# Does entropy increase with improbability?

Well, maybe that's a simple question, but it has been pissing me off for some time ... Does entropy increase with improbability?

Nugatory
Mentor
What is the definition of entropy (the real one that has an equation in it, not the vague English-language bit about "disorder")? The answer to that question will suggest an answer to your question.

Does entropy increase with improbability?
Entropy is defined for all systems (whether or not the are at equilibrium) by the Gibbs entropy equation
$$S=-k\sum_{i}^{l}p_{i}\ln p_{i}$$
where ##p_{i}## is the probability for the system to enter the ##i##th state. Notice that the entropy has a minimum value of zero for the delta function distribution ##p_{i}=\delta_{ij}## and a maximum value for the uniform distribution ##p_{i}=1/l##. Thus the entropy is how spread out the probability distribution ##p_{i}## is. A system which has many equal probability states is at higher entropy than a system with only a few probable states.