Analytical solution of the generalized Langevin equation with hydrodynamic interactions: Subdiffusion of heavy tracers

Denis S. Grebenkov and Mahsa Vahabi

Phys. Rev. E 89, 012130 – Published 21 January 2014

Abstract

We consider a generalized Langevin equation that can be used to describe thermal motion of a tracer in a viscoelastic medium by accounting for inertial and hydrodynamic effects at short times, subdiffusive scaling at intermediate times, and eventual optical trapping at long times. We derive a Laplace-type integral representation for the linear response function that governs the diffusive dynamics. This representation is particularly well suited for rapid numerical computation and theoretical analysis. In particular, we deduce explicit formulas for the mean and variance of the time averaged (TA) mean square displacement (MSD) and velocity autocorrelation function (VACF). The asymptotic behavior of the TA MSD and TA VACF is investigated at different time scales. Some biophysical and microrheological applications are discussed, with an emphasis on the statistical analysis of optical tweezers' single-particle tracking experiments in polymer networks and living cells.

Received 7 November 2013

DOI:https://doi.org/10.1103/PhysRevE.89.012130

Authors & Affiliations

Denis S. Grebenkov^* and Mahsa Vahabi

Laboratoire de Physique de la Matière Condensée (UMR 7643), CNRS–Ecole Polytechnique, 91128 Palaiseau, France

^*denis.grebenkov@polytechnique.edu

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Vol. 89, Iss. 1 — January 2014

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Images

Figure 1
For normal diffusion ( $α = 1$ ), function $G_{c} (s)$ from Eq. (71) (blue solid line) is compared to its limit at $k = 0$ from Eq. (81) (red full circles). Two vertical dotted lines show the approximate positions $s_{1} \approx k / γ_{1}$ and $s_{2} \approx γ_{1} / M$ of two maxima. In the limit $k = 0$ , the first maximum disappears.
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Figure 2
MSD and VACF as functions of lag time $t$ for a heavy tracer in water ( $α = 1$ ). (a) Exact MSD from Eq. (69) (full circles), the short-time ballistic $\frac{k_{B} T}{M} t^{2}$ behavior (solid line), the intermediate normal scaling $\frac{k_{B} T}{γ_{1}} t$ , and the long-time behavior $\frac{2 k_{B} T}{k} (1 - e^{- k t / γ_{1}})$ from Eq. (74) without power law corrections (dashed blue line). Three dotted vertical lines indicate the characteristic time scales $τ_{h} = {(M / γ_{h})}^{2}$ , $τ = M / γ_{1}$ , and $τ_{k} = γ_{1} / k$ . (b) Exact VACF from Eq. (70) (full circles), the short-time plateau $k_{B} T / M$ , and the long-time behavior (75) without power law corrections (dashed blue line). The power law behaviors $\propto t^{- 3 / 2}$ and $\propto t^{- 7 / 2}$ are also shown by solid line. Note that the VACF changes the sign from positive to negative at $t \approx 0.3$ ms.
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Figure 3
Linear response function $G (t)$ (multiplied by $k_{B} T$ ) for normal diffusion ( $α = 1$ , red dashed curve) and subdiffusion ( $α = \frac{7}{8}$ , blue solid curve) with optical trapping. For comparison, symbols show $G (t)$ for the same parameters but without optical trapping ( $k = 0$ ). Two plots show semilogarithmic and log-log scales.
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Figure 4
MSD (a), (b) and VACF (c), (d) of a heavy tracer in a viscoelastic medium for different scaling exponents $α$ (ranging progressively as $\frac{4}{8}$ , $\frac{5}{8}$ , $\frac{6}{8}$ , $\frac{7}{8}$ , and 1), the other parameters being kept fixed as described in the text. Plots (b) and (d) represent the situation without optical trapping ( $k = 0$ ). Dashed lines show the short- and long-time asymptotics. For MSD plots (a), (b), dotted lines indicate the subdiffusive behavior $D_{α} t^{α} / Γ (1 + α)$ at intermediate times, with $D_{α} = k_{B} T / γ_{α}$ . For VACF plot (d), dotted lines indicate the asymptotic behavior $D_{α} t^{α - 2} / Γ (α - 1)$ at intermediate times.
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Figure 5
Function $G_{c} (s)$ from Eq. (89) (blue solid line) and its small $s$ approximation (91) (cyan full squares). For comparison, function $G_{c} (s)$ with $k = 0$ is also shown (red full circles). Three vertical dotted lines show the positions $s_{1, 2}$ of the extrema from Eq. (92) and the root $s_{0}$ from Eq. (90).
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Figure 6
Diagram for the number of the roots $z_{j}$ satisfying the condition $| arg (z_{j}) | < π / q$ , for different values $c_{r}$ and $α$ (here $c_{0} = 0$ , i.e., $k = 0$ ).
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