Hui Dong & C. P. SunHui Dong & C. P. Sun
http://power.itp.ac.cn/~suncp/quantum.htmhttp://power.itp.ac.cn/~suncp/quantum.htm
Institute of Theoretical Physics, CASInstitute of Theoretical Physics, CAS
Quantum MaxwellQuantum Maxwell’’s Demon in s Demon in Quantum Szilard CycleQuantum Szilard Cycle
ContentsContents
1. 1. MaxwellMaxwell’’s Demon in Historys Demon in History
2. 2. LandauerLandauer’’s Principles Principle
3. Quantum Model3. Quantum Model
4. Thermodynamic Cycle4. Thermodynamic Cycle
5. Results5. Results
6. Conclusions6. Conclusions
ContentsContents
1. 1. MaxwellMaxwell’’s Demon in Historys Demon in History
2. 2. LandauerLandauer’’s Principles Principle
3. Quantum Model3. Quantum Model
4. Thermodynamic Cycle4. Thermodynamic Cycle
5. Results5. Results
6. Conclusions6. Conclusions
MaxwellMaxwell’’s Demon in Historys Demon in History
J. C. Maxwell1871
It is impossible to convert heat completely into work in a cycliIt is impossible to convert heat completely into work in a cyclic c process.process.
The Second Law of Thermodynamics (SLoT) by KelvinThe Second Law of Thermodynamics (SLoT) by Kelvin::
MaxwellMaxwell’’s Demons Demon ColdCold HotHot
Single molecule versionSingle molecule version
Leo Szilard1929
(a)(a) (b)(b)
LL
RR
(d)(d) (c)(c)
Total work extracted:Total work extracted:
?
L. Szilard, Z. Phys. 53, 840 (1929).L. Szilard, Z. Phys. 53, 840 (1929).
ContentsContents
1. 1. MaxwellMaxwell’’s Demon in Historys Demon in History
2. 2. LandauerLandauer’’s Principles Principle
3. Quantum Model3. Quantum Model
4. Thermodynamic Cycle4. Thermodynamic Cycle
5. Results5. Results
6. Conclusions6. Conclusions
LandauerLandauer’’s Principle(1)s Principle(1)
Erasing one bit information from the memory in computing processErasing one bit information from the memory in computing process
would would
inevitably accompany an increasing entropy inevitably accompany an increasing entropy ln2ln2 in the environment.in the environment.
(a) (b)
L
R
(d) (c)
?
0
1
P=1/2 P=1/2 ““00””P=1/2 P=1/2 ““11””
MD joins in the cycleMD joins in the cycle
R.Landauer, IBM J.Res.Dev.R.Landauer, IBM J.Res.Dev.55, 183(1961); , 183(1961); C.H. Bennett, Int.J.Theor.Phys. 21, 905(1982)C.H. Bennett, Int.J.Theor.Phys. 21, 905(1982)
LandauerLandauer’’s Principle(2)s Principle(2)
Information Erasing:Information Erasing:
Work disspated:Work disspated:
SystemSystem’’s Bath temperatures Bath temperature
Total work extracted:Total work extracted:
The second law of Thermodynamics is saved!The second law of Thermodynamics is saved!
Contents
1. 1. MaxwellMaxwell’’s Demon in Historys Demon in History
2. 2. LandauerLandauer’’s Principles Principle
3. Quantum Model3. Quantum Model
4. Thermodynamic Cycle4. Thermodynamic Cycle
5. Results5. Results
6. Conclusions6. Conclusions
Quantum Model (I)Quantum Model (I)
ChamberChamber PotentialPotential
Finite size Finite size L L effecteffect
Effective inverse temperatureEffective inverse temperature
General setup:General setup:
Quantum Model (II)Quantum Model (II)
EigenEigen--system in the potential:system in the potential:
0 L 0 Ll
RRLL
Quantum Model (III)Quantum Model (III)
MDMD’’s temperature:s temperature:
Quantum coherence:Quantum coherence:
Effective population:Effective population:
Effective inverse temperature:Effective inverse temperature:
1. 1. MaxwellMaxwell’’s Demon in Historys Demon in History
2. 2. LandauerLandauer’’s Principles Principle
3. Quantum Model3. Quantum Model
4. Thermodynamic Cycle4. Thermodynamic Cycle
5. Results5. Results
6. Conclusions6. Conclusions
ContentsContents
Thermodynamic CycleThermodynamic Cycle
InsertionInsertion MeasurementMeasurement
=
ExpansionExpansion
=
RemovingRemoving
Step 1: InsertionStep 1: Insertion
Prob. of finding molecule on the leftProb. of finding molecule on the left
Work done:Work done:
FiniteFinite--Size EffectSize Effect
Step 2: MeasurementStep 2: Measurement
CNOT:CNOT:
L L : no change : no change R R : flip: flip
MDMD’’s Logics LogicWrong informationWrong information
Finite temperature Finite temperature TTDD Precision of measurementPrecision of measurement
Ideal measurementIdeal measurement
Maximum information obtainedMaximum information obtained
No information obtainedNo information obtained
Worst measurementWorst measurement
Finite TemperatureFinite Temperature
Step 3: Controlled ExpansionStep 3: Controlled Expansion
MD moves piston according to the memory:MD moves piston according to the memory:
The process is done isothermally.The process is done isothermally.
Step 4: RemovingStep 4: Removing
Removing piston from potential:Removing piston from potential:
The process is also done isothermally.The process is also done isothermally.
Return to the initial stateReturn to the initial state
MD factorized outMD factorized out
MD releases heat to its bathMD releases heat to its bath
1. 1. MaxwellMaxwell’’s Demon in Historys Demon in History
2. 2. LandauerLandauer’’s Principals Principal
3. Quantum Model3. Quantum Model
4. Thermodynamic Cycle4. Thermodynamic Cycle
5. Results5. Results
6. Conclusions6. Conclusions
ContentsContents
Total Work Extracted (I)Total Work Extracted (I)
compensates change of MDcompensates change of MD’’s entropys entropy
Work to flip MDWork to flip MD’’s memory in the measurement s memory in the measurement
Critical value for positive work:Critical value for positive work:
Total Work Extracted (II)Total Work Extracted (II)
Maximum work :Maximum work :
EfficiencyEfficiency
SLoT is saved:SLoT is saved:
Two LimitsTwo Limits
Ideal measurement :Ideal measurement :
Worst measurement :Worst measurement :
Different OrdersDifferent Orders Different ResultsDifferent Results
1. 1. MaxwellMaxwell’’s Demon in Historys Demon in History
2. 2. LandauerLandauer’’ss
PrincipalPrincipal
3. Quantum Model3. Quantum Model
4. Thermodynamic Cycle4. Thermodynamic Cycle
5. Results5. Results
6. Conclusions6. Conclusions
ContentsContents
ConclusionsConclusions
A Quantum model is constructed and whole thermodynamic cycle iA Quantum model is constructed and whole thermodynamic cycle is established.s established.
e.g. Removing process e.g. Removing process
Efficiency is evaluated to prove the validity of Efficiency is evaluated to prove the validity of SLoTSLoT..
Quantum feature in the cycleQuantum feature in the cycle
e.g. Quantum coherence helps to low down effective tee.g. Quantum coherence helps to low down effective temperaturemperature
Finite size effect persists, even for high temFinite size effect persists, even for high temperature. perature.
TheThe methods can be utilized to analyze other cycle with MD.methods can be utilized to analyze other cycle with MD.