Macroscopic fluctuation theory of interacting Brownian particles

TL;DR

This work applies Macroscopic Fluctuation Theory to Brownian particles with pairwise interactions, enabling exact access to large-scale dynamics and fluctuations. Central to the approach are the transport coefficients D(ρ) and σ(ρ), with σ(ρ) fixed by Brownian mobility and D(ρ) determined from the equation of state via D(ρ) = μ0 P′(ρ) (or equivalently D(ρ) = μ0 kBT f′′(ρ)). The authors obtain explicit or parametric expressions for D(ρ) in a range of 1D models (Calogero, Riesz, hard and sticky rods, Rouse, tethered chains) and employ virial expansions to treat higher dimensions, including current and tracer statistics such as ⟨X_t^2⟩ ∼ const × √t and ⟨Q_t^2⟩ ∼ const × √t, as well as density–current correlations. The results yield precise predictions for tracer diffusion, integrated currents, and cross-dimensional channel behavior, and establish a scalable framework to study large-scale diffusion beyond 1D, with potential extensions to hydrodynamic interactions and non-pairwise forces.

Abstract

We apply the macroscopic fluctuation theory (MFT) to study the large-scale dynamical properties of Brownian particles with arbitrary pairwise interaction. By combining it with standard results of equilibrium statistical mechanics for the collective diffusion coefficient, the MFT gives access to the exact large-scale dynamical properties of the system, both in- and out-of-equilibrium. In particular, we obtain exact results for dynamical correlations between the density and the current of particles. For one-dimensional systems, this allows us to obtain a precise description of these correlations for emblematic models, such as the Calogero and Riesz gases, and for systems with nearest-neighbor interactions such as the Rouse chain of hardcore particles or the recently introduced model of tethered particles. Tracer diffusion with the single-file constraint (but for arbitrary pairwise interaction) is also studied. For higher-dimensional systems, we quantitatively characterize these dynamical correlations by relying on standard methods such as the virial expansion.

Figures (6)

• Figure 1: Diffusion coefficient $D(\rho)$ for some of the one-dimensional models presented in Sec. \ref{['sec:1d']}. (a) The Calogero gas with potential \ref{['eq:CaloPot']}; the solid line is the analytical expression (\ref{['eq:DCalo']},\ref{['eq:MuCalo']}). (b) The dipole gas with potential \ref{['eq:DipPot']}. (c) The WCA gas with potential \ref{['eq:WCAPot']}, with $l = 1$. In all panels, the red symbols are obtained from numerical simulations, with $g=1$, $T=1$, $\mu_0 = 1$ and $k_{\mathrm{B}} = 1$ (see Appendix \ref{['app:NumDSigma']}). The dashed lines are different approximations: the virial expansion (see Section \ref{['sec:Virial']}), the nearest-neighbor approximation \ref{['eq:DiffNN']} (see Section \ref{['sec:NearNeigh']}), and the hypernetted-chain (HNC) approximation (see Section \ref{['sec:PfromGr']}).
• Figure 2: (a) Diffusion coefficient for the Rouse chain with nearest-neighbour interaction potential (\ref{['eq:PotRouse']}) with $g=\frac{1}{2}$, for different values of the particle size $\ell$. We have performed simulations for the case $\ell=0$ only, since the arbitrary size can be deduced from it, see Section \ref{['sec:ParticSize']}. (b) Diffusion coefficient for the chain of tethered particles with nearest-neighbour interaction potential (\ref{['eq:PotTethered']}), for different values of the particle size $\ell$ and length $\Delta$ of the tether. We have not performed numerical simulations in this case, because the sharp potential \ref{['eq:PotTethered']} is tricky to implement numerically. (c) Mobility $\sigma(\rho)$ for the different interaction potentials considered in this article. The points are obtained from numerical simulations.
• Figure 3: Prefactor of the asymptotic behaviour of $\left\langle X_t^2 \right\rangle_c \sim \sqrt{t}$ as a function of the mean density $\bar{\rho}$, given in \ref{['eq:Xt2']} at $T=1$, in the case of (a) pointlike particles $\ell=0$, or (b) extended particles $\ell = 1$, for different interaction potentials considered in Section \ref{['sec:TrCoefs']}: (i) the Calogero potential \ref{['eq:CaloPot']} with $g=1$; (ii) the WCA potential \ref{['eq:WCAPot']} with $g=1$ and $l = 1$ for nearest-neighbour interaction (which is a good approximation, see Fig. \ref{['fig:DWCA']}); (iii) the Rouse chain \ref{['eq:PotRouse']} with $g=\frac{1}{2}$; (iv) the gas of tethered particles \ref{['eq:PotTethered']} with $\Delta = 8$. In (a), the black solid line represents the case of hardcore Brownian particles, for which $D(\rho)=\mu_0 k_{\mathrm{B}} T$, see Eq. \ref{['eq:Dfree']}. In (b), the black solid line represents the case of hardcore Brownian rods of length $\ell = 1$, for which $D(\rho)$ is given by \ref{['eq:DiffHR']}, while the dotted line corresponds to pointlike ($\ell=0$) hardcore Brownian particles.
• Figure 4: Prefactor of the asymptotic behaviour of (a) $\left\langle X_t^2 \right\rangle_c \sim \sqrt{t}$ as a function of the temperature $T$, given in \ref{['eq:Xt2']}, and (b) $\left\langle X_t^4 \right\rangle_c \sim \sqrt{t}$ as a function of the mean density $\bar{\rho}$, given in \ref{['eq:Xt4']}. In both panels we considered the case of pointlike particles $\ell = 0$ at $\bar{\rho}=1$, for different interaction potentials considered in Section \ref{['sec:TrCoefs']}: (i) the Calogero potential \ref{['eq:CaloPot']} with $g=1$; (ii) the WCA potential \ref{['eq:WCAPot']} with $g=1$ and $l = 1$ for nearest-neighbour interaction (which is a good approximation, see Fig. \ref{['fig:DWCA']}); (iii) the Rouse chain \ref{['eq:PotRouse']} with $g=\frac{1}{2}$; (iv) the gas of tethered particles \ref{['eq:PotTethered']} with $\Delta = 4$ or $\Delta = 8$ (see the legend). In (a), the dotted line represents the low-temperature behaviour of the Calogero gas computed in Touzo:2024Touzo:2024a. In (b), the inset is a zoom to show the variations of the prefactor for the Rouse chain, which is orders of magnitude below the other models. In both panels, the black solid line represents the case of hardcore Brownian particles.
• Figure 5: Scaled correlation profile $\left\langle \rho_0(X_t + x, t)X_t \right\rangle_c$, given by \ref{['eq:XtRho']}, as a function of the scaled variable $x/\sqrt{4 D(\bar{\rho}) t}$, at $T=1$, compared to numerical simulations for: (i) the Calogero potential \ref{['eq:CaloPot']} with $g=1$, at $t=200$; (ii) the WCA potential \ref{['eq:WCAPot']} with $g=1$ and $l = 1$, at $t=200$ (the diffusion coefficient is obtained from the nearest-neighbour approximation, see Fig. \ref{['fig:DWCA']}).