Force Fluctuations

red ball All forces discussed so far are averages of a fluctuating quantity.

yellow ballAs a simple consequences of such fluctuations consider the wanderings of

a classical particle of mass m in a quantum bath.

yellow ballThere is no net force on the particle, but it disperses due to force fluctuations:

yellow ballLangevin equation for an otherwise free particle,

yellow ball 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.

yellow ball 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

yellow ball 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:

yellow ballPotential realizations for solitons in Bose-Einstein condensate.