A Phase-Volume Based
Exorcism of Maxwell’s Demon
John D. Norton
Department of History and Philosophy of Science
University of Pittsburgh
Based on “All Shook Up: Fluctuations, Maxwell’s Demon and the
Thermodynamics of Computation,” Section 4.
www.pitt.edu/~jdnorton
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Any Closed Thermodynamic System
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Its Phase Space
total phase
volume
V
equilibrium
states
small fraction
a
intermediate
states
of total volume
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Assumption (a)
(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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Assumptions (b), (c)
(b) The device returns to its initial
state at the completion of the process;
and it operates successfully for every
microstate in that initial state.
(c) The device and thermal system do
not interact with any other systems.
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Assumption (a)
b
demon degrees
of freedom
V
equilibriu
m states
demon phase
space volume v in
initial = final state
intermediate
states
a
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Assumptions (d), (e) and (f)
(d) The time evolution of the total system is
Hamiltonian with a time-reversible, timeindependent Hamiltonian.
(e) The final equilibrium state occupies all but a
tiny portion α of the thermal system’s phase
space, V, where α is very close to zero.
(f) The intermediate states are all within the
small remaining volume of phase space, αV.
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Contradiction
Hamiltonian
evolution of
system +
demon
Liouville
theorem
phase volume
v(1-a)(1-b)V
evolves
much smaller
phase volume
vaV
Phase volume is conserved under
Hamiltonian time development.
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Nota Bene
(i) The second law of thermodynamics is not
assumed to hold, so that the circularity troubling
the information theoretic exorcisms does not arise
(ii) The volumes of phase space representing
states are not coarse grained; they consist simply
of all those microstates upon which the demon
can successfully act and the states that result.
(iii) Thermodynamic entropy is not
included in the argumentation.
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