Why does time run forward and never backward?

Why is it you can break an egg, but not make the pieces spring back together again? To find out, we have to go back to the birth of the universe.

There's egg on your face, literally. You tried to juggle some eggs, it all went wrong, and now you've got to shower and change your clothes.
Wouldn't it be faster to just un-break the egg? Breaking it only took a few seconds, so why not do that again, but in reverse? Just reassemble the shell and throw the yolk and the white back inside. You'd have a clean face, clean clothes, and no yolk in your hair, like nothing ever happened.
Sounds ridiculous — but why? Why, exactly, is it impossible to un-break an egg?
It isn't. There's no fundamental law of nature that prevents us from un-breaking eggs. In fact, physics says that any event in our day-to-day lives could happen in reverse, at any time. So why can't we un-break eggs, or un-burn matches, or even un-sprain an ankle? Why don't things happen in reverse all the time? Why does the future look different from the past at all?
It sounds like a simple question. But to answer it, we've got to go back to the birth of the universe, down to the atomic realm, and out to the frontiers of physics.
Like many stories about physics, this one starts with Isaac Newton. In 1666, an outbreak of bubonic plague forced him to leave the University of Cambridge, and move back in with his mother in the Lincolnshire countryside. Bored and isolated, Newton threw himself into the study of physics.
He came up with three laws of motion, including the famous maxim that every action has an equal and opposite reaction. He also devised an explanation of how gravity works.
Newton's laws are astonishingly successful at describing the world. They explain why apples fall from trees and why the Earth orbits the Sun. But they have an odd feature: they work just as well backwards as forwards. If an egg can break, then Newton's laws say it can un-break.
This is obviously wrong, but nearly every theory that physicists have discovered since Newton has the same problem. The laws of physics simply don't care whether time runs forwards or backwards, any more than they care about whether you're left-handed or right-handed.
But we certainly do. In our experience, time has an arrow, always pointing into the future. "You might mix up east and west, but you would not mix up yesterday and tomorrow," says Sean Carroll, a physicist at the California Institute of Technology in Pasadena. "But the fundamental laws of physics don't distinguish between past and future."
The first person to seriously tackle this problem was an Austrian physicist named Ludwig Boltzmann, who lived in the late 19th century. At this time, many ideas that are now known to be true were still up for debate. In particular, physicists were not convinced – as they are today - that everything is made up of tiny particles called atoms. The idea of atoms, according to many physicists, was simply impossible to test.
Boltzmann was convinced that atoms really did exist. So he set out to use this idea to explain all sorts of everyday stuff, such as the glow of a fire, how our lungs work, and why blowing on tea cools it down. He thought he could make sense of all these things using the concept of atoms.
A few physicists were impressed with Boltzmann's work, but most dismissed it. Before long he was ostracised by the physics community for his ideas.
He got into particularly hot water because of his ideas about the nature of heat. This may not sound like it has much to do with the nature of time, but Boltzmann would show that the two things were closely linked.
At the time, physicists had come up with a theory called thermodynamics, which describes how heat behaves. For instance, thermodynamics describes how a refrigerator can keep food cold on a hot day.
Boltzmann's opponents thought that heat couldn't be explained in terms of anything else. They said that heat was just heat.
Boltzmann set out to prove them wrong. He thought heat was caused by the random motion of atoms, and that all of thermodynamics could be explained in those terms. He was absolutely right, but he would spend the rest of his life struggling to convince others.
Boltzmann started by trying to explain something strange: "entropy". According to thermodynamics, every object in the world has a certain amount of entropy associated with it, and whenever anything happens to it, the amount of entropy increases. For instance, if you put ice cubes into a glass of water and let them melt, the entropy inside the glass goes up.
Rising entropy is unlike anything else in physics: a process that has to go in one direction. But nobody knew why entropy always increased.
Once again, Boltzmann's colleagues argued that it wasn't possible to explain why entropy always went up. It just did. And again, Boltzmann was unsatisfied, and went searching for a deeper meaning. The result was a radical new understanding of entropy — a discovery so important that he had it engraved on his tombstone.
Boltzmann found that entropy measured the number of ways atoms, and the energy they carry, can be arranged. When entropy increases, it's because the atoms are getting more jumbled up.
According to Boltzmann, this is why ice melts in water. When water is liquid, there are far more ways for the water molecules to arrange themselves, and far more ways for the heat energy to be shared among those molecules, than when the water is solid. There are simply so many ways for the ice to melt, and relatively few ways for it to stay solid, that it's overwhelmingly likely the ice will eventually melt.