Overview
This lesson is structured in three parts:
- Self-averaging and disorder in statistical systems
Disordered systems are characterized by a random energy landscape, however, in the thermodynamic limit, physical observables become deterministic. This property, known as self-averaging, does not always hold for the partition function which is the quantity that we can compute. When it holds the annealed average
and the quenched average
coincides otherwiese we have
We study the Random Energy Model (REM) introduced by Bernard Derrida. In this model at each configuration is assigned an independent energy drawn from a Gaussian distribution of extensive variance. The model exhibits a freezing transition at a critical temperature, below which the free energy becomes dominated by the lowest energy states.
- Extreme value statistics and saddle-point analysis
The results obtained from a saddle-point approximation can be recovered using the tools of extreme value statistics.
Part I
Random energy landascape
In a system with
degrees of freedom, the number of configurations grows exponentially with
. For simplicity, consider Ising spins that take two values,
, located on a lattice of size
in
dimensions. In this case,
and the number of configurations is
.
In the presence of disorder, the energy associated with a given configuration becomes a random quantity. For instance, in the Edwards-Anderson model:
where the sum runs over nearest neighbors
, and the couplings
are independent and identically distributed (i.i.d.) Gaussian random variables with zero mean and unit variance.
The energy of a given configuration is a random quantity because each system corresponds to a different realization of the disorder. In an experiment, this means that each of us has a different physical sample; in a numerical simulation, it means that each of us has generated a different set of couplings
.
To illustrate this, consider a single configuration, for example the one where all spins are up. The energy of this configuration is given by the sum of all the couplings between neighboring spins:
![{\displaystyle E[\sigma _{1}=1,\sigma _{2}=1,\ldots ]=-\sum _{\langle i,j\rangle }J_{ij}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d2521466e1db3f90c1710e2a847fc5347b7e42f7)
Since the the couplings are random, the energy associated with this particular configuration is itself a Gaussian random variable, with zero mean and a variance proportional to the number of terms in the sum — that is, of order
. The same reasoning applies to each of the
configurations. So, in a disordered system, the entire energy landscape is random and sample-dependent.
Self-averaging observables
A crucial question is whether the macroscopic properties measured on a given sample are themselves random or not. Our everyday experience suggests that they are not: materials like glass, ceramics, or bronze have well-defined, reproducible physical properties that can be reliably controlled for industrial applications.
From a more mathematical point of view, it means that the free energy
and its derivatives (magnetization, specific heat, susceptibility, etc.), in the limit
, these random quantities concentrates around a well defined value. These observables are called self-averaging. This means that,
Hence
becomes effectively deterministic and its sample-to sample fluctuations vanish in relative terms:
The partition function
The partition function
is itself a random variable in disordered systems. Analytical methods can capture the statistical properties of this variable. We can define to average over the disorder realizations:
- The annealed average corresponds to the calculation of the moments of the partition function. The annealed free energy is
- the quenched average corresponds to the average of the logarithm of the partition function, which is self-averaging for sure.
Do these two averages coincide?
If the partition function is self-averaging in the thermodynamic limit, then
As a consequence, the annealed and the quenched averages coincide.
If the partition function is not self-averaging, only typical partition function concentrates, but extremely rare configurations contribute disproportionately to its moments:
There are then two main strategies to determine the deterministic value of the observable :
- Compute directly the quenched average using methods such as the replica trick and the Parisi solution.
- Determine the typical value
and evaluate 
Part II
Random Energy Model
The Random energy model (REM) neglects the correlations between the
configurations. The energy associated to each configuration is an independent Gaussian variable with zero mean and variance
. The simplest solution of the model is with the microcanonical ensemble.
Microcanonical calculation
Step 1: Number of states .
Let
the number of states of energy in the interval (E,E+dE).
It is a random number and we use the representation
with
if
and
otherwise.
We can cumpute its average
Here
is the energy density and the annealed entropy density in the thermodynamic limit is
Step 2: Self-averaging.
Let compute now the second moment
We can then check the self averaging condition:
A critical energy density
separates a self-averaging regime for
and a non self-averaging regime where for
. In the first regime,
is exponentially large and its value is determinstic (average, typical, median are the same). In the secon regime,
is exponentially small but nonzero. The typical value instead is exactly zero,
: for most disorder realizations, there are no configurations with energy below
and only a vanishingly small fraction of rare samples gives a positive contribution to the average. As a result, the quenched average on the entropy density is:
Back to canonical ensemble: the freezing transition
The annealed partition function is the average of the partition function over the disorder:
Using the saddle point for large N we find
and thus
The quenched partition function is obtained replacing the mean with the typical value:
Using the saddle point for large N we find a critical inverse temperature
separating two phases:
- For
,
and the annealed calculation works
- For
,
and the free energy freezes to a temperature independent value. As a result, the quenched average on the free energy density is:
Part III
Detour: Extreme Value Statistics
Consider the REM spectrum of
energies
drawn from a distribution
. It is useful to introduce the cumulative probability of finding an energy smaller than E
We also define:
The statistical properties of
are derived using three key relations:
This relation is exact but depends on M and the precise form of
.
This is an estimation of the typical value of the minimum. It is a crucial relation that will be used frequently.
This is an approximation valid for large M and around the typical value of the minimum energy. It allows to extract the universal scaling.
Back to REM
Let us analyze in detail the case of a Gaussian distribution with zero mean and variance
. Using integration by parts, we can write :
The asymptotic expansion for
is :
In this case it is convenient to introduce the function
defined as
. hence we have:
From the second relation we impose
. For large
we get:
We look for a scaling form of the random variable
such that
,
are deterministic and M dependent while
is random and M independent.
In the Gaussian case, we start from the third relation introduced earlier and expand
around
:
Our goal is obtained by setting
In the REM the variance is
. Then we have:
Key Observations:
- the ground state energy is self-averaging with an extensive deterministic part
.
- Its fluctuations are very small (N independent) with a standard deviation
.