Systematic finite-sampling inaccuracy in free energy differences and other nonlinear quantities

Systematic inaccuracy is inherent in any computational estimate of a non-linear average, such as the free energy difference ΔF between two states or systems, because of the availability of only a finite number of data values, N. In previous work, we outlined the fundamental statistical description of this "finite-sampling error." We now give a more complete presentation of (i) rigorous general bounds on the free energy and other nonlinear averages, which generalize Jensen's inequality; (ii) asymptotic N → ∞ expansions of the average behavior of the finite-sampling error in ΔF estimates; (iii) illustrative examples of large-N behavior, both in free-energy and other calculations; and (iv) the universal, large-N relation between the average finite-sampling error and the fluctuation in the error. An explicit role is played by Lévy and Gaussian limiting distributions.