Air pressure is actually a force experienced by the surface of the container when air molecules hit it in high speeds.
An air valve opens and allows air to move in one direction and blocks the the flow of air in the opposite direction. So in the direction in which air can flow, the air doesn't loose much kinetic energy and there is very less change in the momentum of air. But in the other direction, air is prevented from flowing through, so the air loses some of it's kinetic energy and also momentum changes as the air is reflected back. This must in theory accelerate the valve.
So if we place a valve which works for pressure gradient of say 0.1 atm, on a low friction surface. Then the air which can flow through the valve will flow through, but the air in opposite direction will transfer some energy and must cause the valve to move. The air molecules actually be moving in random directions, but it can be broken into horizontal and vertical components. The vertical components of all air molecules will have no effect. But the Right-to-Left horizontal components will pass through and Left-to-Right components will be reflected back contributing to the movement of the valve.
This is not free energy, the valve if at all it moves, it extracts energy from air and air loses some kinetic energy which will decrease it's overall temperature. So even if this works, it converts the heat energy into kinetic energy of the valve. This is more like a wind turbine, but it tries to make use of regular air, when there is no wind.
I'm pretty certain that this doesn't work, because someone would have by now already discovered it and it would have become popular. But what I don't understand is, why doesn't it work?
For context: A pressure gradient of 1 atm (101325 Pa) can generate a huge force. When applied on an object with dimensions of 10 cm * 10 cm, the object experiences approx 1000 N. This is amount of weight you feel when you try to lift 100 Kgs. And this is the force experienced by an object with 100 cm^2. And a car with width of 1m and height of 1m, if it's flat on one end and 1 atm of pressure is applied on it, it experiences a 100 KN of force, and typical car weighs around 2000 Kgs, so it can experience an acceleration from 0 to 180 KPH in a second. So this is by no regard a small force.