Deformed Fokker-Planck equation: Inhomogeneous medium with a position-dependent mass

Costa, Bruno G. da; Gomez, Ignacio S.; Borges, Ernesto P.

doi:10.1103/physreve.102.062105

Cited by 22 publications

(21 citation statements)

References 61 publications

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“…In the linearization process, nevertheless, and as a byproduct of our reverse engineering strategy, we have started from the general solutions for the GMEE ( 17) and (24) to come out with the most general solutions for the HO (18), DHO (25) and (31). The solution U ± (t) in (18) for the HO in (10) turned out to be a textbook one, provided that a = Cη and b = ±C., that provides a consistent nonlocal connection, through the nonlocal transformation in (4), to…”

Section: Discussionmentioning

confidence: 99%

“…1: For A + = 0.35 and ω = 1 we show (a) q + (t) of ( 17) for η = 0.1 (under-damping), η = 1 (critical-damping), and for η = 1.5 (over-damping), (b) q + (t) of ( 17) for different values of η < 1, and (c) {q + (t), p + (t) = q+ (t)} of ( 17) for different values of η < 1. For η = 0.2 and ω = 1 we show (d) q + (t) of ( 17) for different values of A + , (e) phase-space trajectories for {q + (t), p + (t) = q+ (t)} of (17) for different values of A + , and (f) phase-space trajectories {U + (t), P + (U + ) = U+ (t)} of (18) for…”

Section: Linearization Of Gmee-type Into Ho and Dhomentioning

confidence: 99%

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Effect of nonlocal transformations on the linearizability and exact solvability of the nonlinear generalized modified Emden type equations

Mustafa

2021

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The nonlinear generalized modified Emden type equations (GMEE) are known to be linearizable into simple harmonic oscillator (HO) or damped harmonic oscillators (DHO) via some nonlocal transformations. Hereby, we show that the structure of the nonlocal transformation and the linearizability into HO or DHO determine the nature/structure of the dynamical forces involved (hence, determine the structure of the dynamical equation). Yet, a reverse engineering strategy is used so that the exact solutions of the emerging GMEE are nonlocally transformed to find the exact solutions of the HO and DHO dynamical equations. Consequently, whilst the exact solution for the HO remains a textbook one, the exact solution for the DHO (never reported elsewhere, to the best of our knowledge) turns out to be manifestly the most explicit and general solution that offers consistency and comprehensive coverage for the associated under-damping, critical-damping, and over-damping cases (i.e., no complex settings for the coordinates and/or the velocities are eminent/feasible). Moreover, for all emerging dynamical system, we report illustrative figures for each solution as well as the corresponding phase-space trajectories as they evolve in time.

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Section: Discussionmentioning

confidence: 99%

Section: Linearization Of Gmee-type Into Ho and Dhomentioning

confidence: 99%

Effect of nonlocal transformations on the linearizability and exact solvability of the nonlinear generalized modified Emden type equations

Mustafa

2021

Preprint

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“…This would keep the longstanding gain-loss balance correlation between the kinetic and potential energies of the system and, consequently, the total energy remains an integral of motion (i.e., the standard structure the textbook Lagrangians/Hamiltonians for conservative systems of course). Such a transformation/deformation recipe would in effect introduce the so called position-dependent effective mass concept (or PDM in short) into classical and quantum mechanics (c.f., e.g., sample of references [3][4][5][6][7][8][9][10][11][12][13][14][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39] ). It is, therefore, interesting to study and investigate PDM settings on such DDOs.…”

Section: Introductionmentioning

confidence: 99%

PDM Damped-Driven Oscillators: Exact Solvability, Classical States Crossings, and Self-Crossings

Mustafa

2021

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Within the standard Lagrangian and Hamiltonian setting, we consider a position-dependent mass (PDM) classical particle performing a damped driven oscillatory (DDO) motion under the influence of a conservative harmonic oscillator force field $V\left( x\right) =\frac{1}{2}\omega ^{2}Q\left( x\right) x^{2}$ and subjected to a Rayleigh dissipative force field $\mathcal{R}\left( x,\dot{x}\right) =\frac{1}{2}b\,m\left( x\right) \dot{x}^{2}$ in the presence of an external periodic (non-autonomous) force $F\left( t\right) =F_{\circ }\,\cos \left( \Omega t\right) $. Where, the correlation between the coordinate deformation $\sqrt{Q(x)}$ and the velocity deformation $\sqrt{m(x)}$ is governed by a point canonical transformation $q\left( x\right) =\int \sqrt{m\left( x\right) }dx=\sqrt{%Q\left( x\right) }x$. Two illustrative examples are used: a non-singular PDM-DDO, and a power-law PDM-DDO models. Classical-states $\{x(t),p(t)\}$ crossings are analysed and reported. Yet, we observed/reported that as a classical state $\{x_{i}(t),p_{i}(t)\}$ evolves in time it may cross itself at an earlier and/or a later time/s.

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Section: Introductionmentioning

confidence: 99%

PDM damped-driven oscillators: exact solvability, classical states crossings, and self-crossings

Mustafa¹

2021

Preprint

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Within the standard Lagrangian and Hamiltonian setting, we consider a positiondependent mass (PDM) classical particle performing a damped driven oscillatory (DDO) motion under the influence of a conservative harmonic oscillator force fieldx 2 and subjected to a Rayleigh dissipative force field R (x, ẋ) = 1 2 b m (x) ẋ2 in the presence of an external periodic (non-autonomous) force F (t) = F• cos (Ωt). Where, the correlation between the coordinate deformation Q(x) and the velocity deformation m(x) is governed by a point canonicalTwo illustrative examples are used: a non-singular PDM-DDO, and a power-law PDM-DDO models. Classical-states {x(t), p(t)} crossings are analysed and reported. Yet, we observed/reported that as a classical state {xi(t), pi(t)} evolves in time it may cross itself at an earlier and/or a latter time/s.

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Deformed Fokker-Planck equation: Inhomogeneous medium with a position-dependent mass

Cited by 22 publications

References 61 publications

Effect of nonlocal transformations on the linearizability and exact solvability of the nonlinear generalized modified Emden type equations

Effect of nonlocal transformations on the linearizability and exact solvability of the nonlinear generalized modified Emden type equations

PDM Damped-Driven Oscillators: Exact Solvability, Classical States Crossings, and Self-Crossings

PDM damped-driven oscillators: exact solvability, classical states crossings, and self-crossings

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