1 / 5

Principles of Equal a Priori Probability

Principles of Equal a Priori Probability. All distributions of energy are equally probable If E = 5 and N = 5 then. 5. 5. 5. 4. 4. 4. 3. 3. 3. 2. 2. 2. 1. 1. 1. 0. 0. 0. All configurations have equal probability, but possible number of way (weight) is different.

seanf
Download Presentation

Principles of Equal a Priori Probability

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Principles of Equal a Priori Probability • All distributions of energy are equally probable • If E = 5 and N = 5 then 5 5 5 4 4 4 3 3 3 2 2 2 1 1 1 0 0 0 All configurations have equal probability, but possible number of way (weight) is different.

  2. 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}

  3. Dominating Configuration 5 5 5 4 4 4 3 3 3 2 2 2 1 1 1 0 0 0 W = 1 (5!/5!) 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

  4. 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.

  5. 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

More Related