I am a layman interested in information, and wondered how far off this idea is:
Imagine 2 levels and 3 particles (in the physics sense, indistinguishable), and the different permutations:
- 1 arrange $$ L_{2} \hspace{1em} \_ \hspace{1em} \_ $$
$$ L_{1} \hspace{1em} x \hspace{1em} x\hspace{1em} x $$
- 6 eq. arrages $$ L_{2} \hspace{1em} \_ \hspace{1em} x $$
$$ L_{1} \hspace{1em} x \hspace{1em} \_ \hspace{1em} x $$
- 3 arranges
$$ L_{2} \hspace{1em} x \hspace{1em} x $$
$$ L_{1} \hspace{1em} x \hspace{1em} \_\hspace{1em} \_ $$
Numbers
The total number of states is 10. Also can be get using the permutations formula.
Entropy
The Shannon (or Boltzmann) formula would be
$$= − _1\,(_1) − _2\,(_2) − _3\,(_3)$$
$$= − 1/10\,_{2}(1/10) − 6/10\,_2(6/10) − 3/10\,_2(3/10)=1.3$$
This means that most of the time it is enough to specify a single bit of information to determine the state.
Is this idea correct or am I mislead?