1549380323-Statistical Mechanics Theory and Molecular Simulation

266 Grand canonical ensemble

for an arbitrary parameter,λ. We call such a function ahomogeneous function of degree
nin the variablesx 1 ,...,xk. The functionf(x) =x^2 , for example, is a homogeneous
function of degree 2. The functionf(x,y,z) =xy^2 +z^3 is a homogeneous function
of degree 3 in all three variablesx,y, andz. The functionf(x,y,z) =x^2 (y^2 +z) is
a homogeneous function of degree 2 inxbut not inyandz. The functionf(x,y) =
exy−xyis not a homogeneous function in eitherxory.
Euler’s theorem states the following: Letf(x 1 ,...,xN) be a homogeneous function
of degreeninx 1 ,...,xk. Then,

nf(x 1 ,...,xN) =

∑k

i=1

xi

∂f

∂xi

. (6.2.2)

The proof of Euler’s theorem is straightforward. Beginning with eqn. (6.2.1), we dif-
ferentiate both sides with respect toλto yield:

d

dλ

f(λx 1 ,...,λxk,xk+1,...,xN) =

d

dλ

λnf(x 1 ,...,xk,xk+1,...,xN)

∑k

i=1

xi

∂f

∂(λxi)

=nλn−^1 f(x 1 ,...,xk,xk+1,...,xN). (6.2.3)

Sinceλis arbitrary, we may freely chooseλ= 1, which yields

∑k

i=1

xi

∂f

∂xi

=nf(x 1 ,...,xk,xk+1,...,xN) (6.2.4)

and proves the theorem.
What does Euler’s theorem have to do with thermodynamics? Consider, for exam-
ple, the Helmholtz free energyA(N,V,T), which depends on two extensive variables,
NandV. SinceAis, itself, extensive,A∼N, and sinceV∼N,Amust be a homo-
geneous function of degree 1 inNandV, i.e.A(λN,λV,T) =λA(N,V,T). Applying
Euler’s theorem, it follows that

A(N,V,T) =V

∂A

∂V

+N

∂A

∂N

. (6.2.5)

From the thermodynamic relations of the canonical ensemble for pressure and chemical
potential, we haveP=−(∂A/∂V) andμ= (∂A/∂N). Thus,

A=−PV+μN. (6.2.6)

We can verify this result by recalling that

A(N,V,T) =E−TS. (6.2.7)

From the first law of thermodynamics,

E−TS=−PV+μN, (6.2.8)