Monday, April 7, 2008

[MT 2] Relations between PL exponents

Realistically, the effective medium of the cell should be described by a frequency dependent elastic modulus

G(\omega)\propto \omega^{x-1}

For simplicity we assume a purely elastic medium with x=1, so that the bead displacement is directly proportional to the momentary sum of forces. Then the PL exponents are related as follows:

Mean squared displacement:

\overline{x^2}(\tau)\propto \tau^{\beta}

Velocity Autocorrelation:

C_{vv}(\tau)\propto \tau^{-\gamma}\propto \tau^{\beta-2}

Velocity power spectral density:

P_v(\omega)\propto \omega^{-\alpha} \propto \omega^{1-\beta}

Force power spectral density:

P_F(\omega)\propto \omega^{-\lambda} \propto \omega^{-\beta-1}

In our case we get for diffusive motion lambda=2 and for ballistic motion lambda=3. According to

P.Bursac et al., Nature Materials 4, 557 (2005)

a lambda of 2 is characteristic for force fluctuations that are finite but discontinuous (a series of force steps), whereas lambda=4 corresponds to uniformly continous fluctuations.

This could be checked by evolving the random phases in v(w) and then analyzing the resulting x(t) ~ F(t).

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