Summary 23fromU=
∑
nnjεεj,but this sum can be neatlyperformed bylookingback at (2.15),
which can be re-expressed as
(U/N)=( 1 /Z)dZ/dβ=d(lnZ)/dβ (2. 27 )Note thatitis usually convenient to retainβas thevariablehere rather than to useT.
Method 3: The ‘royal route’ This one never fails, so it is worth knowing! The route
uses theHelmholtzfree energyF,definedasF =U−TS.The reasonfor the
importance of the method is twofold. First, the statistical calculation ofFturns out to
be extremely simple from the Boltzmann distribution. The calculation goes as follows
F=U−TS definition=∑
nnjεεj−kkkBTlnt∗ statisticalUandS=∑
nnjεεj−kkkBT(NlnN−∑
nnjlnnnj) using (2.6) with∑
nnj=N
=−NkkkBTlnZ,simply using(2.23) (2.28)Inthelast step one takes thelogarithmof(2.23) to obtainlnnnj=lnN−lnZ−εεj/kkkBT.
Everythingbut thelnZterm cancels,givingthe memorableandsimple result (2.28).
The second reason for following this route is that an expression forFinterms
of(T,V,N)isofimmediate useinthermodynamics since (T,V,N)are the natural
co-ordinatesforF(e.g.seeThermal PhysicsbyFinn, Chapter 10). In factdF =
−SdT−PdV+μdN, so that simple differentiation can giveS,Pandthechemical
potentialμat once; andmost other quantities can alsobederivedwithlittleeffort.
We shall see how these ideas work out in the next chapter.
2 6 Summary
Thischapterlays the groundworkfor the statisticalmethodwhichisdevelopedin
later chapters.
1 .Wefirst consider an assemblyofdistinguishableparticles, whichmakes counting
of microstates a straightforward operation. This corresponds to several important
physical situations, two of which follow in Chapter 3.
- Countingthemicrostatesleads to (2.3), a result worthknowing.
- The statistics of large numbers ensures that we can accuratelyapproximate the
average distribution by the most probable. - Use of‘undeterminedmultipliers’demonstrates that the resultingBoltzmann
distributionhas theformnnj=exp(α+βεεj).
5 .αrelates to the numberNofparticles,leading to thedefinition ofthe partition
function,Z.