ch4 - Advanced Topics in Equilibrium Statistical Mechanics...

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Advanced Topics in Equilibrium Statistical Mechanics Glenn Fredrickson Lecture 5 Reading: Chandler 7.2, 7.3, 7.4, 7.5 Recap: dilute to moderately dilute gases treated by virial expansions about ρ = 0( z = 0) state. easily extended to molecular fluids and dilute mixtures. Also applicable to dilute solutions of solute in solvents, including polymers and colloids, but not electrolyte solutions as we shall see. 4. Liquid State Theory A. Distribution functions and radiation scattering It is clear that virial expansions, i.e. expansions about the ρ = 0 ideal gas reference state, are not very useful in condensed liquid phases. Instead, we make note of the following central idea in liquid state theory known as the “van der Waals picture of liquids” and exploited heavily by Chandler, Andersen, and Weeks in the 1970’s: * Due to the harsh repulsions between molecules at close range, the structure and thermodynamics of liquids are determined primarily by packing eFects that depend on molecular shape. We shall see that in liquids of nearly spherical molecules like A or CH 4 , this suggests new expansion about a dense hard sphere reference system , where the expansion is in the strength of the relatively weak attractive interactions. 1
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To discuss “liquid structure”, we need to introduce the concept of reduced distribution functions. For simplicity, we work in the canonical ensemble. Re- call that at equilibrium, particle positions and momenta are independently dis- tributed, the former described by: P ( r N )= e βU ( r N ) R d r N e ( r N ) = 1 Q c e ( r N ) This is the joint probability distribution of observing particle 1 at r 1 , ..., and particle N at r N . Suppose we now integrate over r 3 , r 4 , ..., r N to create a “reduced distribution function” P (2) ( r 1 , r 2 Z d r 3 ... Z d r N P ( r N ) This is the joint probability distribution of observing particle 1 at r 1 and par- ticle 2 at r 2 . However, if we were observing a fluid, we would not be able to distinguish 1 and 2 from any other particles. Thus, ρ (2) ( r 1 , r 2 N ( N 1) P (2) ( r 1 , r 2 ) gives the joint probability distribution of observing any particle at r 1 and any other at r 2 . We can now generalize this to de±ne the n - particle reduced distribution func- tion by: ρ ( n ) ( r 1 ,..., r n N ! ( N n )! Z d r n +1 Z d r n +2 Z d r N P ( r N ) The functions are normalized such that Z d r n ρ ( n ) = N ! ( N n )! n = 1 : ρ (1) ( r 1 ) should clearly be independent of position r 1 in a fluid that is ho- mogeneous (uniform) on average. This is true in the gas and liquid states, but clearly not in a crystal phase. To be normalized properly, ρ (1) must correspond to the average density ρ (1) ( r 1 ρ N V n = 2 : ρ (2) ( r 1 , r 2 ) should depend only on | r 1 r 2 |≡ r 12 in a homogeneous, isotropic fluid (liquid or gas state). Hence ρ (2) ( r 1 , r 2 ρ (2) ( r 12 ).
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ch4 - Advanced Topics in Equilibrium Statistical Mechanics...

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