Statistical Mechanics - Physics at Oregon State University
Statistical Mechanics - Physics at Oregon State University
Statistical Mechanics - Physics at Oregon State University
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3.4. GRAND PARTITION FUNCTION. 51<br />
The partial deriv<strong>at</strong>ives are − µR<br />
TR<br />
and 1<br />
TR<br />
for the reservoir. Since we assume<br />
thermal and diffusive equilibrium, both µ and T are the same for system and<br />
reservoir, TR = TS = T and µR = µS = µ, and hence we the probability of<br />
finding the system with temper<strong>at</strong>ure T and chemical potential µ in a quantum<br />
st<strong>at</strong>e s is equal to<br />
P rob(s) = 1 1<br />
k e B T<br />
Z (µns−ɛs)<br />
(3.36)<br />
The exponential factor is called the Gibbs factor in this case and is an obvious<br />
generaliz<strong>at</strong>ion of the Boltzmann factor. The normaliz<strong>at</strong>ion factor Z is called the<br />
grand partition function or grand sum and is defined by<br />
Z(T, V, µ) = <br />
st<strong>at</strong>es<br />
e 1<br />
k B T (µns−ɛs)<br />
(3.37)<br />
The probabilities are again a r<strong>at</strong>io of an exponential factor and a normaliz<strong>at</strong>ion<br />
sum. We saw before th<strong>at</strong> the partition function contains all inform<strong>at</strong>ion needed<br />
to describe a system. The grand partition function also contains all inform<strong>at</strong>ion<br />
needed. Sometimes it is easier to work with the partition function, sometimes<br />
with the grand partition function.<br />
Since both partition functions contain all inform<strong>at</strong>ion, it must be possible<br />
to construct one from the other. The grand partition function is rel<strong>at</strong>ed to the<br />
partition function by a sum over particle numbers:<br />
Z(T, V, µ) = <br />
ˆN<br />
e µ ˆ N<br />
k B T Z(T, V, ˆ N) (3.38)<br />
This follows when we split the summ<strong>at</strong>ion over all st<strong>at</strong>es in the grand sum into<br />
two parts, a summ<strong>at</strong>ion over all possible number of particles and for each term in<br />
th<strong>at</strong> summ<strong>at</strong>ion a sum over all st<strong>at</strong>es with the corresponding number of particles.<br />
The l<strong>at</strong>ter gives the partition function. In a l<strong>at</strong>er chapter we will give a more<br />
formal justific<strong>at</strong>ion why this is allowed even in systems where the number of<br />
particles is not conserved. It is rel<strong>at</strong>ed to the invariance of the trace oper<strong>at</strong>or in<br />
quantum mechanics. The transform<strong>at</strong>ion above is a standard Laplace transform,<br />
and can be inverted. This requires, however, a analytic continu<strong>at</strong>ion of the<br />
grand partition function in the complex µ plane. M<strong>at</strong>hem<strong>at</strong>ically, th<strong>at</strong> is not a<br />
problem, but wh<strong>at</strong> does th<strong>at</strong> mean in terms of physics?<br />
How do we extract inform<strong>at</strong>ion from the grand partition function? One case<br />
is easy. The average number of particles in the system, < ns >= nsP rob(s):<br />
<br />
∂Z<br />
∂µ<br />
T,V<br />
= 1<br />
kBT<br />
<br />
s<br />
nse 1<br />
k B T (µns−ɛs) = 1<br />
kBT Z < ns > (3.39)<br />
Therefore, the average number of particles ( now described by the function<br />
N(T, V, µ) ), is directly rel<strong>at</strong>ed to the grand partition function by<br />
<br />
∂ log(Z)<br />
N(T, V, µ) = kBT<br />
(3.40)<br />
∂µ<br />
T,V