LBan-1: Difference between revisions

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=Part II=
=Part II=
==Random Energy Model==
==Random Energy Model==
This model simplifies the problem by neglecting correlations between the <math>M=2^N</math> configurations and assuming that the energies <math>E_{\alpha}</math> are(i.i.d.) Gaussian variables with zero mean and variance <math>E_N</math>.  
The Random energy model (REM) neglects the correlations between the <math>M=2^N</math> configurations. The energy associated to each configuration is an independent Gaussian variable with zero mean and variance <math>N</math>. The simplest solution of the model is with the microcanonical ensemble:
'''Number of states''' is given by: <center><math> p(E_\alpha) = \frac{1}{\sqrt{2 \pi N}} \exp\left(-\frac{E_{\alpha}^2}{2 N}\right) </math></center>
* '''Step 1''': Let {\cal N}(E) the number of states of energy in the interval (E,E+dE). It is a random number of average
 
  is given by: <center><math> p(E_\alpha) = \frac{1}{\sqrt{2 \pi N}} \exp\left(-\frac{E_{\alpha}^2}{2 N}\right) </math></center>

Revision as of 16:16, 4 August 2025


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

  • 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 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:

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.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_\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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{N \to \infty} Z_N = \lim_{N \to \infty} Z_N^{\text{typ}} =\lim_{N \to \infty} \overline{Z_N} = e^{-\beta N f_\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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{N \to \infty} Z_N^{\text{typ}} = e^{-\beta N f_\infty} < \lim_{N \to \infty} \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 and evaluate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M=2^N} 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:

  • Step 1: Let {\cal N}(E) the number of states of energy in the interval (E,E+dE). It is a random number of average

is given by:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(E_\alpha) = \frac{1}{\sqrt{2 \pi N}} \exp\left(-\frac{E_{\alpha}^2}{2 N}\right) }