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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6.3. MAXIMUM ENTROPY PRINCIPLE. 131<br />
for a closed system we get the same answers, which shows th<strong>at</strong> the procedure is<br />
OK.<br />
How to deal with discrete eigenvalues.<br />
So far we have described our st<strong>at</strong>e vectors in a Hilbert space with st<strong>at</strong>es<br />
<strong>at</strong> a given energy, volume, and particle number. Since the Hamiltonian is the<br />
oper<strong>at</strong>or th<strong>at</strong> governs all quantum mechanics, volume and particle number are<br />
ingredients of the Hamiltonian. But energy is the outcome! Therefore, it is<br />
much better to work in the more n<strong>at</strong>ural Hilbert space of st<strong>at</strong>e vectors th<strong>at</strong><br />
correspond to the Hamiltonian. The density oper<strong>at</strong>or in this case is<br />
ρ = Cδ(U − H) (6.73)<br />
where C is some constant, determined by 1 = CTr δ(U − H).<br />
This form of the density oper<strong>at</strong>or in the microcanonical ensemble implicitly<br />
assumes th<strong>at</strong> we take the thermodynamic limit. For a finite system the energy<br />
eigenvalues are discrete and the δ -function will yield a number of infinitely high<br />
peaks. Therefore we are not able to take the deriv<strong>at</strong>ive <br />
∂S<br />
∂U which is needed<br />
to define the temper<strong>at</strong>ure!<br />
We can, however, rewrite the delta function. Using a standard property we<br />
find:<br />
ρ = C U<br />
δ(<br />
V V<br />
H<br />
− ) (6.74)<br />
V<br />
and in the thermodynamic limit the spacing in the energy density U<br />
V goes to<br />
zero. But th<strong>at</strong> is the wrong order of limits, which can lead to problems!<br />
M<strong>at</strong>hem<strong>at</strong>ical details about delta function limits.<br />
In order to be more precise m<strong>at</strong>hem<strong>at</strong>ically, we need a limiting procedure<br />
for the delta-function. For example, one can choose a Gaussian approxim<strong>at</strong>ion<br />
ρ = C 1<br />
ɛ √ (U−H)2<br />
e− ɛ<br />
π 2 (6.75)<br />
which gives the correct result in the limit ɛ → 0. In order to evalu<strong>at</strong>e the trace<br />
of ρlogρ we now have to sum over st<strong>at</strong>es with all possible values of the energy.<br />
Clearly we will get a value for the entropy which is defined for all values of U,<br />
and the partial deriv<strong>at</strong>ive of the entropy with respect to U will give a good value<br />
for the inverse temper<strong>at</strong>ure. But now we have two limits to deal with: N → ∞<br />
and ɛ → 0. The thermodynamic limit N → ∞ has to be taken after all physical<br />
st<strong>at</strong>e variables have been set equal to the values needed in the problem. For<br />
example, if needed the limit T → 0 has to be taken first. The limit ɛ → 0<br />
is not rel<strong>at</strong>ed to a physical st<strong>at</strong>e variable. This limiting procedure is a purely<br />
m<strong>at</strong>hem<strong>at</strong>ical trick. Therefore, the order of taking limits is first to assign all