Thursday, December 2, 2010

Ambiguities about tunneling at non-zero temperature?

An important question came up in the seminar I gave today. It concerns something I have been confused about for quite a while and I was encouraged that some of the audience got animated about it.

Consider the problem of quantum tunneling out of the potential minimum on the left in the Figure below
If one considers a path integral approach than one says that tunneling occurs when there is an "instanton" solution (i.e., a non-trivial solution) to the classical equations of motion in imaginary time with a period determined by the temperature. This only occurs when the temperature is less than 
which is defined by curvature of the top of the potential barrier. At temperatures above this there is only one solution to the classical equations of motion, the trivial one x(tau)=x_b, corresponding to the minima of the inverted potential. One can then calculate the quantum fluctuations about this minimum, at the Gaussian level. This gives a total decay rate 
which is well defined, provided the temperature T is larger than T_0. What do these quantum fluctuations represent? I would say they represent tunneling just below the top of the barrier and reflection from just above the top of the barrier. In my talk I said this situation  of "no instantons" represents "no tunneling" which several people disagreed with. I corrected myself with "no deep tunneling."

However, there is an alternative way to obtain the same expression, which seems to me to involve assuming lots of tunneling! Eli Pollak [who is also visiting Berkeley] reminded me of this today. I think this derivation was originally due to Bell, following earlier work by Wigner, and is implicit in Truhlar and collaborators treatment of tunneling and implemented in their POLYRATE code. One considers the tunneling transmission amplitude as a function of energy and integrates over all energies with a Boltzmann weighting factor.

How does one reconcile these two derivations, which seem to me to involve very different physics? I welcome comments.

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