Replica trick

The replica trick is a mathematical trick that is used in particular in statistical mechanics and statistical physics to statistical sums, or more specifically to calculate the logarithm of the partition function and hence the free energy, if the direct determination is much more difficult or impossible. It was first used in statistical mechanics by Mark Kac and independently rediscovered in 1975 by Edwards and Anderson, Grinstein and Luther and Emery in connection with the so-called spin-glass problem. It is based on the mathematical identity

Wherein Z represents the partition function and n is the number of identical systems ( replicas ). then the partition function of n replicas, that is, at first it looks as if would go, but the truth is treated the limit (note that this is the norm of p- adic numbers match exactly ). The bar denotes the average over the statistical disorder. Based on the weighting of the replicas, a distinction between replica -symmetric solutions in which all replicas play a symmetric role, and cases where replica symmetry breaking (RSB ) occurs.

Applications in the spin glass theory

The trick is particularly used in the spin glass theory, with especially the Italian Giorgio Parisi has excelled through a fundamental, in a hierarchical manner, the replica - symmetry -breaking mathematical solution.

Mathematical

Nevertheless, no general theorem on the mathematical correctness of the method exists, so one has to rely on specific comparisons with exact results, which were gathered at more complicated ways with other methods. However, if the function of the point set can be extended to a complex - analytic function defined in a point ∞ enclosing open around, then this function is determined according to a known set of function theory by the values ​​to complete because the said amount has in a cluster point. Also, all derivations in are completely determined in this case. Again enters both the behavior at 0 and, indirectly, the behavior at ∞ here.

In practice, however, does not help that result.