The direction of time is something most of us take for granted because it is so central and consistent in all of our experiences, but why does time seem to flow from past to future? Why doesn’t our universe allow experiencing everything at once? Why can’t we reverse the flow of time and go into the past?

Well the answer to these riddles lies in a surprisingly simple statement known as the 2nd law of thermodynamics:

In a closed system, heat flows from hot to cold.

Yup, that’s it. This statement is pretty clear in every day life, if you drop an ice cube in hot water, the ice cube will melt and cool the water. What we never experience is the heat from the ice cube flowing into the water, which would make the ice cube grow and cause the rest of the water to get hotter. Of course adding work to a system can cause the 2nd law to locally go in reverse. For example a freezer causes an ice cube to grow, but it does so by heating the the air outside the freezer. The system as a whole still heats up.

He’s a chart of some equivalent ways to describe the 2nd law in other domains:

Domain |
Statement of the 2^{nd} law |
Example “special” state |
Example equilibrium state |

Heat | Heat flows from hot to cold | Ice cube in hot water | Water all the same temperature |

Entropy | Disorder increases | A person | Ashes |

Information | Information is lost | Beethoven’s 5th Symphony | Static noise |

The first thing to notice, is that these “special” states can be considered start states, while the equilibrium states can be considered end states. That is, if some how the system is put into a special state, it will after a series of random and natural processes end in the equilibrium state.

So why does the universe work this way, and what does this have to do with the flow of time? Let’s look at the ice cube in hot water example in more detail because it is the most tangible of the above examples, but the other examples can be explained in a similar fashion.

**Why heat flows from hot to cold**

As heat flows from hot to cold, the system will eventually reach a equilibrium, where everything is the same temperature. At that point there is no more heat flow. Dropping an ice cube in a cup of warm water will eventually lead to a cup of water which is all the same temperature somewhere in between the 2 starting temperatures.

If you think about all of the water molecules in the cup, let’s say for this example there are 2 billions molecules (1 billion hot and 1 billion cold), we can count all the ways in which these molecules can be distributed. There are total possible states in this system. These state spaces are so large it’s impossible to even begin to imagine their enormity. These make other “large” numbers like the number of elementary particles in the universe () and 1 googol seem minuscule in comparison. The vast majority of these states are equilibrium states, while the special starting states are insignificant.

If you pick a state at random from the total space of water molecules in a cup, the state where some of the molecules are ice and some are hot, is effectively 0. (To do a quick calculation try a binomial distribution with something like Binomial(2 billion, 2 million) = .)

**Maxwell’s Demon**

Many physicists thought the 2nd law was purely statistical and could be easily be violated by intelligent manipulation of states. One example of this is called Maxwell’s demon, where a theoretical “demon” reverses the flow of heat without using any work. However, physicists have shown that no matter how hard you try, the system as a whole cannot violate the 2nd law. For example Maxwell’s demon must either conduct work, which will causes heat in another part of the system, or lose information, which we learned is equivalent to the 2nd law. So the 2nd law survives even with small state spaces, however, the huge numbers of states in everyday molecular interactions is what makes time so indomitable.

**Time**

So the answer to why heat flows from hot to cold is essential a question of probability. The equilibrium states dominate the special states, so unless you start in one of the special states you will never reach it. The same can be said about our universe. The Big Bang is the ultimate start state and Heat Death is the ultimate end state. The big bang represents time = 0 while heat death represents the end of time.

The flow of time from past to future can therefore be defined as special states of a system transitioning to their equilibrium. The sheer magnitude of the state space is what makes time so dominant in the Universe.

Tags: entropy, heat, Maxwell's Demon, time

July 31, 2011 at 10:17 am

It’s an interesting to consider the thermodynamic perspective but taking it as an answer to why time flows forward is of course circular reasoning. Recognizing that entropy increases with time presumes a timeline going from past to future.

September 8, 2011 at 12:44 am

Yeah i was trying to understand this myself, but I think the difference is that you can take any 2 entropy measurements (where the entropies are not equal and at unknown times), and you can always define a direction of time between them (from low entropy to high). However if you take any 2 measurements of time (at 2 different times and unknown entropy), you cannot always define a direction of increasing entropy, because at the boundaries entropy does not change (e.g. thermal equilibrium). So because of these boundary cases, entropy is asymmetric while time is still inherently symmetric.

April 17, 2012 at 6:09 am

you must take 1 time to measure

October 27, 2012 at 5:41 pm

I’m not sure if this is true, but that’s fine either way. The time of each measurement is arbitrary. What’s important that you can make two distinct measurements. In the first case, the times are unknown, yet you will be able calculate which measurement came first by the entropy measurements. In the second case the time of the measurements is known, but you will not always be able to tell which entropy level is higher.