Equal a priori probabilities postulate

The basic assumption in statistical theories is that the initially prepared state, in an indirect (true or apparent) unimolecular reaction A (E) —> products, prior to reaction has relaxed (via IVR) such that any distribution of the energy E over the internal degrees of freedom occurs with the same probability. This is illustrated in Fig. 7.3.1, where we have shown a constant energy surface in the phase space of a molecule. Note that the assumption is equivalent to the basic equal a priori probabilities postulate of statistical mechanics, for a microcanonical ensemble where every state within a narrow energy range is populated with the same probability. This uniform population of states describes the system regardless of where it is on the potential energy surface associated with the reaction. 

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. 

This supposition is somewhat distinct from our previous postulate of equal a priori probabilities ( 11 6) but may be combined with it to form a single postulate, in a more concrete form, as follows ihe equilibrium properties of a system of constant energy and volume are obtained by averaging over aU the accessible quanJlum states, each of these being given equal weight. 

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. 

An underlying idea in the statistical mechanical analysis of chemical systems is that all quantum states with the same energy are equally probable. For any one molecule, or any one quantum mechanical system, there is no a priori reason to favor one state of a given energy over another. This is a postulate we take it to hold so long as there are no violations in the predictions that follow from it. This idea was invoked in Chapter 1 in the statistical analysis that lead to the Maxwell-Boltzmann distribution law (Equation 1.11), and in Chapter 9 we found one direct experimental confirmation of the distribution law in the... 

There are two ways of constructing the probability distributions, of microscopic states in different ensembles. One approach is based on conservative, Hamiltonian dynamics and the fundamental postulate (in addition to energy conservation) that, in an isolated system, all states that have the same energy are equally likely. This a priori uniform measure on the canonical variables is justified by the Liouville theorem in classical systems [41]. Starting from these postulates, one can construct Py for the different ensembles by positing one additional assumption that for two systems in contact with each other, the effects of boundaries can be neglected [20]. This assumption implies that the density of states of the combined system is = Ei + 2) = i( i)f 2( 2)- The entropy is defined as S( ) = lnn( ). 

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