I set out to write a post about what information  and entropy are.  Let's see if it comes out  
  
A cup of coffee is made up of a huge number of molecules.  Most of them are water, but there are also essences, cellulose and sugar.  If we wanted to fully describe what happens in the cup, we would have to know the position and speed of each of these molecules.  
  
![](https://cdn.pixabay.com/photo/2017/08/06/07/09/coffee-2589759_960_720.jpg)  
([CC0](https://pixabay.com/es/caf%C3%A9-cafe-caliente-taza-copa-2589759/))
  
That knowledge would give us the _physical state_ of the cup of coffee.  Of course, the positions and velocities of each molecule change very quickly.  The cup goes from one state to another in a quick succession.  Some states are repeated after a short time, others almost never.  When the molecules return to approximately the same positions and velocities very often, this is a _likely_ state.  When, on the other hand, they almost never have those positions and velocities, the state is _unlikely_.    
  
For example, states where the cream is not mixed with coffee are very unlikely.  The system might eventually spontaneously return to one of them, the cream separating itself from the coffee, but it would take a very long time.  On the other hand, the states in with the cream is dissolved into the whole cup are much more probable, the system goes back to them assiduously.  
  
If we took the trouble to look at the positions and velocities of each of the coffee molecules at any given moment, how much would we learn? Is there any way to quantify the _amount of information_ that we would gain if we knew the state completely?  If we found a very probable state, we would not learn anything we could not have imagined beforehand.  We wouldn't be surprised to find the cream dissolved in coffee.  If instead we found a very unlikely state, we would learn a lot.  If the cream was separated, surprise!   
  
That is to say that the information we gain by looking one by one at the positions and velocities of the coffee molecules, is greater the less frequent those positions and velocities are, or in other words, the lower their probability is.  Information grows when probability decreases.    
 
This idea of quantifying the information we gain by inspecting a given object in detail, forms the basis of the _Information Theory_ and is extremely useful in physics.  So much so that there is the so-called _Wheeler program_ that tries to reformulate all physics based on information.   
  
We could look in detail at the molecules of coffee repeatedly.  Sometimes we will learn a lot, gaining much information.  Other times we will find a very probable state, and gaining little information.  A natural question is how much information would we gain *on average*?  The information we gain on average by repeatedly looking at the system in detail is a good measure of how much could we expect to learn the next time we do.  In other words, it tells us how much we don't know _a priori_ about what happens there.  That average information is called _entropy_.  
  
With this definition of entropy as information gained on average, the second law of thermodynamics becomes easy to understand:  

> _Entropy never decreases._

Of course not: we would learn less information by looking at a system only if we had gained knowledge _a priori_ about it *without having looked at it*!  Moreover: for entropy to increase, we have to somehow lose information.  This would only be possible if, for example, we were able to see coffee molecules, but not water molecules.  The information that visible molecules lose gets hidden into invisible molecules.  
  
Entropy and the theory of information are today a very active field of research in theoretical physics worldwide.  
 
So now you have something to think about while you stir your coffee in the morning.  

  
![](https://cdn.pixabay.com/photo/2017/05/30/22/24/coffee-2358388_960_720.jpg)  
([CC0](https://pixabay.com/es/caf%C3%A9-taza-de-caf%C3%A9-caf%C3%A9-caliente-2358388/))