  In a recent paper (V\'an, P.; Abe, S. Physica A 2022, 588, 126505), it has
been discovered that a scalar field coupled to a fluid and allowed to be a
thermodynamic variable in consistency with the second law of thermodynamics is
only gravitational and accordingly emergence of extended Newtonian gravity has
been predicted. The resulting field equation for the potential of this emergent
force is nonlinear and admits the logarithmic potential as a singular solution,
suggesting its relevance to the dark matter conundrum. Here, a general analysis
of the nonlinear field equation is performed. It is found that the emergent
force field exhibits the unsharp crossover between the 1/r and 1/r^2 forces
outside the fluid, depending on a spatial scale characteristic of the present
theory to be observationally tested in the context of the dark matter
conundrum. Then, the action functional is constructed for the potential of the
emergent field and the field energy is shown to be free from an infrared
divergence. A comment is also made on the difference of the present theory from
MOND (modified Newtonian dynamics).
