59-441/541 Final Exam

1. 59-441/541 Final Exam • Time: December 13, from 2 to 5 PM • Location: DH room 253 (our lecture room) • Content: It covers chapters 12, 13, 14, 15, 16, 18, 19 and 20. • Formula sheet and constants will be provided. • No problem requiring integration by parts.

2. Tentative format of the final exam • Concepts (10%). • Analysis and qualitative discussion without necessary calculations (20%). • Problems involving accurate calculations (70%). There are possibly 6 questions in this category. • Bonus question (10%**). • ** For the individual who missed a midterm exam, your final grade will be calculated as 100* (40+midterm)% + bonus*40%

3. 20.5 The connection to statistical thermodynamics • The entropy is define as • Then

4. A disordered system is likely to be in any number of different quantum states. If Nj = 1 for N different states and Nj= 0 for all other available states, • The above function is positive and increases with increasing N. • Associating Nj/N with the probability pj • The expected amount of information we would gain is a measure of our lack of knowledge of the state of the system. • Negative entropy (negentropy)

5. The boltzmann distribution for non-degenerate energy state • Where

6. Summary • Information theory is an extension of thermodynamics and probability theory. Much of the subject is associated with the names of Brillouin and Shannon. It was originally concerned with passing messages on telecommunication systems and with assessing the efficiency of codes. Today it is applied to a wide range of problems, ranging from the analysis of language to the design of computers. • In this theory the word ‘information’ is used in a special sense. Sup pose that we are initially faced with a problem about which we have no ‘information’ and that there are P possible answers. When we are given some ‘information’ this has the effect of reducing the number of possible answers and if we are given enough ‘information’ we may get to a unique answer. The effect of increased information is thus to reduce the uncertainty about a situation. In a sense, therefore, information is the antithesis of entropy since entropy is a measure of the randomness or disorder of a system. This contrast led to the coining of the word negentropy to describe information. • The basic unit of information theory is the bit—a shortened form of ‘binary digit’.

7. As we are given more information, the situation becomes more certain. • For example, if one is given a playing card face down without any information, it could be any one of 52; if one is then told that it is an ace, it could be any one of 4; if told that it is also a spade, one knows for certain which card one has. • In general, to determine which of the P possible outcomes is realized, the required information is defined as H = K ln P

8. 20.4) consider a loaded die. Let the probabilities of throwing a number 1, 2,3,4,5, and 6 be 0.1, 0.1, 0.1, 0.1, 0.1 and 0.5 respectively. Calculate the uncertainty H in bits for a throw of the die. What would H be if the die were ``honest``? • Solution • For honest case

9. Fermions and Bosons • Calculate the total number of elementary particles making up this molecule or atom. If the summation of electron, proton and neutron is even, then the molecule or atom is Boson, obeying Bose-Einstein statistics.

10. Overview (I) • Statistics for distinguishable particles • Statistics for non-distinguishable particles Bose-Einstein statistics Fermi-Dirac statistics • Calculation of number of quantum states as a function of energy. • Calculation of partition function for different systems.

11. Overview (II) • Distribution functions for different types of statistics, becoming the same in diluted gas systems. • Calculation of various thermal quantities, U, F, S, Cv. • Recognize the importance of physical properties, say, whether the system is in a solid, liquid or gas state. • Diatomic or monatomic system.