Hyperballistic transport in dense systems of charged particles under ac electric fields

TL;DR

This paper develops a Caldeira-Leggett-based model in which both a tagged charged particle and the bath oscillators respond to an external AC field, producing a non-Markovian generalized Langevin description with a field-modified fluctuation-dissipation relation. It derives closed-form expressions for the time-dependent diffusivity $D(t)$ under two initial conditions, revealing damped oscillations and transient giant enhancements; notably, short-time hyperballistic transport ($MSD \,\sim\, t^4$) arises when the field is switched on at $t=0$ after DC-equilibrated bath conditions. In the long-time limit, $D(t)$ reverts to the Stokes–Einstein value, independent of the field, while the polarization-induced renormalized charge $Q$ controls the transient amplification and a resonance-like but interference-suppressed local Lorentz field. These results point to a new nonequilibrium diffusion regime with potential implications for transport in dense plasmas and electrolytes under AC driving, and suggest extensions to include magnetic effects and linear-response frameworks.

Abstract

The Langevin equation is ubiquitously employed to numerically simulate plasmas, colloids and electrolytes. However, the usual assumption of white noise becomes untenable when the system is subject to an external AC electric field. This is because the charged particles in the system, which provide the thermal bath for the particle transport, become themselves responsive to the AC field and the thermal noise is field-dependent and non-Markovian. We theoretically study the particle diffusivity in a Langevin transport model for a tagged charged particle immersed in a dense system of charged particles (plus also, possibly, other neutral particles) that act as the thermal bath, under an external AC electric field. This is done by properly accounting for the effects of the AC field on the thermal bath statistics. We analytically derive the time-dependent generalized diffusivity $D(t)$ for different initial conditions. The generalized diffusivity exhibits damped oscillatory-like behaviour with initial very large peaks, where the generalized diffusion coefficient is enhanced by orders of magnitude with respect to the infinite-time steady-state value. The latter coincides with the Stokes-Einstein diffusivity in the absence of external field. For initial conditions where the external field is already on at $t=0$ and the system is thermalized under DC conditions for $t \leq 0$, the short-time behaviour is hyperballistic, $MSD \sim t^4$ (where MSD is the mean-squared displacement), leading to giant enhancement of the particle transport. Finally, the theory elucidates the role of medium polarization on the local Lorentz field, and allows for estimates of the effective electric charge due to polarization by the surrounding charges.

Figures (6)

• Figure 1: Different initial conditions for the particle-bath Hamiltonian \ref{['eq:bathHamiltonian1']} under the action of an external AC electric field $E(t)$. (a): Field-off initial conditions, where the field is switched on at $t=0$ (Eq. (\ref{['eq:caseOne']})). (b): Field-on initial conditions, where the field is already on at $t=0$ (Eq. (\ref{['eq:caseTwo']})).
• Figure 2: Plot of the effective (Lorentz) field Eq. \ref{['eq:campoEffettivo']} (see also the more explicit form Eq. \ref{['eq:lorentzFieldExplicit']}, with $i=0$) as a function of time. Panel (a): field-off ICs. Panel (b): field-on ICs. Blue, yellow and green lines correspond to values $\Omega/\omega_0=0.8, 0.99, 1.13$. The oscillations are giantly enhanced at the resonant frequency $\Omega\simeq\omega_0$, and their amplitude increases with time (Eq. \ref{['eq:campoEffettivo']}).
• Figure 3: Plot of the maximum (with respect to $\Omega$) of the effective (Lorentz) field \ref{['eq:campoEffettivo']} (see also the more explicit form \ref{['eq:lorentzFieldExplicit']}, with $i=0$) as a function of the dimensionless time $\omega_0 t$. Panel (a): field-off ICs. Panel (b): field-on ICs. The maximum of $E_0'(\omega_0 t)$ is reached when the driving frequency $\Omega\sim\omega_0$. This maximum is rapidly increasing in time.
• Figure 4: Schematic depiction of the physical system (dense ionized matter) considered in the theoretical model. The tagged particle (green) moves in a thermal bath (dense medium) of other charges schematically represented as charged harmonic oscillators in the theoretical model. The frequencies of these fictitious oscillators schematically represent the eigenfrequencies of the sound modes of the system. These oscillators are coupled to the tagged particle by different coupling strengths, represented as red lines. The effective charge of the tagged particle, $Q$ (represented as a shaded area), accounts for the local charge distribution and polarizing Lorentz field around the tagged particle. The system is under an AC electric field (depicted in figure at a certain instant of time) exerted by the electrodes (represented by two opposite charged plates in the graph) and might be subjected to an overall static field $V(x)$, which is not explicitly taken into account in our derivations but the presence of which would not qualitatively change our results.
• Figure 5: Time evolution of the generalized diffusivity $\tilde{D}=D\frac{m\gamma_0}{k_BT}$ for (a) field-off \ref{['eq:diffusivity1']} and (b) field-on \ref{['eq:diffusivity2']} initial conditions, obtained with different field amplitudes $\mathcal{E}_0$ (Eq. \ref{['eq:E1']}). Yellow, green and orange lines correspond to $\mathcal{E}_0=1,2,3$, whereas the blue line is the Stokes-Einstein result obtained for $\mathcal{E}_0=0$. In the inset, the red line shows the diffusivity's behavior for small times. For comparison, the green line in the inset of panel (a) is $\sim t^3$, while the green line in the inset of panel (b) is $\sim t$.