Neural optimization of the most probable paths of 3D active Brownian particles

TL;DR

The results show that neural optimization combined with the Onsager-Machlup variational principle provides an efficient and versatile framework for exploring optimal transition pathways in active and nonequilibrium systems.

Abstract

We develop a variational neural-network framework to determine the most probable path (MPP) of a 3D active Brownian particle (ABP) by directly minimizing the Onsager-Machlup integral (OMI). To obtain the OMI, we use the Onsager-Machlup variational principle for active systems and construct the Rayleighian of the ABP by including its active power. This approach reveals geometric transitions of the MPP from in-plane I- and U-shaped paths to 3D helical paths as the final time and net displacement are varied. We also demonstrate that the initial and final boundary conditions have a significant impact on the MPPs. Our results show that neural optimization combined with the Onsager-Machlup variational principle provides an efficient and versatile framework for exploring optimal transition pathways in active and nonequilibrium systems.

Figures (3)

• Figure 1: (a) A trajectory (red curve) of a 3D ABP described by a time-dependent position vector $\mathbf{r}(t)$. The self-propulsion velocity is $\mathbf{U}(t)=U\mathbf{e}(t)$, where $U$ is a constant speed and $\mathbf{e}(t)$ is the orientational unit vector parameterized by the polar angle $\theta$ and azimuthal angle $\phi$. We consider the conditional path probability starting from $\mathbf{r}_{\mathrm{i}}$ with $\mathbf{e}_{\mathrm{i}}$ at $t=0$ and terminating at $\mathbf{r}_{\mathrm{f}}$ with $\mathbf{e}_{\mathrm{f}}$ at $t=t_{\rm f}$. The initial and the final self-propulsion velocities $\mathbf{U}$ are shown by blue arrows. (b) Schematic diagram of NN representation of position $\mathbf{r}(t)$ and orientation functions $\mathbf{e}(t)$. The fully connected neural network consists of an input layer, two hidden layers (each with 30 nodes), and output layers for the respective functions. Activation functions are applied sequentially as tanh (first hidden layer), tanh (second hidden layer), and linear (output layer) for $\mathbf{r}(t)$, and tanh (first hidden layer), linear (second hidden layer), and linear (output layer) for $\mathbf{e}(t)$.
• Figure 2: NN solutions of the MPP of a 3D ABP for different final times $\overline{t}_{\rm f}$, with fixed initial and final boundary conditions $\overline{x}_{\rm i}=\overline{y}_{\rm i}=\overline{z}_{\rm i}=\phi_{\rm i}=0$, $\theta_{\rm i}=\pi/2$ and $\overline{x}_{\rm f}=5$, $\overline{y}_{\rm f}=\overline{z}_{\rm f}=\phi_{\rm f}=0$, $\theta_{\rm f}=\pi/2$, respectively. (a) I-path when $\overline{t}_{\rm f}=8$, (b) U-path when $\overline{t}_{\rm f}=10$, and (c) H-path when $\overline{t}_{\rm f}=14$. The red curves represent the MPPs $\mathbf{r}^\ast(t)$, and the gray curves show their projections onto the $x$-$y$, $x$-$z$, and $y$-$z$ planes. The blue arrows indicate the optimized orientations $\mathbf{e}^\ast(t)$ of the ABP. (d) The average curvature $\overline{\kappa}$ (left axis) and the torsion $\overline{\tau}$ (right axis) as functions of the final time $\overline{t}_{\rm f}$ (see Eq. (\ref{['geometry']}) and the text). The blue, green, and red background regions correspond to I-, U-, and H-paths, respectively. (e) The dimensionless OMI $\overline{O} = 2 O/(k_{\rm B}T {\rm Pe})$ [left axis, see Eq. (\ref{['OMI_eq']})] and the entropy change $\overline{S}=\Delta S_{\rm b}/(k_{\rm B}{\rm Pe})$ [right axis, see Eq. (\ref{['entropy']})] as functions of the final time $\overline{t}_{\rm f}$. The blue triangles, green circles, and red squares represent I-, U-, and H-paths, respectively. (f) Phase diagram of the MPPs in the parameter space spanned by $\overline{x}_{\rm f}/\overline{t}_{\rm f}$ and $\overline{t}_{\rm f}$. The symbols are the same as in (e). The solid black line represents the analytical line (see the text) that separates the I- and U-paths in 2D, while the black dashed line is a guide for the eye.
• Figure 3: NN solutions of the MPP of a 3D ABP for different final times $\overline{t}_{\rm f}$, with fixed initial and final boundary conditions $\overline{x}_{\rm i}=\overline{y}_{\rm i}=\overline{z}_{\rm i}=\phi_{\rm i}=0$, $\theta_{\rm i}=\pi/2$ and $\overline{x}_{\rm f}=5$, $\overline{y}_{\rm f}=\overline{z}_{\rm f}=\phi_{\rm f}=0$, $\theta_{\rm f}=5\pi/2$, respectively. (a) H-path when $\overline{t}_{\rm f}=8$, (b) H-path when $\overline{t}_{\rm f}=14$, and (c) $\ell$-path when $\overline{t}_{\rm f}=18$. (d) The average curvature $\overline{\kappa}$ (left axis) and the torsion $\overline{\tau}$ (right axis) as functions of the final time $\overline{t}_{\rm f}$ (see Eq. (\ref{['geometry']}) and the text). The red and purple background regions correspond to H- and $\ell$-paths, respectively. (e) The dimensionless OMI $\overline{O} = 2 O/(k_{\rm B}T {\rm Pe})$ [left axis, see Eq. (\ref{['OMI_eq']})] and the entropy change $\overline{S}=\Delta S_{\rm b}/(k_{\rm B}{\rm Pe})$ [right axis, see Eq. (\ref{['entropy']})] as functions of the final time $\overline{t}_{\rm f}$. The red squares and purple triangles represent H- and $\ell$-paths, respectively. (f) Phase diagram of the MPPs in the parameter space spanned by $\overline{x}_{\rm f}/\overline{t}_{\rm f}$ and $\overline{t}_{\rm f}$. The black dashed line is a guide for the eye.