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: $\bar{Z}$

• 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:

$E=-\sum _{⟨i,j⟩}{J}_{ij}{\sigma }_i{\sigma }_j,$

where the sum runs over nearest neighbors $⟨i,j⟩$, 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:

$E[{\sigma }_1=1,{\sigma }_2=1,\dots ]=-\sum _{⟨i,j⟩}{J}_{ij}.$

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 $F_N(\beta )=N{f}_N(\beta )$ and its derivatives (magnetization, specific heat, susceptibility, etc.) — are self-averaging. This means that, in the limit $N\to \mathrm{\infty }$, the distribution of the observable concentrates around its average:

$\underset{N\to \mathrm{\infty }}{\lim }{f}_N=\underset{N\to \mathrm{\infty }}{\lim }\overline{f_N},\overline{f_N}=\int df{P}_{f_N}(f)f$

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

$\underset{N\to \mathrm{\infty }}{\lim }\frac{\overline{f_N^2}}{\stackrel{2}{\stackrel{‾}{f_N}}}=1.$

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:

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