  Molecular Dynamics (MD) simulations of standard systems of interacting
particles ("atoms") give excellent agreement with the equipartition theorem for
the average energy, but we find that these simulations exhibit finite-size
effects in the dynamics that cause local fluctuations in energy to deviate
significantly from the analogous energy fluctuation relation (EFR). The main
conclusion of our study is that systems separated into nanometer-sized "blocks"
inside much larger simulations exhibit excess fluctuations in potential energy
(pe) that diverge inversely proportional to T in a manner that is strongly
dependent on the range of interaction. Specifically, at low T with long-range
interactions pe fluctuations exceed the EFR by at least an order of magnitude,
dropping abruptly to below the EFR when interactions include only 1st-neighbor
atoms. A simplistic model that includes 2nd-neighbor interactions matches the
behavior of the excess pe fluctuations, but only if the 2nd-neighbor terms are
not included in Boltzmann's factor, attributable to energy localization due to
anharmonic effects. Characterizing energy correlations as a function of time
and distance reveals that excess pe fluctuations in a block coincide with
negative pe correlations between neighboring blocks, whereas reduced pe
fluctuations coincide with positive pe correlations. Indeed, anomalous pe
fluctuations in small systems at low T can be quantified by using the net
energy in Boltzmann's factor that includes the pe from a surrounding shell of
similarly small systems, or equivalently an effective local temperature. Our
analysis elucidates the source of non-Boltzmann fluctuations, and the need to
include mesoscopic thermal effects from the local environment for a consistent
theoretical description of the equilibrium fluctuations in MD simulations of
standard models with long-range interactions.
