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Pitt-Tsinghua Summer School for Philosophy of Science Institute of Science, Technology and Society, Tsinghua University

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Information and Thermodynamic EntropyJohn D. NortonDepartment of History and Philosophy of ScienceCenter for Philosophy of ScienceUniversity of Pittsburgh

Pitt-Tsinghua Summer School for Philosophy of Science

Institute of Science, Technology and Society, Tsinghua University

Center for Philosophy of Science, University of Pittsburgh

At Tsinghua University, Beijing June 27- July 1, 2011

Philosophy and Physics

=

Information

ideas and concepts

Entropy

heat, work,

thermodynamics

And why not?

Mass = Energy

Particles = Waves

Geometry = Gravity

….

Time = Money

This Talk

Foreground

Failed proofs of Landauer’s Principle

Thermalization, Compression of phase space

Information entropy, Indirect proof

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

Background

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

Exorcismby principle

Szilard’s Principle,

Landauer’s principle

The original conception

Divided chamber with a kinetic gas.

Demon operates door intelligently

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.”

“[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.”

Maxwell’s demon 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

“One can almost see Maxwell’s demon at work.”

Poincaré, 1905

Could these momentary, miniatureviolations of the second law be accumulated to large-scale violations?

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

Simplest case of fluctuations

Many molecules

A few molecules

One molecule

Can a demon exploit these fluctuations?

The One-Molecule Engine

Szilard 1929

A partition is inserted to trap the molecule on one side.

Initial state

Work kT ln 2

gained in raising the weight.

It comes from the

heat kT ln 2,

drawn from the heat bath.

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

Net effect of the completed cycle:

Heat kT ln 2 is drawn from the heat bath and fully converted to work.

The total entropy of the universe decreases by k ln 2.

The Second Law of Thermodynamics is violated.

The One-Molecule Engine

Szilard 1929

A partition is inserted to trap the molecule on one side.

Initial state

Work kT ln 2

gained in raising the weight.

It comes from the

heat kT ln 2,

drawn from the heat bath.

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

Net effect of the completed cycle:

Heat kT ln 2 is drawn from the heat bath and fully converted to work.

The total entropy of the universe decreases by k ln 2.

The Second Law of Thermodynamics is violated.

Szilard’s Principle

Landauer’s Principle

versus

Von Neumann 1932

Brillouin 1951+…

Landauer 1961

Bennett 1987+…

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

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

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: …???...

Direct Proofs that model the erasure processes in the memory device directly.

or

1.Thermalization

An inefficiently designed erasure procedure creates entropy.

No demonstration that all must.

2.Phase Volume Compressionaka “many to one argument”

Erasure need not compress phase volume but only rearrange it.

3. Information-theoretic Entropy “p ln p”

Associate entropy with our uncertainty

over which memory cell is occupied.

Wrong sort of entropy.

No connection to heat.

See: "Eaters of the Lotus: Landauer's Principle and the Return of Maxwell's Demon." Studies in History and Philosophy of Modern Physics, 36 (2005), pp. 375-411.

4. Indirect Proof: General Strategy

Process known to reduce entropy

Entropy reduces.

Assume

second law of thermodynamics holds on average.

coupled

to

Arbitrary erasure process

Entropy must increase on average.

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

4. An Indirect ProofOne-Molecule

gas

or

isothermal reversible expansion

insert partition

dissipationlessly detect gas state

Reduces entropy of heat bath by k ln 2.

One-Molecule

memory

or

shift cell to match

perform any erasure

Assume second law of thermodynamics holds on average.

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

Erasure must create entropy k ln 2 on average.

4. An Indirect Proof

Fails

Inventory of admissible processes allows:

Processes that erase dissipationlessly (without passing heat to surroundings) in violation of Landauer’s principle.

Processes that violate the second law of thermodynamics, even in its statistical form.

See: “Waiting for Landauer,” Studies in History and Philosophy of Modern Physics, forthcoming.

Dissipationless Erasure

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.

Exorcism of Maxwell’s demon by fluctuations.

Marian Smoluchowski, 1912The 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

AND

SO

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

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

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

The second law holds on average only over time.

Machines that try to accumulate fluctuations are disrupted fatally by them.

The Intended Process

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

The massive piston…

….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)

Mean

height

= kT/Mg = heq

Standard deviation

= kT/Mg = heq

What Happens.

Fluctuations obliterate the infinitely slow expansion intended

This analysis is approximate. The exact analysis replaces the gravitational field with

piston

energy

= 2kT ln (height)

Bennett’s Machine for Dissipationless Measurement…

FAILS

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).

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

A Measurement Scheme Using Ferromagnets

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

A Measurement Scheme Using Ferromagnets

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

A General Model of Detection

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

The process intended:

The process is isothermal, thermodynamically reversible:

• It proceeds infinitely slowly.

• The driver is in equilibrium with the detector.

The coupling is an isothermal, reversible compression of the detector phase space.

Fluctuation Disrupt All Reversible, Isothermal Processes at Molecular Scales

Intended process

l

l=l2

l=l1

Actual process

l

l=l2

l=l1

Einstein-Tolman Analysis of Fluctuations

Total system of gas-piston

or target-detector-driver is canonically distributed.

p(x, p) = (1/Z) exp(-E(x,p)/kT)

Different stages l

Different subvolumes of the phase space.

Probability density

that system is in stage l

p(l) proportional to Z(l)

Z(l) = ∫l exp(-E(x,p)/kT) dxdp

Free energy of stage l

F(l) = - kT ln Z(l)

p(λ) proportional to exp(-F(λ)/kT)

Probability density for fluctuation to stage λ:

= exp(-)

p(l2)

F(λ2)-F(λ1)

kT

p(l1)

Equilibrium implies uniform probability over l

Condition for equilibrium

= exp(-)

∂F/∂l = 0 F(l) = constant

p(l2)

F(λ2)-F(λ1)

kT

p(l1)

Probability distribution over l

p(l) = constant p(l1) = p(l2)

since

Time evolution over phase space

Expected

Actual

One-Molecule Gas/Piston System

Overlap of subvolumes corresponding to stages

h = 0.5H

h=0.75H

h=H

h=1.25H

Slice through phase space.

What it takes to overcome fluctuations

= exp(-) > exp(3) = 20

p(l2)

F(λ2)-F(λ1)

Enforcing a small probability gradient…

kT

p(l1)

…requires a disequilibrium…

F(λ1) > F(λ2) + 3kT

…which creates entropy.

S(λ2)-S(λ1) – (E(λ2)-E(λ1))/T = 3k

Exceeds the entropy k ln2 = 0.69k tracked by Landauer’s Principle!

No problem for macroscopic reversible processes.

F(λ1) - F(λ2) = 25kT

p(λ2)/p(λ1) = 7.2 x 1010

= mean thermal energy of ten Oxygen molecules

Dissipationless Insertion of Partition?

No friction-based device is allowed to secure the partition.

With a conservative Hamiltonian, the partition will bounce back.

Arrest partition with a spring-loaded pin?

The pin will bounce back.

Feynman, ratchet and pawl.

In Sum… We are selectively ignoring fluctuations.

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.

Why should we believe that…

…the reason for the supposed failure of a Maxwell demon is localizable into some single information theoretic process? (detection? Erasure?)

…the second law obtains even statistically when we deal with tiny systems in which fluctuations dominate?

Conclusions

Is a Maxwell demon possible?

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

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.

Its inventory of processes is assembled inconsistently.

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

http://www.pitt.edu/~jdnorton/lectures/Tsinghua/Tsinghua.htmlhttp://www.pitt.edu/~jdnorton/lectures/Tsinghua/Tsinghua.html

Do information theoretic ideas reveal why the demon must fail?

Total system =

gas + demon + all surrounding.

Earman and Norton, 1998, 1999, “Exorcist XIV…”

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

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

the total system IS canonically thermal.

(sound horn)

Principles need independent justifications which are not delivered.

(…and cannot? Zhang and Zhang pressure demon.)

the total system is NOT canonically thermal.

(profound horn)

OR

Profound

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

Bennett, 1987.

Cannot have both!

Sound

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

and

1.Thermalization

Reversible isothermal compression passes heat kT ln 2 to heat bath.

Initial data

L or R

Irreversible expansion

“thermalization”

Data reset to L

Entropy k ln 2 created in heat bath

!!!

!!!

Entropy created in this ill-advised, dissipative step.

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.

2.Phase Volume Compressionaka “many to one argument”

Boltzmann statistical mechanics

thermodynamic entropy

=

k ln (accessible phase volume)

“random” data

occupies twice the phase volume of

reset data

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”

FAILS

“random” data

DOES NOT occupy twice the phase volume of

reset data

It occupies the same phase volume.

Confusion with

thermalized

data

A Ruinous Sense of “Reversible”

Random data

and

insertion of the partition

removal of the partition

thermalized data

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

DS = 0

random data

thermalized data

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

DS = k ln 2

3. Information-theoretic Entropy “p ln p”

Information

entropy

Sinf = - k Si

Pi ln Pi

“random” data

PL = PR = 1/2

Sinf = k ln 2

reset data

PL = 1; PR = 0

Sinf = 0

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

does NOT equal

Thermodynamic

entropy

Information

entropy

But…

in this

case,

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

What it takes…

Information

entropy

DOES equal

Thermodynamic

entropy

“p ln p”

Clausius dS = dQrev/T

IF…

A system is distributed canonically over its phase space

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

Z normalizes

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

AND

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

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

4. An Indirect Proof

One-Molecule

gas

or

isothermal reversible expansion

insert partition

dissipationlessly detect gas state

Reduces entropy of heat bath by k ln 2.

One-Molecule

memory

or

shift cell to match

Dissipationlessly detect memory state.

If R, shift to L.

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

Net effect is a reduction of entropy of heat bath. Second law violated even in statistical form.

(Earman and Norton, 1999, “no-erasure” demon.)

“…the same bitcannot be both the control and the target of a controlled operation…”

The Most Beautiful Machine2003

Trunk, prosthesis, compressor, pneumatic cylinder

13,4 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

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

Exorcism of Maxwell’s demon by fluctations.

Marian Smoluchowski, 1912The 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

AND

SO

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

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

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

The second law holds on average only over time.

Machines that try to accumulate fluctuations are disrupted fatally by them.

We may…

Inventory read from steps in Ladyman et al. proofs.

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

Insert and remove a partition

Perform reversible, isothermal expansions and contractions

We may…

Detect the location of the molecule without dissipation.

?

?

Shift between equal entropy states without dissipation.

Memory

R

Gas

Trigger new processes according to the location detected.

?

L

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