Statistical Mechanics - Physics at Oregon State University
Statistical Mechanics - Physics at Oregon State University
Statistical Mechanics - Physics at Oregon State University
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
8.4. DENSITY-MATRIX APPROACH (BRAGG-WILLIAMS APPROXIMATION.175<br />
particular direction our extremum is a minimum, and since we only have one<br />
extremum, it is a minimum no m<strong>at</strong>ter how we make changes.<br />
Finally, we find th<strong>at</strong> the entropy is<br />
<br />
1 + m 1 + m 1 − m 1 − m<br />
S = −NkB log( ) + log( ) (8.89)<br />
2 2 2 2<br />
At this point we still have m as an unknown parameter. The thermodynamic<br />
variables which are specified are h, T, and N. The minimiz<strong>at</strong>ion needed to get<br />
to the free energy is now<br />
G(h, T ) min<br />
m<br />
<br />
− 1<br />
2 JNqm2 <br />
1 + m<br />
− hNm + NkBT log(<br />
2<br />
1 + m 1 − m<br />
) +<br />
2 2<br />
(8.90)<br />
where m is restricted between −1 and +1. The value of m which minimizes this<br />
expression is then used to construct ρ and hence ρ. Once we have the density<br />
m<strong>at</strong>rix ρ, we are able to calcul<strong>at</strong>e all thermodynamic quantities of interest.<br />
The expression above looks like Landau theory, and, of course, it is. Landau<br />
theory has indeed a very good microscopic basis. We can use any parametriz<strong>at</strong>ion<br />
of the density m<strong>at</strong>rix, and the Gibbs like free energy has to be the minimum<br />
as a function of these parameters. The general form of the energy then follows,<br />
and for a magnetic model takes the form above.<br />
The slope of G(T, h, N; m) as a function of m is given by<br />
∂G<br />
1<br />
(T, h, N; m) = −NqJm − Nh +<br />
∂m 2 NkBT<br />
1 + m<br />
log (8.91)<br />
1 − m<br />
At m = −1 the slope is −∞ and <strong>at</strong> m = +1 the slope is +∞. Hence<br />
G(T, h, N; m) always has <strong>at</strong> least one minimum as a function of m for a value<br />
of m between −1 and +1. The value of m is found by equ<strong>at</strong>ing the deriv<strong>at</strong>ive<br />
of G to zero, which yields<br />
<br />
mqJ + h 1 + m<br />
= log<br />
(8.92)<br />
kBT 1 − m<br />
It is not hard<br />
<br />
to show th<strong>at</strong> this is equivalent to m = tanhβ(qJm + h). If we<br />
1+m<br />
have y = log 1−m we have (1 − m)e2y = 1 + m, and e2y − 1 = m(e2y + 1), or<br />
ey − e−y = m(ey + e−y ).<br />
If there is only one solution, this has to be the minimum value of G according<br />
to our observ<strong>at</strong>ion of the values of the deriv<strong>at</strong>ives <strong>at</strong> the end-points. Since the<br />
deriv<strong>at</strong>ive of G with respect to m is a continuous function of m we are also<br />
able to deduce th<strong>at</strong> there has to be an odd number of extrema between −1 and<br />
+1 due to the values of the slopes <strong>at</strong> the end-points. They have to be ordered<br />
minimum..maximum..minimum etc. Of course, one can always check this by<br />
calcul<strong>at</strong>ing the second deriv<strong>at</strong>ive<br />
∂2G NkBT<br />
= −NqJ +<br />
∂m2 1 − m2 log(<br />
(8.93)<br />
<br />
1 − m<br />
)<br />
2