Random field fluctuations Introduction

Random field fluctuations Introduction

Field fluctuations …

… are random

… have zero mean

… and small amplitude

= 0

⟨ ^�

❑

2

⟩ ≪ �

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Random field fluctuations Introduction

Random fluctuations …

… can be slow … and fast

How to measure the rapidness of fluctuations?

 correlation function

Random field fluctuations

Correlation function: Auto-correlation

Correlation function

… defined as sliding integral with itself

… is called Auto-correlation function

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Random field fluctuations

Correlation function: Cross-correlation

Correlation function

… defined as sliding integral with another function

… is called Cross-correlation function

Random field fluctuations Auto-correlation

Use sliding integral to judge rapidness of random field fluctuations

… slow fluctiations

… rapid fluctiations

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Random field fluctuations Auto-correlation

… slow fluctiations

… rapid fluctiations

Auto correlation for random field fluctuations

~ exponential decay

Only valid for isotropic tumbling

other motions have different (more complex) correlation function

random fluctuations

… are oscillatory (like an FID)

which frequencies contribute to the oscillations?

 perform Fourier Analysis to obtain spectral density

Random field fluctuations

From Auto-correlation to Spectral density

FT

FT

FT

Lorentzian (= Cauchy distribution)

Intensity scaled x 10

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Transition probabilites ~ Spectral density

Random field fluctuations Spectral density

~ ~

1

H @ 600 MHz