Why are people crowding in the hall and not in the door through which they go out?

You may not like it, but people crowding outside the exit of a concert hall are subject to the Venturi effect. The same physical principle applies to water flowing through a pipe or wind blowing through a canyon.

The fluid flowing through a pipe must obey the laws of conservation of energy. The total energy of the fluid, i.e. the sum of its kinetic, potential and internal energy, must always be the same. If an incompressible fluid passes through a pipe and the cross section of the pipe changes, some characteristic of the flowing fluid must also change. For example, its velocity or pressure. If a fluid were compressible, its temperature might also change as its molecules press more tightly together or move further apart, but with an incompressible fluid, only the pressure and velocity will change. The Bernoulli equation describes this dependence. It says that if we let a liquid of a certain pressure and velocity flow through a tube, and we narrow the passage through the tube, the pressure of the liquid will decrease, but the velocity will increase.

We can picture it nicely with the people crowding out of the hall. A person can be thought of as an incompressible individual, and if they stand shoulder to shoulder in the hall, we have a pretty good model of an incompressible fluid. This human fluid is slowly advancing towards the exit. If the hall is rectangular, we can say that, for instance, a hundred people squeezed from wall to wall will proceed ten metres in a minute. Behind them, more people are crowding in, advancing at exactly the same speed.

When these hundred people reach the door, they face a difficult task. In a minute, they have to walk through the door as another hundred people crowd in to take their place. But the door is narrow, only one or two people can get through at a time. To make it through in one minute, they have to hurry and get through the door very quickly. And because they're in a hurry, they don't have time to push each other. So in a narrow doorway, their speed increases, but the pressure they put on each other decreases.

This simple rule of physics also explains why people cannot leave the hall at a normal speed and a slowly progressing crowd must develop. If the people in the wide foyer were advancing at a normal speed, then they would have to run out of the narrow door faster than Olympic sprinters!