Quantum escape kinetics over a fluctuating barrier

Bidhan Chandra Bag; Debashis Barik; Deb Shankar Ray; Pulak Kumar Ghosh

REVIEW

Quantum escape kinetics over a fluctuating barrier

Pulak Kumar Ghosh, Debashis Barik, Bidhan Chandra Bag, Deb Shankar Ray

Statistical Mechanics · Submitted 23 Feb 2006 · 39 references

Pith's verdict: Plausible quantum extension of resonant activation, but the central quantum correction is never closed or computed from the model, so the advertised quantitative agreement is not independently checkable.

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arXiv IDcond-mat/0602553 (version 1)

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CitationBidhan Chandra Bag, Debashis Barik, Deb Shankar Ray, Pulak Kumar Ghosh. "Quantum escape kinetics over a fluctuating barrier." arXiv:cond-mat/0602553 (2006). https://pith.science/paper/PUBG63EB

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Claim paper

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Editor's note

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Dear colleague,

Quick take: this is a plausible extension of classical resonant activation to a quantum heat bath, but the central quantum correction is underdetermined in both the analytics and the numerics. The big claim — that quantum effects deepen the resonance at low temperature via tunneling — is probably right, but the paper does not give the reader a way to verify it quantitatively.

What is genuinely new: the authors take their earlier quantum Langevin/Fokker-Planck machinery and apply it to the fluctuating-barrier problem, producing the first (as far as they cite) quantum version of resonant activation. The qualitative picture is physically sensible: zero-point diffusion and tunneling reduce the mean escape time at low T, and the effect fades as T rises. The resonance minima in the mean escape time as a function of correlation time appear in all their simulations, and the classical limit is recovered as Dq approaches gamma kT. The writing is clear about the model and the approximations, which is helpful.

The soft spots are real and, for the main quantitative claim, load-bearing. Equation (2.30) sets proportional to [V'(x,t)]^2, which forces the second moment to vanish at the well minima, where Delta_1 and Delta_2 are evaluated. A physical state has nonzero width there, so the quantum dispersion is pushed into uncontrolled higher moments. The moment hierarchy (2.27)–(2.28) is not closed; Eq. (2.28) involves , and the paper never states a factorization or truncation scheme for the simulation. The moment equation (2.26) has no noise operator, so the stationary fluctuations are not tied to the quantum noise whose spectrum is fixed in (2.14). And the analytic MFPT (3.2) depends on Delta_1 and Delta_2, which are never evaluated from the model parameters; in the Fig. 5 comparison, the reader has no way to know how they were set. The Fokker-Planck approximation, exact only at tau -> 0, is used at the finite tau where resonance occurs; the authors acknowledge this but proceed anyway.

I do not think these flaws make the qualitative conclusion wrong. The physics is reasonable, and the simulations look internally consistent. But the paper as written does not allow an independent check of the very comparison it advertises. With the Deltas supplied or a closed hierarchy, this could be a solid contribution.

Who is this for: people working on noise-activated escape in quantum bistable systems, and anyone using this group's quantum Langevin formalism. It deserves a serious referee: the question is timely, and the flaw is fixable in revision. I would send it to review, but with a clear request to close the moment hierarchy, report how Delta_1 and Delta_2 are computed, and test the tau -> 0 approximation against the finite-tau regime where the resonance lives.

Best, [You]

Referee report

Significance

If the central claim were fully supported, the paper would make a useful contribution by showing how zero-point fluctuations and tunneling modify resonant activation. The model setup is physically motivated, and the qualitative expectation that quantum effects lower the escape time at low temperature is plausible. The paper demonstrates that the quantum noise strength Dq retains a nonzero zero-point contribution at T→0, which is a physically important point. However, the quantitative analysis is not self-contained: the quantum correction Q is obtained from a moment hierarchy that is not closed, the analytical formula contains parameters that are never evaluated from the model parameters, and the only analytical-numerical comparison is carried out at T=1.5 where the quantum effect is claimed to be small. Because of these gaps, the paper's main message is not yet supported by a reproducible derivation or a well-defined numerical test.

Major objections (5)

[Section II.B, Eqs. (2.27)–(2.30)] The moment hierarchy that defines the quantum correction Q is not closed. Eq. (2.28) contains ⟨δx̂^4⟩, and the paper states no factorization or truncation scheme for the numerical simulation described in Section III. Eq. (2.29) then drops the V'''⟨δx̂^3⟩ term, and the integration leading to Eq. (2.30) sets ⟨δx̂^2⟩ = Δq[V'(x,t)]². This expression vanishes at every extremum of V, including the well minimum x = −√(a/b), where Δ1 and Δ2 are evaluated in Eqs. (3.3)–(3.4). A physically admissible state has nonzero width at the minimum; forcing ⟨δx̂^2⟩ to zero transfers all dispersion into higher, uncontrolled moments. The authors must provide a closure scheme, justify the truncation, or otherwise show that Q is a controlled approximation; otherwise the quantum-classical comparison in Fig. 4 is not reliable.
[Section III, Eq. (3.2) and Eqs. (3.3)–(3.4)] The central analytical result, Eq. (3.2), depends on Δ1 and Δ2, which are defined in terms of Q(x, ⟨δx̂ⁿ⟩) but are never evaluated in closed form from the model parameters a, b, σ², τ, ℏ, and T. The paper does not state the numerical values of Δ1 and Δ2 used to draw the theoretical curves in Fig. 5, nor does it describe how they were computed. Furthermore, the comparison in Fig. 5 is restricted to T=1.5, a temperature at which the authors themselves say the quantum correction is insignificant; no analytical-simulation comparison is shown for the low-temperature regime (e.g., T=0.1) where the central claim is made. Without explicit values for Δ1 and Δ2 and a low-temperature comparison, Eq. (3.2) is not a predictive result and the 'fairly well' agreement claimed in the text cannot be verified.
[Section II.B, Eq. (2.24) and Section III] The Fokker-Planck equation (2.24) is introduced as an approximation that is exact only in the limit τ→0, yet it is used to compute mean first-passage times for correlation times up to τ=10, where the resonance minima in Figs. 1–5 appear. No validity criterion, error estimate, or comparison with the exact small-τ expansion is provided for these finite values of τ. Since the existence and location of the resonance minimum is the central phenomenon, the analytic curves in Fig. 5 could be an artifact of using the τ→0 approximation outside its regime of validity. The authors need to justify quantitatively that Eq. (2.24) remains accurate at the τ values where the resonance is observed.
[Section II.B, Eq. (2.26)] Eq. (2.26), from which the moment equations are derived, contains no noise operator on the right-hand side. Consequently, the moments ⟨δx̂^2⟩ and ⟨δx̂^3⟩ are not driven by the quantum noise f(t) whose spectrum is fixed by Eq. (2.14), and the stationary values of the moments are not connected to the fluctuation-dissipation relation. The only explicit source in the displayed hierarchy is the ad hoc ⟨δx̂^2⟩² term in Eq. (2.28). This means the quantum correction Q(x, ⟨δx̂ⁿ⟩) is not determined by the thermal state of the bath, so the low-temperature enhancement reported in the simulation may depend on initial conditions or on the particular truncation rather than on the physical zero-point and tunneling contributions.
[Section III, Fig. 4] The classical limit is described in Section II.B as Q=0 and Dq→γkT (the classical limit of Eq. (2.15)). At T=0.1, the classical escape time is therefore controlled by a very small thermal noise, while the 'quantum' curve retains a nonzero zero-point diffusion Dq→ℏω0/2. The observation that the quantum escape time is lower at low temperature then largely reflects the presence of zero-point diffusion, not a subtle tunneling correction in the nonlinear potential. This may be a valid physical effect, but the paper should state it explicitly and separate the trivial zero-point effect from the claim that 'quantization significantly enhances resonant activation.'

Minor objections (4)

[Throughout] There are numerous typographical errors, including 'Godua' for 'Gadoua' in reference [14], 'Ornstein-Unhenbeck' in Section III, 'taylored' at the end of Section II.A, 'relavent' in Section III, and 'algorithom' in Section III; these should be corrected throughout.
[Section II.B, Eq. (2.23) and Eq. (2.24)] The stochastic interpretation of Eq. (2.23) is not specified. The Fokker-Planck equation (2.24) corresponds to a particular interpretation for the multiplicative noise term g'(x)ξ(t) (Stratonovich convention), and the paper should state this explicitly.
[Section II.B, Eq. (2.25)] The expression for Ps(x,τ) is written without an explicit normalization constant; as printed it is not a normalized probability density. The constant should be included or its presence noted.
[Section III, Eq. (3.1)] In Eq. (3.1), the inner integral is written as ∫_{−∞}^{x} Ps(x) dy; the variable in the integrand should be y, not x, i.e., ∫_{−∞}^{x} Ps(y) dy. This appears to be a typographical error.

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The paper

The paper claims

A quantum particle escapes a fluctuating double well faster than a classical one at low temperature, and tunneling deepens the resonance.

The reading

The paper seeks to show that resonant activation, the nonmonotonic dependence of the mean escape time on the correlation time of barrier fluctuations, survives in the quantum regime and is sharpened by quantization. Starting from a system-reservoir quantum Langevin description of a particle in a double well whose barrier height is modulated by Ornstein-Uhlenbeck noise, the authors derive a Fokker-Planck equation and a mean-first-passage-time formula for escape over the fluctuating barrier. Their central numerical and analytical point is that at low temperature the quantum mean escape time is smaller than its classical counterpart, because tunneling and the zero-point contribution to the quantum diffusion coefficient assist escape, while the difference becomes negligible at high temperature. A sympathetic reader would care because, if correct, noise-induced resonance persists even as temperature approaches zero, and quantization acts as a qualitative enhancer of the effect rather than a small correction.

What carries the argument

The load-bearing object is the quantum-corrected, noise-averaged overdamped Langevin equation, Eq. (2.23), together with the associated Fokker-Planck equation (2.24) for the probability density of the quantum mechanical mean position. Quantum effects enter in two places: the thermal quantum noise with diffusion $D_q=\frac{1}{2}\gamma\hbar\omega_0\coth(\hbar\omega_0/2kT)$, and the dispersion-induced force $Q(x,\langle\delta\hat x^n\rangle)$, approximated at leading order by $\langle\delta\hat x^2\rangle=\Delta_q[V'(x,t)]^2$ with quantum correction parameter $\Delta_q$. Applying the standard mean-first-passage-time formula to the stationary distribution gives the central analytical result, Eq. (3.2), where the effective barrier $\Delta V_{\mathrm{eff}}(R,\tau)$ depends on the ratio $R=\sigma^2/D_q$ of barrier-noise variance to quantum diffusion and on the correlation time $\tau$. The resonance itself is understood through three time scales: the barrier fluctuation time, the typical escape time, and the attempt time.

Extended reading notes

Core claim

On the paper's own terms, the discovery is that quantization significantly enhances resonant activation at low temperature: for an overdamped quantum particle in the quartic double well $U(x)=-\frac{a}{2}x^2+\frac{b}{4}x^4$ with barrier fluctuations of correlation time $\tau$, the mean escape time passes through a minimum as a function of $\tau$, and at low temperature that quantum minimum lies below the classical one. The mechanism is the interplay between the quantum diffusion coefficient $D_q=\frac{1}{2}\gamma\hbar\omega_0\coth(\hbar\omega_0/2kT)$ and the quantum correction $Q(x,\langle\delta\hat x^n\rangle)$: at low temperature, tunneling and the zero-point contribution lower the effective barrier, while at higher temperature the growth of $D_q$ and the anharmonic correction compete and the quantum-classical gap closes. The analytical basis is the stationary solution of the approximate Fokker-Planck equation and the closed-form mean escape time, Eq. (3.2), whose effective barrier height $\Delta V_{\mathrm{eff}}(R,\tau)$ carries both the noise-induced renormalization and the quantum corrections.

Load-bearing premise

The paper's central comparison depends on the assumption that the quantum fluctuation hierarchy can be truncated at leading order, treating Eqs. (2.27) and (2.28) as a closed system even though Eq. (2.28) still contains $\langle\delta\hat x^4\rangle$; if higher moments are not negligible near the barrier, the low-temperature quantum speed-up could be an artifact of the closure rather than tunneling.

Editorial extensions

If this is right

The resonance minimum persists in the quantum regime: for fixed barrier-noise strength, the mean escape time as a function of $\tau$ remains nonmonotonic and the minimum moves in the expected directions when noise strength, temperature, or barrier height is changed.
The quantum-classical gap is largest where tunneling dominates: as $T$ drops toward the vacuum limit, $D_q$ saturates at $\gamma\hbar\omega_0/2$ and the quantum mean escape time lies below the classical curve.
At absolute zero the paper expects resonant activation to remain observable, because the zero-point contribution keeps the effective diffusion nonzero.
An increase in barrier height shifts the resonance minimum to larger $\tau$, while stronger thermal or barrier noise shifts it to smaller $\tau$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural testable extension would be the same quantum moment-truncation scheme with dichotomous barrier noise; the resonance minimum should then also depend on the ratio of the two flipping rates, not only on $\tau$.
The leading-order relation $\langle\delta\hat x^2\rangle\propto[V'(x)]^2$ effectively turns the quantum correction into a state-dependent softening of the potential, so the same calculation could be carried over to asymmetric or anharmonic bistable landscapes without changing the basic mechanism.
An experiment in a system with tunable barrier fluctuations, such as a superconducting Josephson junction, could test the predicted low-temperature quantum dip in the mean first passage time versus $\tau$; the prediction is that the dip deepens as the temperature is reduced.

Original abstract

The escape rate of a particle over a fluctuating barrier in a double well potential exhibits resonance at an optimum value of correlation time of fluctuation. This has been shown to be important in several variants of kinetic model of chemical reactions . We extend the analysis of this phenomenon of resonant activation to quantum domain to show how quantization significantly enhances resonant activation at low temperature due to tunneling.

The compiled record

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What the abstract itself states, as probabilities: claims first 0.08 · states limitation 0.07 · human participants 0.01 · controlled comparison 0.05 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.47 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

The escape rate of a particle over a fluctuating barrier in a double well potential exhibits resonance at an optimum value of correlation time of fluctuation. This has been shown to be important in several variants of kinetic model of chemical reactions . We extend the analysis of this phenomenon of resonant activation to quantum domain to show how quantization significantly enhances resonant activation at low temperature due to tunneling.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The escape rate of a particle over a fluctuating barrier in a double well potential exhibits resonance at an optimum value of correlation time of fluctuation.	background 0.94	0.18	0.14	asserted 1.00
s1	This has been shown to be important in several variants of kinetic model of chemical reactions .	background 1.00	0.05	0.03	asserted 1.00
s2	We extend the analysis of this phenomenon of resonant activation to quantum domain to show how quantization significantly enhances resonant activation at low temperature due to tunneling.	contribution 0.80	0.93	0.90	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.82.

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arXiv listing abstract v1

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full text
figures and tables
references and citing sentences
relations to other papers

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Figures

**Figure 4.** Fig.4. This behaviour can be interpreted in terms of an

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References

Works cited (39)

[1]

R. Benzi, G. Parisi, A. Sutera, and A. Vulpiani, Tellus 34, 10 (1982); R. Benzi, A. Sutera, and A. Vulpiani, J. Phys. A 14 L 453 (1981)

View on Pith 1982
[2]

B. McNamara and K. Wiesenfeld, Phys. Rev. A, 39, 4854 (1988); L. Gammaitoni, F. Marchesoni, E. Menichella- Saetta and S. Santucci, Phys. Rev. Lett. 62, 349 (1989)

View on Pith 1988
[3]

B. McNamara, K. Wiesenfeld and R. Roy, Phys. Rev. Lett. 60, 2626 (1988)

View on Pith 1988
[4]

L. Gammaitoni, P. H¨ anggi, P. Jung and F.Marchesoni, Rev. Mod. Phys., 70, 223(1998)

View on Pith 1998
[5]

F. Julicher, A. Ajdari and J. Prost, Rev. Mod. Phys. 69, 1269 (1997)

View on Pith 1997
[6]

M. O. Magnasco, Phys. Rev. Lett. 71, 1477 (1993), C. R. Doering, W. Horsthemke and J. Riordan, Phys. Rev. Lett. 72, 2984 (1994)

View on Pith 1993
[7]

H. Linke, T. E. Humphrey, A. Lfgren, A. O. Sushkov, R. Newbury, R. P. Taylor, and P. Omling, Science 286, 2314 (1999); L. Q. Zhou, X. Jia and Q. Ouyang, Phys. Rev. Lett. 88, 138301(2002)

View on Pith 1999
[8]

J. Garcia-Ojalvo, A. Hernandez-Machado and J. M. San- cho, Phys. Rev. Lett. 71, 1542 (1993)

View on Pith 1993
[9]

A. Becker and L. Kramer, Phys. Rev. Lett. 73, 955 (1994), A. Becker and L. Kramer, Physica D 90, 408(1995)

View on Pith 1994
[10]

J. M. R. Parrondo, C. Van den Broeck, J. Buceta and F. J. de la Rubia, Physica A 224, 153(1996)

View on Pith 1996
[11]

A. A. Zaikin and L. Schimansky-Geier, Phys. Rev. E. 58, 4355(1998)

View on Pith 1998
[12]

J. Buceta, M. Ibanes, J. M. Sancho and K. Lindenberg, Phys. Rev. E 67, 021113(2003)

View on Pith 2003
[13]

S. Dutta, S. S. Riaz and D. S. Ray, Phys. Rev. E 71, 036216 (2005)

View on Pith 2005
[14]

C. R. Doering and J. C. Gadoua, Phys. Rev. Lett. 69, 2318 (1992)

View on Pith 1992
[15]

M. Bier and R. D. Astumian, Phys. Rev. Lett. 71, 1649 (1993); U. Z¨ urcher and C. R. Doering, Phys. Rev. E 47, 3862 (1993)

View on Pith 1993
[16]

C. Van den Broeck, Phys. Rev. E 47, 4579 (1994)

View on Pith 1994
[17]

J. J. Brey and J. Casado-Pascual, Phys. Rev. E. 50, 116 (1994); O. Flomenbom and J. Klafter, Phys. Rev. E. 69, 051109 (2004)

View on Pith 1994
[18]

T. D. Shepherd and R. Hernandez J. Chem. Phys. 115, 2430 (2001); A. M. Berezhkovskii, A. Szabo, G. H. Weiss, and H. Zhou J. Chem. Phys. 111, 9952 (1999)

View on Pith 2001
[19]

M. Boqu¨ n´ a, J. M. Porr ˙ a, J. Masoliver and K. Lindenberg, Phys. Rev. E 57, 003990 (1998)

View on Pith 1998
[20]

M. Marchi, F. Marchesoni, L. Gammaitoni, E. Menichella -Saetta, and S. Santucci Phys. Rev. E 54, 3479 (1995)

View on Pith 1995
[21]

P. H¨ anggi, Chem. Phys.,180 157 (1994); R. F. Fox, Phys. Rev. A. 34, R4525 (1986); Y. Jia and J. R. Li Phys. Rev. E. 53, 5786 (1996); X. Luo and S. Zhu, Phys. Rev. E 67, 021104 (2003)

View on Pith 1994
[22]

R. N. Mantegna and B. Spagnolo, Phys. Rev. Lett. 84, 3025 (2000)

View on Pith 2000
[23]

J. Maddox , Nature(London) 359, 771 (1992)

View on Pith 1992
[24]

J. Wang and P. Wolynes, Comput. Phys. 180 141 (1994)

View on Pith 1994
[25]

J. M. Martinis, M. H. Devoret and J. Clarke, Phys. Rev. Lett. 55, 1543 (1985)

View on Pith 1985
[26]

R. L. Straonovich, Topics in the Theory of Random Noise, Vol. 1 (Gordon and Breach, New York, 1963); K. Lindenberg and B.J. West, J. Stat. Phys. 42, 201 (1986); J. Masoliver, B.J. West, and K. Lindenberg, Phys. Rev. A 35, 3086 (1987); D. Mei, G. Xie, Li Cao, and D. Wu, Phys. Rev. E 59, 3880 (1999)

View on Pith 1986
[27]

D. Banerjee, B. C. Bag, S. K. Banik and D. S. Ray, Phys. Rev. E 65, 021109 (2002)

View on Pith 2002
[28]

D. Barik and D. S. Ray, J. Chem. Phys. 119, 12973 (2003)

View on Pith 2003
[29]

D. Barik, S. K. Banik and D. S. Ray, J. Chem. Phys. 119, 680 (2003)

View on Pith 2003
[30]

D. Banerjee, B. C. Bag, S. K. Banik and D. S. Ray, J. Chem. Phys. 120, 8960 (2004)

View on Pith 2004
[31]

D. Barik and D. S. Ray, J. Chem. Phys. 121, 1681 (2004)

View on Pith 2004
[32]

P. K. Ghosh, D. Barik and D. S. Ray, Phys. Rev. E 71, 041107 (2005)

View on Pith 2005
[33]

P. K. Ghosh, D. Barik and D. S. Ray, Physics Letters A 342 12 (2005)

View on Pith 2005
[34]

D. Barik and D. S. Ray, J. Stat. Phys. (2005) To be published

View on Pith 2005
[35]

R. Zwanzig, J. Stat. Phys. 9, 215 (1973). 7

View on Pith 1973
[36]

A. Mielke, Ann. Phys. 4, 476 (1995); J. M. Sancho, Phys. Rev. A 31, R3523 (1985); J. M. Gutirrez, A. Iglesias, and M. A. Rodrguez, Phys. Rev. E. 48, 2507 (1993)

View on Pith 1995
[37]

E. P. Wigner, Phys. Rev. 40, 749 (1932)

View on Pith 1932
[38]

M. Hillery, R. F. O’Connell, M. O. Scully and E. P. Wigner, Phys. Reps. 106, 121 (1984)

View on Pith 1984
[39]

W. H. Louisell, Quantum Statistical Properties of Radia- tion, (John Wiley and Sons, New York, 1990); Figure Captions Fig.1. A plot of mean escape time vs τ using Gaussian colour noise for diﬀerent temperatures [(i) T = 0. 1 (dot- ted line), (ii) T = 1 . 0 (dashed line), (iii) T = 2 . 0 (solid line)] and the other parameter set a = 0. 5, b = 0. 005 and σ ...

View on Pith 1990

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

R. Benzi, G. Parisi, A. Sutera, and A. Vulpiani, Tellus 34, 10 (1982); R. Benzi, A. Sutera, and A. Vulpiani, J. Phys. A 14 L 453 (1981)

View on Pith 1982

[2] [2]

B. McNamara and K. Wiesenfeld, Phys. Rev. A, 39, 4854 (1988); L. Gammaitoni, F. Marchesoni, E. Menichella- Saetta and S. Santucci, Phys. Rev. Lett. 62, 349 (1989)

View on Pith 1988

[3] [3]

B. McNamara, K. Wiesenfeld and R. Roy, Phys. Rev. Lett. 60, 2626 (1988)

View on Pith 1988

[4] [4]

L. Gammaitoni, P. H¨ anggi, P. Jung and F.Marchesoni, Rev. Mod. Phys., 70, 223(1998)

View on Pith 1998

[5] [5]

F. Julicher, A. Ajdari and J. Prost, Rev. Mod. Phys. 69, 1269 (1997)

View on Pith 1997

[6] [6]

M. O. Magnasco, Phys. Rev. Lett. 71, 1477 (1993), C. R. Doering, W. Horsthemke and J. Riordan, Phys. Rev. Lett. 72, 2984 (1994)

View on Pith 1993

[7] [7]

H. Linke, T. E. Humphrey, A. Lfgren, A. O. Sushkov, R. Newbury, R. P. Taylor, and P. Omling, Science 286, 2314 (1999); L. Q. Zhou, X. Jia and Q. Ouyang, Phys. Rev. Lett. 88, 138301(2002)

View on Pith 1999

[8] [8]

J. Garcia-Ojalvo, A. Hernandez-Machado and J. M. San- cho, Phys. Rev. Lett. 71, 1542 (1993)

View on Pith 1993

[9] [9]

A. Becker and L. Kramer, Phys. Rev. Lett. 73, 955 (1994), A. Becker and L. Kramer, Physica D 90, 408(1995)

View on Pith 1994

[10] [10]

J. M. R. Parrondo, C. Van den Broeck, J. Buceta and F. J. de la Rubia, Physica A 224, 153(1996)

View on Pith 1996

[11] [11]

A. A. Zaikin and L. Schimansky-Geier, Phys. Rev. E. 58, 4355(1998)

View on Pith 1998

[12] [12]

J. Buceta, M. Ibanes, J. M. Sancho and K. Lindenberg, Phys. Rev. E 67, 021113(2003)

View on Pith 2003

[13] [13]

S. Dutta, S. S. Riaz and D. S. Ray, Phys. Rev. E 71, 036216 (2005)

View on Pith 2005

[14] [14]

C. R. Doering and J. C. Gadoua, Phys. Rev. Lett. 69, 2318 (1992)

View on Pith 1992

[15] [15]

M. Bier and R. D. Astumian, Phys. Rev. Lett. 71, 1649 (1993); U. Z¨ urcher and C. R. Doering, Phys. Rev. E 47, 3862 (1993)

View on Pith 1993

[16] [16]

C. Van den Broeck, Phys. Rev. E 47, 4579 (1994)

View on Pith 1994

[17] [17]

J. J. Brey and J. Casado-Pascual, Phys. Rev. E. 50, 116 (1994); O. Flomenbom and J. Klafter, Phys. Rev. E. 69, 051109 (2004)

View on Pith 1994

[18] [18]

T. D. Shepherd and R. Hernandez J. Chem. Phys. 115, 2430 (2001); A. M. Berezhkovskii, A. Szabo, G. H. Weiss, and H. Zhou J. Chem. Phys. 111, 9952 (1999)

View on Pith 2001

[19] [19]

M. Boqu¨ n´ a, J. M. Porr ˙ a, J. Masoliver and K. Lindenberg, Phys. Rev. E 57, 003990 (1998)

View on Pith 1998

[20] [20]

M. Marchi, F. Marchesoni, L. Gammaitoni, E. Menichella -Saetta, and S. Santucci Phys. Rev. E 54, 3479 (1995)

View on Pith 1995

[21] [21]

P. H¨ anggi, Chem. Phys.,180 157 (1994); R. F. Fox, Phys. Rev. A. 34, R4525 (1986); Y. Jia and J. R. Li Phys. Rev. E. 53, 5786 (1996); X. Luo and S. Zhu, Phys. Rev. E 67, 021104 (2003)

View on Pith 1994

[22] [22]

R. N. Mantegna and B. Spagnolo, Phys. Rev. Lett. 84, 3025 (2000)

View on Pith 2000

[23] [23]

J. Maddox , Nature(London) 359, 771 (1992)

View on Pith 1992

[24] [24]

J. Wang and P. Wolynes, Comput. Phys. 180 141 (1994)

View on Pith 1994

[25] [25]

J. M. Martinis, M. H. Devoret and J. Clarke, Phys. Rev. Lett. 55, 1543 (1985)

View on Pith 1985

[26] [26]

R. L. Straonovich, Topics in the Theory of Random Noise, Vol. 1 (Gordon and Breach, New York, 1963); K. Lindenberg and B.J. West, J. Stat. Phys. 42, 201 (1986); J. Masoliver, B.J. West, and K. Lindenberg, Phys. Rev. A 35, 3086 (1987); D. Mei, G. Xie, Li Cao, and D. Wu, Phys. Rev. E 59, 3880 (1999)

View on Pith 1986

[27] [27]

D. Banerjee, B. C. Bag, S. K. Banik and D. S. Ray, Phys. Rev. E 65, 021109 (2002)

View on Pith 2002

[28] [28]

D. Barik and D. S. Ray, J. Chem. Phys. 119, 12973 (2003)

View on Pith 2003

[29] [29]

D. Barik, S. K. Banik and D. S. Ray, J. Chem. Phys. 119, 680 (2003)

View on Pith 2003

[30] [30]

D. Banerjee, B. C. Bag, S. K. Banik and D. S. Ray, J. Chem. Phys. 120, 8960 (2004)

View on Pith 2004

[31] [31]

D. Barik and D. S. Ray, J. Chem. Phys. 121, 1681 (2004)

View on Pith 2004

[32] [32]

P. K. Ghosh, D. Barik and D. S. Ray, Phys. Rev. E 71, 041107 (2005)

View on Pith 2005

[33] [33]

P. K. Ghosh, D. Barik and D. S. Ray, Physics Letters A 342 12 (2005)

View on Pith 2005

[34] [34]

D. Barik and D. S. Ray, J. Stat. Phys. (2005) To be published

View on Pith 2005

[35] [35]

R. Zwanzig, J. Stat. Phys. 9, 215 (1973). 7

View on Pith 1973

[36] [36]

A. Mielke, Ann. Phys. 4, 476 (1995); J. M. Sancho, Phys. Rev. A 31, R3523 (1985); J. M. Gutirrez, A. Iglesias, and M. A. Rodrguez, Phys. Rev. E. 48, 2507 (1993)

View on Pith 1995

[37] [37]

E. P. Wigner, Phys. Rev. 40, 749 (1932)

View on Pith 1932

[38] [38]

M. Hillery, R. F. O’Connell, M. O. Scully and E. P. Wigner, Phys. Reps. 106, 121 (1984)

View on Pith 1984

[39] [39]

W. H. Louisell, Quantum Statistical Properties of Radia- tion, (John Wiley and Sons, New York, 1990); Figure Captions Fig.1. A plot of mean escape time vs τ using Gaussian colour noise for diﬀerent temperatures [(i) T = 0. 1 (dot- ted line), (ii) T = 1 . 0 (dashed line), (iii) T = 2 . 0 (solid line)] and the other parameter set a = 0. 5, b = 0. 005 and σ ...

View on Pith 1990