Random walks on combs

Durhuus, Bergfinnur; Jónsson, Thórdur; Wheater, J.F.

doi:10.1088/0305-4470/39/5/002

Cited by 41 publications

(116 citation statements)

References 22 publications

(51 reference statements)

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“…The transition between the two regimes varies depending on the model but it is continuous in general. Examples include causal dynamical triangulations [5][6][7], asymptotically safe quantum gravity [8,9], loop quantum gravity and spin foams [10][11][12], Hořava-Lifshitz gravity [7,9,13], noncommutative geometry [14][15][16] and κ-Minkowski spacetime [17,18], nonlocal quantum gravity [19], Stelle's gravity [20], spacetimes with black holes [21][22][23], fuzzy spacetimes [24], random combs [25,26], random multigraphs [27,28], and causal sets [29].…”

Section: A Dimensional Flow and Multiscale Theoriesmentioning

confidence: 99%

Standard Model in multiscale theories and observational constraints

2016

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We construct and analyze the Standard Model of electroweak and strong interactions in multiscale spacetimes with (i) weighted derivatives and (ii) q-derivatives. Both theories can be formulated in two different frames, called fractional and integer picture. By definition, the fractional picture is where physical predictions should be made. (i) In the theory with weighted derivatives, it is shown that gauge invariance and the requirement of having constant masses in all reference frames make the Standard Model in the integer picture indistinguishable from the ordinary one. Experiments involving only weak and strong forces are insensitive to a change of spacetime dimensionality also in the fractional picture, and only the electromagnetic and gravitational sectors can break the degeneracy. For the simplest multiscale measures with only one characteristic time, length and energy scale t Ã , l Ã and E Ã , we compute the Lamb shift in the hydrogen atom and constrain the multiscale correction to the ordinary result, getting the absolute upper bound t Ã < 10 −23 s. For the natural choice α 0 ¼ 1=2 of the fractional exponent in the measure, this bound is strengthened to t Ã < 10 −29 s, corresponding to l Ã < 10 −20 m and E Ã > 28 TeV. Stronger bounds are obtained from the measurement of the fine-structure constant.(ii) In the theory with q-derivatives, considering the muon decay rate and the Lamb shift in light atoms, we obtain the independent absolute upper bounds t Ã < 10 −13 s and E Ã > 35 MeV. For α 0 ¼ 1=2, the Lamb shift alone yields t Ã < 10

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Section: A Dimensional Flow and Multiscale Theoriesmentioning

confidence: 99%

Standard Model in multiscale theories and observational constraints

2016

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“…In one dimension, if 6 Often it is also described by another transport equation, called a bifractional or fractional Fick equation, where the diffusion equation is "redistributed," (∂ σ − ∂ 1−β σ ∂ 2 x )P = 0 [58,61]. For Caputo derivatives these two formulations coincide, while for the RiemannLiouville derivative a source term must be added to the first equation.…”

Section: B General Solution In One Dimension For S =mentioning

confidence: 99%

Diffusion in multiscale spacetimes

Calcagni

2013

Phys. Rev. E

140

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We study diffusion processes in anomalous spacetimes regarded as models of quantum geometry. Several types of diffusion equation and their solutions are presented and the associated stochastic processes are identified. These results are partly based on the literature in probability and percolation theory but their physical interpretation here is different since they apply to quantum spacetime itself. The case of multiscale (in particular, multifractal) spacetimes is then considered through a number of examples, and the most general spectral-dimension profile of multifractional spaces is constructed.

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“…In the present work, we shall focus on the problem of diffusion on a comb displaying a random variability in the lengths of its teeth, a subject that has already been discussed to some extent in previous works [7,10,16,[26][27][28][29][30][31][32]. Specifically, the length ℓ of each tooth is drawn from a distribution whose large-ℓ behavior is given by the asymptotic form P (ℓ) ∼ ℓ −(1+α) .…”

Section: Introductionmentioning

confidence: 99%

Anomalous diffusion and dynamics of fluorescence recovery after photobleaching in the random-comb model

2016

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We address the problem of diffusion on a comb whose teeth display a varying length. Specifically, the length ℓ of each tooth is drawn from a probability distribution displaying the large-ℓ behaviorOur method is based on the mean-field description provided by the well-tested CTRW approach for the random comb model, and the obtained analytical result for the diffusion coefficient is confirmed by numerical simulations. We subsequently incorporate retardation effects arising from binding/unbinding kinetics into our model and obtain a scaling law characterizing the corresponding change in the diffusion coefficient. Finally, our results for the diffusion coefficient are used as an input to compute concentration recovery curves mimicking FRAP experiments in comb-like geometries such as spiny dendrites. We show that such curves cannot be fitted perfectly by a model based on scaled Brownian motion, i.e., a standard diffusion equation with a time-dependent diffusion coefficient. However, differences between the exact curves and such fits are small, thereby providing justification for the practical use of models relying on scaled Brownian motion as a fitting procedure for recovery curves arising from particle diffusion in comb-like systems. PACS numbers: 05.40.Fb,

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Random walks on combs

Cited by 41 publications

References 22 publications

Standard Model in multiscale theories and observational constraints

Standard Model in multiscale theories and observational constraints

Diffusion in multiscale spacetimes

Anomalous diffusion and dynamics of fluorescence recovery after photobleaching in the random-comb model

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