# Usefulness in microstates that belong to the same macrostate?

The number of microstates can be huge, numbers with millions of digits, but they can all represent the same macrostate.

IS there any information that can be associated with each microstate individually or is it all statistics? i.e. you can count microstates and calculate probabilities, but the difference between one microstate and another has no use.

Yes, microstates possess individual information, as they are genuine physical objects and not just tools of statistics.

Individual microstates may differ in positions and momenta of individual particles, or even in the total energy. The fact that you are treating microstates statistically is simply a tool for simplifying calculations.

• my question was a bit vague. One microstate is different from another and a huge number of microstates can all represent the same macrostate, but is that useful in any way to know which is which? Oct 4, 2015 at 23:16
• @0kelvin The definition of a macrostate for a specific statistical problem is largely defined precisely by which microstate distinguishing features you care about and which you don’t. Jun 26, 2018 at 21:24

In some cases the information about the microstate elements is more important or the macrostate variables suffice. Eg with $\Omega(E,N,V)$, for the microcanonical, because you assume that the aggregate 'de-labeled' effect on the system is all that you need. Consider the Ising model where the configurations over the lattice do not matter, only the aggregate total, $H(\sigma)$. You could say that if there are particular spatial arrangements that would matter, but you assume they don't.

A particular case is when there are microstates which are inaccessible, eg. some patterns are not feasible for whatever reason. You may look at the number of macrostate $(E,N,V)$ and say that not all of the number of $\Omega(E,N,V)$ are possible.

A silly example; imagine you are tossing a coin and are scared of large consistent sequences of heads (H) or tails (T) (inspired from Hamlet), you toss it 5 times, and as a rule you do not throw a 5th trial if the rest were all of the same random variable outcome from fear. So, -HHHH*- and -TTTT*- prevent the set of $R={HHHHH,HHHHT,TTTTT,TTTTH}$ from being realized but a sequence such as $THHHH$ is feasible to be seen because it is known that the 5 consecutive trials is no longer possible although in terms of the macrostate there are a member of $R$ which has the same macrostate; therefore making it inaccessible. So it would have to be looked not just at the macrostate level. So the macrostate size for $\Omega(five same)=0$ (none accessible), and $\Omega(foursame)=8$ (partial accessible for this macrostate).

Tracking individual particles in the case of an ideal gas is not typically considered and ignored usually by default.