 |
Phys.org
27 August
2009
|
Loschmidt’s paradox,
that time-symmetric dynamics should not lead to irreversible processes, is an
illusion that originates from quirks of human definition.
Firstly, “observable”
versus “absolute” reversibility should not be conflated. A whirlwind in a
library is far more likely to randomly scatter the 60 volumes of the Oxford
Dictionary of National Biography out of alphabetical order than it is to randomly
scatter them back into alphabetical order. While the latter scenario is
expected to essentially never happen (the number of ways of arranging 60 books,
approximately 1082, may exceed the number of atoms in the observable
universe), the probability is still greater than zero. Therefore “observably”
irreversible processes may still be “absolutely” reversible. (Poincaré
recurrence theorem is a related concept, whose popularity in cosmology,
however, is unjustified due to reliance upon non-physical premises.)
Secondly, physical processes may appear asymmetrical only because of asymmetrical categorisation. Of the 1082 ways of arranging 60 volumes, 1 has been categorised as “alphabetical order” (volumes 1, 2, 3…) while 1082 – 1 have been categorised as “jumble;” as described in Boltzmann’s entropy formula, macrostates with greater entropy contain more microstates, S = kB ln W. However, each “jumble” microstate, a particular configuration of “jumble” (eg, volumes 24, 60, 1…), is actually of equal likelihood as “alphabetical order.”
As a rough
demonstration, the situation between “wine glass” and “mess” appears
asymmetrical at first glance:
However, when one
considers that “mess” comprises more microstates than “wine glass” (ie, of all
the ways of arranging glass molecules, more result in a “mess” than in a “wine
glass”), the asymmetry disappears (in reality, the difference in number of
microstates is greater by many orders of magnitude):
The arrow of time,
that entropy only increases over time despite time-symmetric dynamics, is closely
related but nevertheless a separate paradox. This phenomenon is an illusion
resulting from low-entropy initial conditions. (Just as Newton’s laws
describe the motion of a falling apple but not the initial height of the apple,
so cosmological initial conditions may be separate from other laws of physics.)
The action of S = k
ln W over time can be analogised in how a carbon dioxide molecule originating
in an opened can of soft drink will diffuse into the air of the room, then the
airspace of the street, then suburb, then city, then state, et cetera (the
position of the molecule analogises, in phase space, the current
microstate of a system and the volume of each body of air analogises W for each
macrostate). While the actual law is that atmospheric carbon dioxide diffuses
randomly throughout the atmosphere, the apparent law that carbon dioxide only
diffuses from smaller to larger bodies of air only arises from the remarkable
statistical unlikelihood of the particular molecule initially chosen (inside a
can of soft drink, out of all the carbon dioxide on Earth).
While superficially appearing
to be time-asymmetric phenomena emerging from time-symmetric dynamics, neither
Loschmidt’s paradox nor the arrow of time are paradoxical. The former arises
from the fact that random rearrangements of glass molecules are far more likely
to correspond to “mess” than to “wine glass” and the latter from the
remarkableness of random glass molecules being in the form of a wine glass in
the first place. The true conundrum then is not of thermodynamics but of
initial conditions.
Unde venistis?
