Dynamics of Aligning Active Matter: Mapping to a Schrödinger Equation and Exact Diagonalization

Abstract

There has been recent interest in the relaxational modes of small-scale fully connected systems of aligning self-propelled particles (Spera et al., Phys. Rev. Lett. {\bf 132}: 078301 (2024)). We revisit the classical connection between Fokker-Planck and Schrödinger equations to address this by means of exact diagonalization, allowing for rigorous analytical insight into the full spectrum. This allows us to extract exact results which we compare to the existing result from linearized statistical field theory. We derive asymptotically correct analytical results that improve upon the prior approximations. We show that this methodology can fruitfully be extended to the case of non-reciprocal interactions which gives rise to a non-Hermitian Schrödinger problem akin to those in open quantum mechanics. While the non-reciprocity can be chosen such as not to alter the stationary distribution, it fundamentally changes the nature of the steady state which we quantify via the entropy production. We discuss the case of low particle numbers as well as the emergence of mean-field dynamics at large numbers.

Figures (10)

• Figure 1: Empirical stationary energy distribution $p_\text{sim}(E)$ as obtained for $\bar{\Gamma}=0.5$ from agent based simulations. The main panels compare $p_\text{sim}(E)/g_\text{est}(E)$ to the Boltzmann weight $\exp(-\beta E)/\cal Z$, where $g_\text{est}(E)$ is the density of states, estimated by uniform sampling of the configuration space. In the insets we show the empirical density directly without normalization to the density of states. The ensembles consist of $10^6$ systems with $N=3$ particles each, with a runtime of $10D ^{-1}$. a): Balanced-tournament non-reciprocity as given by \ref{['eq:three-particle-nonreciprocity']}. The stationary state is Boltzmann distributed, slight tail errors can be attributed to the sampling of $g_\text{est}(E)$. Note that the reciprocal case, $\gamma=0$, is trivially Boltzmann distributed. b): Same as a), for more arbitrary non-reciprocity as indicated within the panel. The reciprocal energy is no longer a valid potential function in this case.
• Figure 2: Stationary probability densities (top row, heatmap corresponding to the colorbar at the bottom) and currents (bottom row) for $\bar{\Gamma}/D =0.3$ (left row) and $\bar{\Gamma}/D=1.2$ (right row) respectively. The periodicity depends on the symmetry number $m=1,2,\ldots$ of the interaction. The currents are always along $\bm S$, see \ref{['eq:current']}; red and blue colors are used as a guide to the eye to distinguish the sign. For $\gamma=0$, interaction is reciprocal and the current vanishes. For the case of negative $\bar{\Gamma}$, the distributions are shifted relative to the positive $\bar{\Gamma}$ case by $\pi/m$.
• Figure 3: Relaxation of an exemplary initial $P_2(\theta_1,\theta_2)$ for $\bar{\Gamma}/D=1$ and $\gamma_{ij}\equiv 0$. The panels show a time series for $t/D^{-1}=0,0.2,0.5,5.0$. Angle-resolved heatmaps where obtained using the mapping method and \ref{['eq:time-evolution']} (colors as in \ref{['fig:stationary-current']}). Insets show the marginals $P_1(\bm 1)$ (top) and $P_1(\bm 2)$ (right) in blue. Reconstructed marginals from empirical Fourier components, obtained from agent-based simulation as $\hat{p}_n^{(i)} = \expectationvalue{\exp(-\textrm{i} n \theta_i)}$, are drawn in orange dashed-dotted lines, showing excellent quantitative agreement. In agent-based simulations, the ensemble average $\expectationvalue{\ldots}$ contains $10^6$ systems, the time step is fixed at $0.01/D$. The stationary state, despite pronounced correlations, has uniform marginals. The color scheme used here is the same as in \ref{['fig:stationary-current']}.
• Figure 4: Left: Spectrum of $\mathcal{H}$ for reciprocal interactions in the $N=2$ case. We show the eigenvalues $E_n/D$ as a function of the reduced symmetric coupling strength $\bar{\Gamma}/D$. The flat line at $E_0/D \equiv 0$ corresponds to the stationary state. The masses of the polar and nematic fields are given by the eigenvalues of particular eigenmodes, distinguished by symmetry w.r.t. particle permutation. We compare these to the approximate field theory given by spera2024 (orange dashed line: polar mode, red dashed line: nematic mode). In between the physical relevant symmetric modes, lie genuine two-particle modes that do not contribute to the order parameter. The two stars mark modes whose relaxation in time is shown in the other panels. There is an apparent deviation of the field theoretic results at larger values of $\bar{\Gamma}/D$, this is explored in more detail in \ref{['fig:projection-approximation-with-spera']}. Right: Relaxation of $\hat{p}_{1,2}(t)$ for the two eigenmodes marked in the spectrum with stars ($\bar{\Gamma} = 0.5D$, $m=2$) which are associated with nematic order as a function of reduced time. We find excellent quantitative agreement with direct agent-based simulations (using a time step of $0.01 D ^{-1}$ and ensemble size of $10^7$) which were initialized to be close to the relevant eigenmodes by inversion sampling from the cumulative distributions corresponding to the eigenmode. In the respective insets, we visualize the modes in the $\theta_1$-$\theta_2$-plane using the same color scheme as in \ref{['fig:stationary-current']}.
• Figure 5: Numerically exact eigenvalues (blue solid lines) associated with nematic order in the case of nematic interactions ($m=2$), along with the projective approximation (orange dashed lines) given by \ref{['eq:projection_new']} and the spera2024 approximation (green dash-dotted line) for the nematic mass, as given by \ref{['eq:spera-nematic']}. Also shown are the mean-field results, see \ref{['eq:mean-field-mass']}, in the thermodynamic limit (light gray dashed line), over which all methods are a considerable improvement, as well as specifically for $N=2$ (dark gray dashed line).