520 The thermodynamic transition scenario
glassy solid in terms of the basic interaction potential between the constituent particles. As
has already been indicated above in the discussion of the ergodicity-breaking transition, the
configurational entropy S
c
remains zero below T
K
. This is ensured by keeping the effective
temperature (βm)
−1
(where m is the number of replicas) of the pinning field ψ (see eqn.
(10.4.6) of earlier section) as T
K
. This indicates the possibility of understanding the glassy
state by analytically continuing the above model to the region m < 1.
For T < T
K
, the saddle-point free energy f
∗
corresponding to the optimum contribution
in the partition function sticks at the extreme value f
min
of the range of variability of f .
There are only subexponentially many such minima at f
min
. We can control the pinning
of f
∗
at f
min
by working with a system consisting of m identical copies of the original
N -particle system. On applying the above formulation of eqn. (10.4.3) for the partition
function of the composite system of m replicas we obtain
Z
m
=
f
max
f
min
e
β N{TS
c
( f,T )−mf}
. (10.5.67)
The saddle point f
∗
for the cloned system is obtained from the solution of the equation
dS
c
( f )
df
*
*
*
f =f
∗
=
m
k
B
T
. (10.5.68)
For T > T
K
the corresponding value of the saddle point f
∗
lies in between the two limits
at this temperature, i.e., f
min
(T )< f
∗
< f
max
(T ).Form = 1, the saddle point occurs at
f
∗
= f
min
, corresponding to temperature T = T
K
. Let us now consider the situation for
T < T
K
. Equation (10.5.68) now can be used as the defining equation for f
∗
and by treat-
ing m as a tuning parameter we can make the saddle point coincide with the corresponding
f
min
(T ). This requires making an analytic continuation to a region where the parameter m
can be less than 1. Depending on the temperature T < T
K
, a critical value m = m
∗
(T )<1
is obtained. At this value of m for the composite system the saddle point reaches f
∗
= f
min
and configurational entropy S
c
= 0.
Alternatively, we can also describe the region m < 1 as follows. For fixed positive values
of m less than unity, the corresponding temperature at which eqn. (10.5.68) is satisfied for
f
∗
= f
min
is T
mK
which is less than T
K
. If the saddle point reaching the corresponding
f
min
(T ) (and hence vanishing of S
c
) is considered synonymous with the transition point
then the same occurs in an m-replicated system (m < 1) at a temperature T
mK
. Thus, for
this chosen value of m, the replicated system has positive S
c
until temperature T
mK
, below
which S
c
= 0.
The m-replicated system considered above corresponds to a special liquid state of
molecules of m atoms. In fact, for understanding the physics for T < T
K
, it is the cloned
system with m values less than m
∗
(T ) that is of interest to us. This represents a liquid
state with the various replicas forming a molecular bound state. Physically, at low temper-
atures the attraction between the different replicas forces all the m members to fall within
the same free-energy minimum. However, the strength of the interaction between one such