The Boltzmann Entropy Formula

So just two equations left in 'Month of Equations' series this year. In the penultimate equation, I will discuss the meaning of entropy, not in thermodynamics but in statistical mechanics. We will see that the definition of entropy in statistical physics gives more information about the system. The Boltzmann entropy formula is the equation that connects statistical mechanics to thermodynamics. So let us see what this equation really means.

Boltzmann entropy formula

The Boltzmann Entropy Formula
The Boltzmann Entropy Formula

The Boltzmann's law states that the entropy of a system is the measure of the number of possible microscopic states (microstates) of the system corresponding to a particular macrostate.

The terms microstate and macrostate are mind boggling. But understanding them is very important for statistical mechanics. So I will try to explain these two terms by taking a very simple day to day life example.

Microstates and macrostates

Forget thermodynamics and statistical mechanics for a while. Let us play a game. Take two fair and unbiased coins and start tossing them both simultaneously. Make a table of all the possible outcomes. The possible outcomes are:

A. Both heads (HH)
B. both tails (TT)
C. one head and one tail (HT or TH).

A and B are possible only with one configuration i.e. both the coins show same side at the end. C has two possible configurations: coin 1 shows H and coin 2 shows tail or coin 1 shows tail and coin 2 shows head. Well, that's it. A, B and C are known as the macrostates of the system while the possible configurations are the microstates of the system. So the macrostates A and B have only one microstate whereas C has two microstates. Thus microstates are the possible configurations at the microscopic level that make up one complete state at the macroscopic level.

Consider another example. Suppose you put a drop of blue ink in a glass of water. Within a few seconds, the ink dissolves in water and the water attains a uniform blue color. At this point, we can depict the state of the water by just defining the color of it so this becomes a macrostate of the system. However, the color is actually a result of the diffusion of particles of blue ink within the water and attaining particular positions. Hence, the configuration of individual particles within the system indicates the microstate of that water-ink system.

Now it will be easier to understand these two terms in the statistical picture in physics.

Microstates and macrostates in statistical mechanics

Suppose you have a huge collection of particles in a box. They can be in trillions. Each particle will have a different position, velocity, momentum, spin etc. Such states are known as microstates. A particle has an access to every possible microstate. It can have any speed, momentum etc. Also at equilibrium, the probability of accessing any microstate is equally likely. Now the collection of particles, which we call a gas, will have a particular temperature, volume, pressure etc. These are known as the macrostates. Statistical mechanics assumes that for a given configuration of macrostate, all the microstates are equally likely.

Also Read: All the 30 articles of Basics of Astrophysics series

Microstates and Macrostates in Physics
The concept of microstates and macrostates in statistical physics. If I am talking about the position, momentum etc of each particle, I am referring to microstate of the system. But if I am talking about the volume, temperature etc. of the system as a whole, I am referring to macrostate.

The last fact that should be mentioned is that if you leave a system undisturbed, the particles will try to take all the possible microstates and if you observe the system at any time, you'll always get a random configuration that is evenly spread out. All the combinations are possible. Say, it is possible that all the trillion particles occupy one half of the box and other half remains empty. However, the statistics tell that the probability of such a configuration is so low that it will never happen in an isolated system. But what entropy has to do with microstates? Well, Boltzmann figured out that too.

Entropy and microstates | The boltzmann equation

Boltzmann's law signifies that the entropy of a thermodynamic system in equilibrium is defined as the probability of a macrostate for some probability distribution of possible microstates. For a system in thermodynamic equilibrium, the probability of each microstate is equal. Therefore, it becomes equal to the total number of possible microstates of the system. The term depicted by capital omega in the equation is obtained by the permutations of particles within for a given macrostate.

It is interesting that the entropy 'S' is itself a thermodynamic property of the system and hence, is a bridge between the macroscopic and the microscopic properties if the system. Thus this equation connects two important branches of physics: statistical mechanics and thermodynamics.

Previous in series: The Fermi Dirac Statistics and Their Applications In Physics

Author's message

My aim in this article was to introduce you to the concept of microstates and macrostates. These two terms are very important for statistical mechanics. When you study something, try to understand it deeply. Teach it to your friends. If you grab the concept deeply, physics will be cake walk for you. There is no use of memorizing long derivations and puking them on the exam next day. If you have understood the concept of the problem, the derivations will not be a burden on you!

Also Watch: How was quantum mechanics born?

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