At finite temperature, an attractive pair potential does not by itself define a bond between two particles. Specifically, two particles initialized near a potential minimum can remain localized over one observation window and escape over another; therefore, bonding is an observation-time-dependent localization problem. Here, we treat bonding as finite-time localization and ask when thermal fluctuations delocalize this state. We measure localization by a fluctuation-derived stiffness, $k_{\mathrm{eff}}=1/\left(\beta \langle u^2\rangle \right)$, where $\langle u^2\rangle$ is the variance of the particle separation. For isotropic particle–particle and one-dimensional particle–wall interactions, we derive a nonequilibrium theory based on the finite-time escape problem that yields the nonequilibrium probability density function for the separation coordinate, drawing on ideas from Becker-Döring theory, and thus $k_{\mathrm{eff}}$ as a function of temperature and observation time ${\tau }_{\mathrm{obs}}$. We find that as temperature increases or as the sampled region grows, the configuration-space volume outside the well dominates the fluctuations and $k_{\mathrm{eff}}$ drops rapidly, signaling the melting of finite-time bonds at a temperature $T^{*}$. Langevin simulations show the same loss of stiffness for particle–particle and particle–wall potentials, and the theory captures this behavior without any adjustable parameters. The temperature $T^{*}$ marks finite-time escape from the localized state and therefore depends on the observation window. Although the nonequilibrium probability density function is obtained exactly, evaluating $k_{\mathrm{eff}}$ from its moments requires numerical quadrature. To provide a simplified analytical representation of this behavior, we introduce a constrained Boltzmann distribution restricted to a finite region of configuration space of size $R_{\mathrm{eq}}$, where $R_{\mathrm{eq}}$ is determined from the nonequilibrium theory. We show that both the nonequilibrium theory and constrained Boltzmann-like representation quantitatively capture the melting of finite-time bonds, with the latter providing an approximate coarse-grained description of bond interactions.

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• Brownian motion
• Classical statistical mechanics
• Escape problems
• Nonequilibrium statistical mechanics
• Random walks
• Stochastic processes