html5-img
1 / 8

Information Theory

Information Theory. PHYS 4315 R. S. Rubins, Fall 2009. Lack of Information. Entropy S is a measure of the randomness (or disorder) of a system. A quantum system in its single lowest state is in a state of perfect order: S=0 (3 rd law).

kyrie
Download Presentation

Information Theory

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. Information Theory PHYS 4315 R. S. Rubins, Fall 2009

  2. Lack of Information • Entropy S is a measure of the randomness (or disorder) of a system. • A quantum system in its single lowest state is in a state of perfect order: S=0 (3rd law). • A system at higher temperatures may be in one of many quantum states, so that there is a lack of information about the exact state of the system. • The greater the lack of information, the greater is the disorder. • Thus, a disordered system is one about which we lack complete information. • Information theory (Shannon, 1948) provides a mathematical measure of the lack of information, which may be linked to the entropy.

  3. Missing Information H • For an experiment with n possible outcomes p1, p2,… pn, Shannon introduced a function H(p1, p2,… pn), which quantitatively measures the missing information associated with the set of probabilities. • Three conditions are needed to specify H to within a constant factor. • 1. H is a continuous function of pi. • 2. If all the pi are equal, then pi = 1/n, and H is a monotonically increasing function of n, since the number of possibilities increases with n. • 3. If the possible outcomes of an experiment depend on the outcomes of n subsidiary experiments, then H is the sum of the uncertainties of the subsidiary experiments. • With these assumptions, H was found to be proportional to the entropy S = – k r pr ln pr.

  4. Example of Sum of Uncertainties Single experiment, using H = – K(pr lnpr). H(1/2,1/3,1/6) = – K[(1/2) ln(1/2) + (1/3) ln(1/3) + (1/6) ln(1/6)] = K[(1/2)ln2 + (1/3)ln3 + (1/6)ln6] = K[(2/3)ln2 + (1/2)ln3] = 1.01 K. Two successive experiments H(1/2,1/2) = – K[(1/2) ln(1/2) + (1/2) ln(1/2)] = K ln2. (1/2)H(2/3,1/3) = K[(– (1/3)ln2 + (1/3)ln3 + (1/6)ln3] Thus, H(1/2,1/2) + (1/2)H(2/3,1/3) = K{[1 – (1/3)]ln2 + [(1/3) + (1/6)]ln3} = K[(2/3)ln2 + (1/2)ln3] = 1.01 K.

  5. Shannon’s Calculation 1 The simplest choice of continuous function (Condition 1)is For simplicity, consider the case of equal probabilities; i.e. pi = 1/n. Since H is a monotonically increasing function of n (Condition 2), For two successive experiments (Condition 3)

  6. Shannon’s Calculation 2 Since H(1/n,…1/n) = n f(1/r), . Letting R = 1/r and S =1/s,

  7. Shannon’s Calculation 3 Since g(R) + g(S) = g(R,S) and , so that Thus, so that Since R = 1/r, .

  8. Shannon’s Calculation 4 For r =1, the result is certain, so that H(r) = f(1/r) = 0. Thus, C =0, so that Now d/dn(– A ln n) must be positive (Condition 2), so that A must be negative. Letting K = – A, and p = pi =1/n, Thus, the missing information function H is given by . With K replaced by Boltzmann’s constant k, H equals S (entropy).

More Related