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Symplectic integrators are widely used for the study of planetary dynamicsand other $N$-body problems. In a study of the outer Solar system, wedemonstrate that individual symplectic integrations can yield biased errors inthe semi-major axes and possibly other orbital elements. The bias is resolvedby studying an ensemble of initial conditions of the outer Solar system. Suchstatistical sampling could significantly improve measurement of planetarysystem properties like their secular frequencies. We also compared thedistributions of action-like variables between high and low accuracyintegrations; traditional statistical metrics are unable to distinguish thedistribution functions.

Significant improvement in planetary system simulations from statistical averaging