Inferring intermediate states by leveraging the many-body Arrhenius law

TL;DR

The paper addresses inferring metastable states in escape dynamics of interacting diffusive particles within deep traps by generalizing the Arrhenius law to a many-body regime, yielding $\Phi_{\mathrm{MP}} \asymp \exp[- \beta \Delta U\, g(\bar{\rho})]$ with $g(\bar{\rho})=1-U_{\rm top}(\bar{\rho})/\Delta U$. It shows that non-monotonic traps generate kinks in $g(\bar{\rho})$ that count and locate metastabilities via the MEC, and introduces a thermodynamic-like formalism with a free-energy $\mathcal{F}=-\frac{1}{\beta \Delta U}\log \Phi_{\mathrm{MP}}$ and response functions $\mathcal{R}_n$ to characterize these transitions. Finite-$\beta \Delta U$ scaling is developed using macroscopic fluctuation theory applied to the SEP, establishing observable precursors to the kinks through $\mathcal{R}_2$ and $\mathcal{R}_3$, and providing analytic and numerical support for both monotone and non-monotone traps. The approach offers a practical route to quantify underlying energy landscapes from escape statistics, with potential experimental validation in colloidal transport and biological pore translocation, and sets the stage for extensions to higher dimensions and more complex interactions.

Abstract

Metastable states appear as long-lived intermediate states in various natural transport phenomena which are governed by energy landscapes. As such, these intermediate metastable states dominate the system's dynamics at coarse grained times. Moreover, they can strongly influence the overall pathways through which the energy landscape is explored. Thus, quantifying these metastabilities is crucial for uncovering the key details of the underlying landscape. Here, we introduce a robust method based on a generalized many-body Arrhenius law to identify metastable states in escape problems involving interacting particles with excluded volume. Experimental platforms such as colloidal transport or macromolecular translocation through biological pores can offer promising settings to validate our predictions.

Figures (8)

• Figure 1: (a) The escape problem of a single particle. The bottom figure shows the traditional inference of the activation barrier height from Arrhenius law Eq. \ref{['eq:AL sp']} (b) Inference of the potential minima and maxima by employing the many-body Arrhenius law Eq. \ref{['eq:gen AL']}. The number of kinks in the bottom figure corresponds to local maxima and minima of the potential $U(x)$ in the top figure. Thus, the many-body escape dynamics (for instance, activated dynamics of a stochastic lattice gas model with volume exclusion as shown in the top) allow to infer the metastabilities of the single particle escape problem.
• Figure 2: A visualization of the MEC for particles with excluded volume on an arbitrary non-monotonic potential. Particles occupy the potential for $U(x)<U_{\rm{top}}$. (a) Here $U_{\rm{top}}$ is smaller then the lowest value of the potential extrema. Thus, $U_{\rm{top}}$ has a single intersection with $U(x)$, at position $x_1$, and we find $\overline{\rho}(U_{\rm{top}} = U(x_1) )$, leading to $U_{\rm{top}}(\overline{\rho}) = U(x=\overline{\rho})$ as in the monotonous potential. (b) Here $U_{\rm{top}}$ is in the range where there are three intersection points with $U(x)$, denoted by $x_{1,2,3}$. In this case $\overline{\rho}(U_{\rm{top}}) = x_1 + x_3-x_2$. Notice that in this case $U_{\rm{top}}(\overline{\rho}) \neq U(x=\overline{\rho})$.
• Figure 3: (Top) The piecewise linear potential $U_{\rm{pwl}}(x)$ is plotted (solid thick brown) together with the MEC prediction for $1-g$ (solid magenta). The black circles highlight the positions of the kinks in $g$. The dashed lines reveal the convergence of the finite $\beta \Delta U$ approximation scheme to the MEC $1-g$ curve. Noticeably, the convergence, marked by the black arrows, is faster away from the kinks.
• Figure 4: (a) The "free energy" $\mathcal{F}$ is calculated for finite $\beta \Delta U$ values, and compared with the MEC result ($\beta \Delta U=\infty$) for SEP particles subjected to the $U_{\rm{pwl}}$ trapping potential. It is evident that the convergence is better away from the critical point at $\delta \rho =0$. However, to infer the criticality, we need to consider the response function $\mathcal{R}_2$. (b) The rescaled response function $\mathcal{R}_2/(\beta\Delta U)$ for different $\beta \Delta U$ as found from Eq. \ref{['eq:F analytic']}. The precursor of the kink manifest as a scalable peak, at distance $1/ \beta \Delta U$ from the expected kink. (c) The distance $d$ of the peaks from the expected critical point is shown to scale like $1/\beta \Delta U$. We remind that in finite systems, the singularities characteristic of critical phenomena are rounded and shifted: response functions exhibit smooth peaks rather than divergences. As the system size increases, these peaks grow sharper and their location approaches the true critical point in the thermodynamic limit. Here, we observe the same for $d$ as $\beta \Delta U$ replaces the system size kardar2007statistical.
• Figure 5: The coarse-graining process: The top panel illustrates the microscopic dynamics of an exclusion process on a lattice. The middle panel depicts the process of division of the system into boxes of size $\ell$, which is necessary to define the local density(see Eq. \ref{['Eq:Local density']}). After coarse graining via diffusive rescaling, the system is described in terms of macroscopic density $\rho(x)$. The bottom panel shows a representative density profile.