formulasearchengine - User contributions [en]

Roller screw

2014-04-20T20:44:35Z

87.254.72.114: /* Configuration */

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Pitch angle (particle motion)

2013-10-30T02:33:16Z

87.254.72.51:

{{Inadequate lead|date=December 2013}}
In [[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

: <math>\lim_{n\to 0} {Z^n-1\over n}=\ln Z.</math>

== Mathematical Trick ==
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>.

A particular case which is of great use in physics is in averaging the free energy, or <math>-\ln Z[J_{ij}]</math>, over values of <math>J_{ij}</math> with a certain probability distribution, typically Gaussian,<ref name=nishimori_book group="books on spin glasses"></ref> and the function <math>Z[J_{ij}] \sim e^{-\beta J_{ij}}</math>. Notice that if it were <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) would be of the form <math>\int dJ_{ij} e^{-\beta J - \alpha J^{2}}</math>, which can be performed by completing squares and carrying out the standard [[Gaussian integral|gaussian integration]]. But we have the special property or the limit form expression for the logarithm function, given by:

<math>\ln Z_{ij} = \lim_{n\to 0}\dfrac{Z^{n}-1}{n}</math>

which clearly reduces the task of averaging to solving a relatively simpler Gaussian integral.<ref>{{cite journal|last=Hertz|first=John|title=Spin Glass Physics|date=March–April 1998}}</ref>
The replica trick involves extending this argument to the case where <math>n</math> is no longer constrained to be an integer, by positing 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.

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.).
When using [[mean field theory]] to perform one's calculations, taking this limit often requires introducing extra order parameters, in consequence of '[[replica symmetry breaking]]' which is closely related to [[ergodicity breaking]] and slow dynamics within disorder systems.

== Physical Applications ==
The replica trick is used in determining [[ground state]]s of statistical mechanical systems, in the mean field approximation. Typically, for systems in which the determination of ground state is easy, one can analyze fluctuations near the ground state. But in cases where, for some reason the determination of ground state is hard, one uses the replica method.<ref name=replica_approach group="papers on spin glasses">{{cite journal|last=Parisi|first=Giorgio|title=On the replica approach to spin glasses|date=17 January 1997|url=http://chimera.roma1.infn.it/P_COMPLEX/pa_1997d.ps}}</ref> An example is the case of a [[Order and disorder (physics)#Quenched disorder|quenched disorder]] in a spin system [[Spin glass]] with different types of magnetic links between spin sites, thereby causing many configurations to have the same energy. Hence finding a particular ground state is hard.<ref name=replica_approach group="papers on spin glasses"></ref>

In statistical physics of quenched disorder systems, any two states (set of configurations) with the same realization of the disorder, on 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|date=1 January 1970|year=2005|month=May|doi=10.1088/1742-5468/2005/05/P05012|url=http://iopscience.iop.org/1742-5468/2005/05/P05012|accessdate=3 April 2011|arxiv = cond-mat/0505032 |bibcode = 2005JSMTE..05..012C }}</ref> For systems with quenched disorder, one typically expects that macroscopic quantities will be [[self-averaging]], whereby any macroscopic quantity for a specific realization of the disorder will be indistinguishable from the same quantity calculated by averaging over all possible realizations of the disorder. Hence replicas are introduced for \emph{integrating out the disorder}<ref name=nishimori_book group="books on spin glasses">{{cite book|last=Nishimori|first=Hidetoshi|title=Statistical physics of spin glasses and information processing : an introduction|year=2001|publisher=Oxford Univ. Press|location=Oxford [u.a.]|isbn=0-19-850940-5|url=http://cdn.preterhuman.net/texts/science_and_technology/physics/Statistical_physics/Statistical%20physics%20of%20spin%20glasses%20and%20information%20processing%20an%20introduction%20-%20Nishimori%20H..pdf|page=13|chapter=2}}</ref> in a system.

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}]} = -k_{B}T\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 <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.

==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.

==Proof of initial formula==

We prove the formula

: <math>\lim_{n\to 0} {x^n-1\over n}=\ln x.</math>

Start from the observation

: <math>\exp y = 1 + y + {1 \over 2!} y^2 + {1 \over 3!} y^3 + \dots = \lim_{N\to \infty} \sum_{r=0}^N {N! \over r! (N-r)!} ({y \over N})^r = \lim_{N\to \infty} (1 + {y \over N})^N.</math>

We write <math>y=\ln x</math>, then

: <math> x = \lim_{N\to \infty} (1+{\ln x\over N})^N</math>

or rearranged

: <math>\ln x = \lim_{N\to \infty} {(x^{1/N} - 1) \over 1/N} = \lim_{n\to 0} {x^n-1\over n}.</math>

A much easier and straighforward proof can be obtained by using the [[L'Hôpital's rule]].

==See also==

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]]

==References==

* M Mezard, G Parisi & M Virasoro, "Spin Glass Theory and Beyond", World Scientific, 1987
Papers on Spin Glasses
{{Reflist|group=papers on spin glasses}}
Books on Spin Glasses
{{Reflist|group=books on spin glasses}}
{{Reflist}}

{{DEFAULTSORT:Replica Trick}}
[[Category:Statistical mechanics]]

Logic alphabet

2013-10-08T04:28:02Z

87.254.72.193: /* Significance */

{{E (mathematical constant)}}
{{refimprove|date=December 2007}}
The [[mathematical constant]] [[E (mathematical constant)|{{math|''e''}}]] can be represented in a variety of ways as a [[real number]]. Since {{math|''e''}} is an [[irrational number]] (see [[proof that e is irrational]]), it cannot be represented as a [[fraction (mathematics)|fraction]], but it can be represented as a [[continued fraction]]. Using [[calculus]], {{math|''e''}} may also be represented as an [[infinite series]], [[infinite product]], or other sort of [[limit of a sequence]].

==As a continued fraction==

[[Leonhard Euler|Euler]] proved that the number {{math|''e''}} is represented as the infinite [[simple continued fraction]]<ref>{{cite web|url=http://www.maa.org/editorial/euler/How%20Euler%20Did%20It%2028%20e%20is%20irrational.pdf|title=How Euler Did It: Who proved ''e'' is Irrational?|last=Sandifer|first=Ed|date=Feb. 2006|publisher=MAA Online|accessdate=2010-06-18}}</ref> {{OEIS|id=A003417}}:

:<math>e = [2; 1, \textbf{2}, 1, 1, \textbf{4}, 1, 1, \textbf{6}, 1, 1, \textbf{8}, 1, 1, \ldots, \textbf{2n}, 1, 1, \ldots]. \,</math>

Its convergence can be tripled by allowing just one fractional number:

:<math> e = [ 1 ; \textbf{0.5} , 12 , 5 , 28 , 9 , 44 , 13 , 60 , 17 , \ldots , \textbf{4(4n-1)} , \textbf{4n+1} , \ldots]. \,</math>

Here are some infinite [[generalized continued fraction]] expansions of {{math|''e''}}. The second is generated from the first by a simple [[generalized continued fraction#The equivalence transformation|equivalence transformation]].

:<math>
e= 2+\cfrac{1}{1+\cfrac{1}{2+\cfrac{2}{3+\cfrac{3}{4+\cfrac{4}{5+\ddots}}}}} = 2+\cfrac{2}{2+\cfrac{3}{3+\cfrac{4}{4+\cfrac{5}{5+\cfrac{6}{6+\ddots\,}}}}}
</math>

:<math>e = 2+\cfrac{1}{1+\cfrac{2}{5+\cfrac{1}{10+\cfrac{1}{14+\cfrac{1}{18+\ddots\,}}}}} = 1+\cfrac{2}{1+\cfrac{1}{6+\cfrac{1}{10+\cfrac{1}{14+\cfrac{1}{18+\ddots\,}}}}}</math>

This last, equivalent to [1; 0.5, 12, 5, 28, 9, ...], is a special case of a general formula for the [[exponential function]]:

:<math>e^{x/y} = 1+\cfrac{2x} {2y-x+\cfrac{x^2} {6y+\cfrac{x^2} {10y+\cfrac{x^2} {14y+\cfrac{x^2} {18y+\ddots}}}}}</math>

==As an infinite series==
The number {{math|''e''}} can be expressed as the sum of the following [[infinite series]]:

:<math>e^x = \sum_{k=0}^\infty \frac{x^k}{k!} </math> for any real number ''x''.

In the special case where ''x'' = 1, or −1, we have:

:<math>e = \sum_{k=0}^\infty \frac{1}{k!}</math>,<ref>{{cite web|url=http://oakroadsystems.com/math/loglaws.htm|title=It’s the Law Too — the Laws of Logarithms|last=Brown|first=Stan|date=2006-08-27|publisher=Oak Road Systems|accessdate=2008-08-14}}</ref> and

:<math>e^{-1} = \sum_{k=0}^\infty \frac{(-1)^k}{k!}</math>

Other series include the following:

:<math>e = \left [ \sum_{k=0}^\infty \frac{1-2k}{(2k)!} \right ]^{-1}</math> <ref>Formulas 2–7: [[Harlan J. Brothers|H. J. Brothers]], [http://www.brotherstechnology.com/docs/Improving_Convergence_(CMJ-2004-01).pdf Improving the convergence of Newton's series approximation for ''e''], ''The College Mathematics Journal'', Vol. 35, No. 1, (2004), pp. 34–39.</ref>

:<math>e = \frac{1}{2} \sum_{k=0}^\infty \frac{k+1}{k!}</math>

:<math>e = 2 \sum_{k=0}^\infty \frac{k+1}{(2k+1)!}</math>

:<math>e = \sum_{k=0}^\infty \frac{3-4k^2}{(2k+1)!}</math>

:<math>e = \sum_{k=0}^\infty \frac{(3k)^2+1}{(3k)!}</math>

:<math>e = \left [ \sum_{k=0}^\infty \frac{4k+3}{2^{2k+1}\,(2k+1)!} \right ]^2</math>

:<math>e = \left [ -\frac{12}{\pi^2} \sum_{k=1}^\infty \frac{1}{k^2} \ \cos \left ( \frac{9}{k\pi+\sqrt{k^2\pi^2-9}} \right ) \right ]^{-1/3} </math>

:<math>e = \sum_{k=1}^\infty \frac{k^n}{B_n(k!)}</math> where <math>B_n</math> is the <math>n^{th}</math> [[Bell number]]. Some few examples: (for ''n''=1,2,3)

:<math>e = \sum_{k=1}^\infty \frac{k}{k!} = \sum_{k=1}^\infty \frac{1}{(k-1)!} = \sum_{k=0}^\infty \frac{1}{k!}</math>

:<math>e = \sum_{k=1}^\infty \frac{k^2}{2(k!)}</math>

:<math>e = \sum_{k=1}^\infty \frac{k^3}{5(k!)}</math>

:<math>e = \sum_{k=1}^\infty \frac{k^4}{15(k!)}</math>

:<math>e = \sum_{k=1}^\infty \frac{k^5}{52(k!)}</math>

:<math>e = \sum_{k=1}^\infty \frac{k^6}{203(k!)}</math>

:<math>e = \sum_{k=1}^\infty \frac{k^7}{877(k!)}</math>

==As an infinite product==
The number {{math|''e''}} is also given by several [[infinite product]] forms including [[Nick Pippenger|Pippenger]]'s product

:<math> e= 2 \left ( \frac{2}{1} \right )^{1/2} \left ( \frac{2}{3}\; \frac{4}{3} \right )^{1/4} \left ( \frac{4}{5}\; \frac{6}{5}\; \frac{6}{7}\; \frac{8}{7} \right )^{1/8} \cdots </math>

and Guillera's product <ref>J. Sondow, [http://arxiv.org/abs/math/0401406 A faster product for pi and a new integral for ln pi/2,] ''Amer. Math. Monthly'' 112 (2005) 729–734.</ref><ref>J. Guillera and J. Sondow, [http://arxiv.org/abs/math.NT/0506319 Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent,]''Ramanujan Journal'' 16 (2008), 247–270.</ref>
:<math> e = \left ( \frac{2}{1} \right )^{1/1} \left (\frac{2^2}{1 \cdot 3} \right )^{1/2} \left (\frac{2^3 \cdot 4}{1 \cdot 3^3} \right )^{1/3}
\left (\frac{2^4 \cdot 4^4}{1 \cdot 3^6 \cdot 5} \right )^{1/4} \cdots ,</math>
where the ''n''th factor is the ''n''th root of the product
:<math>\prod_{k=0}^n (k+1)^{(-1)^{k+1}{n \choose k}},</math>

as well as the infinite product

:<math> e = \frac{2\cdot 2^{(\ln(2)-1)^2} \cdots}{2^{\ln(2)-1}\cdot 2^{(\ln(2)-1)^3}\cdots }.</math>

==As the limit of a sequence==
The number {{math|''e''}} is equal to the [[limit of a sequence|limit]] of several [[infinite sequences]]:

:<math> e= \lim_{n \to \infty} n\cdot\left ( \frac{\sqrt{2 \pi n}}{n!} \right )^{1/n} </math> and

:<math> e=\lim_{n \to \infty} \frac{n}{\sqrt[n]{n!}} </math> (both by [[Stirling's formula]]).

The symmetric limit,<ref>[[Harlan J. Brothers|H. J. Brothers]] and J. A. Knox, [http://www.brotherstechnology.com/docs/Closed-Form_Approximations_(MI-1998-12).pdf New closed-form approximations to the Logarithmic Constant ''e'',] ''The Mathematical Intelligencer'', Vol. 20, No. 4, (1998), pp. 25–29.</ref><ref>{{cite web|url=http://ans.hsh.no/home/skk/Publications/Lobatto/PRIMUS_KHATTRI.pdf|title=From Lobatto Quadrature to the Euler constant e|last=Khattri|first=Sanjay}}</ref>

:<math>e=\lim_{n \to \infty} \left [ \frac{(n+1)^{n+1}}{n^n}- \frac{n^n}{(n-1)^{n-1}} \right ]</math>

may be obtained by manipulation of the basic limit definition of {{math|''e''}}. Another limit is<ref>S. M. Ruiz 1997</ref>

:<math>e= \lim_{n \to \infty}(p_n \#)^{1/p_n} </math>

where <math> p_n </math> is the ''n''th [[prime number|prime]] and <math> p_n \# </math> is the [[primorial]] of the ''n''th prime.

:<math>e= \lim_{n \to \infty}n^{\pi(n)/n} </math>

where <math> \pi(n) </math> is the prime counting function. This definition is a direct corollary of the [[prime number theorem]].

Also:
:<math>e^x= \lim_{n \to \infty}\left (1+ \frac{x}{n} \right )^n.</math>

In the special case that <math>x = 1</math>, the result is the famous statement:

:<math>e= \lim_{n \to \infty}\left (1+ \frac{1}{n} \right )^n.</math>

== In trigonometry ==
Trigonometrically, {{math|''e''}} can be written as the sum of two [[hyperbolic functions]]:

<math>e = \sinh(1) + \cosh(1)\,</math>

==Notes==
{{reflist}}

[[Category:Transcendental numbers]]
[[Category:Mathematical constants]]
[[Category:Exponentials]]
[[Category:Logarithms]]
[[Category:E (mathematical constant)]]