ps6 - Statistical Physics I(8.044 Spring 2009 Assignment 6...

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Statistical Physics I (8.044) Spring 2009 Assignment 6 March 18, 2009 Due April 1, 2009 Please remember to put your name and section number at the top of what you turn in. Readings The reading for Chapter IV of 8.044 is Baierlein, Chapters 2 and 4.3 and the Notes on the Microcanonical Ensemble available on the web page. The reading for Chapter V of 8.044 is Adkins, Chapters 4, 5.1-5.5 and 7, and Baierlein Chapter 3.1-3.4 and 10. Of these, Adkins Chapter 7 and Baierlein Chapter 10 are not needed for this problem set. Although Adkins Chapters 8 and 9 are not required reading for 8.044, they are rec- ommended as a source of good examples. Problem Set 6 1. A Strange Chain (12 points) l F F A one dimensional chain is made up of N identical elements, each of length l . The angle between successive elements can be either 0 0 or 180 0 , but there is no difference in internal energy between these two possibilities. For the sake of counting, one can think of each element as either pointing to the right (+) or to the left (-). Then one has N = n + + n - L = l ( n + - n - ) = l (2 n + - N ) (a) Use the microcanonical ensemble to find the entropy as a function of N and n + , S ( N, n + ). 1
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(b) Find an expression for the tension in the chain as a function of T , N , and n + , F ( T, N, n + ). Notice the strange fact that there is tension in the chain even though there is no energy required to reorient two neighboring elements! The “restoring force” in this problem is generated by entropy considerations alone. This is not simply an academic oddity, however. This system is used as a model for elastic polymers such as rubber. (c) Rearrange the expression from b) to give the length as a function of N , T , and F . (d) Use the result for the high temperature behavior from c) to find an expression for the thermal expansion coefficient α L - 1 ( ∂L/∂T ) F . Note the sign. Find a stout rubber band. Hang a weight from it so that its length is extended by about a factor of two. Now heat the rubber band (a hair drier works well here) and see if the weight goes up or down. 2. Classical Harmonic Oscillators (12 points) Consider a collection of N identical harmonic oscillators with negligible (but non-zero) interactions. In a microcanonical ensemble with energy E, the system is on a surface in phase space given by N X i =1 p 2 i 2 m + 2 q 2 i 2 !
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