LBan-1

Overview

This lesson is structured in three parts:

• Self-averaging and disorder in statistical systems

Disordered systems are characterized by a random energy landscape, where the microscopic details vary from sample to sample. However, in the thermodynamic limit, physical observables become deterministic. This property is known as self-averaging. This is in general not the case for the partition function: ${\overline {Z}}$

• The Random Energy Model

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. 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. In the REM, the low-temperature phase is governed by the minimum of a large set of independent energy values.

Part I

Random energy landascape

In a system with $N$ degrees of freedom, the number of configurations grows exponentially with $N$. For simplicity, consider Ising spins that take two values, $\sigma _{i}=\pm 1$, located on a lattice of size $L$ in $d$ dimensions. In this case, $N=L^{d}$ and the number of configurations is $M=2^{N}=e^{N\log 2}$.

In the presence of disorder, the energy associated with a given configuration becomes a random quantity. For instance, in the Edwards-Anderson model:

$E=-\sum _{\langle i,j\rangle }J_{ij}\sigma _{i}\sigma _{j},$

where the sum runs over nearest neighbors $\langle i,j\rangle$, and the couplings $J_{ij}$ 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 $J_{ij}$.

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:

$E[\sigma _{1}=1,\sigma _{2}=1,\ldots ]=-\sum _{\langle i,j\rangle }J_{ij}.$

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 $N$. The same reasoning applies to each of the $M=2^{N}$ configurations. So, in a disordered system, the entire energy landscape is random and sample-dependent.

Self-averaging observables

A crucial question is whether the physical 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 tha physical observables — such as the free energy $F_{N}(\beta )=Nf_{N}(\beta )$ and its derivatives (magnetization, specific heat, susceptibility, etc.) — are self-averaging. This means that, in the limit $N\to \infty$, the distribution of the observable concentrates around its average:

$\lim _{N\to \infty }f_{N}=\lim _{N\to \infty }{\overline {f_{N}}},\quad \quad {\overline {f_{N}}}=\int \,df\,P_{f_{N}}(f)\,f$

Hence macroscopic observables become effectively deterministic and their fluctuations from sample to sample vanish in relative terms:

$\lim _{N\to \infty }{\frac {\overline {f_{N}^{2}}}{{\overline {f_{N}}}^{2}}}=1.$

The partition function

The partition function

$Z_{N}=\exp(-\beta Nf_{N})$

is itself a random variable in disordered systems. What we are interested in computing is the quenched average of the free energy: However, what is usually easier to compute is the annealed average: $-{\frac {1}{\beta N}}\ln \,{\overline {Z_{N}}}$

Do these two averages coincide?

There are then two main strategies:

• One can directly compute the quenched average using methods such as the Parisi solution.

• Alternatively, one may determine the typical value $Z_{N}^{\text{typ}}$ and evaluate $f_{N}^{\text{typ}}=-{\frac {1}{\beta N}}\ln Z_{N}^{\text{typ}}$ in the limit $N\to \infty$.

Part II

Random Energy Model

This model simplifies the problem by neglecting correlations between the $M=2^{N}$ configurations and assuming that the energies $E_{\alpha }$ are(i.i.d.) Gaussian variables with zero mean and variance $E_{N}$.

Number of states is given by:

$p(E_{\alpha })={\frac {1}{\sqrt {2\pi N}}}\exp \left(-{\frac {E_{\alpha }^{2}}{2N}}\right)$