Quantitative Analysis of the Stochastic Approach to Quantum Tunneling
  Recently there has been increasing interest in alternate methods to compute
quantum tunneling in field theory. Of particular interest is a stochastic
approach which involves (i) sampling from the free theory Gaussian
approximation to the Wigner distribution in order to obtain stochastic initial
conditions for the field and momentum conjugate, then (ii) evolving under the
classical field equations of motion, which leads to random bubble formation.
Previous work showed parametric agreement between the logarithm of the
tunneling rate in this stochastic approach and the usual instanton
approximation. However, recent work [1] claimed excellent agreement between
these methods. Here we show that this approach does not in fact match
precisely; the stochastic method tends to overpredict the instanton tunneling
rate. To quantify this, we parameterize the standard deviations in the initial
stochastic fluctuations by $\epsilon \sigma$, where $\sigma$ is the actual
standard deviation of the Gaussian distribution and $\epsilon$ is a fudge
factor; $\epsilon = 1$ is the physical value. We numerically implement the
stochastic approach to obtain the bubble formation rate for a range of
potentials in 1+1-dimensions, finding that $\epsilon$ always needs to be
somewhat smaller than unity to suppress the otherwise much larger stochastic
rates towards the instanton rates; for example, in the potential of [1] one
needs $\epsilon \approx 1/2$. We find that a mismatch in predictions also
occurs when sampling from other Wigner distributions, and in single particle
quantum mechanics even when the initial quantum system is prepared in an exact
Gaussian state. If the goal is to obtain agreement between the two methods, our
results show that the stochastic approach would be useful if a prescription to
specify optimal fudge factors for fluctuations can be developed.
