N-body force calculation and softening

Let the displacement from particle i to particle j point toward j. A fixed positive Plummer softening length replaces the inverse separation in the pair potential by the inverse square root of squared separation plus squared softening. Differentiating the potential with respect to the first position gives an attractive force along that displacement. Swapping particle labels reverses the displacement, so the forces are opposite. The accelerations are generally not opposite when the masses differ.

Calculating a pair once and applying equal and opposite momentum increments enforces the pair symmetry up to floating-point accumulation. A central pair force also has zero net torque about a common origin. These statements concern the specified force law, not exact energy conservation by an arbitrary integrator. If the softening changes with particle configuration or time, differentiating the resulting potential may require additional terms; simply substituting changing lengths into a fixed-softening force is not automatically energy consistent.

For a smooth collisionless target, a finite set of particles samples a distribution rather than resolving every microscopic encounter. Too little smoothing can make nearby numerical particles produce large fluctuating forces; too much removes real spatial structure and systematically weakens attraction at resolved separations. The best compromise depends on particle number, density, kernel and the observable. There is no single universal optimal softening length independent of the physical system or desired resolution.

Plummer smoothing is noncompact: its force differs from Newton's at every finite radius, although the relative change becomes small far outside the softening scale. Other kernels can recover Newton's law beyond a finite support. Softening studies should therefore vary the length and particle sampling, rather than interpret the time-step convergence of one softened model as convergence to the unsoftened continuum. Statistical sampling fluctuations also need ensembles or justified estimators; deterministic integrator error is not the same uncertainty.

Direct summation evaluates all unordered pairs, requiring a number of pair interactions proportional to N squared. Tree methods approximate distant groups and fast multipole methods organize higher-order spatial expansions. Their force tolerances are separate from integration time steps. Some approximate evaluations do not preserve exact pair antisymmetry, so total momentum drift can diagnose an implementation choice rather than a defect of Newtonian mechanics. Boundary treatment also matters: isolated and periodic gravitational systems have different long-range force problems.

A practical comparison fixes the target distribution and boundaries, measures force error against a more accurate evaluation, and then studies time integration with that force prescription. Close binary dynamics may demand a collisional method rather than stronger smoothing. The fields lab displays a softened planar source with an optional rotational term. Its radial power and sign convention differ from three-dimensional gravity, and it contains neither particle pairs nor their backreaction. Only its linear response to source strength is the intended comparison.