Information and Thermodynamic

Entropyor, Waiting for Landauer

John D. NortonDepartment of History and Philosophy of Science

Center for Philosophy of ScienceUniversity of Pittsburgh

CARL FRIEDRICH VON WEIZSÄCKER LECTURESUNIVERSITY OF HAMBURG

June 2010

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Philosophy and Physics

Informationideas and concepts

Entropyheat, work,thermodynamics

=And why not?

Mass = EnergyParticles = Waves

Geometry = Gravity….

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Time = Money

This Talk

Background

Maxwell’s demon and the molecular challenge to the second law of thermodynamics.

Exorcism by fluctuations

Exorcism by principleSzilard’s Principle,Landauer’s principle

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Foreground

A dilemma for information theoretic exorcisms

Failed proofs of Landauer’s PrincipleThermalization, Compression of phase

spaceInformation entropy, Indirect proof

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

Maxwell’s demon

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The original conception

J. C. Maxwell in a letter to P. G. Tait, 11th December 1867

“…the hot system has got hotter and the cold system colder and yet no work has been done, only the intelligence of a very observant and neat-fingered being has been employed.”

Divided chamber with a kinetic gas.

Demon operates door intelligently

“[T]he 2nd law of thermodynamics has the same degree of truth as the statement that if you throw a tumblerful of water into the sea you cannot get the same tumblerful of water out again.”

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The Devil livesin the details of Brownian motion and other fluctuations

“…we see under out eyes now motion transformed into heat by friction, now heat changed inversely into motion, and that without loss since the movement lasts forever. That is the contrary of the principle of Carnot.”

Poincaré, 1907

Could these momentary, miniature

violations of the second law be accumulated to large-scale violations?

Guoy (1888), Svedberg (1907) designed mini-machines with that purpose.

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Exorcism by Fluctuation

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Marian Smoluchowski, 1912

The second law holds on average only over time.Machines that try to accumulate fluctuations are disrupted fatally by them.

The best known of many examples.

Trapdoor hinged so that fast molecules moving from left to right swing it open and pass, but not vice versa.

BUT

The trapdoor must be very light so a molecule can swing it open.

AND

The trapdoor has its own thermal energy of kT/2 per degree of freedom.

SO

The trapdoor will flap about wildly and let molecules pass in both directions.

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Szilard’sOne-Molecule

Engine

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Simplest case of fluctuations

Many molecules

A few molecules

One molecule Can a demon exploit these fluctuations?

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The One-Molecule Engine

Initial stateA partition is inserted to trap the molecule on one side.

The gas undergoes a reversible, isothermal expansion to its original state.

Heat kT ln 2 is drawn from the heat bath and

converted fully to work.Cycle is completed and second law is violated?

Szilard 1929

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The heat bath entropy decreases by k ln 2.

There must be an entropy creation of of k ln2 somewhere else.

Entropy k ln 2 must be created by the demon in the process of measuring which side holds the molecule.

Proof by “working

backwards”

Szilard’s proposal: Second Law is protected by the entropy cost of the operation of demon.

…&*%#???!!!

Exorcism by principle

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Szilard’s Principle

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Landauer’s Principleversus

Acquisition of one bit of information creates k ln 2

of thermodynamic entropy.

Erasure of one bit of information creates k ln 2 of thermodynamic entropy.

Von Neumann 1932Brillouin 1951+…

Landauer 1961Bennett 1987+…

Proof:By “working backwards.”

By suggestive thought experiments.

(e.g. Brillouin’s torch)

Szilard’s principle is false.

Real entropy cost only taken when naturalized demon erases the memory of the position of the molecule

Proof: …???...

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).

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

FAILS

A dilemmafor information

theoretic exorcisms

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EITHER

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the total system IS canonically thermal.(sound horn)

the total system is NOT canonically thermal.(profound horn)

OR

Earman and Norton,

1998, 1999, “Exorcist XIV…”

Total system =gas + demon + all surrounding.

Canonically thermal = obeys your favorite version of the second law.

Cannot have both!

Profound“ …the real reason Maxwell’s demon cannot violate the second law …uncovered only recently… energy requirements of computers.”Bennett, 1987.

andSoundDeduce the principles (Szilard’s, Landauer’s) from the second law by working backwards.

Demon’s failure assured by our decision to consider only system that it cannot breach.

Principles need independent justifications which are not delivered.(…and cannot? Zhang and Zhang pressure demon.)

Do information theoretic ideas reveal why the demon must fail?

Failed proofs of Landauer’s

Principle

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1. 18

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

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2. Phase Volume Compressionaka “many to one argument”

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Boltzmann statistical

mechanics

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.

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 data

have 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

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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…

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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.

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4. An Indirect Proof

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Ladyman et al., “The connection between logical

and thermodynamic irreversibility,” 2007.

gas

memory

One-Molecule

One-Molecule

Reduces entropy of heat bath by k ln 2.

isothermal reversible expansion

insert partition

or

shift cell to match

dissipationlessly detect gas state

or

perform any erasure

Assume second law of thermodynamics holds on

average.

Erasure must create entropy k ln 2 on average.

Original proof given only in terms of quantities of heat passed among components.

4. An Indirect Proof

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gas

memory

One-Molecule

One-Molecule

Reduces entropy of heat bath by k ln 2.

isothermal reversible expansion

insert partition

or

shift cell to match

dissipationlessly detect gas state

or

Dissipationlessly detect memory state.

If R, shift to L.

Net effect is a reduction of entropy of heat bath. Second law violated even in statistical form.(Earman and Norton, 1999, “no-erasure” demon.)

Final step is a dissipationless erasure built out of processes routinely admitted in this literature.

Fails

The standard inventory of

processes

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We may…

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Exploit the fluctuations of single molecule in a chamber at will.

Insert and remove a partition

Perform reversible, isothermal expansions and contractions

Inventory read from steps in Ladyman et al. proofs.

We may…

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Detect the location of the molecule without dissipation.

?

?

Trigger new processes according to the location detected.

Shift between equal entropy states without dissipation.

?

Gas

Memory

R

L

We are selectively ignoring fluctuations.

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Dissipationless detection disrupted by fluctuations.

Reversible, isothermal expansion and contraction does not complete due thermal motions of piston.

Inserted partition bounces off wall unless held by… what?Friction?? Spring loaded pin??...

Need to demonstrate that each of these processes is admissible. None is primitive.

Inventory assembled inconsistently.It concentrates on fluctuations when convenient; it ignores them when not.

Conclusion

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Conclusions

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Is a Maxwell demon possible?

The best analysis is the Smoluchowski fluctuation exorcism of 1912. It is not a proof but a plausibility argument.

Information principle based exorcisms are

troubled.

They either presume the result or base the exorcism on principles posited without good foundation.

Efforts to prove Landauer’s Principle have failed.

…even those that presume a form of the second law. It is still speculation and now looks dubious.

Thermodynamics of computation has incoherent

foundations.

The standard inventory of processes admits composite processes that violate the second law and erase without dissipation.

It selectively considers and ignores fluctuation phenomena according to the result sought.

Its inventory of processes is assembled inconsistently.

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Readall about it.

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Commercials

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Finis