1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Reif Ch7, Decomposition of partition function

  1. May 5, 2012 #1
    1. The problem statement, all variables and given/known data

    For a system A consists of two parts A' and A'' which interact only weakly with each other, if the states of A' and A'' are labeled respectively by r and s, then a state of A can be specified by the pair of numbers r,s and its corresponding energy [itex]E_{rs}[/itex] is simply additive, i.e.,
    [itex]E_{rs}[/itex] = [itex]E^{'}_{r}[/itex] + [itex]E^{''}_{s}[/itex]

    The partition function Z for the total system A is a sum over all states labeled by rs, i.e.,

    Z=[itex]\sum_{r,s}e^{-\beta(E^{'}_{r}+E^{''}_{s})}[/itex] = [itex]\sum_{r,s}e^{-\beta E^{'}_{r}} \ e^{E^{''}_{s}}[/itex] = ([itex]\sum_{r}e^{-\beta E^{'}_{r}}[/itex])([itex]\sum_{r}e^{-\beta E^{''}_{s}}[/itex]) = [itex]Z^{'}Z^{''}[/itex]

    My question is: how the sum of product [itex]\sum ()()[/itex] is converted to product of sum ([itex]\sum[/itex])([itex]\sum[/itex]), they are not generally equal

    Last edited: May 5, 2012
  2. jcsd
  3. May 5, 2012 #2


    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    Do you understand why you can do this?
    $$\sum_r \big[\sum_s e^{-\beta E_r}e^{-\beta E_s}\big] = \sum_r \big[ e^{-\beta E_r}\sum_s e^{-\beta E_s}\big] $$
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Similar Threads for Reif Decomposition partition Date
Decomposition of linearly polarized field MRI Feb 27, 2018
Mathematical Issues with Reif Thermodynamics Problem (BYO Math Chops) Apr 8, 2013
Reif 2.9 Feb 10, 2011
Partition function problem (Reif) Apr 24, 2008
Reif 1.20 Apr 7, 2008