The total volume available for small spheres increases when the excluded volumes around large spheres overlap. The increased volume allotted for small spheres allows them greater translational freedom which increases their entropy. Because the canonical ensemble is an athermal system at a constant volume the Helmholtz free energy is written
where  is the Helmholtz free energy,  is the entropy and  is the temperature. The system's net gain in entropy is positive from increased volume, thus the Helmholtz free energy is negative and depletion flocculation happens spontaneously.
The free energy of the system is obtained from a statistical definition of Helmholtz free energy
where  is the partition function for the canonical ensemble. The partition function contains statistical information that describes the canonical ensemble including its total volume, the total number of small spheres, the volume available for small spheres to occupy, and the de Broglie wavelength. If hard-spheres are assumed, the partition function  is
The volume available for small spheres, was calculated above. is the number of small spheres and  is the de Broglie wavelength. Substituting  into the statistical definition, the Helmholtz free energy now reads
The magnitude of the depletion force,  is equal to the change in Helmholtz free energy with distance between two large spheres and is given by
The entropic nature of depletion forces was proven experimentally in some cases. For example, some polymeric crowders induce entropic depletion forces that stabilize proteins in their native state.
Other examples include many systems with hard-core only interactions.