Random field fluctuations Introduction … are random … have zero mean … and small amplitude 𝐵 𝑥,𝑦,𝑧 = 0 𝐵 𝑥,𝑦,𝑧 2 ≪ 𝐵 0 Arrow representation taken from SpinDynamics (M. Levitt)

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 … defined as sliding integral with itself … is called Auto-correlation function 𝑮 𝝉 =𝑭 𝟎 𝑭 𝝉 = 𝑭 𝟎 𝑭 𝝉

Random field fluctuations Correlation function: Cross-correlation … defined as sliding integral with another function … is called Cross-correlation function 𝑮 𝝉 =𝑭 𝟎 𝑯 𝝉 = 𝑭 𝟎 𝑯 𝝉

Random field fluctuations Auto-correlation Use sliding integral to judge rapidness of random field fluctuations … slow fluctiations … rapid fluctiations

Random field fluctuations Auto-correlation Auto correlation for random field fluctuations ~ exponential decay Only valid for isotropic tumbling other motions have different (more complex) correlation function 𝑮 𝝉 = … slow fluctiations … rapid fluctiations

Random field fluctuations From Auto-correlation to Spectral density random fluctuations … are oscillatory (like an FID) which frequencies contribute to the oscillations?  perform Fourier Analysis to obtain spectral density Lorentzian (= Cauchy distribution) FT FT FT Intensity scaled x 10

Random field fluctuations Spectral density Transition probabilites ~ Spectral density ~ ~ 1H @ 600 MHz