Misplaced Idealizations

Entropy, Information and Maxwell's Demon

John D. NortonDepartment of History and Philosophy of Science

Center for Philosophy of ScienceUniversity of Pittsburgh

1

4th Tuebingen Summer Schoolin History and Philosophy of Science, July 2015

This Lecture

The thermodynamics of computation presumes it is possible to…

Chain molecular-scale computational steps that are thermodynamically reversible or nearly so.

Bad Idealization

Detection of memory device states.

Moving data from one location to another.

…

Compression and expansion of components spaces.steps

This Lecture

No-Go result

Thermal fluctuations (noise) prevent completion of any individual, molecular scale step.

Thermodynamic entropy must be created to complete each step.

€

ΔStot = k lnPfin

Pinit

⎛

⎝ ⎜

⎞

⎠ ⎟= k lnO fin

Minimum entropy creation not set by the logical specification of the computation, but by the number of steps chained.

Maxwell’s Demon

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The Maxwell Era1867-1905

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Theory of Heat,1871, first ed.

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AlsoLetter to Tait, 1867;Rayleigh 1871

Theory of Heat

7Better scan from 1872, 2nd ed.

Maxwell’s Proposal

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“He will thus, without expenditure of work,

raise the temperature of B

and lower that of A,

in contradiction to the second law of thermodynamics.”

air initially atuniform temperature

Maxwell’s Moral: The Demon Wins

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“This is only one of the instances in which conclusions which we have drawn from our experience of bodies consisting of an immense number of molecules may be found not to be applicable to the more delicate observations and experiments which we may suppose made by one who can perceive and handle the individual molecules which we deal with only in large masses.

In dealing with masses of matter, while we do not perceive the individual molecules, we are compelled to adopt what I have described as the statistical method of calculation, and to abandon the strict dynamical method, in which we follow every motion by the calculus.”

Theory of Heat.

No compulsion to exorcise the demon to protect the Second Law.

The demon illustrates that Second Law would fail if we could manipulate individual molecules.

…. Nanotechnology has not yet overturned the Second Law.

The Fluctuation Era1905-1929

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Einstein’s Brownian Motion Paper

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"On the motion of small particles suspended in liquids at rest required by the molecular-kinetic theory of heat.” Annalen der Physik, 17(1905), pp. 549-560. (May 1905; received 11 May 1905)

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“…no longer strictly valid…”“If it is really possible to observe the motion discussed here …”

“… then classical thermodynamics can no longer be viewed as strictly valid even for microscopically distinguishable spaces....”

“… … and an exact determination of the real size of atoms becomes possible.”

Maxwell’s demon livesin the details of Brownian motion and other fluctuations

Could these momentary, miniature

violations of the second law be accumulated to large-scale violations? A real Maxwell’s demon?

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

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“[…] we see under our eyes now motion transformed into heat by friction, now heat changed inversely into motion, and that without loss since the movement lasts forever. This is the contrary of the principle of Carnot. If this be so, to see the world return backward, we no longer have need of the infinitely keen eye of Maxwell's demon; our microscope suffices.”

Poincaré, 1904

Casing heats

Colloid cools

Svedberg’s Proposal

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Svedberg, The. “Über die Bedeutung der Eigenbewegung der Teilchen in kolloidalen Lösungen für die Beurteilung der Gültigkeitsgrenzen des zweiten Haupsatzes der Thermodynamik”.Annalen der Physik, 59 (1907) pp. 451–458.

Charged colloid particles radiate

their thermal energy.

Tuned lead casing absorbs the radiation.

…plus many more layers, details designed to prevent return of heat.

Marian Smoluchowski, 1912

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Exorcism of Maxwell’s demon by fluctuations.

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

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

disrupted fatally by them.

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.

Marian Smoluchowski, 1912

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Other examples of defeated demons.

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

disrupted fatally by them.

Later popularized by Feynman

The Information Era1929- ????

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“On the decrease of entropy in a thermodynamic system by the intervention of intelligent beings.”

Zeitschrift für Physik, 53 (1929), pp. 840-856.

Szilard 1929

1The One-Molecule Engine

Initial state 2A partition is

inserted to trap the molecule on one

side.

3The gas undergoes a reversible, isothermal

expansion to its original state.4

Work kT ln 2gained in raising the weight.

It comes from theheat kT ln 2,

drawn from the heat bath.

Szilard 1929

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.

Net effect of the completed cycle:

Szilard’s Principle

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Acquisitionof one bit of information

by the demon creates k ln 2 of thermodynamic

entropy.

Szilard 1929Von Neumann 1932

Brillouin 1951+…

Landauer’s Principleversus

Landauer 1961Bennett 1987+…

Erasureof one bit of information by the demon creates k ln 2 of thermodynamic entropy.

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

Szilard’s principle is false.

Process is thermodynamically reversible if data is “random”; not if “known” data.

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(e.g. erasure)

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

k ln 2 per bit erased

The Standard Erasure Procedure

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Model of binary memory.One molecule gas in a divided chamber. Heat kT ln 2

Entropy k ln 2passes to environment.

No-Go Result

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No-Go Result

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NO molecular-scale process that completes is thermodynamically reversible.

Thermodynamic entropy must be created to complete each step.

€

ΔStot = k lnPfin

Pinit

⎛

⎝ ⎜

⎞

⎠ ⎟= k lnO fin

ΔSΔS ΔS ΔS ΔS

ΔS ΔS ΔS

No-Go ResultIllustrated

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Fluctuations disrupt

Reversible Expansion and

Compression

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The Intended Process

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

Mass M of piston continually adjusted so its weight remains in near 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 surroundings 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 very slow expansion intended

A better analysis (elsewhere) does not need external adjustment of weight during expansion. It replaces the gravitational field withpiston

energy = 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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Measurement is compression of detector phase space

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

The process is isothermal, thermodynamically reversible:

• It proceeds very 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:

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

No-Go Result

Preparatory notions

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Thermodynamically Reversible Processes

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For…Two systems interacting isothermallyin thermal contact with constant temperature surroundings 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

Thermo-dynamically

reversible process

Set of irreversible processes that approach a perfect balance of all thermodynamic forces in the (unrealized) limit.

Condition approached arbitrarily closely in the limit:Total entropy of universe is

constant.

Total generalized forces vanish.

X1+X2=0

Total free energyF=U-TS is constant.

F1+F2=constant

Self-contained thermodynamically reversible processes

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No interventions fromnon-thermal orfar-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

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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 −E

kT

⎛

⎝ ⎜

⎞

⎠ ⎟

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.

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Combine 1. and 2.

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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

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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.ΔFsys

Net creation of thermodynamic entropy.ΔStot = -ΔFsys/T

odds of final statePinit = probability that fluctuation throws

the system back to the initial state.

What it takes to overcome fluctuations

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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.

€

ΔStot = k lnPfin

Pinit

⎛

⎝ ⎜

⎞

⎠ ⎟= k lnO fin

€

Pfin

Pinit

= exp −Ffin − Finit

kT

⎛

⎝ ⎜

⎞

⎠ ⎟= exp

ΔStot

k

⎛

⎝ ⎜

⎞

⎠ ⎟

Doing the sums…

44

Macroscopic Scale

Odds of completionOfin = 20Pfin = 0.95

ΔStot = k ln 20 = 3k

compareLandauer’s principle

k ln2 = 0.69 k

Odds of completionOfin = 7.2x1010

ΔStot = k ln (7.2x1010) = 25k

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

Molecular Scale

Bead on a Wire

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46

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

47

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

48

More complicated

cases

49

50

Electric field moves a charge through a channel.

Two state dipole measures sign of

target charge.

Computed in “All Shook Up…”

Conclusion

51

Thermal Fluctuations Cannot be Idealized Away

No-Go result

Thermal fluctuations (noise) prevent completion of any individual, molecular scale step.

Thermodynamic entropy must be created to complete each step.

€

ΔStot = k lnPfin

Pinit

⎛

⎝ ⎜

⎞

⎠ ⎟= k lnO fin

Minimum entropy creation not set by the logical specification of the computation, but by the number of steps chained.

53

The End

54

Appendices

A Measurement Scheme Using Ferromagnets

55

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

A Measurement Scheme Using Ferromagnets

56

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

Thermodynamically reversible processes are NOT…

57

…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

Computing Fluctuations

58

Isolated, microcanonically distributed system

probability P that system is in non-equilibrium state

with phase volume V

∝ phase volume

V

give equilibrium, macroscopic description of non-equilibrium state

S = k ln V

S = k ln P + constant P∝ exp(S/k)