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Principle of equal a priori probability

Principle of equal a priori probabilities: An isolated system, with N,V,E, has equal probability to be in any of the W (N,V, E ) quantum states or Each and every one of the W (N,V, E ) quantum states is represented with equally probability. Principle of equal a priori probability.

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Principle of equal a priori probability

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  1. Principle of equal a priori probabilities: An isolated system, with N,V,E, has equal probability to be in any of the W(N,V,E) quantum states or Each and every one of the W(N,V,E) quantum states is represented with equally probability Principle of equal a priori probability

  2. Temperature

  3. A I II III heat bath(T) E V=AxV N=AxN One possible state of ensemble{ 1 2 3 l El E1 E2 … El al a1 a2…al A=Sjaj E=SjajEj Canonical N,V,T There are A identical replicas with same N,V, and T Canonical ensembles Each microsystem has an energy value Ej(N,V). Each Ejis W(Ej) times degenerate alnumber of systems in statel , occupation number Set of {al } = distribution = a describes the state of the Ensemble

  4. heat bath(T)EB N,V,E E=Ej+EB Canonical as subsystem of microcanocal Ej(N,V)

  5. Principle of equal priori probability  every {a} is equally probable Distributions How many times a particular distribution can be found in the ENSEMBLE? Since there are Asystems (each with energy Ej)  there are A number distinguishable particles that can be distributed according to their alvalue  a1in group 1, a2 in group 2, alin group l… We know the answer to this one, right?

  6. 3 systems Example (Nash)

  7. Example (Nash)II

  8. systems Example (Nash)III For E=1000 and A=1000 SW=10600 there are 1070 atoms in the galaxy For larger and larger A values, the ratio Wn/ Wmax=An

  9. Probability of finding a system in Ej is obtained by Probability

  10. Distribution for max W(a)

  11. Maximazing lnW(a)

  12. Evaluating the multiplier Probability of finding the quantum state with Ej at a given N,V From here we can calculate all other mechanical thermodynamic properties

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