Introduction ot the toy model in the framework of statistical mechanics

Abstract 

This chapter presents a simple “toy model” to illustrate the concepts of energy, entropy, and free energy. In this model, multiple microstates are grouped together into a single macrostate through a process of coarse-graining. The system tends to go into the macrostate that minimizes the free energy. At low temperature, this is the ordered state with the lowest energy. At high temperature, it is the disordered state with the highest entropy or multiplicity.

                                To see a world in a grain of sand, 
                                      And a heaven in a wild flower, 
                                             Hold infinity in the palm of your hand, 
                                                       And eternity in an hour. 
                                                                           ... 
                                                               William Blake, Auguries of Innocence

Physicists are always looking for the Big Ideas—ideas that are fundamentally simple in concept but can be applied to understand many phenomena throughout nature. In my field of statistical mechanics and phase transitions, the Big Idea is the balance between order and disorder, a balance that is controlled by temperature. In this chapter, I would like to introduce you to this Big Idea through a toy model, i.e., a model which is not a serious theory but is just meant as a simple way to illustrate a basic point. I want to see the world not in a grain of sand, but in a speck of dust.

Let us begin with the situation shown in Fig. 1.1. Here, a speck of dust can have two states: it can be either on the floor or on the table. Each of these states has some energy Efloor or Etable. Based on the  Boltzmann distribution,2 these two states have the probabilities pfloor = (e−Efloor/kB T )/Z and ptable = (e−Etable/kB T )/Z, where kB is Boltzmann’s constant,3 T is the temperature, and Z = e−Efloor/kB T + e−Etable/kB T is the partition function that makes the probabilities add up to 1.

Which state is more likely? Let us compare the two probabilities: The speck of dust is more likely to be on the floor if

Hence, the speck is more likely to be on the floor than the table if the floor energy is lower than the table energy. That is not a big surprise!

Now let us consider a slightly more interesting problem. Suppose there are N tables, where N is some large number, as shown in Fig. 1.2. The probability of being on the floor is (e−Efloor/kB T )/Z. The probability of being on a particular table is (e−Etable/kB T )/Z. The probability of being on any of the N tables is N(e−Etable/kB T )/Z. Here, the partition function that normalizes the probabilities must be Z = e−Efloor/kB T + Ne−Etable/kB T .

In this new problem, let us compare the probabilities again. The speck is more likely to be on the floor than on any of the tables if

This little calculation shows that we should not just compare the energy Efloor and Etable. Instead, we should compare some adjusted energy that takes into account the multiplicity of the states. The adjustment (without the factor of T ) is called the entropy, and the adjusted energy is called the free energy. For the group of all table states, the entropy is

                                                   Stable = kB log N,                                       (1.3)

and the free energy is

                                            Ftable = Etable − T Stable = Etable − kB T log N.            (1.4)

Likewise, because there is only one floor state, the entropy of the floor is

                                               Sfloor = kB log(1) = 0,                                (1.5)

and the free energy of the floor

                                   Ffloor = Efloor − T Sfloor = Efloor − kB T log(1) = Efloor.            (1.6)

Hence, the speck is more likely to be on the floor than on any of the tables if

                                                         Ffloor < Ftable.                                                              (1.7)

From this toy model, I think we learn three general lessons. First, we see the importance of grouping states together. If we care about whether the speck is on any table, and we do not care which table, then it is appropriate to group all of the table states together. In this grouping, we are treating a collection of microstates as a single macrostate. This macrostate has an energy E and a multiplicity N (and hence an entropy S = kB log N). This grouping is our first example of the concept of coarse-graining, which will be fundamental throughout statistical mechanics.

Second, we see the concept of free energy, which combines the energy and entropy of a macrostate into the quantity F = E − T S. This quantity is essential for understanding what macrostate is most likely to occur at a nonzero temperature T .

Third, we see that the relative importance of energy and entropy depends on temperature. At low temperature, energy is the dominant part of the free energy, and the system is most likely to go into whatever macrostate has the lowest energy. That is usually a single, special state, which might be called an ordered state. By contrast, at high temperature, entropy is the dominant part of the free energy, and the system is most likely to go into whatever macrostate has the highest entropy. That is usually a big collection of many microstates. They each have a high energy, so they are individually unlikely, but there are a lot of them! That collection might be called a disordered state.

Now, dear students, I imagine that some of you might have an objection. You might say “What do you mean by defining the free energy of a state? I have already learned the definition of free energy in a class on thermal physics or statistical thermodynamics, and it was different! In that class, we did not talk about the free energy of a state; we just talked about THE free energy. It was defined in terms of the partition function Z as”

                                                                F = −kB T log Z.                                                (1.8)

“So what’s going on here?!?”
I am glad you asked that question. Let us look at Eq. (1.8) for THE free energy, and rewrite it as

             e−F/kB T = Z = e−Efloor/kB T + e−Etable 1/kB T +···+ e−Etable N /kB T .            (1.9)

Now we group the N table microstates together into a single macrostate, and rewrite the equation as

                                          e−F/kB T = Z = e−Efloor/kB T + Ne−Etable/kB T                      (1.10)
                                                                 = e−Ffloor/kB T + e−Ftable/kB T .                        (1.11)

Look at this: When we combine many microstates together in the partition function, we get the free energy of the combined macrostate. We can keep combining macrostates together to make super-macrostates, and combining super-macrostates to make super-duper-macrostates, and each of them has a free energy. When we eventually finish combining all possible states together, we reach THE free energy of the system. That is coarse-graining!

At this point, I think we have learned everything we can from this toy model, and we need to move on to something more physical. Onward!


Reference: https://b-ok.cc/book/2617724/522dcf

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