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, 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 $\ln \bar{Z}$ and the quenched average $\ln Z$ coincides otherwiese we have

$\ln Z<\ln \bar{Z}$

• 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 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 $N$ 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 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 $F_N(\beta )=N{f}_N(\beta )$ and its derivatives (magnetization, specific heat, susceptibility, etc.), in the limit $N\to \mathrm{\infty }$, these random quantities concentrates around a well defined value. These observables are called self-averaging. This means that,

$\underset{N\to \mathrm{\infty }}{\lim }{f}_N=\underset{N\to \mathrm{\infty }}{\lim }{f}_N^{\text{typ}}=\underset{N\to \mathrm{\infty }}{\lim }\overline{f_N}=f_{\mathrm{\infty }}$

Hence $f_N$ becomes effectively deterministic and its sample-to sample fluctuations vanish in relative terms:

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

The partition function

The partition function

$Z_N=\exp (-\beta N{f}_N)$

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

$f^{\text{ann.}}-\frac{1}{\beta N}\ln \overline{Z_N}$

• the quenched average corresponds to the average of the logarithm of the partition function, which is self-averaging for sure.

$f_{\mathrm{\infty }}\sim \overline{f_N}=-\ln Z_N/(\beta N)$

Do these two averages coincide?

If the partition function is self-averaging in the thermodynamic limit, then

$\underset{N\to \mathrm{\infty }}{\lim }{Z}_N=\underset{N\to \mathrm{\infty }}{\lim }{Z}_N^{\text{typ}}=\underset{N\to \mathrm{\infty }}{\lim }\overline{Z_N}=e^{-\beta N{f}_{\mathrm{\infty }}}$

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:

$\underset{N\to \mathrm{\infty }}{\lim }{Z}_N^{\text{typ}}=e^{-\beta N{f}_{\mathrm{\infty }}}<\underset{N\to \mathrm{\infty }}{\lim }\overline{Z_N}=e^{-\beta N{f}^{\text{ann.}}}$

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 $Z_N^{\text{typ}}$ and evaluate $f_{\mathrm{\infty }}=-\frac{1}{\beta N}\ln Z_N^{\text{typ}}$

Part II

Random Energy Model

The Random energy model (REM) neglects the correlations between the $M=2^N$ configurations. The energy associated to each configuration is an independent Gaussian variable with zero mean and variance $N$. The simplest solution of the model is with the microcanonical ensemble:

• Step 1: Let ${{𝒩}}_N(E)dE$ the number of states of energy in the interval (E,E+dE).

It is a random number and we use the representation

${{𝒩}}_N(E)dE=\sum _{\alpha =1}^{2^N}{\chi }_{\alpha }(E)dE$

with ${\chi }_{\alpha }(E)=1$ if $E_{\alpha }\in [E,E+dE]$ and ${\chi }_{\alpha }(E)=0$ otherwise. We can cumpute its average

$\overline{{{𝒩}}_N(E)}=\frac{2^N}{\sqrt{2\pi N}}\exp (-\frac{E^2}{2N})\sim \exp [N(\ln 2-{ϵ}^2/2)]$

Here $ϵ=E/N$ is the energy density and the annealed entropy density in the thermodynamic limit is

$s^{\text{ann.}}(ϵ)=\ln 2-{ϵ}^2/2$

• Step 2: Self-averaging. Let compute now the second moment

$\overline{{{𝒩}}_N^2(E)}=\sum _{\alpha =1}^{2^N}\overline{{\chi }_{\alpha }}\left(\sum _{\beta \ne \alpha }\overline{{\chi }_{\beta }}\right)+\sum _{\alpha =1}^{2^N}\overline{{\chi }_{\alpha }^2}\sim {{𝒩}}_N(E)({{𝒩}}_N(E)-\exp (-\frac{E^2}{2N}))+{{𝒩}}_N(E)$

We can then check the self averaging condition:

$\frac{\overline{{{𝒩}}_N^2(E)}}{\stackrel{2}{\overline{{{𝒩}}_N(E)}}}\sim 1+\frac{1}{\overline{{{𝒩}}_N(E)}}$

A critical energy density ${ϵ}^{*}=\sqrt{2\ln 2}$ separates two distinct phases:

• Self-averaging phase for $|ϵ|<{ϵ}^{*}$

In this regime, $\overline{{{𝒩}}_N(E)}$ is exponentially large and becomes self-averaging, i.e., its value is determinstic (average, typical, median are the same). The annealed entropy density coincides with the quenched entropy density:

$s_{\mathrm{\infty }}(ϵ)=s^{\text{ann.}}(ϵ)=\ln 2-\frac{{ϵ}^2}{2}$

• Non self-averaging phase for $|ϵ|>{ϵ}^{*}$

In this regime, $\overline{{{𝒩}}_N(E)}$ is still exponentially small but nonzero. However, for most disorder realizations, there are no configurations with energy below $-{ϵ}^{*}N$. Only a vanishingly small fraction of rare samples gives a positive contribution to the average. As a result, the typical number of such configurations is zero, ${{𝒩}}_N^{\text{typ}}(E)=0$, and the quenched entropy becomes:

$s_{\mathrm{\infty }}(ϵ)=-\mathrm{\infty }<s^{\text{ann.}}(ϵ)=\ln 2-\frac{{ϵ}^2}{2}$