Liouville's Theorem

Theorem.

Any bounded (holomorphic) entire function is constant.

In fact, considerably more than this is known: the real and imaginary parts of a holomorphic function are harmonic functions; not only do no bounded harmonic functions exist in R^n, but in fact minimal rates of growth may be established.

Actually, proving a discrete version of this theorem is rather more difficult than the continuous versions -- see Z^n admits no bounded harmonic function for more details.

Liouville's theorem in Physics refers to the theorem that under Hamiltonian evolution the volume in phase space occupied by a collection of particles remains constant - that is these particles behave like an incompressible fluid.

For example consider a box full of monoatomic gas molecules. The phase space for this system is 6 dimensional. Now each molecule may be represented by a point in phase space. Liouville's theorem states that as the system evolves the total volume occupied by these particles in phase space will be conserved.

p /\        .
  |        ....
  |        ......
  |      .....
  |       ..
  |       
  |---------------> 
                  x