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Quantitative Analysis of the Stochastic Approach to Quantum Tunneling | | Mark P. Hertzberg
; Fabrizio Rompineve
; Neil Shah
; | | Date: |
31 Aug 2020 | | Abstract: | 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 precise quantitative 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 $epsilonsigma$, where $sigma$ is the actual
standard deviation of the Gaussian distribution and $epsilon$ is a fudge
factor; $epsilon=1$ is the physical value. We show that the choice of initial
conditions to obtain quantitative agreement in [1] corresponds to
$epsilon=1/2$. 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. 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. | | Source: | arXiv, 2009.00017 | | Services: | Forum | Review | PDF | Favorites | |
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