LBan-1

Disordered systems and 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:

$E=-\sum _{\langle i,j\rangle }J_{ij}\sigma _{i}\sigma _{j},$

where the sum runs over nearest neighbors $\langle i,j\rangle$, 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:

$E[\sigma _{1}=1,\sigma _{2}=1,\ldots ]=-\sum _{\langle i,j\rangle }J_{ij}.$

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 and partition function

A crucial question is whether the physical 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 tha physical observables — such as the free energy $F_{N}(\beta )=Nf_{N}(\beta )$ and its derivatives (magnetization, specific heat, susceptibility, etc.) — are self-averaging.

This means that, in the limit $N\to \infty$, its distribution concentrates around the average, collapsing to a deterministic value:

$\lim _{N\to \infty }Y_{N}=\lim _{N\to \infty }{\overline {Y_{N}}}:={Y_{\infty }},\quad \quad {\overline {Y_{N}}}=\int \,dy\,P_{Y_{N}}(y)\,y$

This happens when its fluctuations are small compared to the averagein the thermodynamic limit, their values become independent of the particular realization of disorder. In other words macroscopic observables, which depend on a statistical average over an exponentially large number of configurations, become sharply peaked around their average values when $N\to \infty$. The fluctuations from sample to sample vanish in relative terms, and physical observables become effectively deterministic.