------------------------------------
S1j. Classical and quantum tunneling
------------------------------------

Consider a particle in an external potential.
Assume the potential is everywhere finite, locally constant and positive
near the origin, and decays to zero far away.

There is no force, when the motion is deterministic and classical.

In practice, however, the classical, deterministic setting is an 
approximation only, and the particle makes random motions. 
Thus it moves away from the origin and will sooner or later reach 
the nonconstant part of the potential. With low probability p,
it will even escape over any barrier; roughly, log p is proportional 
to the negative barrier height. For details, you might 
wish to consult my paper 
   A. Neumaier, 
   Molecular modeling of proteins and mathematical prediction of 
   protein structure, 
   SIAM Rev. 39 (1997), 407-460. 
   http://www.mat.univie.ac.at/~neum/papers/physpapers.html#protein
and the references there.

Quantum mechanically, there is always a probability of escaping to 
infinity, without assuming any approximations. This is called 
tunneling. 

In both cases, once the particle is in the infinite region, 
the probability that it returns is zero.

Thus a positive potential drives a particle in the long run off to
infinity (though, in case of a high barrier, one has to wait a long 
time). In particular, in the classical case one also has a form of
(stochastic) tunneling.

Thus it is justified to refer to a potential such as the above as
repelling. However, no one would object if you call a potential 
repelling _only_ in the neighborhood of a strict local minimizer, i.e., 
close to a metastable state. 


Of course, a golf ball sitting on top of a flat hill will not move 
down the hill; because of friction it remains in a metastable state. 

Thus the above is an idealization. But most of physics is idealized, 
and the language is also somewhat idealized (and, as actually used by 
people, not even completely precise).