Chapter 0
Background o f th e proble m
The subjec t o f thi s boo k i s th e stud y o f a clas s o f exactl y solvabl e lattic e
models whic h aris e i n statistica l mechanics . Befor e embarkin g upo n thei r
formulation, w e wish in this chapte r t o giv e the reade r a rough ide a of wha t
statistical mechanic s an d th e lattic e model s are all about. O f course, we are
by no means aiming at somethin g like a rapid cours e in statistical mechanic s
for undergraduat e physic s students . Al l we wish i s provide a minima l back -
ground to motivate the reader s o he can proceed t o the subsequent chapters .
0.1 Statistica l mechanic s
In classica l mechanics , w e stud y th e motion s o f suc h object s a s massiv e
particles o r rigi d bodie s governe d b y Newton' s equatio n o f motion . Th e
states o f suc h system s ar e specifie d b y a finit e numbe r o f quantities . Fo r
instance the motion ofTV particles in our three dimensional space is described
by67 V parameters— th e positio n {xi,y%,Zi) and th e momentu m (pf,pf,pf )
of the i-t h particle , i = 1, •, N. I n thi s sens e w e are dealin g wit h system s
of a finit e numbe r o f degrees o f freedom .
In statistica l mechanics , w e stud y statistica l propertie s o f a huge (typi -
cally of order
numbe r o f particles. B y mathematical idealization , suc h
systems ar e bes t though t o f a s consisting o f a n infinit e numbe r o f particles ;
they ar e system s o f a n infinite numbe r o f degree s o f freedom . Th e motio n
of each o f these particle s i s governed b y quantu m mechanics ; i n som e case s
classical mechanic s i s a goo d enoug h approximation . I n eithe r case , i t i s
hopeless t o kee p trac k o f th e microscopi c behavio r o f th e system , i.e . tha t
of eac h individua l particle . Statistica l mechanic s i s designe d t o stud y th e
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