptosecondvirialco-efficient.Onecangobeyond
thisassumptiontomimicwiththerealisticpictureandconsiderhigherordervirialco-efficientsbutthereisnoend
toit. Itwillobviouslygivechangesinworkextraction. Buttofirstapproximationweareabletoshowanalytically
how inter-particle interactions affect work extraction. Our study reveals that work extraction is less when more
particlesareinteractingthroughhardcoreinteractions. Moreworkisextractedwhentheparticlesareinteracting
via square well interaction. Another important result for the second case(square well interactions) is that as we
increase the particle number the work extraction becomes independent of the initial position of the partition, as
opposedtothefirstcase(hardcoreinteractions). Workextractiondependscruciallyontheinitialpositionofthe
partition. Moreworkcanbeextractedwithlargernumberofparticleswhenpartitionisinsertedatpositionsnear
theboundarywalls.
VI. ACKNOWLEDGEMENT
AMJthanksDST,Indiaforfinancialsupport(throughJ.C.BoseNationalFellowship).
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