Thermal melting of finite-time bonds

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_{eff}=1/(β〈u^{2}〉), where 〈u^{2}〉 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_{eff} as a function of temperature and observation time τ_{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_{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_{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_{eq}, where R_{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.