Summary of the Evolution of the System towards the Equilibrium State

Consider again the simple experiment as described in Fig. (7.2).

We start with N = 10²³ particles in L. A specific configuration is a detailed list of which particle is in L and which is in R. A dim configuration is a description of how many particles are in L and in R. As long as the partition is in place, there is no change in the dim configuration nor in the specific configuration.¹⁴The system will not evolve into occupying more states if these are not accessible.

Now, remove the barrier. The new total number of specific states is now 2^N; each particle can either be in L or R. The total number of states, W(total), is fixed during the entire period of time that the system evolves towards equilibrium.

14There is a subtle point to note here. The information we are talking about is where the particles are; in L or in R. We could have decided to divide the system into many smaller cells, say four cells in R and four cells in L. In such a case, the information of the initial state is different since we shall be specifying which particle is in which cell; so is the information of the final state. However, the difference between the information (as well as the entropy) is independent of the internal divisions, provided we do the same division into cells in the two compartments. [For further details, see Ben-Naim, (2006, 2007).]

Translating from the Dice-World to the Real World 165

Clearly, once the barrier is removed, changes will occur and we can observe them. It could be a change in color, taste, smell or density. These are different manifestations of the same under-lying process. What is the thing that changes and is common to all the manifestations?

The thing that changes is the dim state or dim configura-tion or dim event, and there are various ways of indexing, or assigning to these dim states a number that we can plot so as to be able to follow its numerical change. Why does it change?

Not because there is a law of nature that says that systems must change from order to disorder, or from a smaller MI to a big-ger MI. It is not the probability of the dim states that changes (these are all fixed).¹⁵It is the dim state itself that changes from dim states having lower probabilities to dim states having higher probabilities.

Let us follow the system immediately after the removal of the barrier, and suppose for simplicity, that we open a very small door that allows only one particle to pass through in some short span of time. At the time we open the door, the dim state consists of only one specific state belonging to the post-removal condition (i.e., when there are zero particles in R). Clearly, when things start moving at random (either in the dice game or in the real gas colliding with the walls and once in a while hitting the hole), the first particle that passes through the hole is from L to R, resulting in the new state, dim-1. As we have seen in great detail in Chapter 4, there is a high probability of the system either moving up or staying at that level, and there is a very small probability of the system going downwards to a lower dim state. The reason is quite simple. The probability of any

15There is an awkward statement in Brillouin’s book (1962) that “the probability tends to grow.” That is probably a slip-of-the-tongue. What Brillouin probably meant is that events of low probability are evolving into events of high probability. The probabilities themselves do not change.

particle (any of the N) crossing the border between L and R is the same. Let us call that probability p1 (which is determined by the speed of motion, the size of the door, etc.). Whatever p1 is, the probability of moving from dim-1 to dim-0 is the probability of a single particle in R crossing over to L, which is p1. On the other hand, the probability of crossing from dim-1 to dim-2 is (N−1) times larger than p1 simply because there are (N − 1) particles in L, and each has the same chance of crossing from L to R. The same argument can be applied to rationalize why the system will have a higher probability of proceeding from dim-1 to dim-2, from dim-2 to dim-3, etc. Each higher dim state has a larger number of specific states and therefore a larger probability. As we have seen in Chapters 3 and 4, this tendency of a system going upwards is the strongest initially, becoming weaker and weaker as we proceed towards dim-N/2, which is the equilibrium line.

This equilibrium line is the dim state with highest probability since it includes the largest number of specific states.

One should be careful to distinguish between the number of specific states belonging to dim-N/2 and the total number of states of the system which is W(total). These are two different numbers. The latter is the total of all the specific states included in all the possible dim states. For instance, for N = 10, we have¹⁶

W(Total)= W( dim-0) + W( dim-1) + W( dim-2) + · · ·

= 1 + 10 + 45 + 120 + 210 + 252 + 210 + 120 + 45 + 10 + 1 = 2¹⁰

16In general, this equality can be obtained from the identity (1+ 1)^N = 2^N =

_N n=0 N!

n!(N−n)!.

Translating from the Dice-World to the Real World 167

As we have discussed in Chapters 2 and 3, the probability of the dim event is just the sum of the probabilities of the specific events that comprise the dim event.

For example, for dim-1 of the case N = 10, we have 10 specific events. The probabilities of each of these 10 specific events are equal to (¹/2)¹⁰. The probability of the dim-event is simply 10 times that number, i.e.,

Prob(dim-1)= 10 × (¹/2)¹⁰

For dim-2, we have 10× 9/2 = 45 specific events. All of these have the same probability of (¹/2)¹⁰. Therefore, the probability of dim-2 is:

Prob(dim-2)= 45 × (¹/2)¹⁰

In the table below, we have listed the probabilities of all the dim events for the case N = 10. Note again the maximum value at dim-^N/2 or dim-5.

Dim-Event Number of Specific Events Probability

dim-0 1 1/2¹⁰

dim-1 10 10/2¹⁰

dim-2 45 45/2¹⁰

dim-3 120 120/2¹⁰

dim-4 210 210/2¹⁰

dim-5 252 252/2¹⁰

dim-6 210 210/2¹⁰

dim-7 120 120/2¹⁰

dim-8 45 45/2¹⁰

dim-9 10 10/2¹⁰

dim-10 1 1/2¹⁰

Figure (7.10) shows the number of specific events for each dim event, for different N (N = 10, N = 100 and N = 1000).

In the lower panel, we show the same data but in terms of prob-abilities of the dim events.

It is clear from Fig. (7.10) that as N increases, the number of specific states belonging to the maximal dim states also becomes very large. However, the probability of the maximal dim state decreases with N. Thus, as N increases, the probability distri-bution of the dim events is spread over a larger range of values.

The apparent sharpness of the distribution as shown in the lower panel of Fig. (7.10) means that deviations from the maximal dim state, in absolute values, are larger, the larger N is. However, deviations from the maximal dim states, relative to N, become smaller, the larger N is.

For instance, the probability of finding deviations of say

±1% from the maximal dim state becomes very small when N is very large.

We next calculate the probability of finding the system in any of the dim states between, say, ^N/2−^N/100 to ^N/2+^N/100, i.e., the probability of the system being found around the maximal dim state, allowing deviations of ±1% of the number N. This probability is almost one for N= 10,000 (see Fig. (7.11)). With

Fig. (7.10)

Translating from the Dice-World to the Real World 169

2000 4000 6000 8000 10000

N

0.2 0.4 0.6 0.8 1

probability

Fig. (7.11)

N on the order of 10²³, we can allow deviations of 0.1% or 0.001% and still obtain probabilities of about one, of finding the system at or near the equilibrium line.

What we have found is very important and crucial to under-standing the Second Law of Thermodynamics. The probability of the maximal dim-^N/2 decreases with N. However, for very large N, on the order of 10²³, the system is almost certain (i.e., probability nearly one) to be in one of the dim states within very narrow distances from the equilibrium dim state. When N = 10²³, we can allow extremely small deviations from the equilibrium line, yet the system will spend almost all the time in the dim states within this very narrow range about the equilib-rium line.

We stress again that at the final equilibrium state, the total number of specific states is 2^N, and all of these have equal probabilities and therefore each of these will be visited with an extremely low frequency (¹/2)^N. However, the dim states (alto-gether there are N+ 1 dim states) have different probabilities since they include different numbers of specific states. The dim states at and near the maximal dim-^N/2 will have a probability