10.10 Fluctuations
501
These second derivatives are evaluated at equilibrium. Close to equilibrium, the
higher order terms in the Taylor expansion beyond the quadratic will be small. As
they enter as a small term in an exponential factor, their effect near equilibrium is
dramatically reduced compared to the quadratic term.
Also, close to equilibrium, the conjugate variables y i will be linearly related to
the δY j : From Eq. (10.229), 64
y i = −
1
k B
∂S e
∂Y i
+
1
k B
∂S
∂(δY i )
= y ei + β ij δY j
(10.231)
so that fluctuations in Y i are related to fluctuations in its conjugate by
δy i = β ij δY j .
(10.232)
With Einstein’s insight, averages of fluctuations for any thermodynamic variable
can be simply calculated. For example, the mean square of the fluctuations in Y i ,
(δY i ) 2 , will have the expression
(δY k )
2
= A
exp
−
1
2
β ij δY i δY j
(δY k )
2 dY 1 dY 2 · · · ,
(10.233)
with A a normalization factor.
The integrand is significant only for small δY i , so the range of the integral can
be extended to {−∞, ∞}, making Gaussian integrals. These can be simplified by
diagonalizing the matrix 65 with a linear transformation of the integration variables.
Let
Y i = Y i + a ij Z i
(10.234)
making
β kl = a ki β ij a lj ≡ δ kl .
(10.235)
The Jacobian of this transformation is det(a). We see from Eq. (10.235) in matrix
form: aβa T = I that the Jacobian satisfies
det(a) = (det (β))
−1/2 .
(10.236)
64 Care must be taken to distinguish the values of the entropy as a function of the extensive
variables in quasi-equilibrium states and the entropy in its dependence on fluctuations in these
same variables. The fluctuations are not restricted to be on the thermodynamic surface S =
S(U, V , N, · · · ).
65 Any real symmetric matrix B can be diagonalized by a specific similarity transformation A.
ABA −1 =
B = diag(b 1 , b 2 , · · · ). In our case, AA T = I : If this transformation is followed by a
scale transformation S = diag(b
−1/2
1
, b
−1/2
2
, · · · ), the result is a unit matrix: SABA T S T = I .
501
These second derivatives are evaluated at equilibrium. Close to equilibrium, the
higher order terms in the Taylor expansion beyond the quadratic will be small. As
they enter as a small term in an exponential factor, their effect near equilibrium is
dramatically reduced compared to the quadratic term.
Also, close to equilibrium, the conjugate variables y i will be linearly related to
the δY j : From Eq. (10.229), 64
y i = −
1
k B
∂S e
∂Y i
+
1
k B
∂S
∂(δY i )
= y ei + β ij δY j
(10.231)
so that fluctuations in Y i are related to fluctuations in its conjugate by
δy i = β ij δY j .
(10.232)
With Einstein’s insight, averages of fluctuations for any thermodynamic variable
can be simply calculated. For example, the mean square of the fluctuations in Y i ,
(δY i ) 2 , will have the expression
(δY k )
2
= A
exp
−
1
2
β ij δY i δY j
(δY k )
2 dY 1 dY 2 · · · ,
(10.233)
with A a normalization factor.
The integrand is significant only for small δY i , so the range of the integral can
be extended to {−∞, ∞}, making Gaussian integrals. These can be simplified by
diagonalizing the matrix 65 with a linear transformation of the integration variables.
Let
Y i = Y i + a ij Z i
(10.234)
making
β kl = a ki β ij a lj ≡ δ kl .
(10.235)
The Jacobian of this transformation is det(a). We see from Eq. (10.235) in matrix
form: aβa T = I that the Jacobian satisfies
det(a) = (det (β))
−1/2 .
(10.236)
64 Care must be taken to distinguish the values of the entropy as a function of the extensive
variables in quasi-equilibrium states and the entropy in its dependence on fluctuations in these
same variables. The fluctuations are not restricted to be on the thermodynamic surface S =
S(U, V , N, · · · ).
65 Any real symmetric matrix B can be diagonalized by a specific similarity transformation A.
ABA −1 =
B = diag(b 1 , b 2 , · · · ). In our case, AA T = I : If this transformation is followed by a
scale transformation S = diag(b
−1/2
1
, b
−1/2
2
, · · · ), the result is a unit matrix: SABA T S T = I .