Inequalities for overdamped fluctuating systems

In many ambits of the chemical sciences it happens to deal with complex systems udergoing thermal fluctuations in the overdamped regime of the motion (i.e., multidimensional diffusive processes). Although such stochastic dynamics are well specified in terms of the Fokker–Planck–Smoluchowski equation for the time-dependent probability density, the solution becomes rapidly unfeasible as the number of degrees of freedom increases beyond a few units. Here we present a strategy, based on inequalities for “completely monotone decreasing” functions viewed as convex functions of time, to by-pass such a difficulty and aimed to achieve only bounds (but with low computational effort) on some quantities that pertain the system’s dynamics. Namely, we derive (i) a lower bound for the maximum value of the probability density that develops from a given initial condition, and (ii) a lower bound on the correlation time for a generic self-correlation function. The former bound is quantified by means of simple operations on the initial condition, while the latter is gained by the knowledge of an initial “piece” of correlation function to be supplied, for instance, by molecular or Brownian dynamics simulations. Some practical applications are discussed.