Exact solution to the classical equation of motion for a charged particle in external electric and magnetic fields

Donkov, A. A.; Donkov, A. D.; Grancharova, E. I.

doi:10.1134/1.1490102

Cited by 5 publications

(9 citation statements)

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“…In this approximation we can ignore the coefficients D (n) for n ≥ 3. To perform a quantitative test of the result with these coefficients, we solve the FokkerPlanck equation for the PDF at scales ∆x ≪ L with a given distribution at sample size L [6,7]. Fig.…”

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confidence: 99%

Stochastic Analysis and Regeneration of Rough Surfaces

et al. 2003

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We investigate Markov property of rough surfaces. Using stochastic analysis we characterize the complexity of the surface roughness by means of a Fokker-Planck or Langevin equation. The obtained Langevin equation enables us to regenerate surfaces with similar statistical properties compared with the observed morphology by atomic force microscopy.Studying the growth, formation and morphology of interfaces has been one of the recent interesting fields of study because of its high technical and rich theoretical advantages [1]. One of the main problems in this area is the scaling behaviour of the moments of height difference ∆h = h(x 1 ) − h(x 2 ) and the evolution of the probability density function (PDF) of ∆h, i.e. P (∆h, ∆x) in terms of the length scale ∆x. Recently Friedrich and Peinke have been able to obtain a Fokker-Planck equation describing the evolution of the probability distribution function in terms of the length scale, by analyzing some stochastic phenomena, such as turbulent free jet, etc. [2][3][4]. They noticed that the conditional probability density of field increments (velocity field, etc.) satisfies the ChapmanKolmogorov equation. Mathematically this is a necessary condition for the fluctuating data to be a Markovian process in the length scales [5].In this letter using the method proposed by Friedrich and Peinke, we measure the Kramers-Moyal's (KM) coefficients for the fluctuating fields ∆h and h(x) of a deposited copper film. It is shown that the first and second KM's coefficients have well-defined values, while the third and fourth order coefficients tend to zero. Therefore, by addressing the implications dictated by the theorem [5] a Fokker-Planck evolution operator has been found. The Fokker-Planck equation for P (∆h, ∆x) is used to give information on changing the shape of PDF as a function of the length scale ∆x. By using this strategy the information of the observed intermittency of the height fluctuation is verified [6]. The first and second KM's coefficients for the fluctuations of h(x), enables us to write a Langevin equation for the evolution of height with respect to x. Using this equation we regenerate the surface with similar statistical properties, compared with the observed morphology by atomic force microscopy. The regeneration of a surface is known as the inverse method. There are other inverse method approaches introduced in the literature [13]. In the previous attempts, to regenerate the surface, an evolution equation for h(x, t) vs t has been evaluated. Here we do this by an evolution equation for h(x) vs x, for a certain time.For this purpose, a copper film was deposited on a polished Si(100) substrate by the resistive evaporation method in a high vacuum chamber. The pressure during evaporation was 10 −6 Torr. The thickness of the growing films was measured in situ by a quartz crystal thickness monitor. We performed all depositions at room temperature, with a deposition rate about 20 − 30nm/min. The substrate temperature was determined using a chromel/alumel thermocouple ...

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Stochastic Analysis and Regeneration of Rough Surfaces

et al. 2003

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“…where the drift and diffusion coefficients D (1) and D (2) are derived by analysis of experimental data of a fluid dynamical experiment. In this paper, we consider the application of the FPE to obtain the Kolmogorov scaling with simplified assumption that D (1) and D (2) are time-independent.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we consider the application of the FPE to obtain the Kolmogorov scaling with simplified assumption that D (1) and D (2) are time-independent. D (1) is linear in v and D (2) is quadratic in v.…”

Section: Introductionmentioning

confidence: 99%

An Analytical Solution for the Stochastic Model of a Turbulent Cascade

Masoudi

Anaraki

2006

Mod. Phys. Lett. B

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“…Our method may be regarded as a combination of the disentangling techniques of R. Feynman [22] with the operational methods developed in the functional analysis and in particular in the theory of pseudodifferential equations with partial derivatives [23]− [27]. As we have emphasized in [20] and [21] this approach is an extension and generalization of the M. Suzuki's method [18] for solving the one-dimensional linear FPE (1).…”

Section: Introductionmentioning

confidence: 99%

“…have been exactly solved ( here a 4 and a 4 (t) are arbitrary non-negative constant and function of t respectively). In the paper [21] we have found the exact solutions of the following Cauchy problems:…”

Section: Introductionmentioning

confidence: 99%