### Abstract

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.

Original language | English (US) |
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Pages (from-to) | 1303-1323 |

Number of pages | 21 |

Journal | Journal of Statistical Physics |

Volume | 114 |

Issue number | 5-6 |

DOIs | |

State | Published - Mar 2004 |

### Keywords

- Finite-sampling error
- Free energy computation
- Jensen's inequality
- Nonlinear averages
- Systematic error

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

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## Cite this

*Journal of Statistical Physics*,

*114*(5-6), 1303-1323. https://doi.org/10.1023/b:joss.0000013961.84860.5b