# Physical implications of no. Of microstate

I was studying the Statistical mechanics and what I have understood is that if there is a large number of particles in a system , and if we want to study the system then we have to calculate the position and momentum of all particles in every instant of time, and this will be the complete information about the system , but this is impossible to do . So in statistical mechanics people use the total no. Of microstate $$\Omega$$ to calculate the thermodynamic veriables.

But my question is that how can I argue that to calculate the thermodynamic veriables we only need the no. Of microstate not the detailed information about those microstate state ( information about those microstate state means say the position and momentum at every insrant)?

We actually have Boltzmann hypothesis $$S=k_B ln(\Omega)$$ which tells that entropy is fully related to only no. Of microstate not to the full details of those microstate.

But this is a hypothesis we can not explicitly prove this. But I want a proper explanation that why I need only the no. Of microstate not the detailed information about those microstate.

Typically, we measure bulk quantities like energy, temperature, volume, pressure etc. which do not depend on the details of individual particles (to reasonable precision). Because we don't have access to that information, the thermodynamic variables won't depend on the details of the microstates.

The measurements we make are average quantities, where the time scale for averaging is much longer than the time scale of microscopic dynamics. So we average out the statistical fluctuations.

The ergodic hypothesis then says that the probability distribution for the detailed information of the microstates is uniform in equilibrium. It's this crucial step that allows us to forget all knowledge of the microstates, since their statistical distribution is so trivial. It's this hypothesis which makes thermodynamics so simple yet so powerful! We can forget microscopic details, yet still determine macroscopic behaviour.

The ergodic hypothesis is extremely difficult to prove. Actually it's a rather subtle affair. Thermodynamics is actually incorrect. An isolated system doesn't actually stay in equilibrium forever, but returns to its initial state due to the Poincare recurrence theorem (also see Kac ring for a toy model). The recurrence time for realistic systems is $$\sim$$ age of the universe.

To mathematically derive the stability of equilibrium (Boltzmann's H-theorem), one needs actually to make an assumption called molecular chaos, which puts in an arrow of time by hand.

But maybe I've wandered too far from the question. Hope I could clear things up!

Flip 4 coins: A microstate is any configuration of the coins. One microstate is when all are tails TTTT. Another is when the first two are tails and the last two are heads TTHH. Another is when the first is heads, the second is tails, the third is heads and the fourth is tails HTHT. Etc. Now, several of such microstates will give the same overall outcome (the same number of heads and tails):

• The no. of microstates giving the outcome of four tails is 1, TTTT.
• The no. of microstates giving the outcome of three tails is 4, TTTH TTHT THTT HTTT.
• The no. of microstates giving the outcome of two of each is 6, TTHH THTH HTTH THHT HTHT HHTT.
• The no. of microstates giving the outcome of three heads is 4, HHHT HHTH HTHH THHH.
• The no. of microstates giving the outcome of four heads is 1, HHHH.

Since all microstates are equally likely, the no. of microstates corresponding to a specific outcome (a macrostate) tells us the propability of that outcome. And we are usually only interested in the macroscopic outcome. In this coin example, the most probable outcome is two heads and two tails, while it is quite unlikely that all coins are equal - such outcome-prediction is usually our goal.

In reactions and alike, you therefor know the dominant behaviour from the no. of microstates. In statistical physics you use statistical means to analyse or predict what will be the dominant behaviour or expected outcome.

For that you only need to know the no. of microstates with the assumption that they are all equally likely (or a reason to include a weighed influence), because that shows the overall trend and tendency. You don't need individual knowledge about single particles/events/microstates to predict such outcome, since it would make no difference for you to know that the outcome was TTHH rather than HTHT. With many, many, many - basically infinitely many - particles in a system, you can mathematically expect such system to behave according to statistical probability.