How Does the Most Probable Energy Minimize Helmholtz Free Energy?

[captainjack2000]

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)

 

[Count Iblis]

The probability 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.

 

[captainjack2000]

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)?

 

[Welshy]

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.

 

[Count Iblis]

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.

 

[gravityandlev]

I wrote a post on this subject fairly recently that explains minimization of free energy slowly, and in detail, using a specific example. Maybe you'll find it useful:
http://gravityandlevity.wordpress.c...e-plays-skee-ball-the-meaning-of-free-energy/