ch2 - Advanced Topics in Equilibrium Statistical Mechanics...

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Advanced Topics in Equilibrium Statistical Mechanics Glenn Fredrickson 2. Classical Fluids A. Coarse-graining and the classical limit For concreteness, let’s now focus on a fluid phase of a simple monatomic sub- stance, e.g. Argon. We can write generally that Q ( N,V,T )= X ν e βEν where the sum is over all quantum states ν that the electrons and nuclei of the atoms can be in. This is a hard sum to do, since it requires solving a many-body quantum mechanics problem to ±nd the states. Instead, we use our physical intuition that the light electrons are moving much faster than the heavy nuclei, so that for a given nuclear con±guration, the quantum states of the electrons are nearly at local equilibrium (Born-Oppenheimer approximation). 1
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This suggests the rewriting Q = ν e βE ν R X i e R,i | {z ± R e β ˜ E R = e β ˜ E R where R is a sum over nuclear states (i.e. position and momentum – treat with classical mechanics) and i e R,i is a sum over electronic states (use quantum mechanics). The second line is a defnition of ˜ E R , the eFective energy (actually free energy, since it is β dependent) of just the nuclear states with the electronic states averaged out. This is an example of the notion of “coarse-graining” in statistical mechanics small objects (electrons, nuclei) “fundamental” interactions −→ coarse- grain larger objects (atoms) “eFective” interactions We have now made a lot of progress in getting rid oF the quantum mechanics in our problem, assuming that we can evaluate the eFective interactions between atoms that enter ˜ E R . Indeed, this is what “ab initio quantum chemistry” tries to do. There are now a large number of user-friendly software packages for computing the eFective interactions between atoms and molecules that are very useful in deducing classical descriptions of fluids. B. Classical phase space averages What now are the “classical states” R that we are supposed to sum over? In classical mechanics, the state of a particle is determined by specifying its position r =( x, y, z ) and momentum p p x ,p y z ). (We can then ±gure out its future state by integrating Newton’s equation F = m a = ˙p .) Thus, for an N -atom gas, R ( r 1 ,..., r N | {z ± ; p 1 p N | {z ± ) and R is a 6 N -dimensional “phase space”. We expect that Q = X R e β ˜ E R C Z d r 1 ... Z d r N Z d p 1 Z d p N e β ˜ E R where C is a constant prefactor. We adopt the shorthands: ( r 1 r N ; p 1 p N )=( r N , p N ) 2
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Note that we use a slightly diferent notation than that oF Chandler. d r 1 ...d r N d p 1 p N = d r N d p N Also, ˜ E R = ˜ E ( r N , p N ) H ( r N , p N ) where H is known as the “Hamiltonian”. The Hamiltonian (don’t conFuse with enthalpy!) is the sum oF the efective kinetic energy oF the atoms (nuclei) and the efective potential energy associated with their mutual interactions. We can separate the two as H ( r N , p N )= K ( p N )+ U ( r N ) ±inally, what is C ? There are various ways oF deriving it, the simplest oF
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ch2 - Advanced Topics in Equilibrium Statistical Mechanics...

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