1549380323-Statistical Mechanics Theory and Molecular Simulation

Legendre transforms 17

x

x 0

slope = f ’(x 0 )

f(x) + c

f(x)

Fig. 1.6Depiction of the Legendre transform.

clear, is thats 0 , being the slope of the line tangent tof(x) atx 0 , is also the slope
off(x) +catx=x 0 for any constantc. Thus,f(x 0 ) cannot be uniquely determined
froms 0. However, if we specify both the slope,s 0 =f′(x 0 ), and they-intercept,b(x 0 ),
of the line tangent to the function atx 0 , thenf(x 0 ) can be uniquely determined. In
fact,f(x 0 ) will be given by the equation of the line tangent to the function atx 0 :

f(x 0 ) =f′(x 0 )x 0 +b(x 0 ). (1.5.2)

Eqn. (1.5.2) shows how we may transform from a description off(x) in terms ofxto
a new description in terms ofs. First, since eqn. (1.5.2) is valid for allx 0 , it can be
written generally in terms ofxas

f(x) =f′(x)x+b(x). (1.5.3)

Then, recognizing thatf′(x) =g(x) =sandx=g−^1 (s), and assuming thats=g(x)
exists and is a one-to-one mapping, it is clear that the functionb(g−^1 (s)), given by

b(g−^1 (s)) =f(g−^1 (s))−sg−^1 (s), (1.5.4)

contains the same information as the originalf(x) but expressed as a function ofs
instead ofx. We call the functionf ̃(s) =b(g−^1 (s)) theLegendre transformoff(x).
The functionf ̃(s) can be written compactly as

f ̃(s) =f(x(s))−sx(s), (1.5.5)

wherex(s) serves to remind us thatxis a function ofsthrough the variable transfor-
mationx=g−^1 (s).