All forces discussed so far are averages of a fluctuating quantity.
As a simple consequences of such fluctuations consider the wanderings of
a classical particle of mass m in a quantum bath.
There is no net force on the particle, but it disperses due to force fluctuations:
Langevin equation for an otherwise free particle,
In the commonly used Caldeira-Legget formalism, the force is linearly coupled to the bath,
However, a linear coupling is forbiddent for a neutral, but polarizable, particle.
With linear coupling, a free field theory implies a Gaussian distribution for φ,
hence the force and the displacement are also Gaussian distributed, entirely described by the variance
For a neutral particle (no linear coupling), the quadratic force is not Gaussian distributed; leading to cumulants
consistent with a power-law probability distribution, reminiscent of Levy flights:
Potential realizations for solitons in Bose-Einstein condensate.