Das Sh.P. Statistical Physics of Liquids at Freezing and Beyond
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Appendix to Chapter 10
A10.1 Matrix identity
We prove here for the general square matrix G the following identity:
Tr
#
G δG
−1
$
= δ
Tr ln G
−1
. (A10.1.1)
Let us first define a matrix A of the same size as G as A = ln G
−1
. We denote the ijth
element of G and that of A as G
ij
and A
ij
, respectively,
Tr
#
G δG
−1
$
=
i, j
G
ij
δ[exp(A)]
ji
=
i, j
G
ij
∞
n=0
1
n!
δ A
n
ji
=
i, j
G
ij
∞
n=1
1
n!
j
1
,..., j
n−1
×
n
r=1
A
jj
1
···{A
j
r−1
j
r
[δ A
j
r
j
r+1
]A
j
r+1
j
r+2
}···A
j
n−1
i
. (A10.1.2)
We now redefine the dummy variables i and j interchanging with j
r
and j
r+1
, respectively,
Tr
#
G δG
−1
$
=
i, j, j
1
,..., j
n−1
{δ A
ij
}
∞
n=1
1
n!
n
r=1
A
j
r
j
1
...{A
j
r−1
i
G
j
r
j
r+1
A
jj
r+2
}...A
j
n−1
j
r+1
=
i, j
{δ A
ij
}
∞
n=1
1
n!
j
1
,..., j
n−1
n
r=1
A
j
r
j
1
...{A
j
r−1
i
G
j
r
j
r+1
A
jj
r+2
}...A
j
n−1
j
r+1
=
ij
{δ A
ij
}
∞
n=1
1
n!
n
r=1
j
1
,..., j
n−1
A
jj
r+2
...A
j
n−1
j
r+1
G
j
r+1
j
r
A
j
r
j
1
...A
j
r−1
i
.
(A10.1.3)
531
532 Appendix to Chapter 10
The quantity within curly brackets represents the jith element of the product of n matrices
and this is the same for all of the r =1,...,n terms in the sum. Hence we obtain
Tr
#
G δG
−1
$
=
ij
{δ A
ij
}
∞
n=1
n
r=1
1
n!
A
n−1−r
GA
r
ji
=
ij
{δ A
ij
}
∞
n=1
A
n−1
(n − 1)!
G
ji
=
ij
{δ A
ij
}[exp[A]G]
ji
=
i
{δ A
ij
}δ
ji
= δ{Tr A}=δ{Tr log G
−1
}. (A10.1.4)
A10.2 Matrix identity II
We prove here identities (10.4.33) and (10.4.34) for matrices of the form A = a
1
I + a
2
E
where I and E are both m × m matrices respectively represents the identity matrix and the
matrix with all elements equal to unity. Let us first consider the i-th eigenvalues μ
i
for
the matrix with the corresponding eigenvector {x
i
1
,...,x
i
m
} (i = 1,...m). The eigenvalue
equation is obtained as
x
i
1
+···+x
i
m
= μ
i
x
i
1
= μ
i
x
i
2
=···=μ
i
x
i
m
(A10.2.1)
The above equation can be satisfied either
(a) For μ
i
= 0, i.e.,
x
i
1
+ x
i
2
+···+x
i
m
= 0 (A10.2.2)
(b) For x
i
1
= x
i
2
=···=x
i
m
, i.e, μ
i
= m.
Since the condition (A10.2.1) can be satisfied with (m − 1) independent choices of x
i
α
for
α = 1,...,m − 1, we have m − 1 of the eigenvalues of m × m matrix E equal to 0 and
one eigenvalue equal to m. For these choices the corresponding eigenvalues of the matrix
B will be
λ = b
1
, b
1
,...b
1
B CD E
, {b
1
+ mb
2
} (A10.2.3)
respectively with the same eigenvectors described in cases (a) and (b) above. Therefore the
trace of this matrix B is equal to m(b
1
+ b
2
) and hence the identity (10.4.33) follows.
Next, for the matrix A defined as above, the eigenvalues for the inverse matrix A
−1
are
the reciprocals of those for the direct matrix. We denote them as {γ
−1
i
},
γ
−1
≡
1
a
1
,
1
a
1
,...
1
a
1
B CD E
,
3
1
a
1
+ ma
2
4
.
A10.3 Computation of the vibrational contribution I
v
533
Hence eigenvalues for the matrix ln A
−1
are
1
ln a
1
,
1
ln a
1
,...
1
ln a
1
,
3
1
ln(a
1
+ ma
2
)
4
. (A10.2.4)
Therefore we obtain for the trace of this matrix ln A
−1
Tr(A
−1
) = ln
1
a
1
+ ma
2
+ (m −1) ln
1
a
1
(A10.2.5)
Hence, assuming a
1
and a
2
are not m dependent
∂
∂m
1
m
Tr ln A
−1
*
*
*
*
m=1
=
3
ln
1
a
1
−
a
2
a
1
+ ma
2
(A10.2.6)
− (m − 1) ln
1
a
1
− ln
1
a
1
+ ma
2
4
m=1
= ln
1
a
1
−
a
2
a
1
+ a
2
− ln
1
a
1
+ a
2
=−
a
2
a
1
+ a
2
− ln
1 −
a
2
a
1
+ a
2
(A10.2.7)
A10.3 Computation of the vibrational contribution I
v
Here we compute the vibrational contribution I
v
to the partition function Z
m
of the repli-
cated liquid defined in eqn. (10.5.81). We evaluate the expression defined in eqn. (10.5.83)
with the Gaussian approximation
I
v
= m
Nd
du
N
m
δ
N
u
N
m
exp
⎧
⎨
⎩
−
β
4
N
i, j=1
m
a=1
d
μ,ν=1
u
a
i
− u
a
j
μ
u
a
i
− u
a
j
ν
× ∂
μ
∂
ν
v(r
i
−r
j
) −
4
N
i=1
m
a,b=1
u
a
i
− u
b
i
2
⎫
⎬
⎭
.
(A10.3.1)
By replacing the delta functions on the RHS by a corresponding integral representation,
δ
N
u
N
m
≡
N
%
i=1
δ
a
u
a
i
=
N
%
i=1
(2π)
−d
dJ
i
exp
⎡
⎣
i
i,a
J
i
· u
a
i
⎤
⎦
, (A10.3.2)
we obtain the vibrational contribution I
v
as
I
v
=
(
m
2π
)
Nd
du
N
m
dJ exp
!
−A
v
u
a
i
, J
i
"
. (A10.3.3)
534 Appendix to Chapter 10
We have used the notation dJ ≡
<
N
i=1
dJ
i
above. The exponent A
v
u
a
i
, J
i
is a quadratic
function of the u
a
i
fields and a linear coupling to the current field J
i
. Ignoring the cross-
coupling terms in the →0 limit, we obtain the “action” A
v
in the exponent as
A
v
=
1
2
m
a=1
N
i, j=1
d
μ,ν=1
M
iμ, jν
u
a
iμ
u
a
jν
+i
i,a
J
i
· u
a
i
. (A10.3.4)
The matrix M has the form
M
iμ, jν
=
β
&
δ
ij
k
v
(μν)
(r
i
−r
k
) − v
(μν)
(r
i
−r
j
)
'
+ δ
ij
δ
μν
, (A10.3.5)
where we have used the abbreviation v
(μν)
≡ ∂
μ
∂
ν
v(r
i
−r
j
). From the “action” functional
obtained in eqn. (A10.3.4) we identify from each of the m replica indices a a contribution
A
a
v
, such that
A
v
=
m
a=1
A
a
v
. (A10.3.6)
A particular replica component A
a
v
is obtained in terms of the Nd-dimensional field φ
a
i
(i =1,...,Nd) and Nd×Nd symmetric matrix M
ij
as
A
a
v
=
1
2
Nd
i, j=1
φ
a
i
M
ij
φ
a
j
+i
Nd
i=1
J
i
φ
a
i
=
1
2
Nd
ijkl=1
#
φ
a
i
+ J
l
M
−1
li
$
M
ij
#
φ
a
j
+ J
k
M
−1
kj
$
+
1
2
Nd
ij=1
J
i
M
−1
ij
J
j
=
1
2
Nd
i, j=1
˜
φ
a
i
M
ij
˜
φ
a
j
+
1
2
Nd
i, j=1
J
i
M
−1
ij
J
j
. (A10.3.7)
In the last equality we made a change of variables,
˜
φ
i
=φ
a
i
+
5
j
J
j
M
−1
ij
.
Using the results (A10.3.7) for each of the replica indices of A
v
in the expression
(A10.3.3), we obtain the following result for I
v
:
I
v
=
m
2π
Nd
d
˜
φ
Nd
m
dJ exp
⎡
⎣
1
2
m
a=1
Nd
i, j=1
˜
φ
a
i
M
ij
˜
φ
a
j
+
m
2
Nd
i, j=1
J
i
M
−1
ij
J
j
⎤
⎦
=
√
m
2π
Nd
d
˜
φ
Nd
m
d
˜
J exp
⎡
⎣
1
2
m
a=1
Nd
i, j=1
˜
φ
a
i
M
ij
˜
φ
a
j
+
1
2
Nd
i, j=1
˜
J
i
M
−1
ij
˜
J
j
⎤
⎦
.
(A10.3.8)
A10.3 Computation of the vibrational contribution I
v
535
In reaching the last equality we used for the differential of
˜
φ
a
i
for all the mNd variables
d
˜
φ
Nd
m
≡
Nd
%
i=1
m
%
a=1
d
˜
φ
a
i
(A10.3.9)
and for all Nd of the J
i
variables a transformation
√
mJ
i
=
˜
J
i
.
The integrals over the
˜
φ
a
i
and J
i
are all Gaussian and straightforward to do. To evaluate
these integrals, we use the following general result for set n of variables {ψ
i
}:
%
i
+∞
−∞
dψ
i
exp
⎡
⎣
−
1
2
n
i, j=1
ψ
i
A
ij
ψ
j
⎤
⎦
= (2π)
n/2
exp
−
1
2
Tr ln A
. (A10.3.10)
Let us consider the Gaussian integral for set n of variables {ψ
i
},
I
n
=
%
i
+∞
−∞
dψ
i
exp
⎡
⎣
−
1
2
n
i, j=1
ψ
i
A
ij
ψ
j
⎤
⎦
. (A10.3.11)
By means of a similarity transform
˜
A =S
−1
AS with the n × n matrix S constructed from
the eigenvectors of the matrix A (assumed nonsingular), A is reduced to a diagonal form
˜
A
with the eigenvalues λ
i
of A on the diagonal, i.e., with
˜
A
ij
=δ
ij
λ
i
. With a corresponding
linear transformation from {ψ
i
} to {
¯
ψ
i
} such that
¯
ψ = Sψ, I
n
reduces to (apart from a
Jacobian) a diagonal form
I
n
∼
%
i
d
¯
φ
i
exp
−
λ
i
2
¯
φ
2
i
=
2π
λ
i
n/2
= (2π)
n/2
exp
−
1
2
i
ln λ
i
. (A10.3.12)
In order to further analyze the above result, we define a matrix A
,
A = exp[A
], (A10.3.13)
or, equivalently, A
= ln A. It is easy to show that the matrix S also diagonalizes A
and
that the diagonal elements are λ
i
= exp
λ
i
. Therefore we have
i
ln λ
i
=
i
λ
i
= Tr A
= Tr ln A. (A10.3.14)
Hence we obtain from eqn. (A10.3.12) the useful result for the Gaussian integral in terms
of the trace of the matrix ln A that
I
n
= (2π)
n/2
exp
−
1
2
Tr ln A
. (A10.3.15)
536 Appendix to Chapter 10
Using this result, we obtain from eqn. (A10.3.8) the vibrational contribution as
I
v
=
√
m
2π
Nd
d
˜
φ
Nd
m
d
˜
J exp
⎡
⎣
1
2
m
a=1
Nd
ij=1
˜
φ
a
i
M
ij
˜
φ
a
j
+
1
2
Nd
i, j=1
˜
J
i
M
−1
ij
˜
J
j
⎤
⎦
=
√
m
2π
Nd
√
2π
(m+1)Nd
exp
−
m
2
Tr ln M −
1
2
Tr ln M
−1
= m
Nd/2
(2π)
(m−1)Nd/2
exp
−
m −1
2
Tr ln M
= C
m
exp
−
m −1
2
Tr ln M
. (A10.3.16)
The factor C
m
in front of the exponential is important for keeping track of the m dependence
of the partition function Z
m
for the replicated liquid with m clones of the original system.
The trace of the matrix M in the limit →0 is determined from
M
iμ, jν
= β
&
δ
ij
k
v
(μν)
(r
i
−r
k
) − v
(μν)
(r
i
−r
j
)
'
. (A10.3.17)
A10.4 Computation of
Tr ln M
∗
We present here an approximate calculation of Tr ln M
∗
, which appears on the RHS
of eqn. (10.5.89), after making the quenched approximation. In general the matrix M
involves the Hessian matrix for the amorphous structure and constitutes a random matrix.
Evaluation of the above term will involve computing the corresponding eigenvalues. These
eigenvalues are also involved in analyzing the instantaneous normal modes (Keyes, 1997)
in the disordered state. Indeed, the harmonic approximation made above makes sense only
if M does not have any negative eigenvalue. The number of negative eigenvalues becomes
very small for temperatures below T
c
(Sciortino and Tartaglia, 1997). Hence the harmonic
expansion is applicable for temperatures below T
c
and close to T
g
.
We proceed here by adopting a simple resummation scheme of Mézard and Parisi,
(1999a). This is based on the superposition approximation of multi-particle correlations
in the amorphous solid. Let us first consider the diagonal element of the matrix M given
by eqn. (10.5.86). We replace the first term in this expression for the matrix M by its
average to obtain the simplified expression
k
v
(μν)
(r
i
−r
k
) ≈
1
N
dr
dr
/
i,k
δ(r − r
i
)δ(r
−r
k
)
0
v
(μν)
(r −r
)
=
n
2
0
N
dr
dr
g
∗
cm
(r −r
)∂
μ
∂
ν
v(r −r
)
= n
0
δ
μν
d
drg
∗
cm
(r)∇
2
v(r) ≡ δ
μν
n
0
r
0
β
, (A10.4.1)
A10.4 Computation of
Tr ln M
∗
537
where the length r
0
is defined as
r
0
=
1
d
drg
∗
cm
(r)∇
2
[βv(r )]. (A10.4.2)
To keep the notation simple we will now assume that v(r ) is scaled with the factor β and
write βv(r) as v(r ). In reaching eqn. (A10.4.1), we have used the isotropy of the system
which is described in terms of the center-of-mass coordinates of the molecules. g
∗
cm
is the
pair correlation function at density n
0
and temperature T
∗
for this center-of-mass system.
It should be noted that, with the present notation of writing the position vector as r,the
differential dr actually denotes a d-dimensional elementary volume. With this approxima-
tion the diagonal elements of the matrix are of the form
M
iμ, jν
= n
0
r
0
δ
ij
δ
μν
−
1
n
0
r
0
∇
μ
∇
ν
v(r
i
−r
j
)
. (A10.4.3)
For computing the Tr ln M term we expand the logarithms in powers of 1/(n
0
r
0
).We
obtain the following expansion for Tr ln M:
+
Tr ln M
,
∗
= Ndln(n
0
r
0
) +
∞
p=2
+
Tr ln M
,
( p)
∗
, (A10.4.4)
where the pth-order term in the expansion is obtained in the form
+
Tr ln M
,
( p)
∗
=−
1
p(n
0
r
0
)
p
N
{i}=1
d
{ν}=1
+
v
(ν
1
ν
2
)
(r
i
1
−r
i
2
)...v
(ν
p
ν
1
)
(r
i
p
−r
i
1
)
,
. (A10.4.5)
We have used above the notation {i }≡i
1
, i
2
,...i
p
and {ν}≡ν
1
,ν
2
,...ν
p
. The prime in
the sum indicates that the terms with only different particles r
i
1
, r
i
2
,...,r
i
p
are involved.
On introducing the delta functions for each of the particle coordinates, we obtain
+
Tr ln M
,
( p)
∗
=−
1
p(n
0
r
0
)
p
dr
1
...
dr
p
+
N
{i}=1
δ(r
1
−r
i
1
)...δ(r
p
−r
i
p
)
,
×
d
{ν}=1
v
(ν
1
ν
2
)
(r
1
−r
2
)v
(ν
2
ν
3
)
(r
2
−r
3
)...v
(ν
p
ν
1
)
(r
p
−r
1
)
=−
1
pr
p
0
dr
1
...
dr
p
g
∗
cm
(r
1
...r
p
)
×
d
{ν}=1
v
(ν
1
ν
2
)
(r
1
−r
2
)v
(ν
2
ν
3
)
(r
2
−r
3
)...v
(ν
p
ν
1
)
(r
p
−r
1
). (A10.4.6)
The lowest-p-value contribution on the RHS of eqn. (A10.4.4) comes from the p = 2term
in the expansion,
+
Tr ln M
,
(2)
∗
=−
V
2r
2
0
drg
∗
cm
(r)
d
ν
1
ν
2
=1
{v
(ν
2
ν
1
)
(r)}
2
. (A10.4.7)
538 Appendix to Chapter 10
For p > 2 terms we have to deal with higher-order correlation functions. We now make the
superposition approximation for the multi-particle correlation function g
∗
cm
(r
1
...r
p
) for
p > 2, as n
p
0
times the products of pair correlation functions g
∗
cm
(r
1
−r
2
)...g
∗
cm
(r
p
−r
1
)
to obtain
7
+
Tr ln M
,
( p)
∗
=−
1
pr
p
0
dr
1
...
dr
p
g
∗
cm
(r
1
−r
2
)...g
∗
cm
(r
p
−r
1
)
×
d
{ν}=1
v
(ν
1
ν
2
)
(r
1
−r
2
)v
(ν
2
ν
3
)
(r
2
−r
3
)...v
(ν
p
ν
1
)
(r
p
−r
1
)
=−
1
pr
p
0
dr
1
d
{ν}=1
ξ
ν
1
ν
2
(r
1
−r
2
) ⊗···⊗ξ
ν
p
ν
1
(r
p
−r
1
). (A10.4.8)
The ⊗ on the RHS of eqn. (A10.4.8) represent the convolution of two ξ functions with all
the variables r
2
,...,r
p
being integrated according to the following definition:
ξ
(2)
ν
1
ν
3
(r
1
−r
3
) =
ν
2
dr
2
ξ
ν
1
ν
2
(r
1
−r
2
)ξ
ν
2
ν
3
(r
2
−r
3
)
≡
ν
2
ξ
ν
1
ν
2
(r
1
−r
2
) ⊗ ξ
ν
2
ν
3
(r
2
−r
3
), (A10.4.9)
where we have defined the quantity
ξ
ν
1
ν
2
(r
1
−r
2
) = g
∗
cm
(r
1
−r
2
)v
(ν
1
ν
2
)
(r
1
−r
2
). (A10.4.10)
Since the Fourier transform of the convolution of two functions is equal to the product of
the Fourier transforms of the two functions, we obtain
ξ
(2)
ν
1
ν
3
(k) =
d
ν
2
=1
ξ
ν
1
ν
2
(k)ξ
ν
2
ν
3
(k), (A10.4.11)
where the quantity ξ
μν
(k) for the isotropic system is expressed as
ξ
μν
(k) =
drg
∗
cm
(r)v
(μν)
(r)e
ik.r
≡
ˆ
k
μ
ˆ
k
ν
a
"
(k) + (
1
d
δ
μν
−
ˆ
k
μ
ˆ
k
ν
)a
⊥
(k). (A10.4.12)
ˆ
k
μ
= k
μ
/k is the μth component of the unit vector in the direction of vector k. Note that
it follows from the definition (A10.4.2) that r
0
= a
"
(0) = a
⊥
(0). We extend the definition
(A10.4.11) to include convolutions of p (> 2) numbers of ξ functions:
ξ
( p)
ν
1
ν
p+1
(k) =
ν
2
...ν
p
ξ
ν
1
ν
2
(k)ξ
ν
2
ν
3
(k)...ξ
ν
p
ν
p+1
(k). (A10.4.13)
7
Note that this is an approximation which is usually good at low densities.
A10.4 Computation of
Tr ln M
∗
539
The above result of convoluted ξ functions is useful for evaluating the RHS of
eqn. (A10.4.8). We note here that the latter is expressed as a chain of convolutions in
the following manner:
+
Tr ln M
,
( p)
∗
=−
V
pr
p
0
d
ν
1
=1
ξ
( p)
ν
1
ν
1
(r
1
−r
1
))
=−
Nd
n
0
p
dk
(2π)
3
˜a
p
"
(k) + (d − 1) ˜a
p
⊥
(k)
,
(A10.4.14)
where the functions ˜a
"
(k) and ˜a
⊥
(k) are obtained as
˜a
"
(k) =
a
"
(k)
r
0
, ˜a
⊥
(k) =
a
⊥
(k)
r
0
. (A10.4.15)
Finally, on summing over the different values of p from eqns. (A10.4.7) and (A10.4.14)
for all p values, we obtain the average of the vibrational contributions as
N
−1
+
Tr ln M
,
∗
= d lnr
0
−
1
2n
0
r
2
0
drg
∗
cm
(r)
μν
(v
(μν)
(r))
2
+
d
n
0
dk
(2π)
3
#
L
c
[˜a
"
(k)]+(d − 1)L
c
[˜a
⊥
(k)]
$
, (A10.4.16)
where the function L
c
is defined as L
c
(x) = ln(1 − x) + x +
x
2
2
.
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