Boltzmann entropy postulate

A thermodynamic argument relates the forces / to the entropy change d5 when the network chain is stretched. Entropy is a measure of disorder, evaluated using Boltzmann s postulate that... 

The constant growth of entropy postulated in the second law is closely related to the following principle By the irreversible process the system is spontaneously directed towards the most frequently occurring set of micro states which corresponds to the same macro state of the system. The macroscopic state of equilibrium - where entropy is at its maximum - is the state which is linked to the largest number of different micro states. This basic concept is the basis of the Boltzmann relation that defines the entropy function S in statistical thermodynamics. 

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... 

In other words, every process generates entropy. The best interpretation, in our opinion, of this important law is given by adopting a postulate by Boltzmann ... 

The connection postulated by Boltzmann between entropy S and the thermodynamic probability ft is given by the equation... 

Similarly, if one is interested in a macroscopic thermodynamic state (i.e., a subset of microstates that corresponds to a macroscopically observable system with bxed mass, volume, and energy), then the corresponding entropy for the thermodynamic state is computed from the number of microstates compatible with the particular macrostate. All of the basic formulae of macroscopic thermodynamics can be obtained from Boltzmann s definition of entropy and a few basic postulates regarding the statistical behavior of ensembles of large numbers of particles. Most notably for our purposes, it is postulated that the probability of a thermodynamic state of a closed isolated system is proportional to 2, the number of associated microstates. As a consequence, closed isolated systems move naturally from thermodynamic states of lower 2 to higher 2. In fact for systems composed of many particles, the likelihood of 2 ever decreasing with time is vanishingly small and the second law of thermodynamics is immediately apparent. 

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