TY - JOUR

T1 - Numerical study of metastability due to tunneling

T2 - The quantum string method

AU - Qian, Tiezheng

AU - Ren, Weiqing

AU - Shi, Jing

AU - E, Weinan

AU - Sheng, Ping

N1 - Funding Information:
The authors thank Eric Vanden-Eijnden for helpful discussions. T.Q. was supported by the Hong Kong RGC under Grants 602805 and 602904. J.S. and W.E were supported by DOE under Grant DE-FG02-03ER25587. W.R. was supported by NSF Grant DMS-0604382. P.S. was supported by the Hong Kong RGC under Grants HKUST6073/02P and CA04/05.SC02.

PY - 2007/6/15

Y1 - 2007/6/15

N2 - We generalize the string method, originally designed for the study of thermally activated rare events, to the calculation of quantum tunneling rates. This generalization is based on the formal analogy between quantum mechanics and statistical mechanics in the path-integral formalism. The quantum string method first locates the minimal action path (MAP), which is a smooth curve connecting two minima of the imaginary-time action in the space of imaginary-time trajectories. From the MAP, the saddle point of the action (called "the bounce") associated with the exponential factor for barrier tunneling probability is obtained and the pre-exponential factor (the ratio of determinants) for the tunneling rate evaluated using stochastic simulation. The quantum string method is implemented to calculate the bounce and rate of tunneling for the Mueller potential in two dimensions. The quantum problem is much more difficult than the thermally activated barrier crossing problem for the same potential. The model calculations show the string method to be an efficient numerical tool for the study of barrier tunneling in higher dimension, from the determination of the saddle point to the computation of the pre-exponential factor.

AB - We generalize the string method, originally designed for the study of thermally activated rare events, to the calculation of quantum tunneling rates. This generalization is based on the formal analogy between quantum mechanics and statistical mechanics in the path-integral formalism. The quantum string method first locates the minimal action path (MAP), which is a smooth curve connecting two minima of the imaginary-time action in the space of imaginary-time trajectories. From the MAP, the saddle point of the action (called "the bounce") associated with the exponential factor for barrier tunneling probability is obtained and the pre-exponential factor (the ratio of determinants) for the tunneling rate evaluated using stochastic simulation. The quantum string method is implemented to calculate the bounce and rate of tunneling for the Mueller potential in two dimensions. The quantum problem is much more difficult than the thermally activated barrier crossing problem for the same potential. The model calculations show the string method to be an efficient numerical tool for the study of barrier tunneling in higher dimension, from the determination of the saddle point to the computation of the pre-exponential factor.

KW - Barrier tunneling

KW - Quantum metastability

KW - Rate of decay

KW - String method

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U2 - 10.1016/j.physa.2007.01.005

DO - 10.1016/j.physa.2007.01.005

M3 - Article

AN - SCOPUS:34247530874

VL - 379

SP - 491

EP - 502

JO - Physica A: Statistical Mechanics and its Applications

JF - Physica A: Statistical Mechanics and its Applications

SN - 0378-4371

IS - 2

ER -