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.
50 CHAPTER 3. VARIABLE NUMBER OF PARTICLES<br />
chemical reactions! How do we describe the equilibrium st<strong>at</strong>e of a system where<br />
the values of µ, T, and V are specified. A simple observ<strong>at</strong>ion is based on the<br />
rel<strong>at</strong>ions between the differentials. If we define Ω = F − µN we find<br />
dΩ = −SdT − pdV − Ndµ (3.33)<br />
Hence Ω has as n<strong>at</strong>ural variables T,V, and µ. This is exactly wh<strong>at</strong> we need.<br />
Also, for a process <strong>at</strong> constant µ and T the change in Ω is equal to the work<br />
performed by the system. In other words, Ω measures the amount of energy<br />
available in a system to do work <strong>at</strong> constant chemical potential and temper<strong>at</strong>ure.<br />
The free energy Ω is called the grand potential.<br />
3.4 Grand partition function.<br />
One way of calcul<strong>at</strong>ing Ω is via the Helmholtz free energy F(T,V,N), obtained<br />
from the partition function. If we solve µ = <br />
∂F<br />
∂N for N(µ, T, V ) we can<br />
T,V<br />
construct Ω(µ, T, V ) = F (T, V, N(µ, T, V )) − µN(µ, T, V ). This assumes th<strong>at</strong><br />
we are able to invert the formula for µ(T, V, N) and solve for N uniquely. Th<strong>at</strong><br />
this is possible is by no means obvious from a m<strong>at</strong>hem<strong>at</strong>ical point of view only.<br />
Of course, in thermodynamics we know th<strong>at</strong> the rel<strong>at</strong>ion between µ and N<br />
is never decreasing, and hence if we did thermodynamics there would be no<br />
problem. Here we are doing st<strong>at</strong>istical mechanics, though, and we have to show<br />
th<strong>at</strong> the st<strong>at</strong>istical mechanical description is equivalent to the thermodynamical<br />
one.<br />
We will now describe a procedure to calcul<strong>at</strong>e the grand potential directly,<br />
and hence in this procedure we find the inverse rel<strong>at</strong>ion for N. The procedure<br />
is the analogue of wh<strong>at</strong> we did to construct the partition function. It is a very<br />
general construction, which works for all types of free energies! Consider a<br />
system S in thermal and diffusive equilibrium with a very large reservoir R.<br />
Both energy and particles can be exchange. The total energy U0 of R + S<br />
and the total number of particles N0 in R + S are constant. Therefore, the<br />
probability of finding the system S in a particular quantum st<strong>at</strong>e s with energy<br />
ɛs and number of particles ns is again proportional to the number of quantum<br />
st<strong>at</strong>es gR(U0 − ɛs, N0 − ns) the reservoir has available with energy U0 − ɛs and<br />
number of particles N0 − ns. Or<br />
P rob(1)<br />
P rob(2) = gR(U0 − ɛ1, N0 − n1)<br />
gR(U0 − ɛ2, N0 − n2)<br />
(3.34)<br />
The multiplicity function is the exponent of the entropy. If we assume the the<br />
reservoir is very large, the presence of the system causes only small perturb<strong>at</strong>ions,<br />
and in first order we find<br />
P rob(1) 1<br />
= exp( (n2 − n1)<br />
P rob(2) kB<br />
<br />
∂SR<br />
+<br />
∂N U,V<br />
1<br />
(ɛ2 − ɛ1)<br />
kB<br />
<br />
∂SR<br />
) (3.35)<br />
∂U V,N