Fluctuations - davidar/scholarpedia GitHub Wiki
Fluctuations: Deviations of the value of an observable from its average or, also, deviations of the actual time evolution of an observable from its average evolution in a system subject to random forces or, simply, undergoing chaotic motion.
The law of errors is the first example of a theory of fluctuations. It deals with sums of a large number <math>N</math> of values <math>\sigma_1,\ldots,\sigma_N</math> occurring randomly with probability <math>p(\sigma)</math> equal for each pair of opposite values (i.e. <math>p(\sigma)=p(-\sigma)</math>), hence with<math>0</math> average. If the possible values of each <math>\sigma</math> are finitely many (and at least two) their sum can be of an order of magnitude as large as the number <math>N\ ,</math> however such a large value is very improbable for large <math>N</math> and deviations from the average of the order of the square root of <math>N</math> follows the errors law, also called normal law, of Gauss
- <math>\label{eq:1}
if <math>D=\sum_\sigma \sigma^2 p(\sigma)\ ,</math> up to corrections approaching <math>0</math> as <math>N\to\infty\ ,</math> (<math>[a,b]</math> being any finite interval).
Gauss' application was to control the errors in the determination of an asteroid orbit when observations, of its position in the sky, in excess of the minimum (three) necessary were available (Gauss 1971).
The error law is universal in the sense that it holds no matter which are the values of the variables <math>\sigma</math> as long as
(1) they have finitely many possibilities,
(2) probabilities <math>p(\sigma)</math> give zero average to the expectation <math>\sum_\sigma \sigma \,p(\sigma)=0\ ,</math>
(3) occurrence of any value takes place independently of occurrences of other values. The simplest application is to the sum of equally probable values <math>\sigma=\pm 1\ .</math>
Another important kind of fluctuations are the Poisson's fluctuations describing, for instance, the number of atoms in a region of volume <math>v</math> or the number of radioactive decays in a time interval <math>\tau\ :</math> these are independent events which occur with an average number <math>\nu</math> proportional to <math>v</math> or <math>\tau\ ;</math> the probability that <math>m</math> events are actually observed is <math>P(m)=e^{-\nu}{\nu^m}/{m!}\ .</math> A feature of such fluctuations, also called rare events, is that the mean square deviation is equal to the mean: <math>\sum_{m=0}^\infty (m-\nu)^2P(m)=\nu\ .</math>
The probabilities of values <math>\sum_{i=1}^N \sigma_i</math> of size of order <math>N</math> is called the theory of large fluctuations, because the <math>\sum_{i=1}^N \sigma_i</math> considered in the errors law and often referred to as small fluctuations is comparatively much smaller, being of order <math>\sqrt{N}\ .</math>
Also large fluctuations show universal properties, but to a lesser extent. The analysis is quite simple when the sum <math>\sum_{i=1}^N \sigma_i</math> involves two equally probable independent values <math>\sigma=\pm 1\ :</math> there is a function <math>f(s)</math> such that the probability that <math>\sum_{i=1}^N \sigma_i=s N</math> with <math>s\in [a,b]</math> and <math></math>-1<a<b<1</math></b<1</math> satisfies
- <math>\label{eq:2}
- <math>\label{eq:3}
="&quot;Extensions:&quot;" mean="&quot;&amp;#10;&amp;#10;&amp;#10;The&quot;" ="&quot;&amp;#10;&amp;#10;*&quot;" &lt;math&gt;\sum_&#123;i="&quot;1&amp;amp;#125;^N&quot;" by &lt;math&gt;\sum_&#123;i="&quot;1&amp;amp;#125;^N&quot;" quantities &lt;math&gt;\sum_&#123;i="&quot;1&amp;amp;#125;^N&quot;" &lt;math&gt;\max &#124;\sigma&#124;="&quot;+\infty&amp;amp;lt;/math&amp;amp;gt;&quot;" &lt;math&gt;p(\sigma)="&quot;p(&amp;amp;#45;\sigma)&amp;amp;lt;/math&amp;amp;gt;&quot;" p(\sigma)="&quot;+\infty&amp;amp;lt;/math&amp;amp;gt;&quot;" the &lt;math&gt;\sum_&#123;i="&quot;1&amp;amp;#125;^N&quot;" \sum_&#123;i="&quot;1&amp;amp;#125;^N&quot;" denoted &lt;math&gt;r_+(s)="&quot;\int_s^\infty&quot;" and &lt;math&gt;r_&#45;(s)="&quot;\int_&amp;amp;#123;&amp;amp;#45;\infty&amp;amp;#125;^&amp;amp;#123;&amp;amp;#45;s&amp;amp;#125;&quot;" \frac&#123;r_+(s)+r_&#45;(s)&#125; &#123;r_+(ks)+r_&#45;(ks)&#125;="&quot;k^&amp;amp;#123;\alpha&amp;amp;#125;&amp;amp;lt;/math&amp;amp;gt;&amp;#10;&quot;" 1968).
="&quot;Correlated&quot;" v="&quot;D&quot;" x&#125;="&quot;&amp;amp;#45;\lambda&quot;" \mathrm&#123;def&#125;&#125;&#123;="&quot;&amp;amp;#125;&amp;#10;\,&quot;" &lt;math&gt;t_0="&quot;m/\lambda&amp;amp;lt;/math&amp;amp;gt;&quot;" equation.
="&quot;Schottky&quot;" fluctuations="&quot;&quot;" is &lt;math&gt;i_0="&quot;n&quot;" I+C^&#123;&#45;1&#125;Q="&quot;0\&quot;" Q="&quot;I&amp;amp;#45;i(t)\&quot;" I+C^&#123;&#45;1&#125;(I&#45;i(t))="&quot;0&amp;amp;lt;/math&amp;amp;gt;&amp;#10;&amp;#10;&amp;#10;where&quot;" is
- &lt;math&gt;\label&#123;eq:11&#125;
="&quot;Langevin&quot;" equation="&quot;&quot;" &lt;math&gt;i,j="&quot;x,y,z\&quot;" T="&quot;\frac&amp;amp;#123;3&amp;amp;#125;&amp;amp;#123;2&amp;amp;#125;&quot;" F_&#123;ext&#125;(t)="&quot;\vec&quot;" with susceptibility
- &lt;math&gt;\label&#123;eq:20&#125;
- &lt;math&gt;\label&#123;eq:21&#125;
="&quot;Fluctuation&#45;Dissipation&quot;" theorem="&quot;&quot;" E="&quot;\vec&quot;" \xi_i="&quot;\Gamma_i(\vec\xi)&amp;amp;lt;/math&amp;amp;gt;&amp;#10;is&quot;" i="&quot;1,\ldots,n\&quot;" E)="&quot;\frac&amp;amp;#123;\partial&quot;" setting &lt;math&gt;\Phi="&quot;&quot;" &lt;math&gt;J_i(\vec E)="&quot;\langle&amp;amp;#123;J_i(X,\dot&quot;" E)&#125;="&quot;\int_0^\infty&quot;" &lt;math&gt;L_&#123;ij&#125;="&quot;L_&amp;amp;#123;ji&amp;amp;#125;&amp;amp;lt;/math&amp;amp;gt;&quot;" 0&#125;="&quot;S_&amp;amp;#123;&amp;amp;#45;t&amp;amp;#125;^&amp;amp;#123;\vec&quot;" integrals)
- &lt;math&gt;\label&#123;eq:24&#125;
="&quot;Blue&quot;" sky="&quot;&quot;" &lt;math&gt;\nu="&quot;\omega/2\pi&amp;amp;lt;/math&amp;amp;gt;&quot;" as &lt;math&gt;W\propto\omega^4="&quot;\lambda^&amp;amp;#123;&quot;" precisely &lt;math&gt;W="&quot;\frac23\frac&amp;amp;#123;e^2&amp;amp;#125;&amp;amp;#123;c^3&amp;amp;#125;\omega^4\big(\frac&amp;amp;#123;e&amp;amp;#125;&amp;amp;#123;m\,(\omega^2&amp;amp;#45;&amp;#10;\omega_0^2)&amp;amp;#125;\big)^2&amp;amp;lt;/math&amp;amp;gt;&quot;" Hence &lt;math&gt;\frac&#123;W_&#123;blue&#125;&#125;&#123;W_&#123;red&#125;&#125;="&quot;\Big(\frac&amp;amp;#123;4.5\,10^4&quot;" E_2="&quot;(n_1&amp;amp;#45;n_2)&quot;" &lt;math&gt;\langle&#123;(n_1&#45;n_2)^2&#125;\rangle ="&quot;\langle&amp;amp;#123;n_1&amp;amp;#125;\rangle&quot;" &lt;math&gt;\sigma_\xi="&quot;\pm1&amp;amp;lt;/math&amp;amp;gt;&quot;" \,{ def\atop ="&quot;&#125;\,&amp;lt;table&amp;gt;&amp;gt;&amp;lt;/table&amp;gt;not&quot;" (&lt;math&gt;\alpha="&quot;\frac87&amp;amp;lt;/math&amp;amp;gt;&quot;" &lt;math&gt;d="&quot;2&amp;amp;lt;/math&amp;amp;gt;)&quot;" from &lt;math&gt;\alpha="&quot;2\&quot;" 2000).
="&quot;&amp;#10;&amp;#10;&amp;#10;*A.&quot;" 3(3):1924.
="&quot;&quot;" Mechanics
="&quot;&quot;" href="&quot;http://www.math.rutgers.edu/~giovanni/&quot;" target="&quot;_blank&quot;&gt;Dr."></a<b<1</math></b<1</math>>Gallavottis webpage</a>
="See" also=" Anosov Diffeomorphism,"></a<b<\max></b<\max>>Smooth Dynamics