The significance of the persistence length (contd...)

Now consider, the scaler product of the two unit tangent vectors  which has a maximum value of unity when the two vector are parallel at the ends of the filament. At non-zero temperature the filament samples variety of orientations, so that the ensemble average  has a maximum average value of unity. The function  is called the correlation function of the tangent vector, i.e. it defines the correlation of tangent vectors at different positions along the curve. At low temperature,  is small so that , yielding,

,

(34.5)

Similarly the mean squared difference between the tangent vectors can be obtained as

,

(34.6)

Thus the persistence length measures the distance along the filament over which the orientation of the curves become decorreleted.

For a rigid rod, i.e. rod with contour length , equation 34.5 shows that the correlation decays linearly with increase in contour length. However, if , the filament appears fluffy and  should vanish as the tangent vectors at the extreme ends of the filament becomes uncorrelated. This behavior is not seen in equation 34.5. The correct expression for the tangent correlation function applicable at long and short distances is,

,

(34.7)

The proof of equation 34.7 can be found elsewhere.