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.