Maxwell’s DemonJohn D. Norton

Department of History and Philosophy of ScienceCenter for Philosophy of Science

University of Pittsburgh

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483, 187–189 (08 March 2012)

Consensus reported

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The paradox of the apparent violation of the second law [by a Maxwell’s demon] can be resolved by noting that during a full thermodynamic cycle, the memory of the demon, which is used to record the coordinates of each molecule, has to be reset to its initial state.

Indeed, according to Landauer’s principle, any logically irreversible transformation of classical information is necessarily accompanied by the dissipation of at least kT ln(2) of heat per lost bit (about 3x10-21 J at room temperature (300 K)), where k is the Boltzmann constant and T is the temperature.

…This entropy cost required to reset the demon’s memory to a blank state is always larger than the initial entropy reduction, thus safeguarding the second law.

Landauer’s principle hence seems to be a central result that not only exorcizes Maxwell’s demon, but also represents the fundamental physical limit of irreversible computation.

1867 4

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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1905 6

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)

Thermal fluctuations in small particles suspended in liquids should be microscopically visible and enable an estimation of Loschmidt’s

number.

Thermal process must be analysed by the molecular-kinetic theory of

heat.

There really are atoms.

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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

1907 10

Svedberg, The (1907). “Über die Bedeutung der Eigenbewegung der Teilchen in kolloidalen Lösungen für die Beurteilung der Gültigkeitsgrenzen des zweiten Haupsatzes der Thermodynamik,” Annalen der Physik, 59, pp.451-58.

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1912 12

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.

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.

Smoluchowski’s original version. The one-way valve demon

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Other examples of defeated demons.

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Later popularized by Feynman

Other examples of defeated demons.

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Marian Smoluchowski, 1912

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

disrupted fatally by them.

quote:

Marian Smoluchowski, 1912

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“Indeed you would be just as mistaken if you wanted to warm a certain part of a fluid by friction through the Brownian molecular motion of suspended particles by means of threads.”

“… it appears at present that the construction of a perpetual motion machine that produces work continuously is excluded not by purely technical difficulties, but as a matter of principle.”

“Naturally this brief exposition should only serve to make this assertion physically plausible. For a proper proof one can consult the presentations of statistical mechanics. In any case, the latter turn out still to have some deficiencies…”

What about an intelligently operating demon?

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Smoluchowski’sproposal

It won’t work, since the operating demon is a physical system…

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Smoluchowski opens the door…

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1929Szilard’s

One-Molecule Engine

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Simplest case of fluctuations

Many molecules

A few molecules

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One molecule Can a demon exploit these fluctuations?

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.

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:

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.

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 escape

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The Information

-Theoretic Turn

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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.

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.