system
Given the following post, respond as if you're commenting on Reddit:
human
Does anyone have a good, intuitive explanation of what the word entropy means?

I know that it's related to the amount of information needed to define a system, but I can't find a satisfying definition.

From Wikipedia: "...entropy is defined as the amount of additional information needed to specify the exact physical state of a system, given its thermodynamic specification."

What does it mean by 'given its thermodynamic specification'? What is entropy and why is it higher in more disordered systems?
gpt
This is the purely thermodynamic point of view :

'Given its thermodynamics' implies that quantities like pressure, volume and total internal energy of the system are known. However, this does not constitute complete knowledge of the system because thermodynamic systems are generally composed of 10^23 atoms or so and there are many combinations of the position and velocities of these atoms that will give you the same set of thermodynamics variables.

This gives you some freedom in choosing the exact state of a system given the thermodynamic variables, because you are free to play around with a huge set of variables. Alternatively, given only the thermodynamic variables, you are ignorant of the states of this vast number of atoms, and entropy is a measure of this ignorance.

For example : if you specify a very low temperature and pressure, the range of velocities you can assign to the atoms is constrained and you end up with fewer variables to choose from, leaving you with a low entropy. At larger temperatures, you have a lot more variables to play with, implying larger entropy.

This way of saying this is flawed though, because entropy is a more basic quantity than temperature. You first measure the freedom you have in assigning non-thermodynamic variables and use that a measure of temperature.

*Why is it higher in more disordered systems ?*

Imagine you have a barrel filled with 10^23 red balls and a similarly large number of serial numbered boxes. Given the constraint that on an average, each box contains three balls, how many ways are there of filling up the boxes ? Lets call this number N1. 

Assume now that you add a bunch of blue balls to the mix. You end up with a larger number of distinct ball configurations that satisfies the three balls on an average criterion. This is because you can do things like create new configurations by swapping a blue and a red ball between two boxes.

Disorder is any lack of uniformity. Decreasing uniformity increases the number of possible configurations. Simple example :

* Lets say you have two identical electrons and you know the velocities are v1 and v2. All you can do is assign v1 to one and v2 to the other and be done with it, just 1 configuration possible. 

* You now discover that they belong to atoms located on opposite sides of a wall. One of them on the right (R) and the other on the left (L). This lets you have two configurations: (L:v1,R:v2) and R:v1,L:v2).

* The first part is equal to both electrons being on the same side of the wall. Making them have distinct positions 'creates disorder' and opens up more configurations. More configurations = more ignorance. More ignorance = more entropy.

*Why does more configurations = more entropy ?*

Say you have two systems that you are totally ignorant about, one has 10 and the other has 100 configurations. If someone asked you at gunpoint to predict the configurations that they are in, the best you can do is guess. The probability that your guess works is 1/10 in the first case and 1/100 in the second. Obviously, the first guess is 'better' than the second because its more likely to be correct.

Entropy is a measure of how bad our ability is guess correctly the state of a unknown system. The larger the set of configurations of the system, the worse our guess will be.