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 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, 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 \sigma_i = \pm 1} , located on a lattice of size 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, 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 = L^d} 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 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 \langle i, j \rangle} , 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 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_N(\beta)=N f_N(\beta)} 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 = \lim_{N \to \infty} f_N^{\text{typ}} =\lim_{N \to \infty} \overline{f_N} =f_\infty }
Hence 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_N } 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}}{\overline{f_N}^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 = \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 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}} } 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: 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 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 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)=0} 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 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^* = \sqrt{2 \ln 2}} separates two distinct regime:
- Self-averaging regime for
In this 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 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:
- Non self-averaging regime for
In this 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 still exponentially small but nonzero. However, for most disorder realizations, there are no configurations with energy below 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^* 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, , and the quenched entropy becomes:
- Step 3: The partition function
The annealed partition function is the average of the partition function over the disorder:
Alternatively, we can focus on the typical value of the partition function, which is the value observed with probability one in the thermodynamic limit. This corresponds to the quenched average: