Difference between revisions of "Autocorrelation"

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m ("dt" added to integral)
m (slight rewritting)
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These correlations typically decay exponentially at long times:
 
These correlations typically decay exponentially at long times:
 
:<math>c(t)=\exp(-t/\tau),</math>
 
:<math>c(t)=\exp(-t/\tau),</math>
with a typical decay time <math>\tau</math>. This holds if
+
with a decay, or relaxation, time <math>\tau</math>. This holds if
 
the underlying process is [[Markov chain |Markovian]], and exceptions are known
 
the underlying process is [[Markov chain |Markovian]], and exceptions are known
 
to occur, even in equilibrium classical fluids: the velocity
 
to occur, even in equilibrium classical fluids: the velocity

Revision as of 13:17, 20 June 2008

The autocorrelation of one magnitude refers to the temporal correlation of a magnitude with itself. The magnitude may be scalar:

c(t)=\langle a(0) a(t) \rangle,

or vectorial, in which case the scalar product is taken:

c(t)=\langle \vec{a}(0)\cdot\vec{b}(t) \rangle.

These correlations typically decay exponentially at long times:

c(t)=\exp(-t/\tau),

with a decay, or relaxation, time \tau. This holds if the underlying process is Markovian, and exceptions are known to occur, even in equilibrium classical fluids: the velocity autocorrelation function (see diffusion) is known to present a long tail (power-law) decay.

A different definition of the decay time would be the time integral of c(t):

\tau'=\int_0^\infty c(t) \,dt ,

which coincides with the previous one if the decay is purely exponential. Since this is not the case at short times, the two times will be similar but different. This later definition seems to be more related to times experimentally measurable.