.. _active_brown: .. .. automodule:: active_brown Active-Brownian Dynamics ======================== Consider a particle undergoing random motion over a potential :math:`V`. For our development, we focus on 1-dimensional transport, but the final results are extended to 3 dimensions. The particle is subjected to thermal (\emph{i.e.} Brownian) force :math:`f_{B}` and active forces :math:`f_{A}` that represent transient perturbations from the surrounding enzyme activity. The temporal evolution of the position :math:`x(t)` is governed by a modified Langevin equation .. math:: \xi \frac{d x}{d t} = f_{V} + f_{B} + f_{A}, :label: langevin where :math:`f_{V} = - \frac{\partial V}{\partial x}` is the potential force on the particle at position :math:`x`. The Brownian force :math:`f_{B}` is governed by the fluctuation dissipation theorem, which states that :math:`f_{B}` is a Gaussian-distributed random force that satisfies .. math:: \langle f_{B}(t) \rangle & = & 0, \\ \langle f_{B}(t) f_{B}(t') \rangle & = & \kappa_B ( |t - t'|) = 2 k_{B}T \xi \delta (t - t'). This assumes that the environment is purely Newtonian, leading to the instantaneous decorrelation of Brownian forces (i.e. Gaussian white noise). Diffusive transport in a viscoelastic fluid leads to temporal memory in the Brownian force, reflecting temporal correlation in the frictional stress. In our work, we assume the active force :math:`f_{A}` are also Gaussian-distributed with an arbitrary temporal correlation, such that .. math:: \langle f_{A}(t) \rangle & = & 0, \\ \langle f_{A}(t) f_{A}(t') \rangle & = & \kappa_{A}(|t-t'|), where :math:`\kappa_{A}(t)` represents the temporal correlation between active forces. We develop a path integral formulation of the Active-Brownian particle that results in an expression for the joint probability :math:`\mathcal{P}[x(t)|x_0;t;f_A^{(0)}]`. This function governs the probability that if a particle begins at :math:`x_{0}` experiencing an active force :math:`f_{A}^{(0)}` at time :math:`t = 0`, the particle will be located at position :math:`x` at time :math:`t`. Carrying out the integral over the active forces is performed by noting a Gaussian form of active forces .. math:: \mathcal{P}[f_A(t)]\propto \exp \left\{ -\frac{1}{2}\int_0^t dt_{1}\int_0^t dt_2 f_A(t_1)\kappa_A^{-1}(|t_1-t_2|)f_A(t_2) \right\} This is used in the path integral formulation, and after functional integration over Brownian and Active forces, we arrive at the expression .. math:: & & \mathcal{P}[x|x_0;t,f_A^{(0)}, t_{0}] \nonumber \\ & & =\int_{x_0}^{x} \mathcal{D}[x(t)]\int \mathcal{D}[w(t)] \int d\eta \exp \left\{ -k_{B}T\xi\int_0^t [w(t_{1})]^2dt_{1}+i\xi\int_0^t \dot{w}(t_{1})x(t_{1})dt_{1} \right. \nonumber \\ & & \left. -i\xi w(t)x(t)+i\xi w(0)x(0)+i\int_0^t w(t_{1})f_V[x(t_{1})]dt_{1} -\frac{1}{2} \int_0^t \! \! dt_1 \int_{0}^{t} \! \! dt_2 w(t_1)\kappa_A(|t_1-t_2|) w(t_2) \right. \nonumber \\ & & \left. -i\eta f_A^{(0)} -\eta\int_0^t dt_1 \kappa_A(|t_1-t_0|)w(t_1)-\frac{1}{2}\eta^2\kappa_A(0) \right\} The theoretical development so far is general to any form of the spatially varying potential :math:`V(x)` and does not assume any special form for the conservative forces between particles. Dynamic behavior of an Active-Brownian polymer ---------------------------------------------- We apply our results of an active-Brownian particle in a harmonic potential to the problem of an active-Brownian polymer chain. Since the normal-mode expansion of the Rouse polymer adopts the form of a particle in a harmonic potential, we can frame our results in the previous section to find the behavior of the pth normal mode. We defined the normal-mode correlation function for the pth mode to be .. math:: C_{p}(\tau) & = & \langle \vec{X}_{p}(\tau) \cdot \vec{X}_{p'} (0) \rangle \\ & = & \frac{3 k_{B}T}{k_{p}} \left\{ \exp \left(-\frac{ p^{2} \tau}{N^2} \right) + \frac{\Gamma}{1- p^4/(K_{A}^{2} N^{4})} \left[ \exp \left(-\frac{p^{2} \tau}{N^2} \right) - \frac{ p^{2}}{K_{A} N^{2}} \exp \left( - K_{A} \tau \right) \right] \right\} \delta_{pp'} The dimensionless time :math:`\tau = t/t_{b}` is scaled by the diffusive time :math:`t_{b} = b^{2} \xi/(k_{B}T)`, and we define the dimensionless Active force :math:`F_{A}^{2} = \Gamma K_{A} = f_{A}^{2} b^{2}/(k_{B}T)^{2}` and dimensionless active rate constant :math:`K_{A} = t_{b} k_{A}`. We find the center-of-mass mean-square displacement to be .. math:: \mathrm{MSD}_{\mathrm{com}} = \langle \left( \vec{X}_{0}(t) - \vec{X}_{0}(0) \right)^{2} =\frac{6Nk_BT}{k_b}\Big[ \Big(\frac{1+\Gamma}{N^2}\Big)\tau+\frac{\Gamma}{N^2K_A}\Big(e^{-K_A \tau}-1\Big) \Big] :label: msd-com We note that Eq. :eq:`msd-com` has the short-time asymptotic behavior :math:`\mathrm{MSD}_{\mathrm{com}} \rightarrow 6 [k_{B}T/(\xi N)] t` as :math:`t \rightarrow 0`, which coincides with the mean-square displacement for a Brownian-only polymer. The long-time asymptotic behavior :math:`\mathrm{MSD}_{\mathrm{com}} \rightarrow 6 [k_{B}T/(\xi N)] (1 + F_{A}^{2}/2) t ` as :math:`t \rightarrow \infty` reveals the long-time effective temperature due to Brownian and active fluctuations, given by :math:`T_{AB} = T (1 + F_{A}^{2}/2)`. The mean-square displacement of a segment of the polymer chain is define as .. math:: \text{MSD}(\tau) = \langle ( \vec{r}(n,t) - \vec{r}(n,0))^{2} \rangle :label: msd-active for the chain position n. In this work, we focus on the midpoint motion at :math:`n=N/2`, but this is easily extended to other polymer positions. We insert our normal-mode representation into Eq. :eq:`msd-active`, resulting in the expression .. math:: \text{MSD}(\tau) = \text{MSD}_{\text{com}}(\tau) + 4 \sum_{p=1}^{\infty} \triangle C_{2p}(\tau) where :math:`\triangle C_{p}(\tau) = C_{p}(0) - C_{p}(\tau)`. In addition, we define the mean-square change in distance (MSCD) for a polymer chain. This quantity is defined as .. math:: \mathrm{MSCD} = \langle \left( \Delta \vec{R}(t) - \Delta \vec{R}(0) \right)^{2} \rangle where :math:`\Delta \vec{R}(t) = \vec{r}(N/2 + \Delta, t) - \vec{r}(N/2 - \Delta,t)`. Thus, MSCD is a measure of the change in separation distance between two points on the polymer that are :math:`2 \Delta` segments from each other. This results in the expression .. math:: \text{MSCD}(\tau) = 16 \sum_{p=0}^{\infty} \triangle C_{2p+1}(\tau) \sin^{2} \! \left[ \frac{\pi(2p+1)\Delta}{2N} \right], where :math:`\triangle C_{p}(\tau) = C_{p}(0) - C_{p}(\tau)` Functions contained with the 'active_brown' module -------------------------------------------------- .. automodule:: wlcstat.active_brown :members: