Transcript A Phase-Volume Based Exorcism of Maxwell`s Demon John D

Misplaced Idealizations
Maxwell's Demon
does not Compute
John D. Norton
Department of History and Philosophy of Science
Center for Philosophy of Science
University of Pittsburgh
4th Tuebingen Summer School
in History and Philosophy of Science, July 2015
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This Lecture
Maxwell’s demon is not idealized
well as an information processor or
computing device.
The quickest, simplest,
No information talk.
clearest, most effective exorcism
No entropy cost of erasing information.
of Maxwell’s demon in 150 years.
No entropy cost of gaining information.
No entropy cost of processing information.
No Szilard’s Principle. No Landauer’s Principle.
etc. etc.
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The Exorcism in Brief (for experts)
same
demon
state
demon
what
1. Describe
a naturalized
must demon do.
any equilibrium system
out-of-equilibrium system
most of phase space
obvious
2. Attribute
properties to demon and
system. Hamiltonian
dynamics…
a tiny part of phase space
that the demon
3. Find
must violate Liouville
4. A quantum
version.
theorem (conservation of
phase volume).
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Maxwell demons
of the fluctuation era
1905-1929 were
not computers
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Maxwell’s demon lives
in the details of Brownian motion and other fluctuations
“[…] 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
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 minimachines with that purpose.
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Guoy’s Proposal, 1888
thread
To and fro
Brownian motion of
a small body…
…is rectified by a
ratchet that can only
turn one way.
The first “Brownian ratchet.”
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Svedberg’s Proposal 1907
Colloid
cools
Charged colloid
particles radiate
their thermal
energy.
Casing
heats
Tuned lead casing
absorbs the
radiation.
…plus many more layers, details
designed to prevent return of heat.
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.
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Exorcism of
Maxwell’s demon
by fluctuations.
Marian Smoluchowski, 1912
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.
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Marian Smoluchowski, 1912
Other examples of
defeated demons.
Ambient voltage
fluctuations
electric
current
resistor heated
above surrounding
temperature
Guoy’s Brownian particle replaced by
Brownian motion of the toothed wheel.
Later popularized by Feynman
The second law holds on average only over time.
Machines that try to accumulate fluctuations are
disrupted fatally by them.
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Is a Maxwell
Demon Possible?
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Nanotechnology now manipulates
individual atoms
Molecules of CO on
surface manipulated
individually by
scanning tunneling
microscopes at -260C.
https://www.youtube.com/watch?v=oSCX78-8-q0
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Micron Scale Magnetic
http://physics.aps.org/story/v26/st22
Brownian Ratchet
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Micron Scale Magnetic
Brownian Ratchet
“A micron-scale bead in water
is driven across a magnetic surface by an external
magnetic field (pointing out of the screen and to the
right) oscillating at about 20 Hz. Diffusion in the
vertical direction is about 4000 times that expected
of an undriven particle.”
http://physics.aps.org/story/v26/st22
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15
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“Any physically meaningful model of a
Brownian motor must be compatible
with the limitations that the second
law imposes on its efficiency. In fact, a
major part of the present paper is devoted
to prove this compatibility for wide classes
of models, and to explore how the entropy
production Sprod can be reduced in order to
devise efficient motors.”
p. 181
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Micron-machines designed by evolution
Electron micrograph of ribosome
Schematic of protein synthesis
Powered by ATP
Evolution has not produced micro
Maxwell’s demons in spite of the
obvious evolutionary advantage.
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The New
Exorcism
Based on Section 4 of John D. Norton, "All Shook Up: Fluctuations, Maxwell's Demon
and the Thermodynamics of Computation" Entropy, 15 (2013), pp. 4432-4483.
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Assumptions
used in the
Exorcism
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What a Maxwell Demon Does
same
demon
state
demon
any equilibrium system
out-of-equilibrium system
“intermediate state”
Assumption (a)
A Maxwell’s demon is a device that,
when coupled with a thermal system in
its final equilibrium state,
will, over time, assuredly or very likely lead the
system to evolve to one of the intermediate states;
Succeeds with
all but small
fraction
b
of equilibrium
systems.
and, when its operation is complete, the thermal
system remains in the intermediate state.
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Demon cycles, Isolation
Assumption (c)
demon
demon
“device”
“device”
Assumption (b)
The device and thermal system
do not interact with any other
systems.
The device returns to its initial state at
the completion of the process;
and it operates successfully for every
microstate in that initial state.
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Constitution
of the Demon and the Thermal System
Assumption (d)
The time evolution of the total
system is Hamiltonian
with a time-reversible, timeindependent Hamiltonian.

Loosely speaking:
The thermal system and demon
have “generic” microdynamics.

dpi
H

dt
q i
dqi H

dt pi
for canonical generalized
momenta p1, p2, p3, … and
positions q1, q2, q3, …
Not inviolable… but relaxing it quickly
violates the Second Law.
Zhang and Zhang “pressure demon.”
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Hamiltonian Flow in Phase Space
Each point in phase
space is a complete
system.
Trajectory is the
complete history of
the evolution of the
system from start to
finish.
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Thermal Equilibrium States
…occupy
virtually all of
phase space.
Hence most
trajectories
starting in nonequilibrium states
end up in
equilibrium states.
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Thermal Equilibrium States
V
Assumption (e)
The equilibrium states occupy all
but a tiny portion α of the
thermal system’s phase space, V,
where α is very close to zero.
Assumption (f)
The intermediate states are all
within the small remaining
volume of phase space, αV.
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Liouville
Theorem
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Conservation of Phase Volume under Hamiltonian Flow
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The Contradiction
that completes the
reductio
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Impossible Compression
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Contradiction
large phase volume
v(1-a)(1-b)V
demon
phase
volume
(stays
constant)
fraction over
which demon
succeeds
Hamiltonian
evolution
of system + demon
thermal
system
phase
volume
much smaller
phase volume
vaV
demon
phase
volume
(stays
constant)
thermal
system must
end up in
nonequilibrium
volume.
Liouville Phase volume is conserved under
theorem Hamiltonian time development.
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Robustness
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What the Exorcism Does Not Do…
The second law of
thermodynamics is
not assumed.
The circularity troubling
the information theoretic
exorcisms does not arise.
(Sound vs profound dilemma.)
All is microphysics AND the assumption that
equilibrium states fills the phase space.
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What the Exorcism Does Not Do…
The volumes of phase
space representing states
are not coarse grained.
Showing compression
fails is the easy problem.
(1-a)(1-b)V contains all
microstates upon which the
demon can successfully act.
Recovering the approach to equilibrium is
hard since we must show a small volume of
non-equilibrium states evolves to a large
volume of equilibrium states. Coarse
graining saves the day.
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What the Exorcism Does Not Do…
Thermodynamic
entropy is not
included in the
argumentation.
It could be added if we
are careful. But many are
not and entropy talk more
often confuses than not.
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Quantum
Exorcism
36
Nanoscale objects are
quantum mechanical
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Classical
States form a phase space of
dimension 2n
Momentum coordinates p1, …, pn
Configuration coordinates q1, …, qn
Quantum
The system state |y(t)> is a vector in an
n dimensional Hilbert space, with
orthonormal basis vectors |e1>, …, |en>.
n is very large: ~1024 for ordinary macroscopic systems
Time evolution is governed by
Hamilton’s equations:
dpi
H dqi H


dt
qi dt pi
Time evolution is governed by
Schroedinger’s equation:
d
ih | y(t)   H | y(t) 
dt
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Low Entropy State
State is in small dimensional
subspace of Hilbert space
n molecule
gas confined
to half of
chamber
quantum
classical
State is in
small
1/2nth
volume of
phase
space
Quantum Liouville theorem:
subspace dimension is conserved.
Classical Liouville
theorem:
phase volume is conserved.
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Concluding
Morals
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?How was such an easy
exorcism
overlooked?
Information enthusiasts
were too fixated on making
the information-based
analysis work.
Information skeptics
were too fixated on showing
that the information-based
analysis fails.
Our idealizations have
been tracking fads.
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1950s: Information!
Claude Shannon (1948). "A Mathematical
Theory of Communication” Bell Systems
Technical Journal 27(1948), pp. 379-423,
623-656.
Information
ideas and concepts
=
Entropy
heat, work,
thermodynamics
Worth trying, but how long do try before you give up?
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Szilard’s Principle
versus
Landauer’s Principle
Szilard 1929
Von Neumann 1932
Brillouin 1951+…
Landauer 1961
Bennett 1987+…
Acquisition
Szilard’s principle is false.
of one bit of information
by the demon creates k ln 2
of thermodynamic entropy.
Information
Erasure
of one bit of information
from a computer memory
by the demon creates k ln 2
of thermodynamic entropy.
vs
Computation
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1970s: Computers!
“This essay is concerned with a particular
philosophical view that holds that the
mind
literally is a digital computer (in a
specific sense of “computer” to be developed),
and that thought literally is a kind of
computation.”
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1970s: Computers!
“I will show that the
universe can be
regarded as a giant
quantum computer.”
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1970s: Computers!
http://www.technologyreview.com/view/429561/the-measurementthat-would-reveal-the-universe-as-a-computer-simulation/
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1970s: Computers!
=
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Physics has principles…
… and so does computer science.
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The End
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