11.4 The Generation of Noise

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by lamda squared divided by tauλ²/τ, where lamdaλ is the step length and tauτ is the duration of each step. The random

walk is, of course, an example of a Markov chain.

Problem. Write out the Markovian transition matrix for a random walk in one dimen-

sion.

11.4

The Generation of Noise

It might be thought that “noise” is the ultimate random, uncorrelated process. In

reality, however, noise can come in various “colours” according to the exponent of

its power spectrum.

Let x left parenthesis t right parenthesisx(t) describe a fluctuating quantity. It can be characterized by the two-point

autocorrelation function

Cx(n) =

N

Σ

j=1

xjxj−n

(11.19)

(in discrete form), where nn is the position along a nucleic acid or protein sequence

of upper NN elements, and by the spectrum or amplitude spectral density

Ax(m) =

∞

Σ

j=−∞

xje^−2πim ,

(11.20)

whose square is the power spectrum or power spectral density:

Sx(m) = |Ax(m)|² ,

(11.21)

where mm is sequential frequency. The autocorrelation function and the power spec-

trum are just each other’s Fourier transforms (the Wiener–Khinchin relations, appli-

cable to stationary random processes).

A truly random process [containing all frequencies, hence “white noise”, w left parenthesis t right parenthesisw(t)]

should have no correlations in time. Hence,

Cw(τ ) ∝ δ(τ )

(11.22)

Ifupper DD itself changes with position (e.g., the diffusivity of a protein depends on the local concentration

of small ions surrounding it), then we have

∂c/∂t = ∇ · (D∇c) .

(11.18)