(2005-02-18)
Determinism: Laplace's Demon
Deducing past and future from a detailed snapshot.
[For an intellect knowing all positions and velocities]
nothing
could be uncertain and the future,
just like the past, would be present before its eyes.
Pierre Simon de Laplace (1749-1827)
The "intellect" so introduced by the Marquis de Laplace
(Essai philosophique sur les probabilités, 1814) has since been dubbed
Laplace's Demon.
It is an icon of the concept of determinism,
which can be traced back to Socrates and which was fully entangled with
Science before the dawn of Quantum Theory.
Interestingly, Laplace himself first discussed this in
a treatise about probabilities,
which deal with the very uncertainty his Demon
would never have to face.
In fact, Laplace's Demon cannot possibly exist, within this world or outside of it.
From a philosophical or religious standpoint, its existence would preclude
free will, which is the one thing that makes the Creation qualitatively different
from a lowly clockwork toy.
To be the greatest of creators, it would seem that God had to allow His creation
some life of its own, that could escape even Him.
Thus, He did make a stone so heavy that He couldn't lift it:
The quantum Universe.
From a scientific standpoint, even the theoretical possibility of Laplace's Demon
condradicts our best understanding of fundamental observations.
In particular, quantum logic is incompatible with so-called
hidden variables,
which would make Quantum Theory deterministic and allow the
entire past and future of the Universe to stand "before the eyes" of the Demon.
Actually, what is ruled out by observation is only the existence of local
hidden variables,
since this would violate Bell's inequalities (John Bell, 1964)
which have been confirmed experimentally.
A priori, some set of nonlocal hidden variables could exist
which would make the whole Universe knowable at once.
However, if time can't be separated from the other dimensions
(motion of the observer trades space for time)
we may push the argument of nonlocality to its ultimate conclusion
and state that the entire knowledge of the past and future of the Universe
would require no part of its history to be unknown, to begin with.
Although a formal proof has yet to be devised, we thus believe in an unavoidable
quantum entanglement of the present with a distant past and a distant future.
However, this quantum entanglement is of such a nature that it cannot be used to
convey any kind of information.
Information cannot travel faster than light, nor can it go back and forth in time.
Some "time travel" is unavoidable in a relativistic quantum world,
but time travelers can't possibly carry any information luggage with them.
Similarly, perpetual motion is mandated by quantum laws, but this motion can't be harnessed
to produce useful work for free.
So, there may be a God who knows everything that ever was and ever
will be, but there's no such thing as an entity capable of deducing such knowledge from
anything but prior knowledge of the same.
Laplace's Demon is a fallacy.
Absolute prior knowledge is next to impossible to fathom in a quantum world...
Could this manifest itself as an absolute will
transcending free will without being incompatible with it?
It just seems easier to believe that the very God who created the quantum
stage would simply allow the actors to improvise in a loosely scripted play,
with a miracle now and then for good measure.
(2005-02-18)
Maxwell's Demon / Maxwellian Demon
How information is physically traded for entropy.
In a letter that he wrote in December 1867 to his friend Peter Guthrie Tait, the
physicist James Clerk Maxwell (1831-1879) described how the action of a "being"
controlling a shutter on a microscopic
hole between two gas containers (A and B)
could apparently violate the second law of thermodynamics.
Lord Kelvin (né William Thomson, 1824-1907) is responsible for
the name now universally given to this tiny creature: Maxwell's Demon.
Maxwell's Demon
"looks" at the gas molecules heading toward the hole.
He leaves the hole open only for fast molecules traveling from A to B and slow molecules
traveling from B to A. In Maxwell's own words:
"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."
Smoluchowski's Valve is another version of a
Maxwellian Demon which raises the pressure of B instead,
by letting through only those molecules which travel from A to B.
This was proposed in 1912 by Marian von Smoluchowski,
who also explained why his version of the demon could not be physically
realized as a tiny door with a weak spring:
The door would soon bounce randomly and become useless.
In fact, we simply can't ignore the temperature and/or entropy of
whatever is responsible for controlling the hole...
The "contradiction" stated by Maxwel disappears if the second law is
carefully stated to account for what the Demon is required to do,
which is to physically deal with information about individual molecules.
The resolution of this paradox has helped define the concept of
entropy in terms of information.
Instrumental in this analysis was a streamlined version of Maxwell's demon,
which was first considered by Leo Szilard in 1929.
We discuss it below.
Louis Vlemincq (Belgium. 2005-06-02; e-mail)
Perfect Diode
Wouldn't a perfect diode produce usable power from thermal electrons?
A device which would allow electrons to flow in only one direction along a wire would,
indeed, let thermal electrons gather on one side of it.
This build-up would create a
difference in electric potential that could be used to drive a load.
Thus, you'd have a generator whose energy would be ultimately derived from the heat
of a single source, in direct violation of the second law of thermodynamics.
Such a device, of course, cannot possibly exist.
It would be to electric potential what Smoluchowski' valve is to pressure.
Both devices seem to work, until one realizes that they will eventually
operate at the same temperature as the microscopic entities they
are supposed to control
(electrons in one case, gas molecules in the other).
This makes them completely ineffective at equilibrium.
The characteristic of an actual diode
(i.e., the current I through it, as a function
of the voltage V across it)
is always something like the following expression, known as
Shockley's Ideal Diode Equation.
It involves the absolute temperature T
in a way that prevents violation of the Second Law.
I = Is [ exp ( qV/nkT ) - 1 ]
- Is = saturation current (proportional to the diode's cross-section)
- k = Boltzmann's constant (in J/K)
- T = thermodynamic temperature (in K)
- q = charge of the electron (in coulombs)
- n = emission coefficient (typically between 1 and 2)
kT/q is called the thermal voltage.
It's about 26 mV at room temperature,
increasing at a rate of roughly 0.1 mV/°C
(25 mV at 17°C, 30 mV at 75°C).
(2005-05-24)
Szilard's Engine and Landauer's Principle
At temperature T, an energy k T ln(2)
is required to erase a bit.
In 1929, Leo Szilard (1898-1964) considered an
idealized engine
which converts elementary knowledge
[one bit of information]
into actual work.
The device is immersed in a single heat source at temperature T.
It consists of a single gas molecule in a box,
with pistons at both ends and a shutter in the middle.
A cycle of Szilard's engine consists of the following steps:
- The shutter is closed.
- The demon finds out which half of the box the molecule is in.
- The demon moves the piston in the empty half to the middle of the box.
- The shutter is opened.
- The molecule pushes the piston back in place, doing isothermal work.
- A new cycle starts, by closing the shutter.
The Thermodynamic Cost of Forgetting :
In 1961, Rolf Landauer (1927-1999) put forth what is now known as the "erasure limit", the minimal
amount of energy that must be dissipated to erase a given amount of information at temperature T.
Since 1973, Charles H. Bennett (b. 1943) has been investigating computing procedures
that do not throw away any previously acquired information
and thus entail arbitrarily little energy dissipation.
In 1982, Bennett proposed that the Maxwellian demon's inability to break the second
law of thermodynamics came from the cost of erasing information, rather than acquiring it.
(2005-02-19)
The Power of Knowledge
Laplace's Demon and Maxwell's Demon are related.
