Distinguishable and indistinguishable particles

Consider a system of N particles in volume V. If the macroscopic state of the system does not change when particles exchange their positions and velocities, then particles are called indistinguishable. Otherwise, they are called distinguishable. 

Consider three balls that are distinguishable. For example, one is red, another is blue, and the third is green. In how many ways can these balls be placed in three boxes. A, B, and C, if only one ball is allowed per box The answer is obviously 3 = 6 different ways. If, on the other hand, the balls are indistinguishable (e.g., they are all blue) then there is only one way to place these balls one in each box (Fig. 2.4). 

Similarly, we can find that there are 4 = 24 ways to place four distinguishable balls in four different boxes, with at most one ball per box, or that there are 5 = 120 ways to place five distinguishable balls in five different boxes. In general, there are M ways to place M distinguishable balls in M boxes, with at most one ball in each box. 

If the balls are indistinguishable we can calculate the number of different ways to place three balls in four boxes to be just four, and the number of ways to place three balls in five boxes to be just 20. 

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At low temperature the situation is complicated by the fact that the difference between distinguishable and indistinguishable particles enters only when they occupy different states. This leads to different statistics between fermions and bosons and to the generalization of (1.174) to... 

MaxweU-Boltzmaim particles are distinguishable, and a partition function, or distribution, of these particles can be derived from classical considerations. Real systems exist in which individual particles ate indistinguishable. Eor example, individual electrons in a soHd metal do not maintain positional proximity to specific atoms. These electrons obey Eermi-Ditac statistics (133). In contrast, the quantum effects observed for most normal gases can be correlated with Bose-Einstein statistics (117). The approach to statistical thermodynamics described thus far is referred to as wave mechanics. An equivalent quantum theory is referred to as matrix mechanics (134—136). 

Give an example of a system containing distinguishable particles and one containing indistinguishable particles. 

In comparing the two experiments with both slits open, we see that interacting with the system by placing a detector at slit A changes the wave function of the system and the experimental outcome. This feature is an essential characteristic of quantum theory. We also note that without a detector at slit A, there are two indistinguishable ways for the particle to reach the detection screen D and the two wave functions xPa(X) and xIjb(x) are added together. With a detector at slit A, the two paths are distinguishable and it is the probability densities PA(x) and Pb(x) that are added. 

You cannot now factor the system partition function into particle partition functions as we did before. Elere s the problem. If one particle occupied energy level 27 and other particle occupied energy level 56, you could not distinguish that from the reverse. Because of this indistinguishability, you would have overcounted by a factor of 2 . ... 

Equations (10.34) for (f> and (10.37) for S apply to distinguishable particles. Compute the corresponding quantities for systems of indistinguishable particles. 

However, in the case of entropy the identity of the particles is a factor. In section 17.2 we assumed that we could tell the difference between individual particles that is, we assumed they were distinguishable. In fact, at the atomic level we cannot distinguish between individual, identical particles atoms and molecules are macroscopically indistinguishable. This means that we are overcounting the total number of possible distributions for El. The factor that fixes this overcounting is a factor of M in the denominator of El. (That is, there are 1/M times fewer distributions for indistinguishable particles than for distinguishable particles.) When this factor is considered, the equations become... 

As discussed in Chapter 9, dense-phase fluidization other than particulate fluidization is characterized by the presence of an emulsion phase and a discrete gas bubble/void phase. At relatively low gas velocities in dense-phase fluidization, the upper surface of the bed is distinguishable. As the gas velocity increases, the bubble/void phase gradually becomes indistinguishable from the emulsion phase. The bubble/void phase eventually disappears and the gas evolves into the continuous phase with further increasing gas velocities. In a dense-phase fluidized bed, the particle entrainment rate is low and increases with increasing gas velocity. As the gas flow rate increases beyond the point corresponding to the disappearance of the bubble/void phase, a drastic increase in the entrainment rate of the particles occurs such that a continuous feeding of particles into the fluidized bed is required to maintain a steady solids flow. Fluidization at this state, in contrast to dense-phase fluidization, is denoted lean-phase fluidization. 

Because for systems, unlike particles, there are no requirements for them to be indistinguishable, we use the thermodynamic formulas analogous to those for distinguishable particles. For systems containing N particles and having volume... 

Boltzon Postulate. Maxwell-Boltzmann (MB) statistics predict that all energies are a priori equally likely, and that all particles in the system are physically distinguishable (labeled by some number, or shirt patch, "color", or whatever, or picked up by "tweezers"). These MB particles can be called boltzons. If, however, we remove this distinguishability, then we have indistinguishable "corrected boltzons (CB)" [2], whose statistics become very roughly comparable to the statistics of fermions or bosons (see Problem 5.3.10 below). 

Localization on distinguishable single sites and gaslike mobility throughout the accessible volume in the crystal are obviously extreme cases intermediate degrees of mobility of the sorbate particles are also conceivable. For example, the crystal may be divided into a large number (z) of equal cells, with each cell able to accommodate up to sorbate particles, which are mobile and therefore indistinguishable within one cell. The total sorption capacity of the solid is then... 

In the classical derivation, there are somehow mysterious assumptions concerning the distribution, that is, in cases when the particles can be distinguished from each other and when they are indistinguishable. 

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