Dynamical density functional theory for dense odd-diffusive fluids

TL;DR

The paper develops a dynamical density functional theory for dense two-dimensional fluids with odd diffusion, modeling the antisymmetric diffusion tensor $\mathbf{D} = D_0(\mathbf{I} + \kappa \boldsymbol{\epsilon})$ and deriving a closed evolution for the one-body density using the adiabatic approximation and a mean-field excess free energy. It demonstrates that odd diffusion qualitatively reshapes relaxation by generating transient circulating currents, while equilibrium density profiles remain unchanged, with repulsive interactions enhancing the transient angular transport. The authors apply the theory to bulk and ring-confined geometries, showing that odd-DDFT predicts circulating currents, angular redistribution, and accelerated relaxation, and that predictions agree quantitatively with Brownian-dynamics simulations. These results provide a self-consistent, field-based framework to study nonequilibrium transport in dense odd-diffusive fluids and open avenues for exploring chirality-driven dynamics in confined and structured environments.

Abstract

Odd diffusion breaks time-reversal symmetry in overdamped systems through transverse probability currents while preserving equilibrium steady states. In this work, we develop a dynamical density functional theory (DDFT) for dense interacting odd-diffusive fluids and apply it to ultrasoft particles in two dimensions. In bulk, odd diffusion qualitatively reshapes collective relaxation by generating transient circulating current patterns which do not exist in normal fluids. Under harmonic ring confinement, the circulation of probability current induces an angular redistribution of density along the ring during relaxation. This unique footprint of odd diffusion opens up a shorter pathway to equilibrium. Repulsive interactions significantly enhance these effects. Excellent agreement with Brownian dynamics simulations confirms that our odd-DDFT framework quantitatively captures all essential nonequilibrium aspects of the nontrivial odd transport and collective redistribution for dense fluids in both bulk and confined geometries.

Figures (7)

• Figure 1: Schematic illustration of the confined odd-fluid in a potential ring. A two-dimensional assembly of ultrasoft Brownian particles is confined by a radially symmetric harmonic ring potential $V_{\mathrm{ext}}(r)$ (gray background), whose minimum defines the preferred radius $R_0$ (white dashed circle). The system is initialized with an off-center Gaussian density distribution (localized blob), which relaxes toward the ring. Superimposed is the trajectory of the center of mass of the density distribution during relaxation for three representative values of the odd-diffusion parameter: $\kappa>0$ (red), $\kappa<0$ (blue), and $\kappa=0$ (green). Odd diffusion induces a transverse drift of the density center of mass, whose direction reverses upon changing the sign of $\kappa$, while purely relaxational motion is recovered for $\kappa=0$. Inset: zoom-in of the initial density distribution, illustrating interacting particles governed by an ultrasoft Gaussian pair potential $V(r)=\varepsilon \exp(-r^2/\sigma^2)$, where $\varepsilon$ controls the interaction strength and $\sigma$ the interaction range.
• Figure 2: Density and flux evolution in a nonconfined interacting fluid for normal and odd diffusion. The system consists of ultrasoft particles interacting via a Gaussian pair potential and is initialized with a centered Gaussian density distribution (leftmost two figures). Shown are snapshots of the density field and probability currents with periodic boundary conditions at four representative times $t=0$, $t=\tau_B$, $t=5\,\tau_B$, and $t=50\,\tau_B$, where $\tau_B=\sigma^2/D_0$ denotes the characteristic diffusion time. The top row corresponds to the normal (even) case $\kappa=0$, while the bottom row shows the odd-diffusive case with $\kappa=4$. The probability current is represented by arrows, whose direction indicates the local current orientation and whose color encodes the current magnitude $|\mathbf{J}|\sigma\tau_B$ (see color bar), whereas the local density $\rho\sigma^2$ is encoded in the background color using the inverse color scale (compare color bar in Fig. \ref{['figure1']}). In the normal case, relaxation proceeds via purely radial currents flowing along density gradients. In contrast, odd diffusion generates pronounced transverse currents and circulating flow patterns during relaxation. Despite these qualitative differences in the transient dynamics, both systems relax toward the same homogeneous steady state at long times.
• Figure 3: Probability-current evolution in a harmonic ring trap for normal and odd diffusion. Time evolution of the one-body current field $\mathbf{J}(\mathbf{r},t)$ obtained from the odd-DDFT equation in a harmonic ring potential $V_{\rm ext}(r)=\tfrac{1}{2} k (r-R_0)^2$, where the particles are initially distributed in an off-center Gaussian blob. Arrows indicate the local direction of the probability current, while their color encodes the current magnitude $|\mathbf{J}|\sigma\tau_B$ (see color bar). The cyan dashed circle marks the ring radius $R_0=6.0\sigma$. Columns correspond to $t=0$, $t=\tau_B$, $t=10\,\tau_B$, and $t=50\,\tau_B$ where $\tau_B=\sigma^2/D_0$ denotes the characteristic diffusion time. The top row corresponds to the normal case $\kappa=0$, for which relaxation is dominated by radial currents directed toward the ring minimum. The bottom row shows the odd-diffusive case $\kappa=4$, where pronounced circulating currents develop along the ring during intermediate stages of relaxation. Despite these transient angular fluxes, the probability current vanishes in the steady state, consistent with relaxation toward an equilibrium density profile.
• Figure 4: Density evolution in a ring trap: normal vs odd diffusion. Time evolution of the one-body density field $\rho(\mathbf{r},t)$ corresponding to the same system as in Fig. \ref{['figure3']} at the same times. The cyan dashed circle marks the ring radius $R_0=6.0\sigma$. The star indicates the instantaneous density center-of-mass position $\mathbf{R}_{\rm CM}(t)$ (computed from the discretized density field). Odd diffusion generates a transverse, Hall-like component of the current, compare Fig. \ref{['figure4']}, leading to a pronounced angular advection of the density along the ring at intermediate times, while both cases ultimately relax towards a ring-shaped steady distribution (see Appendix $C$ for comparison with simulation results).
• Figure 5: Interaction dependence of the normalized angular circulation in a confined odd-diffusive fluid. This figure shows the time evolution of the circulation ratio $C_\varepsilon(t)/C_0(t)$ for the odd-diffusive system ($\kappa=4$) at different interaction strengths $\varepsilon$, where $C_\varepsilon(t)$ denotes the angular circulation in the interacting system, given in Eq. \ref{['eq:circulation']}, and $C_0(t)$ its noninteracting counterpart (odd ideal gas). Normalizing by $C_0(t)$ isolates the effect of interactions on angular transport. Increasing interaction strength leads to a pronounced enhancement of the transient circulation peak, reflecting stronger collective redistribution along the ring during relaxation. At long times, the normalized circulation decays to zero for all $\varepsilon$, confirming that interactions modify only the transient dynamics while the equilibrium state remains free of persistent angular currents. Solid lines correspond to odd-DDFT predictions, while shaded bands represent simulation results.