Entropy, Maximum Entropy Principle
Andrius: I'm trying to understand how energy relates to entropy. I'm taking a combinatorial view and considering the Boltzmann distribution.
Energy
{$p_i = \frac{e^{\frac{-\epsilon_i}{k_BT}}}{\sum_i e^{\frac{-\epsilon_i}{k_BT}}}$}
{$p_i \propto e^{\frac{-\epsilon_i}{k_BT}}$}
where {$p_i$} is the probability that a system is in state {$i$} with energy {$\epsilon_i$} and temperature {$T$}, and {$k_B=1.38\times 10^{-23}$} joules/kelvin is Boltzmann's constant.
The relative probability is
{$\frac{p_i}{p_j}=e^{\frac{\epsilon_i - \epsilon_j}{k_BT}}$}
The two probabilities are the same if and only if their associated energies are the same.