LBan-1: Difference between revisions

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where the sum runs over nearest neighbors <math>\langle i, j \rangle</math>, and the couplings <math>J_{ij}</math> are independent and identically distributed (i.i.d.) Gaussian random variables with zero mean and unit variance.
where the sum runs over nearest neighbors <math>\langle i, j \rangle</math>, and the couplings <math>J_{ij}</math> 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 <math>J_{ij}</math>.

Revision as of 14:12, 2 August 2025

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 .