Question

The concept of number of microstates makes sense for something like magnetisation where each electron has an up or down spin, so we can use counting arguments.

But then they extend this logic to two gas boxes in thermal contact with each other and say that there must be some macrostate with the maximum number of microstates and then use calculus to derive an expression for temperature at equilibrium. But the issue I am having is that energy, atleast classically is a continuous thing, so don’t we have uncountably infinite microstates for each macrostate? Then how does it make sense to compare the number of microstates?

Answer 1

While energy is classically continuous, for statistical analysis, we divide it into countable quanta. The concept of comparing the number of microstates is valid because the most probable state is the one with maximum microstates, leading to Boltzmann's entropy formula. This applies to gases where huge numbers of atoms give rise to an immense number of microstates, each reflecting the same macroscopic properties.

Step-by-step explanation:

The difficulty in grasping the concept of microstates and macrostates in the context of gases, where energy levels are continuous, is a common issue in statistical physics. It is true that classical physics treats energy as a continuous variable, making it seem like there could be an infinite number of microstates for each macrostate. However, in practice, for statistical purposes, energy levels are divided into very small but countable quanta, allowing us to use statistical methods to analyze these systems. The most probable macrostate is the one with the maximum number of microstates, which, in the case of two gas boxes in thermal contact, leads to an equilibrium where the temperature is equalized. This principle is expanded by Ludwig Boltzmann's formulation that the entropy (S) of a system is proportional to the natural logarithm of the number of microstates (W), where S = k ln W. In this statistical analysis, 'k' is the Boltzmann constant, providing a bridge between the microscopic and macroscopic worlds.

When we apply this to a real system, such as a gas, even with a small volume, we are dealing with a huge number of atoms which gives rise to an immense number of ways in which these atoms can be arranged. These arrangements correspond to the system's microstates, all of which reflect the same macroscopic properties such as temperature and pressure. As the system evolves, it will naturally move towards the macrostate with the highest number of microstates, which is also the state of highest entropy.