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Unformatted text preview: © M. S. Shell 2009 1/8 last modified 11/12/2009 Fluctuations ChE210A Distributions in the canonical ensemble In the last lecture, we found that the canonical ensemble gives the probabilities that micro- states are visited by the system at constant temperature: ℘ g = expG−¡¢ g £ ¤G¥,¦,§£ where ¨ is the index of a microstate and ¤ is the canonical partition function, ¤G¥,¦,§£ ≡ © expG−¡¢ ª £ all microstates ª at «,¬ which turns out to be related to the Helmholtz free energy: = −® ¯ ¥ ln ¤ Even though we can use the canonical ensemble to compute and other macroscopic proper- ties like ¥, °, and ± , we actually have access to much more information than we did from a purely macroscopic perspective. We know the microstate probabilities in detail. This enables us to compute all kinds of averages for the system at constant temperature. Consider the case when there is some property ² that can be associated with a microstate. ² might be: • the total energy • the kinetic or potential energy • one component of the potential energy, such as the electrostatic part • the distance between two particular atoms, like the end-to-end distance of a polymer molecule • some metric of geometric structure, such as how tetrahedral a water molecule is ar- ranged with respect to its four nearest neighbors • the number of intermolecular noncovalent bonds formed, such as hydrogen bonds • and so on and so forth… © M. S. Shell 2009 2/8 last modified 11/12/2009 The idea is that, given a particular microstate g , we can compute G for that microstate exactly and denote its value G ¡ . The average value of G at a certain temperature in the canonical ensemble is given by a weighted sum over the microstates using ℘ ¡ : ¢G£ = ¤ G ¡ ℘ ¡ all ¡ at ¥,¦ We already saw that the average total energy is given when G = § , and can also be expressed as a derivative of the partition function. The variance of G can also be computed: σ ¨ © = ¢ªG − ¢G£« © £ = ¢G © £ − 2¢G£¢G£ + ¢G£ © = ¢G © £ − ¢G£ © = ¬ ¤ G ¡ © ℘ ¡ all ¡ at ¥,¦ − ¬ ¤ G ¡ ℘ ¡ all ¡ at ¥,¦ © What if we are interested in the distribution of G , rather than just the average or variance? The probability that a particular value of G will be seen is given by the sum of probabilities of all the microstates that have the desired values of G ¡ . A simple way of expressing this is to sum over all microstates and count up their probabilities if and only if G ¡ = G : ℘ªG« = ¤℘ ¡ ® ¨ ¯ °¨ ¡ Here, the Kronecker delta function ensures that only those microstates with G ¡ = G contribute to the average....
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This note was uploaded on 12/29/2011 for the course CHE 210a taught by Professor Staff during the Fall '08 term at UCSB.
- Fall '08