Editing Autocorrelation
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Latest revision | Your text | ||
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:<math>c(t)=\langle a(0) a(t) \rangle,</math> | :<math>c(t)=\langle a(0) a(t) \rangle,</math> | ||
or vectorial, in which case the scalar product is taken: | or vectorial, in which case the scalar product is taken: | ||
:<math>c(t)=\langle \vec{a}(0)\cdot\vec{ | :<math>c(t)=\langle \vec{a}(0)\cdot\vec{b}(t) \rangle.</math> | ||
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 decay | with a typical decay time <math>\tau</math>. This holds if | ||
the underlying process is | the underlying process is Markovian, and exceptions are known | ||
to occur, even in equilibrium classical fluids: the velocity | to occur, even in equilibrium classical fluids: the velocity | ||
autocorrelation function (see [[diffusion]]) is known to present | autocorrelation function (see [[diffusion]]) is known to present | ||
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A different definition of the decay time would be the time integral | A different definition of the decay time would be the time integral | ||
of <math>c(t)</math>: | of <math>c(t)</math>: | ||
:<math>\tau'=\int_0^\infty c(t) | :<math>\tau'=\int_0^\infty c(t),</math> | ||
which coincides with the previous one if the decay is purely exponential. Since | 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 | this is not the case at short times, the two times will be similar but |