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Principles of Equal a Priori Probability All distributions of energy are equally probable If E = 5 and N = 5 then 5 5 5 4 3 2 1 0 4 3 2 1 0 4 3 2 1 0 All configurations have equal probability, but possible number of way (weight) is different. A Dominating Configuration For large number of molecules and large number of energy levels, there is a dominating configuration. The weight of the dominating configuration is much more larger than the other configurations. Wi Configurations {ni} Dominating Configuration W = 1 (5!/5!) 5 5 5 4 3 2 1 0 4 3 2 1 0 4 3 2 1 0 W = 20 (5!/3!) W = 5 (5!/4!) Difference in W becomes larger when N is increased ! In molecular systems (N~1023) considering the most dominant configuration is enough for average 8.4 The most probable distribution, equilibrium distribution, and Boltzmann distribution 8.4.1 The principle of equal a priori probabilities Statistical thermodynamics is based on the fundamental assumption that all possible configurations of a given system, which satisfy the given boundary conditions such as temperature, volume and number of particles, are equally likely to occur. ( N ,U ,V ) WD D Example Consider the orientations of three unconstrained and distinguishable spin-1/2 particles. What is the probability that two are spin up and one spin down at any instant? Solution Of the eight possible spin configurations for the system, ↑↑↑ ↑↑↓ ↑↓↑ ↓↑↑ ↑↓↓ ↓↑↓ ↓↓↑ ↓↓↓ The second, third, and fourth comprise the subset "two up and one down". Therefore, the probability for this particular configuration is P = 3/8