10 Basic ideas
another wayofcalculatingit)is to appreciate that microstates withUUAB= 3 ε,2ε,ε,
0 are now accessible in addition to 4ε; and correspondingly forUUUCD.Sowecan
observefrom this examplethatinternaladjustments ofanisolatedsystem towards
overallequilibriumincreasethe number ofaccessiblemicrostates.
1 .7 Statistical entropy and microstates
First, a wordabout entropy. Entropyisintroducedinthermodynamics as that rather
shadowyfunction ofthe state ofasystem associatedwiththe secondlaw ofther-
modynamics. The essence of the idea concerns the direction (the ‘arrow of time’) of
spontaneous or naturalprocesses,i.e. processes whicharefoundto occurin practice.
Aprettyexampleisthemixingofsugar andsand. Start withadishcontaininga
discrete pile of each substance. The sugar and sand may then be mixed by stirring,
but theinverse process ofre-separating thesubstancesby un-stirring (stirringinthe
oppositedirection?)doesnotinpracticehappen. Suchun-mixingcanonlybeachieved
with great ingenuity. In a natural process, the second law tells us that the entropySof
theuniverse (or ofanyisolatedsystem) neverdecreases.Andinthemixing process the
entropyincreases. (The‘greatingenuity’ would involve alargerincrease ofentropy
somewhere else in the universe.) All this is a statement of probability, rather than
ofnecessity–itispossibleinprinciple to separate themixedsugar andsandby
stirring, but it is almost infinitelyimprobable. And thermodynamics is the science of
the probable!
Statisticalphysics enables one todiscuss entropyin terms ofprobabilityinadirect
and simple way. We shall adopt in this book a statistical definition for the entropyof
an isolated system
S=kkkBln (1.5)withkkkBequalto Boltzmann’s constant 1.3 8 × 10 −^23 JK−^1 .Thisdefinitionhas an
old history,originatingfrom Boltzmann’s workon thekinetictheoryofgasesin
the last century, and (1. 5 ) appears on Boltzmann’s tombstone. The relationship was
developedfurtherbyPlanckinhis studies ofheat radiation–the start ofthe quantum
theory.
Logically perhaps (1. 5 ) is a derivedresultof statistical physics. However, it is
sucha centralidealthatitis sensibletointroduceitatthis stage, andto treatitasa
definitionof entropyfrom the outset. We shallgraduallysee that theSso definedhas
all the properties of the usual thermodynamic entropy.
Whatwehaveobservedsofar about thebehaviour ofis certainlyconsistent with
this relation to entropy.
- As notedabove,for anisolatedsystem a naturalprocess,i.e. one whichspon-
taneously occurs as the system attains overall equilibrium, is precisely one in
whichthethermodynamic entropyincreases. Andwehave seeninthe exampleof
section 1. 6 thatalsoincreasesinthistype ofprocess. Hence adirect relation