LBan-1: Difference between revisions
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===Microcanonical calculation=== | ===Microcanonical calculation=== | ||
'''Step 1: Number of states '''. | |||
Let <math>{\mathcal N}_N(E) d E</math> the number of states of energy in the interval (E,E+dE). | |||
It is a random number and we use the representation | It is a random number and we use the representation | ||
<center ><math> \mathcal{N}_N(E)dE= \sum_{\alpha=1}^{2^N} \chi_\alpha(E) dE \;</math> </center> | <center ><math> \mathcal{N}_N(E)dE= \sum_{\alpha=1}^{2^N} \chi_\alpha(E) dE \;</math> </center> | ||
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Here <math> \epsilon =E/N </math> is the energy density and the annealed entropy density in the thermodynamic limit is | Here <math> \epsilon =E/N </math> is the energy density and the annealed entropy density in the thermodynamic limit is | ||
<center><math> s^{\text{ann.}}(\epsilon)=\ln 2 -\epsilon^2/2 </math></center> | <center><math> s^{\text{ann.}}(\epsilon)=\ln 2 -\epsilon^2/2 </math></center> | ||
'''Step 2: Self-averaging'''. | |||
Let compute now the second moment | |||
<center ><math> \overline{\mathcal{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 \mathcal{N}_N(E) \left( \mathcal{N}_N(E) - \exp\left(-\frac{E^2}{2 N}\right) \right) + \mathcal{N}_N(E) </math> </center> | <center ><math> \overline{\mathcal{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 \mathcal{N}_N(E) \left( \mathcal{N}_N(E) - \exp\left(-\frac{E^2}{2 N}\right) \right) + \mathcal{N}_N(E) </math> </center> | ||
We can then check the self averaging condition: | We can then check the self averaging condition: |
Revision as of 18:32, 5 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 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 N} degrees of freedom, the number of configurations grows exponentially with 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 N} . For simplicity, consider Ising spins that take two values, , located on a lattice of size 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 L} in 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 d} dimensions. In this case, and the number of configurations is 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 = 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:
where the sum runs over nearest neighbors , and the couplings 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 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 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 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:
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 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 N} . The same reasoning applies to each of 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. 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 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 N \to \infty } , these random quantities concentrates around a well defined value. These observables are called self-averaging. This means that,
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} f_N (\beta)= \lim_{N \to \infty} f_N^{\text{typ}}(\beta) =\lim_{N \to \infty} \overline{f_N(\beta)} =f_\infty(\beta) }
Hence becomes effectively deterministic and its sample-to sample fluctuations vanish in relative terms:
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} \frac{\overline{f_N^2(\beta)}}{\overline{f_N(\beta)}^2}=1. }
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
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(\beta) = \lim_{N \to \infty} Z_N^{\text{typ}}(\beta) =\lim_{N \to \infty} \overline{Z_N(\beta)} = e^{-\beta N f_\infty(\beta)} }
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 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 Z_N^{\text{typ}}(\beta) } 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(\beta) = -\frac{1}{\beta N} \ln Z_N^{\text{typ}}(\beta) }
Part II
Random Energy Model
The Random energy model (REM) neglects the correlations between the configurations. The energy associated to each configuration is an independent Gaussian variable 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 N} . The simplest solution of the model is with the microcanonical ensemble.
Canonical calculation
Microcanonical calculation
Step 1: Number of states .
Let 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 {\mathcal N}_N(E) d E} the number of states of energy in the interval (E,E+dE). It is a random number and we use the representation
with 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 \chi_\alpha(E)=1} if 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_\alpha \in [E, E+dE]} and otherwise. We can cumpute its average
Here 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 \epsilon =E/N } is the energy density and the annealed entropy density in the thermodynamic limit is
Step 2: Self-averaging.
Let compute now the second moment
We can then check the self averaging condition:
A critical energy density separates a self-averaging regime for 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 |\epsilon| < \epsilon^*} and a non self-averaging regime where for 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 |\epsilon| > \epsilon^*} . In the first regime, is exponentially large and its value is determinstic (average, typical, median are the same). In the secon regime, 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 \overline{\mathcal{N}_N(E)}} is exponentially small but nonzero. The typical value instead is exactly zero, 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 \mathcal{N}_N^{\text{typ}}(E) = 0} : for most disorder realizations, there are no configurations with energy below and only a vanishingly small fraction of rare samples gives a positive contribution to the average. As a result, the quenched average on the entropy density is: