I wrote this paper as an undergrad for a philosophy class.


Problems Of A Reductive Analysis Of The Direction Of Time

Rolf Muertter
May 12, 1994

	It takes three numbers to specify the location of a point in euclidean 
space; that is why it is said to be three-dimensional.  However, this will not 
give a complete description of the position of an event - to do this a fourth 
number is needed, called time.  This fourth coordinate differs considerably 
from the other three in that objects can move only in one direction: into the 
future.  One could, of course, argue that future is defined to be any temporal 
direction of events, so that by definition, time always proceeds from past to 
future, and not vice versa.  However, sequences of natural events tend to 
proceed in only one direction.  A swinging pendulum comes to rest; a pendulum 
at rest does not spontaneously start to swing.  We never grow younger, but 
always age.  An ice cube in warm water will always melt, it never forms 
spontaneously in warm water.  Thus, time appears to be asymmetrical with 
respect to past and future.  This asymmetry is also called the arrow of time.

	The usual way to understand natural phenomena is through a reductive 
analysis.  That is, one tries to understand specific phenomena as a 
consequence of more basic principles, or ultimately as a consequence of the 
laws of nature.  The laws of physics are the most basic laws we have.  
Phenomena as diverse as the spherical shape of the sun, tides, the orbit of 
the moon, things falling to the ground, movements of galaxies, or even certain 
features of the human body can all be understood as consequences of a single 
law, Newton's law of gravitation.  Since we find the arrow of time everywhere 
we look, we expect this asymmetry to show up in the laws of physics.  The laws 
of physics are stated mathematically as differential equations.  Differential 
equations are equations that contain derivatives of a dependent variable with 
respect to the independent variable.  In physics, the independent variable is 
often time.  In order to check wether an equation is time-symmetric, or 
T-invariant, t is replaced with -t.  If the equation remains unchanged, it is 
T-invariant, and we conclude that the law that is represented by this 
equation, is also T-invariant and cannot be used to explain the arrow of 
time.  If an equation contains only derivatives of even order, it will be 
T-invariant since (-1)^2 = 1.  For example, Newton's law of motion, F(x) = ma 
is T-invariant since a = d^2x(t)/dt^2, and so for t -> -t it becomes 
md^2x(-t)/d(-t)^2 = md^2x'(t)/dt^2 = F(x').  

F(x) = ma + bx  contains a derivative of odd order and is therefore T-variant 
(1).  However, this is not a fundamental equation of physics, since it is just 
Newton's law of motion including a friction term, bx.  Friction itself is a 
complex phenomenon that is hard to understand on a microscopic level.  There 
are many more examples of equations that contain derivatives of odd order, but 
they are not considered to be fundamental equations since they describe 
phenomena on a macroscopic level.  

	There are, however, exceptions, for example the Schrödinger-
equation,  ihY(t) = HY(t), a fundamental equation of quantum mechanics that 
describes the behavior of particles at a microscopic level, does contain a 
first order time derivative, but the measurable quantity is not Y(t)  but 
Y * Y .  This undoes any change of sign, and the result is always a positive, 
real number.  Therefore, the probability of an elementary quantum mechanical 
transition is equal to the time-reversed process (1).  So even the most 
fundamental of all equations is T-invariant!  It turns out that all 
fundamental equations are T-invariant with one exception (according to 
Vollmer): the equations that describe the decay of neutral K mesons, which is 
due to the weak interaction.  This asymmetry has been found experimentally, 
but indirectly.  Neutral K mesons can decay either into a negative pion, a 
positron, and a neutrino; or into a positive pion, an electron, and an 
antineutrino.  These two decay modes are charge-parity (CP) reverses of each 
other, but the probability of one decay mode is slightly larger than that of 
the other. So empirically, the decay is not CP reversal invariant. However,
according to quantum field theory, all processes should be CPT reversal
invariant (2). That is, any law of physics should be invariant under a
reversal of charge, parity, and time. So in order to preserve CPT reversal
invariance for the neutral K meson decay it must be assumed that T-invariance
does not hold. Since this process is microscopic and irreversible, can it be
used to explain the arrow of time? The answer is probably no, for several
reasons. First, the asymmetry is very slight, on the order of only 1 in 10^9
decays violate the CP-invariance. Second, it is not clear how the decay of
such an exotic particle could have anything to do with commonly observed
irreversible processes such as the diffusion of milk in coffee. Moreover,
according to Horwich, this may be only a de facto asymmetry, not a consequence
of a time-asymmetrical law of nature. He also questions the significance of
the result, since the asymmetry is so slight. But even if the asymmetry is
significant, it might not entail a violation of CP-invariance, since perhaps
other unknown processes accompany the decay, thus restoring CP-invariance.
He even considers the possibility that the CPT-theorem is false.

The second law of thermodynamics, dS/dt >= 0, is another example of a
macroscopic, time-asymmetric law. Regarding the arrow of time, it stands out
from other macroscopic, time-asymmetric laws for two reasons: it is
universally appliccable and it can be derived from a microscopic description
of the behavior of particles, statistical mechanics. The second law states
that the entropy of an isolated system always increases with time.
The entropy remains constant only if the system is in thermodynamic
equilibrium. Roughly speaking, entropy (S) is a measure of how disordered a
system is; the more disordered, the higher the entropy. Consider, for
example, an isolated box separated in two halves by a membrane with gas in
only one half of the box. If the membrane is removed, the gas will expand
until it is evenly distributed throughout the box. The system is now in a
less ordered state, the entropy has increased, and the process is irreversible
and thus time-asymmetric. But how is this possible, since the gas can be
viewed as a collection of molecules that obey Newton's laws of motion, which
have been shown to be time-symmetric? This is the irreversibility paradox
pointed out by Loschmidt in 1876 (3).

In the original formulation of thermodynamics by Carnot, Clausius, and Kelvin,
macroscopic quantities, such as pressure, temperature, or volume are related
to each other. Boltzmann, building on the kinetic theory of gases developed
by Clausius and Maxwell, tried to explain thermodynamics by basing it on
microscopic quantities, i.e. assuming that a gas consists of atoms obeying the
laws of classical mechanics. The problem with this approach is that the
number of molecules is so large (on the order of 10^23), that it is
impractical to keep track of the motion of every single molecule. Therefore,
average quantities must be used, for example temperature is expressed in terms
of the mean kinetic energy of the molecules, and pressure in terms of the
mean rate of transfer of momentum per unit area and time to the wall of the
container due to the collisions of the molecules with the wall. The trick that
Boltzmann used was to distinguish between the macrostate and the microstate of
a gas. The macrostate describes the gas in terms of macroscopic quantities,
such as the pressure, volume, and the total energy. A microstate is a
specification of the position and momentum of each molecule. This
specification is not exact, however, since all molecules whose position and
momentum are within a certain range are lumped together into one compartment
of the phase space. Phase space is an abstract space whose coordinates are
position and momentum. For a point-like particle this space would be
six-dimensional since three numbers are needed to specify the position, and
three numbers to specify the momentum. Each compartment of this phase space
can be occupied by different numbers of molecules as long as two constraints
are satisfied: the number of molecules and the total energy are constant.
One such arrangement corresponds to a microstate. Some macrostates can be
realized with only one microstate, others can be realized by many different
microstates, and if the gas is in equilibrium, there is one macrostate that
can be realized by a vast number of microstates. Boltzmann assumed that every
microstate was equally probable, so that the probability of a particular
macrostate is proportional to the number of microstates that can realize it.
Using this approach and kinetic theory, he was able to show that if a gas in
thermodynamic equilibrium is in its most probable macrostate, then the
macroscopic and microscopic descriptions of the gas are equivalent. Boltzmann
also showed that the entropy of a gas is proportional to the probability of
the macrostate that it is in. Therefore, a gas in equilibrium has maximum
entropy and homogeneity (inhomogeneous states are realized by fewer
microstates). With an additional assumption, the Stosszahlansatz, or
hypothesis with regard to collision numbers, he was also able to prove that a
gas initially not in equilibrium will approach equilibrium, i.e. the most
probable macrostate, monotonically (4).

At this point it seems as if Boltzmann has been successful in basing the arrow
of time on fundamental laws of physics. However, two disillusioning
objections have been made to this conclusion. First, the objection from
reversibility: A gas approaching equilibrium passes through a succession of
microstates. Since every microstate is equally probable, it follows that for
every microstate that takes the gas closer to equilibrium, there is a
time-reverse of that microstate that will take it further away from
equilibrium. As a result, for every entropy increase, there will be an
entropy decrease, which implies a time-symmetric situation. Second, the
objection from recurrence: According to a theorem of Poincaré, a gas that
evolves into a certain microstate, will again evolve as close to that
microstate as we like, if we wait long enough. Therefore, if the gas starts
out in a microstate far from equilibrium, it's entropy will not only increase
as it approaches equilibrium, but also decrease again as it evolves close to
the initial microstate. This implies that the entropy will not increase
monotonically as required by the second law, but will change almost
periodically, resulting, again, in a time-symmetric situation.

With these objections in mind, it is clear that the Stosszahlansatz has to be
abandoned. Boltzmann's reply to these objectons was to change the approach
from the kinetic theory to a purely probabilistic account. In this new view,
a gas that has been isolated for a very long time, will behave in a
time-symmetric fashion. Most of the time it will be close to equilibrium, but
sometimes it will fluctuate away from equilibrium, and very rarely will it be
far from equilibrium. This still does not look like it is compatible with the
second law. It has been assumed that every microstate is equally probable;
however, this assumes that the system has been isolated for a very long time.
But most systems that we observe have recently interacted with their
environment, resulting in what Reichenbach calls a branch system. In a branch
system not every microstate is equally probable. It is an empirical fact that
most branch systems that we observe are in far from equilibrium states. The
part of the universe that we inhabit happens to be in a low entropy state; why
this is so is an interesting cosmological question. But we do owe our
existence to the fact that the earth absorbs low entropy electro-magnetic
radiation (light) from the sun, and emits high entropy radiation (infrared).
This process leads to an entropy increase of the environment. However, the
expansion of the universe ensures that the energy and entropy density do not
increase; in this way outer space serves as an energy and entropy sink,
preventing us from attaining thermodynamic equilibrium with our environment.
So in a way, the anthropic principle gives us the arrow of time, since
observers can only exist in parts of the universe that are far from
thermodynamic equilibrium, with a prevalence of branch systems initially far
from equilibrium and evolving towards equilibrium.

It should also be mentioned that other attempts have been made to explain the
arrow of time based on Boltzmann's initial approach. Gibbs, for example, used
a technique called coarse-graining, which is based on calculating average
motions of molecules in arbitrarily defined sub-compartments of a system.
This approach does succeed in basing the arrow of time on the classical laws
of motion. The problem with this approach is that the coarseness of the
coarse-graining is arbitrary, but will result in different entropy values for
different size-scales of the sub-compartments, in contradiction with
thermodynamics (3).

These considerations imply that the arrow of time is not to be found in a
time-asymmetry of the basic laws of nature, but is a result of initial
conditions. Even T-invariant equations of physics can have time-asymmetric
solutions depending on the applied initial and boundary conditions. However,
not all is lost. First, it is not known why the universe is so
inhomogeneous. Second, there might be a fundamental law of nature that is
T-variant, but that we have not discovered yet.

According to Penrose, the low entropy of the universe is a result of the
space-time geometry of the big bang. The increase in entropy is then a
consequence of the evolution of the cosmic geometry. According to the general
theory of relativity, gravity affects space-time geometry. The general theory
of relativity is T-invariant; however, the final theory of gravity is expected
to be a quantum mechanical theory. Penrose believes that this kind of theory
will be T-variant (1). Therefore, the cosmic arrow of time could have it's
origin in an as yet undiscovered fundamental law of physics.

Sklar, on the other hand, questions whether there should be a theory of the
direction of time at all (4). Even if there were a fundamental theory of
physics which is time-asymmetric, how would we know that it is the explanation
for the arrow of time? Moreover, some relations must be knowable to us
directly, not inferred from other directly observable relations, therefore, a
reductive analysis of the direction of time might not be possible. If
irreversibility is all that is needed to establish an asymmetry in time, then
the cosmic arrow of time is probably the best explanation for the arrow of
time.






References

1. Vollmer, G. 1988. Die Erkenntnis der Natur. Stuttgart: S. Hirzel Verlag.

2. Horwich, P. 1987. Asymmetries in Time. Cambridge: The MIT Press.

3. Coveney, P. and Highfield, R. 1990. The Arrow of Time. New York: Fawcett Columbine.

4. Sklar, L. 1974. Space, Time, and Spacetime. Berkeley: University of California Press.