Kinetic description of one-dimensional stochastic dynamics with small inertia

TL;DR

This work develops a cohesive, multi-representation framework for reducing stochastic dynamics with small inertia to tractable one-variable descriptions. By exploiting moment, cumulant, and Hermite-basis formalisms, it derives adiabatic-elimination results and μ^1 corrections that generalize the Ott–Antonsen reduction to systems with inertia, and it formulates a one-dimensional Fokker–Planck-type equation with a forced drift for active Brownian particles. A key contribution is the explicit construction of a corrected Smoluchowski equation and its stochastic-Langevin counterpart, along with a low-dimensional two-cumulant model for collective oscillator dynamics under weak synchrony. The framework also yields a rigorous diffusion-drift description for ABP in one dimension, highlighting qualitative differences from passive cases. Together, these results provide robust, low-dimensional tools for analyzing noise-driven populations of Brownian particles and phase oscillators with small inertia, with relevance to synchronization, transport in ratchets, and active matter systems.

Abstract

We study single-variable approaches for describing stochastic dynamics with small inertia. The basic models we deal with describe passive Brownian particles and phase elements (phase oscillators, rotators, superconducting Josephson junctions) with an effective inertia in the case of a linear dissipation term and active Brownian particles in the case of a nonlinear dissipation. Elimination of a fast variable (velocity) reduces the characterization of the system state to a single variable and is formulated in four representations: moments, cumulants, the basis of Hermite functions, and the formal cumulant variant of the last. This elimination provides rigorous mathematical description for the overdamped limit in the case of linear dissipation and the overactive limit of active Brownian particles. For the former, we derive a low-dimensional equation system which generalizes the Ott-Antonsen Ansatz to systems with small effective inertia. In the latter case, we derive a Fokker-Planck-type equation with a forced drift term and an effective diffusion in one dimension, where the standard two-/three-dimensional mechanism is impossible. In the four considered representations, truncated equation chains are demonstrated to be utilitary for numerical simulation for a small finite inertia.

Figures (5)

• Figure 1: Hierarchy of smallness of high-order elements for different approaches; $L^1$-norm $||g(\varphi)||\equiv\int_0^{2\pi}|g(\varphi)|\,\mathrm{d}\varphi$. The probability density functions $W_0(\varphi)$ for all approaches coincide with a relative accuracy on the level of the machine calculation accuracy. (a): moments, (b): cumulants, (c): Hermite basis, (d): formal cumulants for the Hermite basis. (a,c): 100 elements are used for simulations, (b,d): 50 elements are used for simulations. Equations are discretized in the $\varphi$-coordinate by means of the central difference schemes for derivatives and the number of nodes $N=100$. The solid lines in panels (b--d) serve as a guide to estimate how faithfully the high-order elements follow a geometric progression.
• Figure 2: Error of calculation of the probability density $W_0(\varphi)$ is plotted vs $\mu$ for different approaches and orders of approximation. (a): moments, (b): cumulants, (c): Hermite basis, (d): formal cumulants for the Hermite basis. The order of approximation: $\mu^0$ (black squares), $\mu^1$ (red diamonds), $\mu^2$ (blue circles).
• Figure 3: Hierarchy of smallness of high-order elements for active Brownian particles with $\alpha=-1$, $\beta=1$ (a) and passive particles with nonlinear friction $\alpha=+1$, $\beta=1$ (b). For convenience of presentation the same rescaling (\ref{['eq:wW']}) is adopted as for the passive particles with linear friction. Dashed line: asymptotic law (\ref{['eq-abp-07']}), dotted line: law (\ref{['eq-abp-08']}); $\mu=0.01$ for both curves. Parameters of force $F(\varphi)$ and discretization in $\varphi$ are the same as in Fig. \ref{['fig1']}. A series of 50 terms is used.
• Figure 4: The dependence of the inverse critical coupling $\varepsilon_\mathrm{cr}$ versus the half-width $\gamma$ of a uniform distribution of natural frequencies $\omega$ is plotted for the corrected Smoluchowski equation (solid line) and for the original Fokker--Planck equation with inertia (dashed line). Parameters: $\mu\sigma^2=0.1$.
• Figure 5: The dependencies of the global Kuramoto order parameter $|R|$ versus $\sigma^2/\varepsilon$ are plotted for a population of phase rotators with a bimodal frequency distribution in the thermodynamic limit. Blue lines: 2CC model (\ref{['eq5200:01']})--(\ref{['eq5200:02']}), orange: exact time-independent solutions of the corrected Smoluchowski equation. Lines are solid (dashed) for stable (unstable) solutions (stability was analyzed only for the blue lines, and for the orange lines it was inferred by analogy with the blue ones). The rms value of $|R|$ for oscillatory regimes is plotted with dotted lines; the shading shows the range of variation of $|R|$ for oscillatory solutions. The red triangles mark the critical values of coupling $\varepsilon_\mathrm{cr}$ calculated with Eq. \ref{['eq:sum_acebron']}. Parameters: $\mu\sigma^2=0.01$ and $\gamma/\sigma^2=0.3$ (a), $1$ (b), $1.1$ (c), $3$ (d), $10$ (e), and $100$ (f).