Coupling an elastic string to an active bath: the emergence of inverse damping

TL;DR

The work develops an effective stochastic field theory for a slow elastic string (Klein-Gordon dynamics) in contact with a fast, persistent bath of Run-and-Tumble Particles. By projecting out the active bath and exploiting a weak-coupling, time-scale separation, the authors derive a Langevin–Klein-Gordon equation with a streaming term, mode-resolved friction, and colored noise, all computed exactly to O(ζ_φ^2). A key finding is the emergence of negative friction (anti-damping) for sufficiently persistent baths, triggering exponential growth of field modes and an inverse-Landau-like instability, which saturates in nonlinear regimes. The results illuminate how activity and persistence in an active bath can actively drive wave-like dynamics in a coupled field and establish connections to nonequilibrium response theory and wave–particle energy transfer phenomena; numerical simulations corroborate the analytical predictions and reveal saturation behavior beyond the weak-coupling limit.

Abstract

We consider a slow elastic string with Klein-Gordon dynamics coupled to a bath of run-and-tumble particles. We derive and solve the induced Langevin-Klein-Gordon string dynamics with explicit expressions for the streaming term, friction coefficient, and noise variance. These parameters are computed exactly in a weak coupling expansion. The induced friction is a sum of two terms: one entropic, proportional to the noise variance as in the Einstein relation for a thermal equilibrium bath, and a frenetic contribution that can take both signs. The frenetic part wins for higher bath persistence, making the total friction negative, and hence creating a wave instability akin to inverse Landau damping. However, this acceleration decreases and eventually disappears when the propulsion speed of the active particles becomes much higher. Detailed simulations confirm the initial growth driven by this anti-damping.

Figures (10)

• Figure 1: A configuration of active particles coupled to the field $\phi$ on a circle. The forces on the particles and string are respectively indicated with red and green arrows: the field $\phi$ is pushed down near the particle position (green arrow), while the $z$-particles perform a gradient descent along $\phi$ (red arrow).
• Figure 2: Showing the dimensionless friction \ref{['nu fourier expansion']} as function of the tumbling rate $\nu_1/\nu_A$vs$L \alpha c/ v_0^2$ with constant $\nu_A = k^2 g^2 \mu m$ for the von Mises distribution \ref{['vmo']} at different $c/v_0$ with $p = 2$,$N = 10$. The passive limit takes $v_0 \to \infty, \quad v_0^2/\alpha \to 2\mu k_B T$, $v_0 \to \infty$. The $x-$axis variable is constructed so that it is dimensionless and well-defined in the passive limit.
• Figure 3: Average of $\Re\{\phi_1(t)\}$ over 100 noise/spin realisations versus time. We show both (a) the negative and (b) the positive friction regimes. In (a), at short times, $t < 1000$ s, the simulation result and Eq. \ref{['average phi n']} overlap almost exactly and exhibit exponential growth due to the negative friction effect. At later times $t > 1500$ s, the simulation amplitude saturates, while \ref{['average phi n']} keeps growing. For (b), both the simulation result and \ref{['average phi n']} decay exponentially. Other figures and the parameter values (Table I) are available in the SM supp.
• Figure 4: Individual $\Re\{\phi_1\}$ trajectory (no average) vs time, simulated over a long time interval. On this timescale, one cannot distinguish the individual oscillations, but we clearly see the negative friction effect due to the initial growth. Eventually, the amplitude saturates and shows pulsations of growth and decay.
• Figure 5: Von Mises distribution for $p = 2$ in Eq. (6) of the main text. The distribution is peaked at $x = 0 = L$.