The roots of this issue go back at least to 1865, when Rudolf Clausius coined the term "entropy" and stated that the entropy of the universe tends to a maximum. This idea is now known as the second law of thermodynamics, which is most often described by saying that the entropy of an isolated system always increases or stays constant, but never decreases. Isolated systems tend to evolve toward the state of maximum entropy, the state of thermodynamic equilibrium. Even though entropy will play a crucial role in this discussion, it will suffice to use a fairly crude definition: entropy is a measure of the "disorder" of the physical system. In terms of the underlying quantum description, entropy is a measure of the number of quantum states that correspond to a given description in terms of macroscopic variables, such as temperature, volume, and density.
The classic example is a gas in a closed box. If we start with all the gas molecules in a corner of the box, we can imagine watching what happens next. The gas molecules will fill the box, increasing the entropy to the maximum. But it never goes the other way: if the gas molecules fill the box, we will never see them spontaneously collect into one corner.
This behavior seems very natural, but it is hard to reconcile with our understanding of the underlying laws of physics. The gas makes a huge distinction between the past and the future, always evolving toward larger entropy in the future. This one-way behavior of matter in bulk is called the "arrow of time." Nonetheless, the microscopic laws that describe collisions of molecules are time-symmetric, making no distinction between past and future.
Any movie of a collision could be played backwards, and it would also show a valid picture of a collision. (To account for some very rare events discovered by particle physicists, the movie is only guaranteed to be valid if it is also reflected in a mirror and has every particle relabeled as the corresponding antiparticle. But these complications do not change the key issue.)
There is an important problem, therefore, which is over a century old, to understand how the arrow of time could possibly arise from time-symmetric laws of evolution.
The arrow-of-time mystery has driven physicists to seek possible causes within the laws of physics that we observe, but to no avail. The laws make no distinction between the past and the future. Physicists have understood, however, that a low entropy state is always likely to evolve into a higher entropy state, simply because there are many more states of higher entropy. Thus, the entropy today is higher than the entropy yesterday, because yesterday the universe was in a low entropy state. And it was in a low entropy state yesterday, because the day before it was in an even lower entropy state. The traditional understanding follows this pattern back to the origin of the universe, attributing the arrow of time to some not well-understood property of cosmic initial conditions, which created the universe in a special low entropy state. As Brian Greene wrote in The Fabric of the Cosmos:
"The ultimate source of order, of low entropy, must be the big bang itself. ... The egg splatters rather than unsplatters because it is carrying forward the drive toward higher entropy that was initiated by the extraordinarily low entropy state with which the universe began."
Based on an elaboration of a 2004 proposal by Sean Carroll and Jennifer Chen, there is a possibility of a new solution to the age-old problem of the arrow of time. This work, by Sean Carroll, Chien-Yao Tseng, and me, is still in the realm of speculation, and has not yet been vetted by the scientific community.
But it seems to provide a very attractive alternative to the standard picture. The standard picture holds that the initial conditions for the universe must have produced a special, low entropy state, because it is needed to explain the arrow of time. (No such assumption is applied to the final state, so the arrow of time is introduced through a time-asymmetric condition.) We argue, to the contrary, that the arrow of time can be explained without assuming a special initial state, so there is no longer any motivation for the hypothesis that the universe began in a state of extraordinarily low entropy. The most attractive feature is that there is no longer a need to introduce any assumptions that violate the time symmetry of the known laws of physics.
The basic idea is simple. We don't really know if the maximum possible entropy for the universe is finite or infinite, so let's assume that it is infinite. Then, no matter what entropy the universe started with, the entropy would have been low compared to its maximum. That is all that is needed to explain why the entropy has been rising ever since!
The metaphor of the gas in a box is replaced by a gas with no box. In the context of what physicists call a "toy model," meant to illustrate a basic principle without trying to be otherwise realistic, we can imagine choosing, in a random and time-symmetric way, an initial state for a gas composed of some finite number of noninteracting particles. It is important here that any well-defined state will have a finite value for the entropy, and a finite value for the maximum distance of any particle from the origin of our coordinate system. If such a system is followed into the future, the particles might move inward or outward for some finite time, but ultimately the inward-moving particles will pass the central region and will start moving outward. All particles will eventually be moving outward, and the gas will continue indefinitely to expand into the infinite space, with the entropy rising without limit. An arrow of time—the steady growth of entropy with time—has been generated, without introducing any time-asymmetric assumptions.
An interesting feature of this picture is that the universe need not have a beginning, but could be continued from where we started in both directions of time. Since the laws of evolution and the initial state are time-symmetric, the past will be statistically equivalent to the future. Observers in the deep past will see the arrow of time in the opposite direction from ours, but their experience will be no different from ours.