Replica trick

The replica trick is a mathematical trick that especially in statistical mechanics and statistical physics is used to partition functions , or more specifically the logarithm of the partition function and thus the free energy to calculate when the direct determination 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

{\ displaystyle \ lim _ {n \ to 0} {{\ overline {Z ^ {n}}} - 1 \ over n} = {\ overline {\ ln Z}}}

where the sum of states and the number of identical systems (replicas) denotes. is then the sum of states of the replicas, d. H. At first it looks as if it would work, but in truth one is dealing with the Limes (note that this fits the norm of the p-adic numbers exactly ). The line indicates the mean value over the statistical disorder. On the basis of the weighting of the replicas, a distinction is made between replica-symmetric solutions, in which all replicas play a symmetrical role, and cases in which replica symmetry breaking (RSB) occurs. ${\ displaystyle Z}$ ${\ displaystyle n}$ ${\ displaystyle Z ^ {n}}$ ${\ displaystyle n}$ ${\ displaystyle n \ to \ infty}$ ${\ displaystyle n \ to 0}$

Applications in spin glass theory

The trick is used particularly in the spin-glass theory , whereby the Italian Giorgio Parisi in particular excelled with a fundamental mathematical solution that breaks the replica symmetry in a hierarchical manner .

Math

Nevertheless, there is no general theorem about the mathematical correctness of the method, so that one has to rely on concrete comparisons with exact results that were obtained in a more complicated way with other methods. If, however, the function can be extended from the point set to a complex-analytical function which is defined in an open neighborhood of enclosing the point , then this function is completely determined by the values according to a known theorem of function theory , because the said set has a cluster point . All derivatives at are also completely determined in this case. Again, both the behavior at 0 and, indirectly, the behavior at . ${\ displaystyle {\ overline {\ ln Z}} (z \ in U)}$ ${\ displaystyle G \ \ equiv \ {n = 0,1,2, ..., \ infty \}}$ ${\ displaystyle \ infty}$ ${\ displaystyle U \ (\ in \ mathbb {C})}$ ${\ displaystyle G}$ ${\ displaystyle G}$ ${\ displaystyle n = \ infty}$ ${\ displaystyle n = 0}$ ${\ displaystyle \ infty}$

In practice this result does not help.

literature

M. Kac, Trondheim Theoretical Physics Seminar, Nordita Publ. 286, 1968 (unpublished); and T.-F. Lin, J. Math. Phys. 11: 1584 (1970).

Edwards and Anderson : Theory of spin glasses. Phys. Q: Met. Phys. 5,965 (1975), doi: 10.1088 / 0305-4608 / 5/5/017 .

Grinstein, G. and Luther, A .: Application of the renormalization group to phase transitions in disordered systems , Phys. Rev. B 13, 1329-1343, doi: 10.1103 / PhysRevB.13.1329 .

VJ Emery, Phys .: Critical properties of many-component systems , Rev. B 11, 239 (1975). doi: 10.1103 / PhysRevB.11.239 .

Individual evidence

^ Giorgio Parisi : On the replica approach to spin glasses. January 17, 1997 online file
↑ Heinrich Behnke , Friedrich Sommer : Theory of the functions of a complex variable. Springer-Verlag, Berlin 1976, ISBN 3-540-07768-5 .

[1] Giorgio Parisi : On the replica approach to spin glasses. January 17, 1997 online file

[2] Heinrich Behnke , Friedrich Sommer : Theory of the functions of a complex variable. Springer-Verlag, Berlin 1976, ISBN 3-540-07768-5 .