LBan-1: Difference between revisions
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\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.}} | \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.}}} | ||
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Revision as of 15:57, 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.
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 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
This model simplifies the problem by neglecting 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 and assuming that the energies are(i.i.d.) Gaussian variables with zero mean and variance 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 E_N} .
Number of states is given by: