Does entropy increase with improbability?

  • #1
Well, maybe that's a simple question, but it has been pissing me off for some time ... Does entropy increase with improbability?
 

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  • #2
Nugatory
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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.
 
  • #3
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.
 

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