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Particle statistics

At the end of the previous section we glimpsed the possibility of applying the relevance quotient to contexts other than induction. In dealing with the foundations of probability, Johnson, Keynes, and Carnap all had inductive statistics in mind, especially Keynes, who in Part V of TP provides an excellent and profound review of the statistical inferences in the nineteenth century. Like Keynes and Carnap, we, too, have inductive statistics in mind. Aiming at determining predictive probabilities, we have considered sequences of individuals intended to describe the statistical units of a sample drawn from a population. Just as a sample may not have a definite number of units, the evidence must not have a definite size. That is to say, new individuals may always be added to the sequence. All the axioms and conditions we have stated involve a finite number of individuals; but we have not fixed the evidence size once and for all. The reason for this is that, if we leave aside the trivial case in which one examines the whole population, samples drawn from a finite population may always be brought up to date adding one or more new units. This is the case with the evidence too. In a few words, we have considered evidences that could be called open. By forcing this scenario a little, it becomes possible to deal with certain problems of statistical mechanics, for instance particles statistics, as we shall briefly show in a highly abstract way, greatly simplifying the problem in order to account for the basic ideas. Furthermore, in this way we are preparing to undertake our study of equilibrium, especially in economics, in the next section.

In a sense, a quantity of gas is like a population. The molecules of the gas, endlessly in motion, can be seen as the statistical units of the population whose attributes are different velocities consistent with the environment surrounding the gas. The mean velocity of the molecules determines the temperature of the gas. Suppose we are interested in the assumptions justifying the actual temperature of the gas. In order to arrive at the mean values of the velocity of the molecules, we must know the statistical distribution of the molecules with respect to velocities. The reason for our interest in the statistical distribution is that a mean value is unaffected by a change in the individual distribution that does not change the statistical distribution. The mean velocity, too, is unaffected by knowledge of which molecules have which velocity. Even if molecules could be distinguished one from the other—something that nobody has really achieved—the distinction would have no value for the search for the mean velocity. Being ignorant of the statistical distribution of the molecules, we can guess a probability distribution whose domain is the set of all possible individual distributions. In its simplest formulation, this was the problem Boltzmann (see Bach, 1990) pointed out in the second half of the eighteenth century. Having focused on individual distributions, he used the probability of these distributions in order to determine the probabilities of all statistical distributions. More exactly, supposing that all possible individual distributions (of the molecules with respect to velocities) have the same probability, he arrived at explaining the temperature of the gas, in general, the macroscopic behavior of gases of molecules. We want show that, by using the condition we have stated for predictive probability (2), it is possible to justify the equiprobability assumptions of Boltzmann as well as similar assumptions later made for quantum particles.

The analogy between a gas of particles—classical, or quantum—and a population leads us to consider a system of N particles and d singleparticle states. The attributes particles may bear are single-particle states that in physics are often called cells (of the ^-space). It is hardly necessary to observe that the name "cell" for attributes we have used comes from statistical mechanics. For simplicity we state that all cells belong to the same energy level and suppose that the system is a void container into which particles are inserted one at the time. This is the simplification and abstraction we have spoken about. X, i = 1,2, ...,N, the ith particle, denotes the particle that has been inserted into the container when i— 1 particles are already in it. Each particle goes in a cell and Xt = j, j e {1, ... ,d} is the event that occurs when the ith particle goes in cell j. Once all the particles has been inserted into the container, the individual description is

It is worth noting that X(N), formally equal to D, does not refer to data but rather to the individual distribution of the particles in the cells. When jt varies over all possible values, (19) takes up all possibilities. For instance,

states that all particles are in cell 1. We are interested in the probability of (19) that can be calculated by using the multiplication rule. This rule ensures that

Now we assume that the probabilities on the right side of this equality satisfies C2 and C3. It follows that for these probabilities the main theorem holds, and this enables us to calculate the probability of all individual distributions of the system. This distribution is

in which the parameters X and p are the same as in (15) while x[n] = x(x + 1)...(x — n + 1) is the Pochhammer symbol. (21) is a probability distribution on the individual distributions of the gas we are considering. It goes without saying that in order to get an actual distribution one must fix the numerical values of the two parameters of (21).

All macroscopic properties of a gas of particles are mean values. Hence in order to determine these values we must have at our disposal a probability distribution on statistical distributions. The sum rule ensures that this probability can be arrived at in a very simple way: summing up the probabilities of all individual distributions consistent with the considered statistical distribution. On the other hand, it is easy to verify that, given an occupation vector (statistical distribution) N = (N ,..., Nj,...,Nd), there are

individual distributions consistent with it. Thanks to exchangeability, all these individual distributions have the same probability. As a consequence, multiplying (21) by (22) we reach the probability of N, that is

This is the (generalized) Polya distribution. It is a probability distribution on statistical distributions.

In order to have a definite probability distribution we must fix the numerical values of the parameters of (23). First of all we consider the Bose-Einstein statistics. Putting pj = d—1, for all j, and X = d in (23) we have

(N + d -1)

у N J is the number of the statistical distributions (occupation vector) of the system. Thus (24) allots the same probability to each occupation vector. Physicists call this uniform probability distribution Bose-Enstein statistics. (24) is the formula governing the behavior of bosons, the particles with integer spin.

The second distribution we take into account is the statistics of Maxwell-Boltzmann. This arises as a limiting case of (23) when pj = d—1, for all j, and X ^ «>. If this is the case (23) becomes

This is again a uniform distribution, not on the occupation vectors but rather on all individual distributions. In fact, that there are dN individual distributions and (25) allots to all them the same probability, that is, dN. The uniform probability distribution (25) is known as Maxwell-Boltzmann statistics. (25) is the formula governing the behavior of classical particles.

The last uniform probability distribution we will consider can be reached using a negative value of X. If we put pj = d-1, for all j, and X = —d, (23) becomes

(d)

^ N J is the number of the statistical distributions whose occupation numbers are either 0 or 1. Thus (26) allots the same probability to all occupation vectors in which no more that one particle is in a cell. Obviously, in this case N < d. Physicists call the probability distribution

(26) Fermi-Dirac statistics. This is the formula governing the behavior of fermions, the particles with half-integer spin.

Before going on we shall make a small change in the symbolism we are using, which will greatly assist our exposition. As we have already noted, the symbolism we have so far used for the (predictive) probability looks at an inductive scenario. In this section dealing with particle statistics, we have continued using the same symbolism. It is now clear in what sense we have forced the inductive scenario. The problem we have tackled did not account for the probability of a succeeding observation. Our problem was: what is the probability that a particle of a sequence inserted into the system will be accommodated in a given cell so that a given distribution comes out? Completely explicitly, at the basis of our calculations there is a system whose size is n = 0,1,...,N — 1, which, as a consequence of the entry of a new particle, increases its size by 1. Such an entry increases the occupation number of the cell j from n to n + 1. Physicists speak of the "creation" of a particle in the cell j. We have determined the probability of this creation in such a way that, after a sequence of N creations, that is, the entry of N particles, the probability distribution on occupation vectors of the resulting system is (23). Because each creation in a cell j changes the system size from n to n + 1, whose occupation vectors are n and nj, the probability we have used in this section can be denoted by P(nj|n). This is the symbol we shall use in what follows.

 
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