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Minimising helmholtz free energy

  • #1

Homework Statement


Show that the most probable energy minimises the Helmholtz free energy.


Homework Equations


F=E-TS(E) where S(E) is the entropy of te system of given energy E.

The Attempt at a Solution


Not sure how you would 'show' is ????

P(E) = 1/Z *weight funciton*exp(-beta E)
 

Answers and Replies

  • #2
1,838
7
The probablity that a system at temperature T is in a state r with energy E_r is given by exp[-E_r/(k T)]/Z

The probability that the system has an energy between
E and E + dE is thus the probability that the system is in any particular state with eneergy E times the number of states inside the energy range from E to E + dE. The probability density P(E) as a function of energy is thus:

P(E) = Omega(E)/(delta E) exp[-E/(k T)]/Z

where delta E is the energy resolution used to define Omega(E). If you take the logarithm, use that S = k Log(Omega), then you find the desired result.
 
  • #3
I'm sorry but I am really not following what you said. Why would you take the logarithm of the probability? and how does this relate to the free energy F=E-TS(E)?
 
  • #4
20
0
I'm sorry but I am really not following what you said. Why would you take the logarithm of the probability? and how does this relate to the free energy F=E-TS(E)?
If you take the log then E - TS pops out. And that's the free energy.
 
  • #5
1,838
7
The probability is maximal if the logarithm of the probability is maximal and vice versa. If you take the logarithm then you see that:

Log[P(E)]= -F(E) + constant

where the constant does not depend on E.
 

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