## 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

$G(\omega)\propto&space;\omega^{x-1}$

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:

$\overline{x^2}(\tau)\propto&space;\tau^{\beta}$

Velocity Autocorrelation:

$C_{vv}(\tau)\propto&space;\tau^{-\gamma}\propto&space;\tau^{\beta-2}$

Velocity power spectral density:

$P_v(\omega)\propto&space;\omega^{-\alpha}&space;\propto&space;\omega^{1-\beta}$

Force power spectral density:

$P_F(\omega)\propto&space;\omega^{-\lambda}&space;\propto&space;\omega^{-\beta-1}$

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).