500
10 The Statistical Basis of Bioenergetics
applies.
The quantities
y i ≡ −
1
k B
∂S
∂Y i
(10.225)
are intensive thermodynamic variables, usually simply related to the temperature,
pressure, and chemical potential. 63 The y i are the conjugate thermodynamic
variables to the Y i , introduced in Eq. (9.240). As intensive variables, each will satisfy
0 =
k
∂y i
∂Y k
Y k = −
1
k B
k
∂ 2 S
∂Y i ∂Y k
Y k .
(10.226)
Except at discontinuities, such as the regions of phase transitions, we expect
∂ 2 S
∂Y i ∂Y j
=
∂ 2 S
∂Y j ∂Y i
(10.227)
which implies that
∂y i
∂Y j
=
∂y j
∂Y i
.
(10.228)
These are generalized Maxwell relations (see Sect. 9.16).
To investigate the size of fluctuations in thermodynamic variables, Einstein
expanded the entropy appearing in Eq. (10.222) into a Taylor series around its
equilibrium value. Since S reaches a maximum (S e ) at equilibrium, the first order
derivatives vanish, giving
S = S e +
1
2
∂ 2 S
∂Y i ∂Y j
δY i δY j + · · · ,
(10.229)
with δY i ≡ Y i − Y ei , and we use the convention to sum over repeated indices,
unless otherwise stated. The average value of the thermodynamic variable Y i near
equilibrium defines its equilibrium value: Y ei = Y i . Since S reaches a peak at S e ,
the second derivative terms must all add to a negative number. We simplify notation
in what follows by defining
β ij ≡ −
1
k B
∂ 2 S
∂Y i ∂Y j
.
(10.230)
63 For simple systems, dS = (1/T )dE + (p/T )dV −
k (μ k /T )dN k , with
l N l = N .