Is the pressure-gradient force an entropic force? A gas flows from an area of high pressure to an area of low pressure when there are no other forces preventing it. From a macrosopic perspective you have to infer that an underlying force is accelerating the gas towards low pressure.
The following animation shows a container filled with air which is at first divided by a barrier in the middle. Both sides have the same temperature but the left side has two times the density of the right side and thus has a higher pressure.

When the barrier is removed, more molecules move from high pressure (left) to low pressure (right) than from low pressure to high pressure. This creates a net movement of molecules towards low pressure which is macroscopically measured as a gas flow. The important part is that the molecules do not move into the low pressure area because they are accelerated by an underlying force; they statistically move there due to their thermal motion. Molecules statistically move where is less resistance by collisions which is generally towards an area of lower density or lower temperature.
The pressure-gradient force seems to accelerate a gas at the macroscopic scale but there is no underlying force that accelerates molecules. In my view the pressure-gradient force is an entropic force; it does not exist at the microscopic scale. It results from the system's tendency to achieve thermodynamic equilibrium which is based on the molecules’ thermal motions tendency to bring the system toward its macroscopic state of maximum entropy.
What do you say?
 A: 
there is no underlying force that accelerates molecules. In my view the pressure-gradient force is an entropic force; it does not exist at the microscopic scale.

There are forces acting on the microscopic scale. These are the forces due to wall acting on the molecules during their impacts on the wall. Without these, there would be no net transport of gas in either direction; the center of mass of the gas would stay still while the red and blue molecules would penetrate to new regions of space and mix. It is only because the walls are supposed to be immovable and acting on the molecules with forces that net transport of gas is happening.
If the molecules do not collide with each other or the collisions are too rare, the mixing cannot be accurately described by continuum theory with one density and one velocity field; the gas is not well described as a fluid and the notion of force density in a gas is not very useful then. Only pressure force due to wall has sense.
If the collisions of the molecules are frequent enough, the gas behaves as a continuum in the sense it can be accurately described by fluid description with one density and one velocity field. For example, ordinary air usually qualifies for this kind of description.
In case two such gases of differing pressure are put to mutual contact (without barrier wall), one can use the notion of force density that accelerates the elements of the gas fluid. It has nothing to do with entropy; it is given by sum of the impact forces acting on the element from the other molecules outside it, divided by the element volume.
