Explain the meaning of the term microstate
State the principle of equal a priori probability
Explain how one can calculate the entropy from the probability distribution
Explain what is meant by phase space
Write an expression that can be used to calculate an ensemble average
Any physical system can adopt one of a number of microstates. For a quantum system these are the various quantum levels. For a classical system these are the various combinations of velocity and positions of the atoms in the system.
The principle of equal a priori probabilities assumes that any system in equilibrium is equally likely to be in any one of the accessible microstates. Consequently, to determine the probability of being in any microstate one has to minimise the information contained in the distribution. However, to make the analogy with classical thermodynamics clearer we will maximise minus the information, which we will assert is equal to the entropy:
$S = - k_B \sum_i p_i\log p_i$
The sum here runs over the set of all microstates.
The set of all the possible microstates the system can adopt is known as phase space. Each of the microstates in phase space will have associated with it values for all the various extensive variables. In addition to these properties, each of the microstates will have a probability associated with it. We can thus determine the ensemble averages of the energy, volume and number of atoms using:
$\langle E \rangle = \sum_i E_i p_i \qquad \langle V \rangle = \sum_i V_i p_i \qquad \langle N \rangle = \sum_i N_i p_i$
where the sums run over all the microstates in phase space and $V_i$, $E_i$, $N_i$ and $p_i$ are the volume, the energy, the number of atoms and the probability of microstate $i$.
These ideas are explained in the following video:
https://www.youtube.com/watch?v=0LdL1LUrE0I&ab_channel=GarethTribello
<aside> 📌 SUMMARY:Any physical system can adopt multiple microstates. These set of of all these microstates is referred to as phase space. There will be values of the extensive variables associated with each of these microstates as well as a probability of being in each of the microstates in the system.
</aside>