Anomalous diffusion in viscoelastic media with active force dipoles

With the use of the “two-fluid model,” we discuss anomalous diffusion induced by active force dipoles in viscoelastic media. Active force dipoles, such as proteins and bacteria, generate nonthermal fluctuating flows that lead to a substantial increment of the diffusion. Using the partial Green's function of the two-fluid model, we first obtain passive (thermal) two-point correlation functions such as the displacement cross-correlation function between the two-point particles separated by a finite distance. We then calculate active (nonthermal) one-point and two-point correlation functions due to active force dipoles. The time correlation of a force dipole is assumed to decay exponentially with a characteristic time scale. We show that the active component of the displacement cross-correlation function exhibits various crossovers from super-diffusive to subdiffusive behaviors depending on the characteristic time scales and the particle separation. Our theoretical results are intimately related to the microrheology technique to detect fluctuations in nonequilibrium environment.

Figure 1

Schematic representation of the two-fluid model. The system consists of an elastic network characterized by the Lamé coefficient $\mu$, and a viscous fluid characterized by the shear viscosity $\eta$. The elastic and fluid components are coupled through the mutual friction. The length scale $\ell$ characterizes the typical internal structure of the elastic network. Orange objects represent stochastic force dipoles which are immersed in the fluid component. Two passive point particles separated by a distance $d$ are embedded in the fluid component. These passive particles undergo correlated random Brownian motion due to thermal fluctuations and active stochastic fluctuations induced by active force dipoles.

Figure 2

Scaling functions (a) ${\mathcal{G}}_1$ and (b) ${\mathcal{G}}_2$ [see Eqs. (10) and (11), respectively] appearing in the partial Green's function of the two-fluid model. The scaling variable is $z=r/\xi$, where $r$ is the distance and $\xi$ is the frequency-dependent characteristic length scale [see Eq. (8)]. The asymptotic behaviors of these scaling functions, as analytically given by Eqs. (12) and (13), respectively, are plotted with dotted lines.

Figure 3

The passive component of the power spectral density (PSD) of the two-point velocity cross-correlation functions (CCFs) (a) ${\langle V_{1x}{V}_{2x}\left(\omega \right)\rangle }_d$ and (b) $|{\langle V_{1y}{V}_{2y}\left(\omega \right)\rangle }_d|$ [see Eq. (19)] as a function of $\omega \tau$ for $d/\ell =1$, 10, 100. Here, $\tau =\eta /\mu$ is the viscoelastic time scale, and $d$ is the distance between the two-point particles immersed in viscoelastic media described by the two-fluid model. Both CCFs are scaled by $k_{\mathrm{B}}T/\left(2\pi \eta d\right)$ in order to make them dimensionless. Since ${\langle V_{1y}{V}_{2y}\left(\omega \right)\rangle }_d$ takes negative values for smaller $\omega \tau$ (shown by the dashed lines), we have plotted in (b) its absolute value. The number “2” in (a) indicates the slope representing the exponent of the power-law behaviors.

Figure 4

The passive component of the two-point displacement cross-correlation functions (CCFs) (a) ${\langle \mathrm{\Delta }{R}_{1x}\mathrm{\Delta }{R}_{2x}\left(t\right)\rangle }_d$ and (b) $|{\langle \mathrm{\Delta }{R}_{1y}\mathrm{\Delta }{R}_{2y}\left(t\right)\rangle }_d|$ [see Eq. (20)] as a function of $t/\tau$ for $d/\ell =1$, 10, 100. Here $d$ is the distance between the two-point particles immersed in viscoelastic media. Both CCFs are scaled by $k_{\mathrm{B}}T\tau /\left(2\pi \eta d\right)$ in order to make them dimensionless. Since ${\langle \mathrm{\Delta }{R}_{1y}\mathrm{\Delta }{R}_{2y}\left(t\right)\rangle }_d$ takes negative values for larger $t/\tau$ (shown by the dashed lines), we have plotted in (b) its absolute value. The numbers indicate the slope representing the exponent of the power-law behaviors.

Figure 5

The active component of the power spectral density (PSD) $\langle V_x^2\left(\omega \right)\rangle$ [see Eq. (32)] as a function of $\omega \tau$ for $\delta /\ell =1$, 10, 100. Here a single point particle is immersed in viscoelastic media described by the two-fluid model, and $\delta$ is the cutoff length corresponding to the particle size. In the plot, the PSD is scaled by $c_0\langle m^2\left(\omega \right)\rangle /\left(2880{\pi }^2{\eta }^2\delta \right)$ in order to make it dimensionless. The number indicates the slope representing the exponent of the power-law behavior.

Figure 6

The active component of the mean squared displacement (MSD) $\langle {\left(\mathrm{\Delta }{R}_x\right)}^2\left(t\right)\rangle$ [see Eq. (36)] as a function of $t/\tau$ for $\delta /\ell =1$, 10, 100. Here a single point particle is immersed in viscoelastic media, and $\delta$ is the cutoff length. (a) The case when the time correlation of the force dipole is $\delta$-correlated [see Eq. (35)]. (b) The case when the time correlation of the force dipole decays exponentially with a characteristic relaxation time ${\tau }_{\mathrm{d}}$ [see Eq. (39)], and we set here ${\tau }_{\mathrm{d}}/\tau =100$. In these plots, $\langle {\left(\mathrm{\Delta }{R}_x\right)}^2\left(t\right)\rangle$ is scaled by $c_{0}S\tau /\left(2880{\pi }^2{\eta }^2\delta \right)$ in order to make it dimensionless. The numbers indicate the slope representing the exponent of the power-law behavior.

Figure 7

The active component of the scaled power spectral density (PSD) (a) ${\langle V_{1x}{V}_{2x}\left(\omega \right)\rangle }_d$ and (b) $|{\langle V_{1y}{V}_{2y}\left(\omega \right)\rangle }_d|$ [see Eq. (44)] as a function of $\omega \tau$ for $d/\ell =1.1$, 10, 100. Here $d$ is the distance between the two-point particles immersed in viscoelastic media. Both PSDs are scaled by $c_0\langle m^2\left(\omega \right)\rangle /\left(960{\pi }^2{\eta }^{2}d\right)$ in order to make them dimensionless. Since ${\langle V_{1y}{V}_{2y}\left(\omega \right)\rangle }_d$ takes negative values for smaller $\omega \tau$ (shown by the dashed lines), we have plotted in (b) its absolute value. The numbers indicate the slope representing the exponent of the power-law behaviors.

Figure 8

The active component of the two-point displacement cross-correlation functions (CCFs) (a) ${\langle \mathrm{\Delta }{R}_{1x}\mathrm{\Delta }{R}_{2x}\left(t\right)\rangle }_d$ and (b) $|{\langle \mathrm{\Delta }{R}_{1y}\mathrm{\Delta }{R}_{2y}\left(t\right)\rangle }_d|$ as a function of $t/\tau$ for $d/\ell =1.1$, 10, 100. Here $d$ is the distance between the two-point particles immersed in viscoelastic media, and the time correlation of the force dipole is $\delta$-correlated [see Eq. (35)]. Both CCFs are scaled by $c_{0}S\tau /\left(960{\pi }^2{\eta }^{2}d\right)$ in order to make them dimensionless. Since ${\langle \mathrm{\Delta }{R}_{1y}\mathrm{\Delta }{R}_{2y}\left(t\right)\rangle }_d$ takes negative values for larger $t/\tau$ (shown by the dashed lines), we have plotted in (b) its absolute value. The numbers indicate the slope representing the exponent of the power-law behaviors.

Figure 9

The active component of the two-point displacement cross-correlation functions (CCFs) (a) ${\langle \mathrm{\Delta }{R}_{1x}\mathrm{\Delta }{R}_{2x}\left(t\right)\rangle }_d$ and (b) $|{\langle \mathrm{\Delta }{R}_{1y}\mathrm{\Delta }{R}_{2y}\left(t\right)\rangle }_d|$ as a function of $t/\tau$ for $d/\ell =1.1$, 10, 100. Here $d$ is the distance between the two-point particles immersed in viscoelastic media. The time correlation of the force dipole decays exponentially with a characteristic relaxation time ${\tau }_{\mathrm{d}}$ [see Eq. (39)], and we set here ${\tau }_{\mathrm{d}}/\tau =100$. Both CCFs are scaled by $c_{0}S\tau /\left(960{\pi }^2{\eta }^{2}d\right)$ in order to make them dimensionless. Since ${\langle \mathrm{\Delta }{R}_{1y}\mathrm{\Delta }{R}_{2y}\left(t\right)\rangle }_d$ takes negative values for larger $t/\tau$ (shown by the dashed lines), we have plotted in (b) its absolute value. The numbers indicate the slope representing the exponent of the power-law behaviors.