Wikipedia

Replica trick: Difference between revisions

Browse history interactively
← Previous edit
Content deleted Content added
VisualWikitext
m Updated expected value notation to be consistent across entire page.
Tags: Reverted Visual edit
Reverting edit(s) by 108.27.222.32 (talk) to rev. 1250689036 by DVdm: Not providing a reliable source (RW 16.1)
 
(2 intermediate revisions by 2 users not shown)
Line 8:
<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>\mathbb E[overline{\ln Z]}</math>, the [[expected value]] of <math>\ln Z</math>, reducing the problem to calculating the disorder average <math>\mathbb E[overline{Z^n]}</math> where <math>n</math> is assumed to be an integer. This is physically equivalent to averaging over <math>n</math> statistically independent copies or '''''replicas''''' of the system, hence the name.
 
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}}]].)
 
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]].
 
== General formulation ==
Line 50:
 
:<math>
F = \mathbb E[overline{F([J_{ij})]} = -k_B T \, \mathbb E[overline{\ln Z([J)]}
</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 we are taking 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 comes 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.
 
==ExampleREM: Randomthe Energyeasiest Modelreplica (REM)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.
Here we consider the [[random energy model]] (REM) to illustrate the replica trick. It a system exhibits quenched disorder in its energy levels: <math>\{E_j\}_{j=1}^{2^N}</math> are picked iid from a [[Normal distribution|Gaussian distribution]] with mean 0 and variance <math>N/2</math>. Thus our partition function is
 
<math>\begin{align}
Z = \sum_{j=1}^{2^N} \exp(-\beta E_j) ~ .
\end{align}</math>
 
Our goal will be to calculate <math>\mathbb E [\log Z]</math>, where we average over the quenched disorder.
 
=== Replica Trick Calculation ===
To begin, we consider <math>n</math> statistical independent replicas of the system.
 
<math>\begin{align}
Z^n & = \sum_{i_1=1}^{2^N} ... \sum_{i_n=1}^{2^N} \exp\left(-\beta (E_{i_1} + ... + E_{i_n})\right)\\
\end{align}</math>
 
We can now compute <math>\mathbb E [Z]</math> by exploiting linearity and noting the joint distribution factorizes due to independence
 
<math>\begin{align}
\mathbb E[Z^n] & = \sum_{i_1=1}^{2^N} ... \sum_{i_n=1}^{2^N} \prod_{j=1}^n\mathbb E [\exp\left(-\beta (E_{i_j} \right) ] \\
& = \sum_{i_1=1}^{2^N} ... \sum_{i_n=1}^{2^N} \prod_{j=1}^n \exp \left(\frac{\beta^2 N}{4}\right) \\
& = 2^{N n} \exp \left(\frac{n \beta^2 N}{4}\right)\\
& = \exp \left(n \left [N \log 2 + \frac{ \beta^2 N}{4} \right]\right)
\end{align}</math>
 
So now we can use the replica trick to finish the calculation. One notes that Taylor expanding yields<math>\exp(n a) = 1 + n a + \mathcal O((an)^2)</math>, so the limit in the replica trick will automatically get rid of higher order terms.
 
<math>\begin{align}
\mathbb E[\log Z ] & = \lim_{n \to 0} \frac{\mathbb E[Z^n] - 1}{n}\\
& = N \log 2 + \frac{ \beta^2 N}{4}
\end{align}</math>
 
=== Discussion ===
We can see the replica trick calculation is only valid in the high temperature phase <math>\beta \leq \beta_c = 2 \sqrt{\log 2}</math>. Meaning, it missed the phase transition.
 
==Alternative methods==
Retrieved from "https://en.wikipedia.org/wiki/Replica_trick"