No-Go Resultfor the Thermodynamics

of Computation John D. Norton

Department of History and Philosophy of ScienceCenter for Philosophy of Science

University of Pittsburgh

1

Warm Up

Computers Generate Lots of

Heat

2

Cray 2 computer (1985)

3From Cray sales brochure.

Cooling chips

4

fan

vanes

heat pipe

What is the Minimum Heat

Generation?

5

To minimize heat generation…

6

Use the smallest systems possible

molecular scale devices

=

Landauer’s Principle

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Rolf Landauer (1961) "Irreversibility and heat generation in the computing process," IBM Journal of Research and Development, 5, pp. 183–191. (p.265)

Logically irreversible operations like erasure pass heat/entropy to the environment.

Erasure0

11or

Erase one bit

PasskT ln 2 heatk ln 2 entropy

Logically reversible operations can be carried out non-dissipatively.

RESTORE TO ONE

The Standard Erasure Procedure

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Model of binary memory.One molecule gas in a divided chamber.

Sketchy… but someone has worked out the details…..??

Credible erasure procedures generate kT ln 2 of heat.

1Erasure compresses the space by a factor of 2.Corresponds to entropy change of k ln 2.

2Information theoretic entropy of ln 2 is associated with distributionP(L) = P(H) = 1/2

3

Heat kT ln 2Entropy k ln 2

passes to environment.

Plausibility of Landauer’s Principle

Bennett’s 2003 statement of Landauer’s Principle

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“Landauer’s principle, often regarded as the basic principle of the thermodynamics of information processing, holds that any logically irreversible manipulation of information, such as the erasure of a bit or the merging of two computation paths, must be accompanied by a corresponding entropy increase in non-information-bearing degrees of freedom of the information-processing apparatus or its environment….”

Bennett, Charles H. (2003). “Notes on Landauer’s Principle, Reversible Computation, and Maxwell’s Demon,” Studies in History and Philosophy of Modern Physics, 34, pp. 501-10.

Logically irreversible

operation

Mustpass entropy to environment

“…Conversely, it is generally accepted that any logically reversible transformation of information can in principle be accomplished by an appropriate physical mechanism operating in a thermodynamically reversible fashion.”

Logically reversible

operation

Can bethermodynamically reversible

Why not…Thermodynamically irreversible process

Thermodynamically reversible process

Bennett’s 2003 statement of Landauer’s Principle

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Logically irreversible

operation

…create entropy in a thermodynamically irrreversible process?

Mustpass entropy to environment

Logically reversible

operation

Can bethermodynamically reversible

Erasing Random versus Nonrandom/Known Data

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“When truly random data (e.g., a bit equally likely to be 0 or 1) is erased, the entropy increase of the surroundings is compensated by an entropy decrease of the data, so the operation as a whole is thermodynamically reversible. … [I]n computations, logically irreversible operations are usually applied to nonrandom data deterministically generated by the computation. When erasure is applied to such data, the entropy increase of the environment is not compensated by an entropy decrease of the data, and the operation is thermodynamically irreversible.”

Bennett, Charles H. (1988). “Notes on the History of Reversible Computation,” IBM Journal of Research and Development, 32 (No. 1), pp. 16-23

?? ? ?

Randomdata

Knowndata

NOentropy created

Entropy created

erase

Erasure procedure on known data…

…DOES use what is known.

Erasure is just rearrangement. No entropy created.

…does NOT use what is known.

No difference in erasing known and random data.

To Come

12

This Presentation

13

Demonstrations of Landauer’s Principle fail.

1. Thermalization2. Compression of phase space3. Information entropy(4. New indirect proof)

No-Go Result

The standard inventory of processes in the thermodynamics of computation neglects fluctuations.

All process must create entropy to overcome them and the quantities created swamp those tracked by Landauer’s principle.

Failed proofs of Landauer’s

Principle

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1. 15

1. Thermalization

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Initial dataL or R

Proof shows only that an inefficiently designed erasure procedure creates entropy.No demonstration that all must.

Mustn’t we thermalize so the procedure works with arbitrary data?No demonstration that thermalization is the only way to make procedure robust.

Entropy created in this ill-advised, dissipative step. !!!!!!

Irreversible expansion

“thermalization”

Reversible isothermal

compression passes heat kT ln 2 to heat

bath.

Data reset to LEntropy k ln 2 created in heat

bath

Dissipationless Erasure

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or

First method.

1. Dissipationlessly detect memory state.

2. If R, shift to L.

Second method.

1. Dissipationlessly detect memory state.

2. If R, remove and reinsert partition and go to 1.Else, halt.

2. 18

2. Phase Volume Compressionaka “many to one argument”

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thermodynamic entropy k ln (accessible phase volume)=

“random” data

reset data

occupies twice the phase volume of

Erasure halves phase volume.

Erasure reduces entropy of memory by k ln 2.

Entropy k ln 2 must be created in surroundings to conserve phase volume.

…if entropy is to connect to heat via Clausius’dS = dqrev/T

2. Phase Volume Compressionaka “many to one argument”

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“random” data

reset data

DOES NOT occupy twice the phase

volume of

thermalizeddata

Confusion with

It occupies the same phase volume.

FAILS

A Ruinous Sense of “Reversible”

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Random data

and

thermalized datahave the same entropy because they are connected by a reversible, adiabatic process???

insertion of the partition

removal of the partition

No. Under this sense of reversible, entropy ceases to be a state function.

S = 0

S = k ln 2

random data thermalized data

Hence confusion over random and

known data.

3. 22

3. Information-theoretic Entropy “p ln p”

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“random” data

reset data

Information

entropy Pi ln PiSinf = - k i

PL = PR = 1/2Sinf = k ln 2

PL = 1; PR = 0Sinf = 0

Hence erasure reduces the entropy of the memory by k ln 2, which must appear in surroundings.

But…in thiscase,

Information

entropyThermodynamic

entropydoes NOT equal

Thermodynamic entropy is attached to a probability only in special cases. Not this one.

What it takes…

24

IF…

Information

entropyThermodynamic

entropyDOES equal“p ln p” Clausius dS = dQrev/T

A system is distributed canonically over its phase space

p(x) = exp( -E(x)/kT) / Z

Z normalizes

All regions of phase space of non-zero E(x) are accessible to the system over time.

AND

For details of the proof and the importance of the accessibility condition, see Norton, “Eaters of the Lotus,” 2005.

Accessibility condition FAILS for “random data” since only half of phase space is accessible.

4. 25

4. Indirect Proof: General Strategy

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Process known to reduce entropy

Arbitrary erasure process

coupledto

Assumesecond law of thermodynamics holds on average.

Entropy must increase on average.

Entropy reduces.

Ladyman et al., “The connection between logical and thermodynamic irreversibility,” 2007.

The Debate is Ongoing

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The Debate is Ongoing

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My concerns:The inventory of processes assumed as possible is…

…survives only with artificial restrictions on what is possible. (“controlled operations”)

…inconsistent with the statistical form of the second law.

…selectively ignores fluctuations present in all molecular scale processes.

No-go result

Ladyman, James; Presnell, Stuart; Short, Anthony J. and Groisman, Berry (2007). “The connection between logical and thermodynamic irreversibility,” Studies in the History and Philosophy of Modern Physics, 38, pp. 58–79.Ladyman, James; Presnell, Stuart and Short, Anthony J. (2008). ‘The Use of the Information-Theoretic Entropy in Thermodynamics’, Studies in History and Philosophy of Modern Physics, 39, pp. 315-324.Ladyman, James and Robertson, Katie (forthcoming). “Landauer Defended: Reply to Norton, Studies in History and Philosophy of Modern Physics.

Norton, John D. (2011). “Waiting for Landauer.” Studies in History and Philosophy of Modern Physics, 42, pp. 184–198.Norton, John D. (2013). “Author's Reply to Landauer Defended” Studies in History and Philosophy of Modern Physics. Available online, May 24, 2013

“…the same bit cannot be both the control and the target of a controlled operation…”

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Every negative feedback control device acts on its own control bit. (Thermostat, regulator.)

The Most Beautiful Machine 2003Trunk, prosthesis, compressor, pneumatic cylinder13,4 x 35,4 x 35,2 in.“…the observers are supposed to push the ON button. After a while the lid of

the trunk opens, a hand comes out and turns off the machine. The trunk closes - that's it!”

http://www.kugelbahn.ch/sesam_e.htm

Failed proofs ofConverse

Landauer’s Principle

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Brownian Computation is the Constructive Proof

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“…Conversely, it is generally accepted that any logically reversible transformation of information can in principle be accomplished by an appropriate physical mechanism operating in a thermodynamically reversible fashion.”

(Bennett, 2003)

Norton, John D. (Manuscript on website). “Brownian Computation is Thermodynamically Irreversible.”

Closer analysis…

Brownian computation is thermodynamically irreversible, the thermodynamic analog of the uncontrolled expansion of a one molecule gas.

No-Go ResultIllustrated

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Fluctuations disrupt

Reversible Expansion and

Compression

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The Intended Process

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Infinitely slow expansion converts heat to work in the raising of the mass.

Mass M of piston continually adjusted so its weight remains in perfect balance with the mean gas pressure P= kT/V.

Equilibrium height is heq = kT/Mg

Heat kT ln 2 = 0.69kT passed in tiny increments from surrounding to gas.

The massive piston…

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….is very light since it must be supported by collisions with a single molecule. It has mean thermal energy kT/2 and will fluctuate in position.

Probability density for the piston at height h

p(h) = (Mg/kT) exp ( -Mgh/kT)

Meanheight

= kT/Mg = heq

Standard deviation

= kT/Mg = heq

Mean energy of gas 3kT/2Standard deviation (3/2)1/2kT = 1.225kT

What Happens.

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Fluctuations obliterate the infinitely slow expansion intended

A better analysis does not need external adjustment of weight during expansion. It replaces the gravitational field with

pistonenergy = 2kT ln (height)

Heat kT ln 2 = 0.69kT passed in tiny increments from surrounding to gas.

Fluctuations disrupt

Measurement and Detection

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Bennett’s Machine for Dissipationless Measurement…

Measurement apparatus, designed by the author to fit the Szilard engine, determines which half of the cylinder the molecule is trapped in without doing appreciable work. A slightly modified Szilard engine sits near the top of the apparatus (1) within a boat-shaped frame; a second pair of pistons has replaced part of the cylinder wall. Below the frame is a key, whose position on a locking pin indicates the state of the machine's memory. At the start of the measurement the memory is in a neutral state, and the partition has been lowered so that the molecule is trapped in one side of the apparatus. To begin the measurement (2) the key is moved up so that it disengages from the locking pin and engages a "keel" at the bottom of the frame. Then the frame is pressed down (3). The piston in the half of the cylinder containing no molecule is able to desend completely, but the piston in the other half cannot, because of the pressure of the molecule. As a result the frame tilts and the keel pushes the key to one side. The key, in its new position. is moved down to engage the locking pin (4), and the frame is allowed to move back up (5). undoing any work that was done in compressing the molecule when the frame was pressed down. The key's position indicates which half of the cylinder the molecule is in, but the work required for the operation can be made negligible To reverse the operation one would do the steps in reverse order.

Charles H. Bennett, “Demons, Engines and the Second Law,” Scientific American 257(5):108-116 (November, 1987).

38

…is fatally disrupted by fluctuations that leave the keel rocking wildly.

FAILS

A Measurement Scheme Using Ferromagnets

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Charles H. Bennett, “The Thermodynamics of Computation—A Review,” In. J. Theor. Phys. 21, (1982), pp. 905-40,

A Measurement Scheme Using Ferromagnets

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Charles H. Bennett, “The Thermodynamics of Computation—A Review,” In. J. Theor. Phys. 21, (1982), pp. 905-40,

A General Model of Detection

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First step: the detector is coupled with the target system.

The process is isothermal, thermodynamically reversible:

• It proceeds infinitely slowly.

• The driver is in equilibrium with the detector.

The process intended:

The coupling is an isothermal, reversible

compression of the detector phase space.

Fluctuations Obliterate Reversible Detection

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What happens:

What we expected:

No-Go Result

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Preparatory notions

44

Isothermal, thermodynamically reversible process

Self-contained isothermal, thermodynamically reversible process.

Computing fluctuations: How to do it.

Thermodynamically Reversible Processes

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For…Two systems interacting isothermallyin thermal contact with constant temperature surrounding at T 1 2

env

T

dU = dq –X dx

internal energy change

heat trans-ferred

generalized force generalized

displacement

X = -∂F/∂for process parameter

Total entropy of universe is

constant.

Total generalized forces vanish.

X1+X2=0

Total free energyF=U-TS is constant.

F1+F2=constant

Condition for thermodynamic

reversibility

All thermodynamic forces are in perfect balance (or minutely removed from it).

Process is a sequence of equilibrium states.

Thermodynamically reversible processes are NOT…

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…merely very slow processes.

capacitor discharges very slowly through resistor

balloon deflates slowly through a pinhole

…merely processes that can go easily in either way.

one molecule gas released

Self-contained thermodynamically reversible processes

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No interventions from non-thermal or far-from-equilibrium systems.

External hand removes shot one at a time to allow piston to rise slowly. Slow compression by slowly

moving, very massive body.

Mass is far from thermal equilibrium of a one-dimensional Maxwell velocity distribution.

Computing Fluctuations

48

Isolated, microcanonically distributed system

probability P that system is in non-equilibrium state

with phase volume V

∝ phase volumeV

give equilibrium, macroscopic description of non-equilibrium state

S = k ln V

S = k ln P + constant P∝ exp(S/k)

Computing Fluctuations

49

Canonically distributed system in heat bath at T.

give equilibrium, macroscopic description of non-equilibrium state

F = -kT ln Z(V)

F = -kT ln P + constant P ∝ exp(-F/kT)

probability system at point with energy E ∝

€

exp −EkT

⎛ ⎝ ⎜

⎞ ⎠ ⎟

probability P that system is in non-

equilibrium state with phase volume V

∝

€

Z(V ) = exp −E(x)kT

⎛ ⎝ ⎜

⎞ ⎠ ⎟

V∫ dx

Z(V)

No-Go ResultIt, at last.

50

Combine 1. and 2.

51

initial finalmiddle

1. Process is thermo-dynamically reversible

Finit = Fmid = Ffin

stages

Pinit∝ exp(-Finit/kT)Pmid∝ exp(-Fmid/kT)Pfin∝ exp(-Ffin/kT)

2. Fluctuations carry the system from one stage to another

any isothermal,

reversible process

Pinit = Pmiddle = PfinNo-Go result

Fluctuation Disrupt All Reversible, Isothermal Processes at Molecular Scales

52

Intended process

=1 =2

Actual process

=1 =2

Beating Fluctuations

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What it takes to overcome fluctuations

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initial final

Downward gradient in free energy

recapture in most likely

state

release from here

..but system can also be found in undesired intermediate states.

Process moves from high free energy state to low

free energy state.Fsys

Net creation of thermodynamic entropy.Stot = -Fsys/T

odds of final statePinit = probability that fluctuation throws

the system back to the initial state.

What it takes to overcome fluctuations

55

initial final

free energy

recapture in most likely state

release from here

Least dissipative case

High free energy mountain makes it unlikely that system is in intermediate stage.

€

Stot = k lnPfin

Pinit

⎛ ⎝ ⎜

⎞ ⎠ ⎟= k lnO fin

€

Pfin

Pinit

= exp −Ffin − Finit

kT ⎛ ⎝ ⎜

⎞ ⎠ ⎟= exp

ΔStot

k ⎛ ⎝ ⎜

⎞ ⎠ ⎟

Doing the sums…

56

Macroscopic ScaleOdds of completionOfin = 20Pfin = 0.95

Stot = k ln 20 = 3k

compareLandauer’s principle

k ln2 = 0.69 k

Odds of completionOfin = 7.2x1010

Stot = k ln (7.2x1010) = 25k

25kT is the mean thermal energy of ten nitrogen molecules.

Molecular Scale

Bead on a Wire

57

58

Each position is an equilibrium position

Slow motion of bead over wire is a thermodynamically reversible process. (Tilt wire minutely.)

Macroscopically…

For 5g bead and T=25C

vrms = 9.071 x 10-10 m/s

Molecular scale…

For 100 amu mass (n-heptane molecule)and T=25C

vrms = 157 m/s

Effect of thermal

fluctuations

Overcome fluctuations by tilting wire

59

Macroscopically…

For 5g bead = 5.8x10-18 radians

For Pfin = 0.999T=25C

stages 1/10th length

Depress by ~10-7 Bohr radius H atom per meter.

n-heptane is volatile!

Molecular scale…

For 100 amu mass (n-heptane molecule),turning the wire vertically has negligible effect!

Least dissipative case

60

More complicated

cases

61

62

Electric field moves a charge through a channel.

Two state dipole measures sign of

target charge.

Computed in “All Shook Up…”

Conclusion

63

Abstract and Concrete

Informationideas and concepts

Entropyheat, work,thermodynamics

=And why not?

Mass = EnergyParticles = Waves

Geometry = Gravity….

64

Time = Money

65

The End