Principle of equal a priori probabilities

As thermodynamics required postulates or laws, so does statistical mechanics. Gibbs postulates which define statistical mechanics are (1) Thermodynamic quantities can be mapped onto averages over all possible microstates consistent with the few macrosopic parameters required to specify the state of the system (here, NVE). (2) We construct the averages using an ensemble . An ensemble is a collection of systems identical on the macroscopic level but different on the microscopic level. (3) The ensemble members obey the principle of equal a priori probability . That is, no one ensemble member is more important or probable than another. 

Note that we have provided theoretical justification for the principle of equal a priori probabilities that was adopted earlier. We see that this is the best guess that can be made on our very limited data. The key point is that rather than asserting equal a priori probabilities as an article of faith we have shown that it is the best distribution that we can possibly assign in the sense that it most faithfully represents our degree of ignorance concerning the various outcomes. 

If dFJdpa 0, then / = 1 is not a solution to the Liouville equation (9). Consequently the principle of equal a priori probabilities is not valid, in the system phase space. This is not surprising equal volumes in the phase space for the universe do not project onto equal volumes in a subspace. In the past there have been some attempts to describe the non-equilibrium steady state by a procedure similar to that used for equilibrium, that is, by treating the steady state as a state of maximum probability... 

There is still the problem that the set of occupation numbers Nj can be anything, according to the principle of equal a priori probabilities. In fact, the total possible arrangements are truly astronomical, but we will ignore all but one the most probable arrangement. 

Principle of equal a priori probabilities Idea that all accessible states are equally probable for an isolated system with fixed volume, energy, and particle numbers. 

According to Eq. (33), all states inside the energy shell are equally probable. In other words, the microcanonical ensemble represents a uniform distribution over the phase points belonging to the energy shell. Such a distribution is often referred to as a mathematical statement of the principle of equal a priori probabilities. According to this principle, the system spends an equal amount of time, over a long time period, in each of the available classical states. This statement is called the ergodic hypothesis. 

Boltzmann s tombstone in Vienna bears the famous formula 5 = k log W (W = Wahrscheinlichkeit—probability) that was a signature of his audacious concepts. The alternative formula (13.69) (which reduces to Boltzmann s in the limit of equal a priori probabilities pa) was ultimately developed by Gibbs, Shannon, and others in a more general and productive way (see Sidebar 13.4), but the key step of employing probability to trump Newtonian determinism was his. Boltzmann was long identified with efforts to establish the //-theorem and Boltzmann equation within the context of classical mechanics, but each such effort to justify the second law (or existence of atoms) in the strict framework of Newtonian dynamics proved futile. Boltzmann s deep intuition to elevate probability to a primary physical principle therefore played a key role in efforts to find improved foundation for atomic and molecular concepts in the pre-quantum era. 

The first assumption, known as the postulate of equal a priori probabilities, sounds quite reasonable, because there is nothing we can think of which favors one particular microstate over another if both have the same energy. The second assumptions corresponds to a principle of least constraint, i.e. a subsystem microstate v is more likely than another if the by comparison huge environment suffers a smaller reduction of its available microstates. 

The idea of equal probabilities has been elevated by Laplace510 to the rank of a philosophical principle, called principle of insufficient reason . Like many philosophical principles it leaves the essential question unanswered How do I select the elementary events to which equal a priori probabilities are to be assigned In textbook problems about tossing dice or drawing cards it is obvious what the author has in mind. One knows that he is concerned with the mathematics of step b and that the dice and cards merely serve as a ritual way of defining an a priori distribution. In actual applications, however, step a cannot be dismissed so cavalierly. 

Worth mentioning is also the associated and intriguing means of irreversibility, or better, of the entropy increase, which are dynamical and which often lie outside the scope of standard edification. Notwithstanding, the entropy as the maximum property of equilibrium states is hardly understandable unless linked with the dynamical considerations. The equal a priori probability of states is already in the form of a symmetry principle because entropy depends symmetrically on all permissible states. TTie particular function of entropy is determined completely then by symmetry over the set of states and by the requirement of extensivity. Consequently it can be even shown that a full thermodynamic (heat) theory can be formulated with the heat, being totally absent. Nonetheless, the familiar central formulas, such as dlS = dQ/T, remains lawful although dQ does not acquire to have the significance of energy. Nevertheless, for the standard thermophysical studies the classical treatises are still of the daily use so that their basic principles and the extent of applicability are worthy of brief recapitulation. 

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