The idea behind the second law of thermodynamics is that thermal energy spreads or disperses until, in equilibrium, the energy is maximally dispersed. How can this idea be quantified? We’ll start with an analogy.

Suppose you toss four coins simultaneously. TABLE 20.2 shows that there are 16 possible outcomes of heads and tails. Any one of these outcomes—say, HTHH—is called a microstate, so there are a total of 16 microstates.

Perhaps you care only about the number of heads, not about the details of how each coin lands. The number of heads ranges from 0 to 4, and each possibility is a macrostate. By examining Table 20.2 you can see that there is only one way (TTTT) the macrostate $H=0$ can occur but six ways to have $H=2$, with an equal number of heads and tails. The number of microstates associated with a particular macrostate is called the multiplicity of that macrostate, denoted by $\text{Ω}$ (uppercase Greek omega). TABLE 20.3 shows the multiplicity of each of the five macrostates when four coins are tossed.

If you assume that each coin is fair, with a 50% probability of giving a head or a tail, then each of the 16 microstates is equally likely to occur. That is, the chances of seeing TTTT are exactly the same—1 in 16—as seeing HTHT. But the probabilities of the macrostates are not equal. There’s only one way to achieve the $H=0$ macrostate, so its probability is $1/16$. There are six ways to have $H=2$, so its probability is $6/16$. If you toss these four coins over and over, you will see the $H=2$ macrostate six times as often as you see $H=0$. Some macrostates are more probable than others, and that is the key to understanding entropy and the second law.

If you toss N coins, there are $2^N$ possible outcomes and thus $2^N$ microstates. There’s always only one way to achieve the $H=0$ macrostate—a toss that gives all tails—so the probability of macrostate $H=0$ is $1/2^N$. You wouldn’t be terribly surprised to see $H=0$ when you toss only four coins because, on average, this macrostate occurs once in every 16 tosses. But if you toss 1000 coins, the odds of $H=0$ (that is, the odds that all 1000 coins are tails) are reduced to $1/2^{1000}\approx {10}^{-300}$, or once in every ${10}^{300}$ tosses. The age of the universe is only $\approx {10}^{17}\text{s,}$ so it would be remotely unlikely to see $H=0$ even if you had tossed 1000 coins once a second since the Big Bang about 14 billion years ago. With 1000 coins, it’s safe to say that the macrostate $H=0$ never occurs.

FIGURE 20.19 shows the macrostate probability graphically. The heights of the histogram bars for $N=4$ in Figure 20.19a are taken from Table 20.3. If N is very large, Figure 20.19b shows that the only macrostates that have any likelihood of occurring are clustered very near $H=N/2$, with equal numbers of heads and tails. It’s not that only $H=N/2$ occurs, but the macrostates that do occur are clustered so close to $H=N/2$ as to be indistinguishable. That’s not yet entirely true for $N=1000$, but think of the situation with $N={10}^{22}$, roughly the number of molecules in a liter of gas at atmospheric pressure. For very large N, the only macrostates that will ever be seen are indistinguishable from $H=N/2\text{;}$ a macrostate that differs from $N/2$ enough to be experimentally noticed, even with the best equipment, will never be seen.

FIGURE 20.19 The number of microstates that show H heads when four coins are tossed and when a very large number of coins are tossed.

Coin flipping may seem to have nothing to do with thermodynamics, but consider a macroscopic system with pressure p, temperature T, thermal energy $E_{\text{th}}$, and so on. The values of the state variables $p\text{,}T\text{,}{E}_{\text{th}}$, and so on define a state of the system—in particular, a macrostate. Just as knowing that $H=2$ tells us nothing about how each coin lands, knowing the state variables of a macroscopic system tells us nothing about what individual molecules are doing.

At the atomic level, each possible arrangement of the atoms or molecules in a system—their positions, their velocities, their rotations and vibrations—is a microstate. The number of microstates that give rise to a particular macrostate is the multiplicity $\text{Ω}$ of that macrostate. There are an enormous number of ways the positions and velocities of the molecules can be shuffled around without changing the observable state variables, so the multiplicity is an unimaginably large number—on the order of ${10}^{{10}^{23}}$ for any real-world macrostate.

Once we recognize that a vast number of microstates (different atomic-level arrangements) correspond to each macrostate, we can reach important conclusions using four steps of logic:

1. Random collisions and atomic-level interactions, like coin flips, constantly change the microstate. Every macrostate has a vast number of microstates, each of which is equally likely to occur.

2. One macrostate is overwhelmingly more probable than the others, which means that its multiplicity $\text{Ω}$ is vastly larger than the multiplicities of other possible macrostates. This idea is illustrated in Figure 20.19b: For large N, almost all the microstates correspond to macrostates that are extremely close to $H=N/2$.

3. The most probable macrostate is the system’s equilibrium state. Once an isolated system is in equilibrium, atomic-level interactions never cause it to change because the probability of being in any other macrostate is essentially zero.

4. If an isolated system is not in equilibrium, atomic-level interactions cause it to evolve from a less-probable macrostate to the most-probable macrostate. That is, a nonequilibrium system evolves toward equilibrium—not because it “knows” where equilibrium is or has any “desire” to be in equilibrium but simply because equilibrium is overwhelmingly the most probable macrostate to be in.

The importance of these key ideas cannot be overstated. They rely on the fact that any macroscopic system has an enormous number of atoms or molecules. We will sometimes use examples that have relatively small numbers of particles to develop a feel for how the statistics work, but it is essential to keep the reality of large N in mind.

An unlikely event