Boltzmann probability 396 Subject

Boltzmann s 5 = In fi can be used by maximizing S subject to maintaining constant energy and the number of particles in order to derive the part of the Boltzmann probability with the help of Stirling s approximation InlV N nN—N. This proves the (1/ T) part of the Boltzmann principle. 

The skeptical reader may reasonably ask from where we have obtained the above rules and where is the proof for the relation with thermodynamics and for the meaning ascribed to the individual terms of the PF. The ultimate answer is that there is no proof. Of course, the reader might check the contentions made in this section by reading a specialized text on statistical thermodynamics. He or she will find the proof of what we have said. However, such proof will ultimately be derived from the fundamental postulates of statistical thermodynamics. These are essentially equivalent to the two properties cited above. The fundamental postulates are statements regarding the connection between the PF and thermodynamics on the one hand (the famous Boltzmann equation for entropy), and the probabilities of the states of the system on the other. It just happens that this formulation of the postulates was first proposed for an isolated system—a relatively simple but uninteresting system (from the practical point of view). The reader interested in the subject of this book but not in the foundations of statistical thermodynamics can safely adopt the rules given in this section, trusting that a proof based on some... 

The Boltzmann law of distribution was obtained, let us repeat ( 6, p. 9), as the most probable distribution of the particles of a gas (in our case the light quantum gas) in the various sheets (called cells in our earlier investigation), subject to the two subsidiary conditions = n and = E when the number of particles and the total energy are given. For the distribution of energy in our light quantum gas we therefore find... 

If droplets of liquid are to form in the midst of vapour, or minute crystals in the midst of liquid, they must grow from nuclei which, in the first instance, have to be produced by the chance encounters of molecules with appropriate velocities and orientations. The incipient nuclei are subject to two opposing influences. In virtue of the attractive forces, they tend to grow, and in virtue of the thermal motion they tend to disperse. These tendencies exist at all temperatures, but above the point of condensation or crystallization they come into balance while the agglomerates are still few, minute, and transitory. The existence of the nuclei amounts to no more than an increased probability of finding small groups of molecules closer together than they would be in the absence of attractive forces, and is a direct consequence of the Boltzmann principle. 

The entropy Sipii,..., pij,... is a function of a set of probabilities. The distribution of p,j s that cause 5 to be maximal is the distribution that most fairly apportions the constrained scores between the individual outcomes. That is, the probability distribution is flat if there are no constraints, and follows the multiplication rule of probability theory if there are independent constraints. If there is a constraint, such as the average score on die rolls, and if it is not equal to the value expected from a uniform distribution, then maximum entropy predicts an exponential distribution of the probabilities. In Chapter 10, this exponential function will define the Boltzmann distribution law. With this law you can predict thermodynamic and physical properties of atoms and molecules, and their averages and fluctuations. How-ever, first we need the machinery of thermodynamics, the subject of the next three chapters. 

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