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Title: Information and Thermodynamic Entropy John D. Norton Department of History and Philosophy of Science Center for Philosophy of Science University of Pittsburgh


1
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

2
Philosophy and Physics

Information ideas and concepts
Entropy heat, work, thermodynamics
And why not? Mass Energy Particles
Waves Geometry Gravity .
Time Money

3
This Talk
Background Maxwells demon and the molecular
challenge to the second law of thermodynamics. Ex
orcism by principle Szilards Principle, Landaue
rs principle

4
Fluctuations and Maxwells demon

5
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.
The 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.

6
Maxwells 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 Maxwells demon at
work. Poincaré, 1905

7
SzilardsOne-Molecule Engine

8
Simplest case of fluctuations
Many molecules

9
The One-Molecule Engine
Szilard 1929
A partition is inserted to trap the molecule on
one side.
Initial state
The gas undergoes a reversible, isothermal
expansion to its original state.
Work kT ln 2 gained in raising the weight. It
comes from the heat kT ln 2, drawn from the heat
bath.

10
The One-Molecule Engine
Szilard 1929
A partition is inserted to trap the molecule on
one side.
Initial state
The gas undergoes a reversible, isothermal
expansion to its original state.
Work kT ln 2 gained in raising the weight. It
comes from the heat kT ln 2, drawn from the heat
bath.

11
Exorcism by principle

12
Szilards Principle
Von Neumann 1932 Brillouin 1951
Acquisition of one bit of information creates k
ln 2 of thermodynamic entropy.

13
Failed proofs of Landauers Principle

14
Direct Proofs that model the erasure processes in
the memory device directly.
or
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.

15
4. Indirect Proof General Strategy
Process known to reduce entropy

16
4. An Indirect Proof
Ladyman et al., The connection between logical
and thermodynamic irreversibility, 2007.
One-Molecule
gas
One-Molecule
memory

17
4. An Indirect Proof
Fails
Inventory of admissible processes allows
See Waiting for Landauer, Studies in History
and Philosophy of Modern Physics, forthcoming.

18
Dissipationless Erasure
or
First method. 1. Dissipationlessly detect memory
state. 2. If R, shift to L.

19
The Importance of Fluctuations

20
Marian Smoluchowski, 1912
Exorcism of Maxwells demon by fluctuations.

21
Fluctuations dispruptReversible Expansion and
Compression

22
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

23
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

24
What Happens.
Fluctuations obliterate the infinitely slow
expansion intended

25
Fluctuations disruptMeasurement and Detection

26
Bennetts 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).

27
A Measurement Scheme Using Ferromagnets
Charles H. Bennett, The Thermodynamics of
ComputationA Review, In. J. Theor. Phys. 21,
(1982), pp. 905-40,

28
A Measurement Scheme Using Ferromagnets
Charles H. Bennett, The Thermodynamics of
ComputationA Review, In. J. Theor. Phys. 21,
(1982), pp. 905-40,

29
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.

30
A General No-Go Result

31
Fluctuation Disrupt All Reversible, Isothermal
Processes at Molecular Scales
Intended process
ll2
l
ll1

32
Einstein-Tolman Analysis of Fluctuations
Total system of gas-piston or target-detector-driv
er 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(?) proportional to exp(-F(?)/kT)
Probability density for fluctuation to stage ?

33
Equilibrium implies uniform probability over l
Condition for equilibrium
?F/?l 0 F(l) constant
Probability distribution over l
p(l) constant p(l1) p(l2)
since

34
One-Molecule Gas/Piston System
Overlap of subvolumes corresponding to stages h
0.5H h0.75H hH h1.25H
Slice through phase space.

35
Fluctuations Obliterate Reversible Detection
What we expected

36
What it takes to overcome fluctuations
exp(- ) gt exp(3) 20
p(l2)
F(?2)-F(?1)
Enforcing a small probability gradient
kT
p(l1)
requires a disequilibrium
F(?1) gt F(?2) 3kT
which creates entropy.
S(?2)-S(?1) (E(?2)-E(?1))/T 3k

37
More Woes

38
Dissipationless Insertion of Partition?
No friction-based device is allowed to secure the
partition.
With a conservative Hamiltonian, the partition
will bounce back.

39
In Sum We are selectively ignoring fluctuations.
Dissipationless detection disrupted by
fluctuations.

40
Conclusions

41
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?

42
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.

43

http//www.pitt.edu/jdnorton/lectures/Tsinghua/Ts
inghua.html

44

Finis

45

Appendix

46
A dilemmafor information theoretic exorcisms

47
EITHER
Earman and Norton, 1998, 1999, Exorcist XIV
Total system gas demon all surrounding.
Do information theoretic ideas reveal why the
demon must fail?
Canonically thermal obeys your favorite version
of the second law.
the total system IS canonically thermal. (sound
horn)
the total system is NOT canonically
thermal. (profound horn)
OR

48
1.

49
1. Thermalization
Initial data L or R

50
2.

51
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

52
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.

53
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???

54
3.

55
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.

56
What it takes
Information entropy
Thermodynamic entropy
DOES equal
p ln p
Clausius dS dQrev/T
IF

57
4.

58
4. An Indirect Proof
Fails
One-Molecule
gas
One-Molecule
memory

59
the same bit cannot be both the control and the
target of a controlled operation
Every negative feedback control device acts on
its own control bit. (Thermostat, regulator.)
The Most Beautiful Machine 2003 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

60
Marian Smoluchowski, 1912
Exorcism of Maxwells demon by fluctations.
The second law holds on average only over
time. Machines that try to accumulate
fluctuations are disrupted fatally by them.

61
The standard inventory of processes

62
We may
Inventory read from steps in Ladyman et al.
proofs.
Exploit the fluctuations of single molecule in a
chamber at will.

63
We may
Detect the location of the molecule without
dissipation.
?
?
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