Physical Limits of ComputingPhysical Limits of ComputingDr. Mike FrankDr. Mike Frank

CIS 6930, Sec. #3753X CIS 6930, Sec. #3753XSpring 2002Spring 2002

Lecture #21Lecture #21Principles of Adiabatic ProcessesPrinciples of Adiabatic Processes

Wed., Feb. 27Wed., Feb. 27

Administrivia & OverviewAdministrivia & Overview• Don’t forget to keep up with homework!Don’t forget to keep up with homework!

– We are We are 7 out of 14 weeks into the course.7 out of 14 weeks into the course.• You should have earned You should have earned ~50 points by now.~50 points by now.

• Course outline:Course outline:– Part I&II, Part I&II, BackgroundBackground, , Fundamental LimitsFundamental Limits - done - done– Part III, Part III, Future of Semiconductor TechnologyFuture of Semiconductor Technology - done - done– Part IV, Part IV, Potential Future Computing TechnologiesPotential Future Computing Technologies - done - done– Part V, Part V, Classical Reversible ComputingClassical Reversible Computing

• Fundamentals of Adiabatic Processes - TODAYFundamentals of Adiabatic Processes - TODAY

– Part VI, Part VI, Quantum ComputingQuantum Computing– Part VII, Part VII, Cosmological Limits, Wrap-UpCosmological Limits, Wrap-Up

Terminology ShiftTerminology Shift• The word “infropy” sounds a bit too goofy.The word “infropy” sounds a bit too goofy.

– Unlikely to be accepted into widespread use.Unlikely to be accepted into widespread use.

• Shift in terminology used in this course:Shift in terminology used in this course:Before today:Before today: After today:After today:“Infropy “Infropy RR”” “Physical Information “Physical Information II””

(“Information”, “Pinformation”?)(“Information”, “Pinformation”?)

“Information “Information II”” “Known information “Known information KK””(“Kinformation”?)(“Kinformation”?)

“Entropy “Entropy SS”” “Entropy “Entropy SS””

Adiabatic Processes - overviewAdiabatic Processes - overview• Adiabatic steps in the reversible Carnot cycleAdiabatic steps in the reversible Carnot cycle• Evolution of the meaning of “adiabatic”Evolution of the meaning of “adiabatic”• Time-proportional reversibility (TPR) of quasi-Time-proportional reversibility (TPR) of quasi-

adiabatic processesadiabatic processes• Adiabatic theorem of quantum mechanicsAdiabatic theorem of quantum mechanics• Adiabatic transitions of a two-state systemAdiabatic transitions of a two-state system• Logic & memory in irreversible and adiabatic Logic & memory in irreversible and adiabatic

processes.processes.

The Carnot CycleThe Carnot Cycle• In 1822-24, Sadi Carnot analyzed the efficiency In 1822-24, Sadi Carnot analyzed the efficiency

of an ideal heat engine all of whose steps were of an ideal heat engine all of whose steps were reversiblereversible, and furthermore proved that:, and furthermore proved that:– AnyAny reversible engine (regardless of details) would reversible engine (regardless of details) would

have the have the samesame efficiency ( efficiency (TTHHTTLL)/)/TTHH..– NoNo engine could have greater efficiency than a engine could have greater efficiency than a

reversible engine w/o producing work from nothing reversible engine w/o producing work from nothing – TemperatureTemperature itself could be defined on a itself could be defined on a

thermodynamic scale based on heat recoverable by thermodynamic scale based on heat recoverable by a reversible engine operating between a reversible engine operating between TTHH and and TTLL

Steps of Carnot CycleSteps of Carnot Cycle• IsothermalIsothermal expansion at expansion at TTHH

• AdiabaticAdiabatic (without flow of (without flow ofheat) expansion heat) expansion TTHHTTLL

• Isothermal compression at Isothermal compression at TTLL

• Adiabatic compression Adiabatic compression TTLLTTHH V

P

TL

TH

Carnot Cycle TerminologyCarnot Cycle Terminology• AdiabaticAdiabatic (Latin): orig. “Without flow of heat” (Latin): orig. “Without flow of heat”

– I.e.I.e., no entropy , no entropy enters or leavesenters or leaves the system the system

• IsothermalIsothermal: “At the same temperature”: “At the same temperature”– Temperature of system Temperature of system remains constantremains constant as entropy as entropy

enters or leaves.enters or leaves.

• Both kinds of steps, Both kinds of steps, in the case of the Carnot cyclein the case of the Carnot cycle, , are examples of are examples of isentropicisentropic processes processes – ““at the same entropy”at the same entropy”– I.e.I.e., no (known) information is transformed into entropy , no (known) information is transformed into entropy

in either processin either process

• But “adiabatic” has But “adiabatic” has mutatedmutated to to meanmean isentropic. isentropic.

OldOld and and NewNew “Adiabatic” “Adiabatic”• Consider a closed system where you just lose track Consider a closed system where you just lose track

of its detailed evolution:of its detailed evolution:– It’s It’s adiabaticadiabatic (no heat flow), (no heat flow),– But it’s not “But it’s not “adiabaticadiabatic” (not isentropic)” (not isentropic)

• Consider a box containing some heat,Consider a box containing some heat,flying ballistically out of the system:flying ballistically out of the system:– It’s not It’s not adiabaticadiabatic,,

• because heat is “flowing” out of the systembecause heat is “flowing” out of the system– But it’s “But it’s “adiabaticadiabatic” (no entropy is generated)” (no entropy is generated)

• Hereafter, we bow to the 20Hereafter, we bow to the 20thth century’s corrupt century’s corrupt usage:usage: let let adiabatic adiabatic :: isentropic isentropic

Box o’ Heat

“The System”

Quasi-Adiabatic ProcessesQuasi-Adiabatic Processes• No real process is No real process is completelycompletely adiabatic adiabatic

– Because some outside system may always have enough energy Because some outside system may always have enough energy to interact with & disturb your system’s evolution - to interact with & disturb your system’s evolution - e.g.e.g., cosmic , cosmic ray, asteroidray, asteroid Evolution of system state is never perfectly knownEvolution of system state is never perfectly known

– Unless you know the exact quantum state of the whole universeUnless you know the exact quantum state of the whole universe

– Entropy of your system always increases.Entropy of your system always increases.• Unless it is Unless it is alreadyalready at a maximum (at equilibrium) at a maximum (at equilibrium)

– Can’t really be at Can’t really be at completecomplete equilibrium with its surroundings equilibrium with its surroundings» unless whole universe is at utterly stable “heat death” state.unless whole universe is at utterly stable “heat death” state.

• Systems at equilibrium are sometimes called “Systems at equilibrium are sometimes called “staticstatic.”.”

• Non-equilibrium, quasi-adiabatic processes are sometimes Non-equilibrium, quasi-adiabatic processes are sometimes also called also called quasi-staticquasi-static– Changing, but near a local equilibrium otherwiseChanging, but near a local equilibrium otherwise

Degree of ReversibilityDegree of Reversibility• The The degree of reversibilitydegree of reversibility (a.k.a. (a.k.a. reversibilityreversibility, a.k.a. , a.k.a.

thermodynamic efficiencythermodynamic efficiency) of any quasi-adiabatic ) of any quasi-adiabatic process is defined as the ratio of:process is defined as the ratio of:– the total free energy at the start of the processthe total free energy at the start of the process by the total energy spent in the processby the total energy spent in the process

• Or, equivalently:Or, equivalently:– the known, accessible information at the startthe known, accessible information at the start by the amount that is converted to entropyby the amount that is converted to entropy

• This same quantity is referred to as the (per-cycle) This same quantity is referred to as the (per-cycle) “quality factor” “quality factor” QQ for any resonant element ( for any resonant element (e.g.e.g., , LCLC oscillator) in EE.oscillator) in EE.

)(

)0(

tE

E

spent

free

tS

K

0

The “Adiabatic Principle”The “Adiabatic Principle”• Claim: Claim: Any Any idealideal quasi-adiabatic process quasi-adiabatic process

performed over time performed over time tt has a thermodynamic has a thermodynamic efficiency that is proportional to efficiency that is proportional to tt, , – in the limit as in the limit as tt0.0.

• We call processes that realize this idealization We call processes that realize this idealization time-proportionally reversibletime-proportionally reversible (TPR) processes. (TPR) processes.

• Note that the total energy spent (Note that the total energy spent (EEspentspent), and the ), and the

total entropy generated (total entropy generated (SS), are both ), are both inverselyinversely proportional to proportional to tt in any TPR process. in any TPR process.– The slower the process, the more energy-efficient.The slower the process, the more energy-efficient.

Proving the Adiabatic PrincipleProving the Adiabatic Principle(See RevComp memo #M14)(See RevComp memo #M14)• Assume free energy is in generalized Assume free energy is in generalized kinetickinetic energy energy

of motion of motion EEkk of system through its configuration of system through its configuration

space.space.EEkk = ½ = ½mvmv22 vv22 = ( = (//tt))22 tt22 for for mm, , const. const.

• Assume that every Assume that every ttff time, on average ( time, on average (mean free mean free

timetime), a constant fraction ), a constant fraction ff of of EEkk is is thermalizedthermalized

(turned into heat)(turned into heat)• Whole process thermalizes energy Whole process thermalizes energy ff((t/tt/tff))EEk k tttt22 = =

tt11. Constant in front is ½ . Constant in front is ½ fmfm22/t/tff : : 22, whereh , whereh

=½=½fm/tfm/tff is the is the effective viscosityeffective viscosity..

Example: Electrical ResistanceExample: Electrical Resistance• We know We know PPspentspent==II22R=R=((QQ//tt))22RR, ,

or or EEspentspent ==PtPt = = QQ22R/tR/t. . Note scaling with 1/ Note scaling with 1/tt– Charge transfer through a resistor obeys the adiabatic Charge transfer through a resistor obeys the adiabatic

principle!principle!

• Why is this so, microscopically?Why is this so, microscopically?– In most situations, conduction electrons have a large thermal In most situations, conduction electrons have a large thermal

velocity relative to drift velocity.velocity relative to drift velocity.• Scatter off of of lattice-atom cross-sections with a mean free time Scatter off of of lattice-atom cross-sections with a mean free time ttff

that is fairly independent of drift velocitythat is fairly independent of drift velocity– Each scattering event thermalizes the electron’s drift kinetic Each scattering event thermalizes the electron’s drift kinetic

energy - a frac. energy - a frac. ff of current’s total of current’s total EEkk

• Therefore assumptions in prev. proof apply!Therefore assumptions in prev. proof apply!

Adiabatic TheoremAdiabatic Theorem• A result of basic quantum theoryA result of basic quantum theory

– proved in many quantum mechanics textbooksproved in many quantum mechanics textbooks

• Paraphrased: A system initially in its ground state Paraphrased: A system initially in its ground state (or more generally, its (or more generally, its nnth energy eigenstate) will, th energy eigenstate) will, after subjecting it to a sufficiently slow change of after subjecting it to a sufficiently slow change of applied forces, remain in the corresponding state, applied forces, remain in the corresponding state, with high probability.with high probability.

• Amount of leakage out of desired state is Amount of leakage out of desired state is proportional to speed of transition at low speedsproportional to speed of transition at low speeds Quantum systems obey the adiabatic principle!Quantum systems obey the adiabatic principle!

Two-state SystemsTwo-state Systems• Consider any system having an adjustable, bistable Consider any system having an adjustable, bistable

potential energy surface in its configuration space.potential energy surface in its configuration space.• The two stable states form a natural The two stable states form a natural bitbit..

– One state represents 0, the other 1.One state represents 0, the other 1.

• Consider now the well having twoConsider now the well having twoadjustable parameters:adjustable parameters:– Height of the potential energy barrierHeight of the potential energy barrier

relative to the well bottomrelative to the well bottom– Relative height of the left and rightRelative height of the left and right

states in the well (bias)states in the well (bias)

0 1

Possible Parameter SettingsPossible Parameter Settings• We will distinguish six qualitatively different We will distinguish six qualitatively different

settings of the well parameters, as follows… settings of the well parameters, as follows…

Direction of Bias Force

BarrierHeight

Possible Adiabatic TransitionsPossible Adiabatic Transitions• Catalog of all the possible transitions in these Catalog of all the possible transitions in these

wells, wells, adiabaticadiabatic & & notnot......

Direction of Bias Force

BarrierHeight

0 0 0

111

10 N

Logicvalue

(Ignoring superposition states.)

leak

leak

Ordinary Irreversible LogicsOrdinary Irreversible Logics• Principle of operation: Lower a barrier, or not, Principle of operation: Lower a barrier, or not,

based on input. Series/parallel combinations ofbased on input. Series/parallel combinations of barriers do logic. Major barriers do logic. Major dissipation in at least one of dissipation in at least one of

the possible transitions.the possible transitions.0

1

0

Example: CMOS logics

Ordinary Irreversible MemoryOrdinary Irreversible Memory• Lower a barrier, dissipating stored information. Lower a barrier, dissipating stored information.

Apply an input bias. Raise the barrier to latch Apply an input bias. Raise the barrier to latch the new informationthe new informationinto place. Remove inputinto place. Remove inputbias.bias.

0 0

11

10 N

Example:DRAM

Input-Bias Clocked-Barrier LogicInput-Bias Clocked-Barrier Logic• Cycle of operation:Cycle of operation:

– Data input applies biasData input applies bias• Add forces to do logicAdd forces to do logic

– Clock signal raises barrierClock signal raises barrier– Data input bias removedData input bias removed

0 0

11

10 N

Input“0”

Input“1”

Retractinput

Retractinput

Clockbarrier

upClock up

Can amplify/restore input signalin clocking step.

Can reset latch reversibly givencopy of contents.

Examples: AdiabaticQDCA, SCRL latch, Rod logic latch, PQ logic,Buckled logic

Input-Barrier, Clocked-Bias RetractileInput-Barrier, Clocked-Bias Retractile

• Cycle of operation:Cycle of operation:– Inputs raise or lower barriersInputs raise or lower barriers

• Do logic w. series/parallel barriersDo logic w. series/parallel barriers– Clock applies bias force which changes state, or notClock applies bias force which changes state, or not

0 0 0

10 N

* Must reset outputprior to input.* Combinational logiconly!

Input barrier height

Clocked force applied

Examples:Hall logic,SCRL gates,Rod logic interlocks

Input-Barrier, Clocked-Bias LatchingInput-Barrier, Clocked-Bias Latching

0 0 0

1

10 N

• Cycle of operation:Cycle of operation:– Input Input conditionally lowersconditionally lowers barrier barrier

• Do logic w. series/parallel barriersDo logic w. series/parallel barriers– Clock applies bias force; conditional bit flipClock applies bias force; conditional bit flip– Input removed, Input removed, raisingraising the barrier & the barrier &

locking in the state-changelocking in the state-change– ClockClock

bias canbias canretractretract

Examples: Mike’s4-cycle adiabaticCMOS logic