Real-time analysis of the telegrapher's equation for tunneling processes

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Volumn 69, Issue 5 1, 2004, Pages

  Ranfagni, A ^a,b   Mugnai, D ^a   Vitali, M A ^b  

^a IFAC CNR   (Italy)

^b UNIVERSITY OF FLORENCE   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

APPROXIMATION THEORY; CIRCULAR WAVEGUIDES; ELECTRIC LINES; LINEAR EQUATIONS; MATHEMATICAL MODELS; MICROWAVES; NATURAL FREQUENCIES; NETWORKS (CIRCUITS); NUMERICAL METHODS; OSCILLATIONS; REAL TIME SYSTEMS; RESONANT CIRCUITS; STOCHASTIC PROGRAMMING; VACUUM;

EXPONENTIAL FUNCTIONS; FREQUENCY REGIONS; STOCHASTIC MODELING; TUNNELING PROCESSES;

ELECTRON TUNNELING;

EID: 42749100332     PISSN: 15393755     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (12)

1. A. Ranfagni et al., Phys. Rev. E 67, 066611 (2003).
2. C. DeWitt-Morette and S. K. Foong, Phys. Rev. Lett. 62, 2201 (1989).
3. D. Mugnai et al., Phys. Rev. Lett. 68, 259 (1992).
4. Handbook of Mathematical Functions, edited by M. A. Abramowitz and I. A. Stegun (Dover Publications, New York, 1971), p. 1020.
5. S. K. Foong, in Developments in General Relativity. A Jubilee Volume in Honour of Nathan Rosen, edited by F. I. Coperstock et al. (Institute of Physics, Bristol, 1990), p. 367.
6. We have to note that the argument of the sinus function should be taken, as usual, kx-ωt, with k=2π/λ as the wave number and ω=2 πv the angular frequency. Therefore, the adopted argument (x-vt) (according to the notations of Ref. [2]) has to be considered as normalized to k, v being ω/k.
7. A procedure of this kind was proposed earlier by D. Mugnai, A. Ranfagni, R. Ruggeri, and A. Agresti in Phys. Rev. E 50, 790 (1994), for the more complicated case of wavelet signals.
8. The approach of Ref. [1] is indeed different, leading to the expressions for the delay time, Eqs. (4)-(7), which are a direct consequence of the stochastic nature of the motion, both for classically allowed and fordidden cases. The latter, however, still rest on the adoption of the analytical continuation into imaginary time, as applied to the distribution function g(t, r). See D. Mugnai, A. Ranfagni, R. Ruggeri, and A. Agresti, Phys. Rev. E 49, 1771 (1994).
9. J. Millman and H. Taub, Pulse and Digital Circuits (McGraw-Hill, New York, 1956), Chap. 2.
10. -2, while v in Eq. (1) is in fact a velocity.
11. The only difference with respect to the results reported in Ref. [1] (Fig. 4 therein) is an unessential factor of two, which can be neutralized in selecting the value of the parameter a.
12. For a completely different approach to this topic, see also M. Saltzer and J. Ankerhold, Phys. Rev. A 68, 042108 (2003), where the problem of real time in tunneling processes is developed on the basis of the Feynman path-integral method. The approach of the present paper, which considers a new way to derive the solution of the telegrapher's equation, is totally different.