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Fluctuations - Fluctuations ChE210A D istributions in the...

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© 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: g3040 = expg4666−g2010g1831 g3040 g4667 g1843g4666g1846,g1848,g1840g4667 where g1865 is the index of a microstate and g1843 is the canonical partition function, g1843g4666g1846,g1848,g1840g4667≡ g3533 expg4666−g2010g1831 g3041 g4667 all microstates g3041 at g3023,g3015 which turns out to be related to the Helmholtz free energy: g1827=−g1863 g3003 g1846lng1843 Even though we can use the canonical ensemble to compute g1827 and other macroscopic proper- ties like g1846,g1842, and g2020 , 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 g1850 that can be associated with a microstate. g1850 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…
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© M. S. Shell 2009 2/8 last modified 11/12/2009 The idea is that, given a particular microstate g1866 , we can compute g1850 for that microstate exactly and denote its value g1850 g3041 . The average value of g1850 at a certain temperature in the canonical ensemble is given by a weighted sum over the microstates using g3041 : g1731g1850g1732= g3533 g1850 g3041 g3041 all g3041 at g3023,g3015 We already saw that the average total energy is given when g1850=g1831 , and can also be expressed as a derivative of the partition function. The variance of g1850 can also be computed: σ g3025 g2870 =g1731g4666g1850−g1731g1850g1732g4667 g2870 g1732 =g1731g1850 g2870 g1732−2g1731g1850g1732g1731g1850g1732+g1731g1850g1732 g2870 =g1731g1850 g2870 g1732−g1731g1850g1732 g2870 =g3437 g3533 g1850 g3041 g2870 g3041 all g3041 at g3023,g3015 g3441−g3437 g3533 g1850 g3041 g3041 all g3041 at g3023,g3015 g3441 g2870 What if we are interested in the distribution of g1850 , rather than just the average or variance? The probability that a particular value of g1850 will be seen is given by the sum of probabilities of all the microstates that have the desired values of g1850 g3041 . A simple way of expressing this is to sum over all microstates and count up their probabilities if and only if g1850 g3041 =g1850 : ℘g4666g1850g4667=g3533℘ g3041 g2012 g3025 g3289 g2880g3025 g3041 Here, the Kronecker delta function ensures that only those microstates with g1850 g3041 =g1850 contribute to the average. The canonical distribution of energies Let’s take as an example the total energy: ℘g4666g1831g4667=g3533℘ g3041 g2012 g3006 g3289 g2880g3006 g3041 Substituting in the canonical expression for g3041 : ℘g4666g1831g4667=g3533 expg4666−g2010g1831 g3041 g4667 g1843g4666g1846,g1848,g1840g4667 g2012 g3006 g3289 g2880g3006 g3041
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© M. S. Shell 2009 3/8 last modified 11/12/2009 Since the only nonzero terms in the sum are those for which g1831 g3041 =g1831 , we can replace g1831 g3041 with
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