ch6 - Advanced Topics in Equilibrium Statistical Mechanics...

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Advanced Topics in Equilibrium Statistical Mechanics Glenn Fredrickson 6. Statistical Mechanics of Classical Fields Up to this point we have talked about how to set up and perform equilibrium statistical mechanical calculations of systems with a Fnite set of phase space coordinates, e.g., { r N , p N } . However, often we are faced with objects, e.g., polymers, that are better described by continuous Felds and have an inFnite number of degrees of freedom. We must learn how to do statistical mechanics for such objects. A. Transverse Oscillations of a Stretched String To illustrate the approach, consider a string of a musical instrument that is stretched between two supports: On a nanoscale , the same situation can be created by pulling on a DNA molecule with optical tweezers. One obvious question is the following: What is the equilibrium distribution of string shapes if the string is equilibrated with its environment at temperature T ? There are two ways to tackle this problem. The Frst would be to approximate a continuous elastic string by a bead-spring discrete chain: Here the bead positions r N are used to approximate the string shape and 1
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each bead has a mass m = σ · a where a is the average bead spacing and σ is the mass per length of string. The tension , τ , in the string would deFne a Hooke’s Law spring constant k = τ a , so U ( r N )= N X i ( r i +1 r i ) 2 1 2 k is the potential energy.
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ch6 - Advanced Topics in Equilibrium Statistical Mechanics...

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