From power law intermittence to macroscopic coherent regime

Bologna, Mauro; Budini, Adrián A.; Giraldi, Filippo; Grigolini, Paolo

doi:10.1063/1.3156807

Cited by 4 publications

(5 citation statements)

References 56 publications

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“…Recent applications of this technique are shown, for example, in Ref. [21] in order to interpret the fluorescence fluctuations in blinking quantum dots.…”

Section: Sion Timesmentioning

confidence: 99%

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Open System Approach to the Internal Dynamics of a Model Multilevel Molecule

Giraldi

Petruccione

2012

Open Syst. Inf. Dyn.

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A model multilevel molecule described by two sets of rotational internal energy levels of different parity and degenerate ground states, coupled by a constant interaction, is considered, by assuming that the random collisions in a gas of identical molecules, provoke transitions between adjacent energy levels of the same parity. The prescriptions of the continuous time quantum random walk are applied to the single molecule, interpreted as an open quantum system, and the master equation driving its internal dynamics is built for a general distribution of the waiting times between two consecutive collisions. The coherence terms and the populations of the energy levels relax to the asymptotics with inverse power laws for relevant classes of non-Poissonian distributions of the collision times. The stable asymptotic equilibrium configuration is independent of the distribution.The long time dynamics may be hindered by increasing the tail of the distribution density. This effect may be interpreted as the appearance of the quantum Zeno effect over long time scales.PACS numbers:

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“…Recent applications of this technique are shown, for example, in Ref. [21] in order to interpret the fluorescence fluctuations in blinking quantum dots.…”

Section: Sion Timesmentioning

confidence: 99%

“…The mean time results to be infinite. This case has been widely studied in literature in both classical [28] and quantum processes [21]. The behavior of the Laplace transform of the distribution of collision times in the origin of the complex plane,…”

Section: B Inverse Power Law Distributionmentioning

confidence: 99%

Open System Approach to the Internal Dynamics of a Model Multilevel Molecule

Giraldi

Petruccione

2012

Open Syst. Inf. Dyn.

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“…The study of this condition is of growing interest in the field of out of equilibrium statistical mechanics, and especially for the linear response of renewal nonergodic systems to an external perturbation. [33][34][35][36] A convenient waiting time distribution for our calculations is given by the negative derivative of ML function, 18,37,38 that is to say,…”

Section: Power Law Waiting Time Distributionmentioning

confidence: 99%

Density approach to ballistic anomalous diffusion: An exact analytical treatment

Bologna

Ascolani

Grigolini

2010

Journal of Mathematical Physics

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This paper addresses the problem of deriving the probability distribution density of a diffusion process generated by a nonergodic dichotomous fluctuation using the Liouville equation (density method). The velocity of the diffusing particles fluctuates from the value of 1 to the value of −1, and back, with the distribution density of time durations τ of the two states proportional to 1/τμ in the asymptotic time limit. The adopted density method allows us to establish an exact analytical expression for the probability distribution density of the diffusion process generated by these fluctuations. Contrary to intuitive expectations, the central part of the diffusion distribution density is not left empty when moving from μ>2 (ergodic condition) to μ<2 (nonergodic condition). The intuitive expectation is realized for μ<μcr, with μcr≈1.6. For values of μ>μcr, the monomodal distribution density with a minimum at the origin is turned into a bimodal one, with a central bump whose intensity increases for μ→2. The exact theoretical treatment applies to the asymptotic time limit, which establishes for the diffusion process the ballistic scaling value δ=1. To assess the time evolution toward this asymptotic time condition, we use a numerical approach which relates the emergence of the central bump at μ=μcr with the generation of the ordinary scaling δ=0.5, which lasts for larger and larger times for μ coming closer and closer to the critical value μ=2. We assign to the waiting time distribution density two different analytical forms: one derived from the Manneville intermittence (MI) theory and one from the Mittag-Leffler (ML) survival probability. The adoption of the ML waiting time distribution density generates an exact analytical prediction, whereas the MI method allows us to get the same asymptotic time limit as the ML one for μ<2 as a result of an approximation. The joint adoption of these two waiting time distribution densities sheds light into the critical nature of the condition μ=2 and into why this is the critical point for the MI process, representing the phase transition from the nonergodic to the ergodic regime. Our main result can be interpreted as a new derivation of Lamperti distribution.

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“…Wilkie [16] previously showed that adding to equation (51) an inhomogeneous term with well-defined characteristic, the positivity properties of ρ(t) are preserved. Recently, Bologna et al [17] demonstrated that it is always possible to build a positive definite density matrix using the discrete version of the Lindblad equation as a starting point. The passage from the natural time n to the continuous time t is obtained by performing the subordination of the density matrix derived from the discrete equation.…”

Section: Physical Applications: the Convoluted Lindblad Equationmentioning

confidence: 99%

“…It is straightforward to show that the convolution of the right-hand side of equation ( 54) with an exponential always produces a positive definite density matrix. We shall test the results of section 4 considering the convolution of the equation containing the full Liouvillian used in [17]:…”

Section: Physical Applications: the Convoluted Lindblad Equationmentioning

confidence: 99%

Asymptotic solution for first and second order linear Volterra integro-differential equations with convolution kernels

Bologna¹

2010

J. Phys. A: Math. Theor.

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This paper addresses the problem of finding an asymptotic solution for first and second order integro-differential equations containing an arbitrary kernel, by evaluating the corresponding inverse Laplace and Fourier transforms. The aim of the paper is to go beyond the tauberian theorem in the case of integral-differential equations which are widely used by the scientific community. The results are applied to the convolute form of the Lindblad equation setting generic conditions on the kernel in such a way as to generate a positive definite density matrix, and show that the structure of the eigenvalues of the correspondent liouvillian operator plays a crucial role in determining the positivity of the density matrix. PACS numbers: 02.50.-r,42.50.Ct Submitted to: J. Phys. A: Math. Gen.where P n (u) or P n (ω) is a n-degree polynomial, ε is a parameter, andK(u) orK(ω) is an arbitrary function. Without loss of generality we shall consider the case of the Laplace transform. With slight changes, the results may be applied to the Fourier transform.The main goal of this paper is to give a prescription to find an asymptotic expression for the function F (t) in the representation of the starting variable, typically the time domain for the Laplace transform, or the space domain for the Fourier transform. This problem is partially solved by the use of the tauberian theorem but, as it is well known, the conditions for correctly applying this theorem are quite strict. We shall focus on the case of P 1 (u) = u + a and P 2 (u) = u 2 ± a 2 giving sufficient conditions onK(u) that

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From power law intermittence to macroscopic coherent regime

Cited by 4 publications

References 56 publications

Open System Approach to the Internal Dynamics of a Model Multilevel Molecule

Open System Approach to the Internal Dynamics of a Model Multilevel Molecule

Density approach to ballistic anomalous diffusion: An exact analytical treatment

Asymptotic solution for first and second order linear Volterra integro-differential equations with convolution kernels

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