The body of observational, experimental, and theoretical knowledge in physics has grown dramatically through the last century. To an outsider examining the current state of physics, the fields appear very deep and specialized. The amount of study needed to attain proficiency in any particular area grows as newer theories build on top of older theories. The limits of current theories have become increasingly apparent, however. A great deal of effort is now being put toward examining the relations between theories that appear to apply to completely separate realms. These relations appear to be of fundamental importance to making progress toward more powerful theories, such as a quantum-mechanical explanation of gravity.

The notion of directionality of time is one that spans the entire range of fields. Evidence of the fundamental directedness of time is seen in properties of the universe as a whole, straight down to the smallest objects for which physical theory exists. There are five basic arrows that summarize our knowledge of time. The time asymmetries we observe appear to be hierarchically related, but the structure of this hierarchy is not universally agreed upon.

Of the various arrows of time physicists have discovered, the thermodynamic arrow in particular is most directly relevant to our perception of time. It guides the physical processes that play such an important role in our lives. Without it, all of the air around you could suddenly condense off in a corner; all of the heat in your body could suddenly flow into the ground. The thermodynamic arrow prevents such things from happening in such a law-like manner that it is reasonable to suspect that this arrow should turn out to be a necessary result of simpler laws of physics. Using only Newtonian mechanics and two unavoidable axioms, the thermodynamic arrow can indeed be derived.

This is not the end of the story, however, as Newtonian mechanics is known to be a fundamentally incomplete description of systems of microscopic particles. Quantum mechanics developed over the course of the last century to replace Newton's ideas of how particles behave. This new framework suffers its own problems. It suggests that nature has some intrinsic randomness to it, and seems to break down whenever measurements are made. Several solutions have been proposed to remedy these problems, many of which add nothing to our scientific knowledge while at the same time carrying a tremendous amount of philosophical baggage. One particular theory, however, allows us to simplify the assumptions necessary to construct a thermodynamic arrow of time, while not introducing a terrible philosophical burden.

The Arrows of Time

The most obvious arrow is that of subjective experience. As conscious beings, we seem to have some a priori knowledge of time. For conscious thought to occur, a mind must be able to take on a variety of different states. Simplistically, we can say that time is the ability to move between these states. But we know empirically that there is some ordering to these states; thought does not seem to move arbitrarily. This ordering reflects the two basic aspects of the subjective arrow of time. First, it is apparent that our ability to remember divides mental states into two very different categories, which we label "future" and "past"; we remember the past but do not remember the future. Second is the more extrinsic, experiential notion of intervention. Being able to choose between courses of action is one of the basic traits of conscious life, but it is somehow intuitively clear that this ability is only applicable to the future. Taking these properties as defining the subjective arrow place it beyond the realm of traditional scientific inquiry, since hypothesizing, experimenting, and observing are processes we have built on top of it. Nonetheless, there is still some reason to believe that the subjective arrow is the least fundamental arrow. Consciousness and memory may simply be ways of making sense of more concrete, scientifically justified arrows of time.

The thermodynamic arrow of time is somewhat less obvious, but still fundamental to our perception of time. Its presence is definitely not available a priori, nor is it inaccessible to science - properties that the remaining three arrows share as well. Colloquially, the thermodynamic arrow is the law-like tendency for isolated systems to become more disordered. Directedness of the arrow implies we should not expect to find systems that, left to themselves, become more orderly over time. Examples abound in nature. The scent of a candle spreads throughout a room instead of concentrating in a corner. A ball rolls and bounces down the street, eventually coming to a rest as it disperses its energy into the pavement, but won't ever jump back up off the ground once it's stopped. The thermodynamic arrow is the physical arrow most directly related to everyday experience. In some sense, our bodies are complicated mechanisms for keeping themselves as organized as possible. Learning and memory are thought to be processes that create specific, orderly relations between neurons in the brain. It is perhaps not surprising that our subjective experience of time would then be a consequence of thermodynamic principles.

The current formulation of electromagnetism allows for two types of solutions: retarded waves, having positive energy and propagating forward in time, and advanced waves, having negative energy and propagating backward in time. The electromagnetic arrow (or radiation arrow) is an asymmetry between observations of retarded and advanced waves. In fact, we do not observe experimentally any advanced waves whatsoever, whereas retarded waves are ubiquitous. Imagine dropping a pebble into a still pond. A series of circular waves will radiate out from where the pebble hit the water, eventually becoming disorderly as they reflect off the edges of the pond. If there were no asymmetry in this situation, we would expect to occasionally see unorganized waves in a pond spontaneously converge, perfectly in phase, and throw drops of water up into the air. In the electromagnetic case, we never see an atom spontaneously gain energy from a series of waves converging on it from all directions. The relationship between the electromagnetic arrow and the thermodynamic arrow has not been determined conclusively. The generally accepted view is that because particle interactions involved in thermodynamics occur at such low (non-relativistic) speeds and small distances, the distinction between retarded and advanced waves is irrelevant. An opposing view is that current thermodynamic theory (specifically Boltzmann's H-theorem, which states roughly that energies of particles in large systems approach a particular bell-shaped distribution) makes implicit time-asymmetric assumptions that can be explained by assuming only retarded waves are involved in electromagnetic interactions.

Cosmologists have found a very striking time asymmetry in the universe as a whole. Edwin Hubble discovered in the 1920's that not only are there many galaxies in the universe, but that there is a particular correlation between the distance of a galaxy from us and its redshift, an indicator of its speed relative to us. The particular correlation, that distance is proportional to redshift, implies that the universe is expanding. Extrapolating this back in time suggests that the universe was once much smaller, and in fact there is very strong evidence that the universe expanded to its current state from an almost unimaginably small, dense point. As the universe expanded, the average energy density decreased, and the universe is thought to have passed through a number of phase transitions (much like a gas condensing into a liquid as it cools) during which the various forces differentiated. The cooling of the universe and the decoupling of forces allowed matter to form and clump together, which in turn allowed for significant deviations from thermal equilibrium. Clearly these conditions are necessary for the electromagnetic and thermodynamic arrows to manifest themselves, regardless of which is more fundamental.

The most recently discovered time asymmetry is perhaps the least well understood. Underlying particle physics are the notions of conjugation and parity symmetry, in addition to time symmetry. The decay of the neutral kaon had been known for some time to violate conjugation and parity symmetry separately, but it was thought that the decay still obeyed the combination of the two. However, in 1964 researchers discovered that a small percentage of the time, the decay did not obey the combined symmetry. This discovery had important implications. It meant that the time-reversed decay process would behave slightly differently, giving rise to the kaon arrow of time. Further analysis and experimentation revealed that neutral kaon decay also favored matter over antimatter, and some believe this could explain the tremendous imbalance in favor of matter that we observe in the universe today. It is suspected that this imbalance could account for the cosmological arrow of time.

Particles, Statistics, and Experience

The directedness of time appears on all scales, from the cosmological to the submicroscopic, but the thermodynamic arrow remains the most relevant to ordinary experience. Even though there is some hope of explaining the thermodynamic arrow in terms of the electromagnetic arrow or the cosmological arrow, it's not obvious that the thermodynamic arrow should exist at all. It is nonetheless possible to derive a thermodynamic arrow given only three major axioms. First, we need a set of rules (Newtonian mechanics, for now) for describing rigorously the motions of particles in a system. Second, we need a methodology (the statistical postulate) that allows us to consider the very large systems of particles that the thermodynamic arrow seems to direct. Third, we need an axiom (the past hypothesis) to provide directedness to the first two.

In the Newtonian picture of the world, any system (large or small) can be described completely at an instant in time by listing the positions and velocities of every particle in the system. Particle systems can be plotted as points in phase space, a space with one dimension for each component of position and velocity for each particle in the system. Combined with knowledge of how properties of the particles determine the forces between them, we have a method of calculating how the system changes with time: let the particles proceed with their given velocities for a short amount of time, compute how the velocities are affected by the known forces, and repeat. Obviously this is an approximation, but in the Newtonian framework, this approximation can be made arbitrarily accurate by simply decreasing the time step. The time evolution of a system of particles can be plotted as a curve in phase space. The deterministic nature of Newtonian mechanics implies that these curves in phase space can never cross each other. (At a point of intersection, the particle described by that point would have two distinct futures and two distinct pasts.)

Considering that any physical process can be represented as a series of instantaneous descriptions of a set of particles, we can now try to determine what the time-reverse of the process should look like. Clearly we cannot simply reverse the series, as this would describe a system in which particles moved in reverse while their velocities continued to point forward. Naturally, we would then reverse the velocities as well as the series itself. In general, in any system of particles, we need to transform the properties of particles in the system as well as the series of snapshots in order to generate a time-reversed picture of the process. The apparent trouble with this reversal is that we will apply precisely the same algorithm for computing the reverse of a process as we do the forward process. If our instantaneous descriptions and algorithm were all we needed to describe the world, we would expect to see things happen in reverse as often as they happen forward, which is manifestly not the case on the macroscopic scale.

To gain some more insight into why Newtonian mechanics fails to generate a realistic picture of the world, we need a systematic way of describing familiar macroscopic processes. With the vast number of particles involved in such processes, it becomes very difficult to capture the instantaneous state of an entire system, and the algorithm for determining the system's evolution becomes unwieldy. Physicists have nonetheless found in large systems of particles empirically law-like behaviors, resulting in the remarkably successful science of thermodynamics and statistical mechanics. In the statistical framework, the Newtonian instantaneous description of a physical system is called its microcondition. The macrocondition of a system is characterized by familiar large-scale properties such as temperature, volume, and pressure. In general, any particular macrocondition of a system corresponds to many different microconditions. For example, in a gas held at a constant temperature and volume, the individual particles will be moving constantly. So at each time, the gas will be in a different microcondition, but the macrocondition of having the same temperature and volume will not have changed. The notion of entropy is central to the directedness of thermodynamic processes. The entropy of a system is defined as a constant times the logarithm of the volume of phase space occupied by the microconditions compatible with the system's macrocondition. Roughly speaking, entropy is a measure of how many ways a system could be organized and still have the same macroscopic appearance. Macroscopic observation picks out a smaller set of microconditions for a low-entropy system than it does for a high-entropy system, so in a sense, entropy is an indicator of how disorganized a system is. In terms of entropy, the thermodynamic arrow of time is equivalent to the observation that the entropy in a closed system does not decrease as time progresses.

Now consider some thermodynamic system and the total possible phase space of it, in any possible macroscopic state. It happens that it can be shown mathematically that one particular state corresponds to an overwhelmingly large volume of phase space compared to other states. In this particular state, thermal equilibrium, the particles in the system are uniformly distributed and have a bell-shaped velocity distribution. Now suppose we know that the system is in some particular macrostate that does not correspond to thermal equilibrium. Without knowledge of the particular microstate of the system, we assume it is equally likely to be in any of the microstates that correspond to the known macrostate. Letting some time pass, the system will progress to some other microstate as would be determined by the Newtonian algorithm if we knew precisely the initial microstate. Not knowing the initial microstate, though, we can only make the probabilistic deduction that the system will end up in one of the vastly more numerous, more uniformly distributed microstates.

Unfortunately, such a probabilistic deduction is not as innocent as it may sound. Newtonian mechanics allows for an uncountably infinity of possible microstates for a system. The only mathematically meaningful probability that can then be assigned to any particular microstate is precisely zero, which is rather blatantly at odds with the assumption that is in fact in some particular microstate. A simple solution is to divide the phase space up into finite (that is non-infinitesimal) blocks and assign a probability to each particular block of phase space that is proportional to the mathematical measure of the points in the block. This partitioning of phase space gives us useful probabilities, but highlights an even more fundamental problem with no clear solution. From the perspective of measure theory, there are an infinite number of ways to assign a consistent type of measure to a set of points. We have chosen arbitrarily to use the standard measure that we commonly apply to describe lengths of string and volumes of containers. The statistical postulate, then, is that the correct way to assign probabilities is to spread the probability uniformly over regions of microconditions, measured in the standard way, that correspond to the known macrocondition of the system.

Only one hole remains in this derivation of thermodynamic directedness. The time symmetry of Newtonian mechanics requires that entropy should increase in both temporal directions in precisely the same law-like manner just described. If this were indeed the case, any system is overwhelmingly likely to have come to its current state from a higher-entropy state, just as we expect it to proceed toward a higher-entropy future. An ice cube in a glass of water must have spontaneously frozen out of the liquid, just as it melts as time goes forward. A term paper is much more likely to have gradually fallen together from particles in the air (as it decomposes into dust toward the future) than to have been organized in some low-entropy past. Allowing higher-entropy past states for isolated systems simply does not allow us to say anything useful about the past (invalidating our belief in any kind of physical laws). Instead, we hypothesize that in one temporal direction (which we call the past), the entropy of isolated systems was lower. If the entropy is assumed to be lower in the past, systems evolved from that state to their current state through an entropy-increasing path and will continue to increase in entropy toward the future, giving us a definite thermodynamic arrow of time. Sufficient cosmological evidence exists to make the past hypothesis quite credible, as applied to the universe as a whole (being itself a very large, isolated thermodynamic system).

The Quantum Interpretation

Although Newtonian mechanics gives us a simple framework in which to derive thermodynamic time asymmetry, it has been known to be incorrect for a century. In particular, the notion that every particle in a system has a definite, theoretically knowable intrinsic properties (mass, charge, and such), position, and velocity is quite wrong. The modern theoretical framework that replaces Newtonian mechanics on the microscopic scale is quantum mechanics. In order to specify the instantaneous physical state of a system, the Newtonian concepts of position and momentum are replaced with a quantum mechanical wave function.

There are two major dissatisfying aspects of quantum mechanics. The first is that it is fundamentally probabilistic. It is possible to set up experiments in which the outcome is one of two possibilities, each of which can occur with equal probability, but for which it is impossible to determine the outcome in advance. Certainly many macroscopic events can be analyzed quite well probabilistically (weather, horse races, etc.). The key difference is that with macroscopic events, our probabilistic treatment of the situation arises from admitting incomplete knowledge of the system being studied. In the quantum mechanical context, this is not the case. The wave function specifies precisely all there is to know about the system, but somehow does not specify the outcome of the experiment, only the probabilities of various possible outcomes. The other troubling aspect of quantum mechanics is the measurement problem. When a particle described by a wave function can exist in any of a number of states, the quantum mechanical description allows the particle to also be in a superposition of states. This is quite at odds with macroscopic experience. A coin toss will come out heads or tails with roughly equal probability, but the outcome of one toss will certainly never be half heads and half tails. The wave function description of particles only applies up to the point at which you attempt to measure some property of the particle and turn the measurement into a macroscopic outcome. At this point, the wave function, which had previously specified the probability of observing a particle in a particular state at a particular time, suddenly collapses. All of the possible outcomes of the experiment disappear except one, the macroscopically observed outcome. After that point, a new wave function will then propagate parameterized by the measurement inflicted upon the particle.

There have been many attempts to clear up the awkwardness in quantum mechanics, of which quite a few lead to no testable hypotheses without simplifying our knowledge of physics at all (and often introducing a great deal of philosophical problems along the way). One theory, due to Ghirardi, Rimini, and Weber, appears to simplify matters significantly with a minimum of outright speculation. The GRW theory is based on the notion of spontaneous localization, originally proposed by Pearle. The theory supposes that wave functions are occasionally "hit" by Gaussian envelopes that cause the wave to localize tightly in a specific region of space. For the wave function of any one particle, these hits occur with a fixed probability over time, somewhere on the order of once in a billion years. Additionally, the probability that a hit centers on a certain location is given by the wave function. In small, isolated systems, the infrequency of these hits makes it unlikely a hit will ever be detected. On the macroscopic scale - the scale at which we are capable of observing and measuring - the tremendous number of wave functions involved will make it overwhelmingly likely that superpositions of macroscopically different states (heads or tails, meter reads "-1" or "+1", etc.) will collapse into purely one state or the other.

The real merit of the GRW theory is in its applicability to the thermodynamic arrow of time. Consider the possible microconditions for a given thermodynamic system with non-maximal entropy. We know that as time progresses, the vast majority of such microconditions will move to higher-entropy states. That is, the vast majority of trajectories through phase space that start at a point within the region of the system's known macrocondition later pass through a point in a region of phase space corresponding to a higher-entropy macrocondition. The persisting problem with this picture is that we have no reason, short of a statistical postulate, to believe that the system is not, in fact, on one of the few trajectories that correspond to the system staying in a low-entropy region or moving to a lower-entropy region. It can be shown mathematically that such trajectories are extremely sparse, in the same sense that rational numbers are very sparse in the continuum of real numbers. Due to this distribution, any decreasing-entropy "abnormal" trajectory is embedded in a surrounding bundle of increasing-entropy "normal" trajectories. Any minute disturbance in a particle's motion along an abnormal path would be guaranteed to knock it onto a normal trajectory. Of course in the Newtonian framework, such disturbances can't occur, but it turns out that the hits involved in the GRW theory are precisely the sort of disturbances needed. Substituting GRW in for Newtonian mechanics allows the removal of the statistical postulate without destroying the thermodynamic arrow of time! The statistical nature of thermodynamics is captured entirely by the GRW theory of dynamics, and is purely a result of the statistical nature of the hits.

The necessity of a past hypothesis is still present in the GRW derivation of the thermodynamic arrow. The hits supposed by GRW are only presumed to happen probabilistically in the future evolution of a wave function. No claim is made whatsoever about the probability of a hit having affected a wave function in the past. Therefore, there is still a requirement to hypothesize that the wave functions in fact have a past, and the same evidence is available to suggest that the universe had a low-entropy past.

Physicists have made a tremendous amount of progress toward understanding the nature of time in the last century. Their work spans all scales of human knowledge, from the dynamics of the universe to the motions of the smallest particles. At all scales, we see evidence of asymmetry in time, and the relations between asymmetries at different scales are not known with certainty. The success of the GRW theory in explaining thermodynamics may be an important step in solidifying our understanding of time's directedness.