Content deleted Content added
mNo edit summary |
Reverting edit(s) by 108.27.222.32 (talk) to rev. 1250689036 by DVdm: Not providing a reliable source (RW 16.1) |
||
| (48 intermediate revisions by 26 users not shown) | |||
Line 1:
{{Short description|Mathematical limit applied in statistical physics}}
In the [[statistical physics]] of [[spin glass]]es and other systems with [[quenched disorder]], the '''replica trick''' is a mathematical technique based on the application of the formula:
The crux of the replica trick is that while the disorder averaging is done assuming <math>n</math> to be an integer, to recover the disorder-averaged logarithm one must send <math>n</math> continuously to zero. This apparent contradiction at the heart of the replica trick has never been formally resolved, however in all cases where the replica method can be compared with other exact solutions, the methods lead to the same results.▼
<math display="block">\ln Z=\lim_{n\to 0} {Z^n-1\over n}</math>
or:
<math display="block">\ln Z = \lim_{n\to 0} \frac{\partial Z^n}{\partial n}</math>
where
<math>Z</math> is most commonly the [[Partition function (statistical mechanics)|partition function]], or a similar thermodynamic function.
It is typically used to simplify the calculation of <math>\overline{\ln Z}</math>, the [[expected value]] of <math>\ln Z</math>, reducing the problem to calculating the disorder average <math>\overline{Z^n}</math> where <math>n</math> is assumed to be an integer. This is physically equivalent to averaging over <math>n</math> copies or '''''replicas''''' of the system, hence the name.
It is occasionally necessary to require the additional property of ''replica symmetry breaking'' (RSB) in order to obtain physical results, which is associated with the breakdown of ergodicity, however the physical existence of RSB is controversial.▼
▲The crux of the replica trick is that while the disorder averaging is done assuming <math>n</math> to be an integer, to recover the disorder-averaged logarithm one must send <math>n</math> continuously to zero. This apparent contradiction at the heart of the replica trick has never been formally resolved, however in all cases where the replica method can be compared with other exact solutions, the methods lead to the same results. (A natural sufficient rigorous proof that the replica trick works would be to check that the assumptions of [[Carlson's theorem]] hold, especially that the ratio <math>(Z^n-1)/n</math> is of [[exponential type]] less than [[pi|{{pi}}]].)
This mathematical trick is used in computation involving functions of a variable that can be expressed as a power series in that variable. The crux of this technique is to reduce the function of a variable, say <math>f(z)</math>, into powers of <math>z</math> or, in other words, replicas of <math>z</math>, and perform the same computation which is to be done on <math>f(z)</math>, using the powers of <math>z</math>.▼
▲It is occasionally necessary to require the additional property of ''replica [[symmetry breaking]]'' (RSB) in order to obtain physical results, which is associated with the breakdown of [[ergodicity
The replica trick postulates that if <math>Z^n</math> can be calculated for all positive integers <math>n</math> then this may be sufficient to allow the limiting behaviour as <math>n\to0</math> to be calculated.▼
== General formulation ==
Clearly, such an argument poses many mathematical questions, and the resulting formalism for performing the limit <math>n\to0</math> typically introduces many subtleties (see Mezard et al.).▼
It is generally used for computations involving [[analytic function]]s (can be expanded in power series).
When using [[mean field theory]] to perform one's calculations, taking this limit often requires introducing extra order parameters, a property known as '[[replica symmetry breaking]]' which is closely related to [[ergodicity breaking]] and slow dynamics within disorder systems.▼
▲
A particular case which is of great use in physics is in averaging the [[thermodynamic free energy]],
:<math>F = -k_{\rm B} T \ln Z[J_{ij}],</math>
over values of <math>J_{ij}</math> with a certain probability distribution, typically Gaussian.<ref name="nishimori_book">{{cite book|title=Statistical physics of spin glasses and information processing : an introduction|last=Nishimori|first=Hidetoshi|publisher=Oxford Univ. Press|year=2001|isbn=0-19-850940-5|location=Oxford [u.a.]}} ''See page 13, Chapter 2.''</ref>
The [[Partition function (statistical mechanics)|partition function]] is then given by
:<math>Z[J_{ij}] \sim e^{-\beta J_{ij}}.</math>
Notice that if we were calculating just <math>Z[J_{ij}]</math> (or more generally, any power of <math>J_{ij}</math>) and not its logarithm which we wanted to average, the resulting integral (assuming a Gaussian distribution) is just
:<math>\int dJ_{ij} \, e^{-\beta J - \alpha J^2},</math>
a standard [[Gaussian integral]] which can be easily computed (e.g. completing the square).
To calculate the free energy, we use the replica trick:<math display="block">\ln Z = \lim_{n\to 0}\dfrac{Z^{n}-1}{n}</math>which reduces the complicated task of averaging the logarithm to solving a relatively simple Gaussian integral, provided <math>n</math> is an integer.<ref>{{cite journal|last=Hertz|first=John|title=Spin Glass Physics|date=March–April 1998}}</ref>
▲The replica trick postulates that if <math>Z^n</math> can be calculated for all positive integers <math>n</math> then this may be sufficient to allow the limiting
▲Clearly, such an argument poses many mathematical questions, and the resulting formalism for performing the limit <math>n\to0</math> typically introduces many subtleties.<ref>{{Cite
▲When using [[mean
== Physical applications ==
The replica trick is used in determining [[ground state]]s of statistical mechanical systems, in the [[mean
In the statistical physics of systems with quenched disorder, any two states with the same realization of the disorder (or in case of spin glasses, with the same distribution of ferromagnetic and antiferromagnetic bonds) are called replicas of each other.<ref name="spin glass pedestrians" group="papers on spin glasses">{{cite journal|last=Tommaso Castellani|first=Andrea Cavagna|title=Spin-glass theory for pedestrians|journal=Journal of Statistical Mechanics: Theory and Experiment|volume=2005|issue=5|pages=P05012|date=May 2005|doi=10.1088/1742-5468/2005/05/P05012
In the case of a spin glass, we expect the free energy per spin (or any self averaging quantity) in the thermodynamic limit to be independent of the particular values of [[ferromagnetic]] and [[antiferromagnetic]] couplings between individual sites, across the lattice. So, we explicitly find the free energy as a function of the disorder parameter (in this case, parameters of the distribution of ferromagnetic and antiferromagnetic bonds) and average the free energy over all realizations of the disorder (all values of the coupling between sites, each with its corresponding probability, given by the distribution function). As free energy takes the form:
:<math>
F = \overline{F[J_{ij}]} = -
</math>
where <math>J_{ij}</math> describes the disorder (for spin glasses, it describes the nature of magnetic interaction between each of the individual sites <math>i</math> and <math>j</math>) and <math>[\cdots ]</math> denotes the average over all values of the couplings described in <math>J</math>, weighted with a given distribution. To perform the averaging over the logarithm function, the replica trick come in handy, in replacing the logarithm with its limit form mentioned above. In this case, the quantity <math>Z^n</math> represents the joint partition function of <math>n</math> identical systems.▼
▲where <math>J_{ij}</math> describes the disorder (for spin glasses, it describes the nature of magnetic interaction between each of the individual sites <math>i</math> and <math>j</math>) and
==REM: The easiest Replica problem==▼
The [[Random Energy Model]] ('''REM''') is one of the simplest models of statistical mechanics of disordered systems, and probably the simplest model to show the meaning and power of the Replica Trick to the level 1 of [[Replica Symmetry Breaking]]. The model is especially suitable for this introduction because an exact result by a different procedure is known, and the Replica Trick can be proved to work by crosschecking of results.▼
▲The [[
==Alternative methods==
The [[cavity method]] is an alternative method, often of simpler use than the replica method, for studying disordered mean
Another alternative method is the [[Supersymmetry#Supersymmetry in condensed matter physics|supersymmetric method]]. The use of the supersymmetry method provides a mathematical rigorous alternative to the replica trick, but only in non-interacting systems. See for example the book: <ref group="other approaches">''Supersymmetry in Disorder and Chaos'', Konstantin Efetov, Cambridge university press, 1997.</ref>
Also, it has been demonstrated <ref group="other approaches">A. Kamenev and A. Andreev, cond-mat/9810191; C. Chamon, A. W. W. Ludwig, and C. Nayak, cond-mat/9810282.</ref> that the [[Keldysh formalism]] provides a viable alternative to the replica approach.
==Remarks==
{{reflist|group=nb}}
The first of the above identities is easily understood via [[Taylor expansion]]:
:<math>\begin{align}\lim_{n \rightarrow 0} \dfrac{Z^n - 1}{n} &= \lim_{n \rightarrow 0} \dfrac{e^{n \ln Z} - 1}{n}\\
&= \lim_{n \rightarrow 0} \dfrac{n \ln Z + {1 \over 2!} (n \ln Z)^2 + \cdots}{n}\\ &= \ln Z ~~.\end{align}</math>
For the second identity, one simply uses the definition of the derivative
▲The [[cavity method]] is an alternative method, often of simpler use than the replica method, for studying disordered mean field problems. It has been devised to deal with models on locally [[tree (graph theory)|tree-like graphs]]
:<math>\begin{align}
\lim_{n \rightarrow 0} \dfrac{\partial Z^n}{\partial n} &= \lim_{n \rightarrow 0} \dfrac{\partial e^{n\ln Z}}{\partial n}\\[5pt]
&= \lim_{n \rightarrow 0} Z^n\ln Z\\[5pt]
&=\lim_{n \rightarrow 0} (1 + n\ln Z +\cdots )\ln Z\\[5pt]
&= \ln Z ~~.\end{align}</math>
==References==
* S Edwards (1971), "Statistical mechanics of rubber". In ''Polymer networks: structural and mechanical properties'', (eds A. J. Chompff & S. Newman). New York: Plenum Press, ISBN 978-1-4757-6210-5.
* {{Cite book |last1=Mézard |first1=Marc |title=Spin glass theory and beyond: an introduction to the replica method and its applications |last2=Parisi |first2=Giorgio |last3=Virasoro |first3=Miguel Ángel |date=1987 |publisher=World scientific |isbn=978-9971-5-0116-7 |series=World Scientific lecture notes in physics |location=Teaneck, NJ, USA}}
* {{cite arXiv |last=Charbonneau |first=Patrick |title=From the replica trick to the replica symmetry breaking technique |date=2022-11-03 |class=physics.hist-ph |eprint=2211.01802}}
===Papers on spin glasses===
{{Reflist|group=papers on spin glasses}}
===Books on
{{Reflist|group=books on spin glasses}}
===References to other approaches===
{{Reflist|group=other approaches}}
{{Reflist}}
| |||