Fluctuation amplitude of a trapped rigid sphere immersed in a near-critical binary fluid mixture within the regime of the Gaussian model

The position of a colloidal particle trapped in an external field thermally fluctuates at equilibrium. As is well known, the ambient fluid is not a simple heat bath and the particle mass appears to increase, which influences the mean square velocity of the particle. In this study, we suppose that the particle is surrounded by a binary fluid mixture in the homogeneous phase near, but not too close to, the critical point. Usually, one component is preferably attracted by the particle surface, and the resultant adsorption layer becomes significant because of the near-criticality. When the particle fluctuates in this situation, its mean square displacement should also be influenced by the ambient fluid because the adsorption layer does not follow the particle motion totally. We calculate the influence in a simple case, where a rigid spherical particle fluctuates with a small amplitude and its surface attracts one component weakly. We utilize the hydrodynamics in the limit of no dissipation to examine the contribution from the ambient mixture to the equal-time correlation. According to our result, the mean square displacement is reduced by an additional stress, including osmotic pressure.