Splitting Methods for Semi-Classical Hamiltonian Dynamics of Charge Transfer in Nonlinear Lattices
We propose two classes of symplecticity-preserving symmetric splitting methods for semi-classical Hamiltonian dynamics of charge transfer by intrinsic localized modes in nonlinear crystal lattice models. We consider, without loss of generality, one-dimensional crystal lattice models described by cla...
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MDPI AG
2022-09-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/19/3460 |
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| author | Jānis Bajārs Juan F. R. Archilla |
| author_facet | Jānis Bajārs Juan F. R. Archilla |
| author_sort | Jānis Bajārs |
| collection | DOAJ |
| description | We propose two classes of symplecticity-preserving symmetric splitting methods for semi-classical Hamiltonian dynamics of charge transfer by intrinsic localized modes in nonlinear crystal lattice models. We consider, without loss of generality, one-dimensional crystal lattice models described by classical Hamiltonian dynamics, whereas the charge (electron or hole) is modeled as a quantum particle within the tight-binding approximation. Canonical Hamiltonian equations for coupled lattice-charge dynamics are derived, and a linear analysis of linearized equations with the derivation of the dispersion relations is performed. Structure-preserving splitting methods are constructed by splitting the total Hamiltonian into the sum of Hamiltonians, for which the individual dynamics can be solved exactly. Symmetric methods are obtained with the Strang splitting of exact, symplectic flow maps leading to explicit second-order numerical integrators. Splitting methods that are symplectic and conserve exactly the charge probability are also proposed. Conveniently, they require only one solution of a linear system of equations per time step. The developed methods are computationally efficient and preserve the structure; therefore, they provide new means for qualitative numerical analysis and long-time simulations for charge transfer by nonlinear lattice excitations. The properties of the developed methods are explored and demonstrated numerically considering charge transport by mobile discrete breathers in an example model previously proposed for a layered crystal. |
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| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T21:28:29Z |
| publishDate | 2022-09-01 |
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| spelling | doaj.art-5cd3305a62a6481aad044c5aef2e51152023-11-23T21:01:53ZengMDPI AGMathematics2227-73902022-09-011019346010.3390/math10193460Splitting Methods for Semi-Classical Hamiltonian Dynamics of Charge Transfer in Nonlinear LatticesJānis Bajārs0Juan F. R. Archilla1Faculty of Physics, Mathematics and Optometry, University of Latvia, Jelgavas Street 3, LV-1004 Riga, LatviaGroup of Nonlinear Physics, Universidad de Sevilla, ETSII, Avda Reina Mercedes s/n, 41012 Sevilla, SpainWe propose two classes of symplecticity-preserving symmetric splitting methods for semi-classical Hamiltonian dynamics of charge transfer by intrinsic localized modes in nonlinear crystal lattice models. We consider, without loss of generality, one-dimensional crystal lattice models described by classical Hamiltonian dynamics, whereas the charge (electron or hole) is modeled as a quantum particle within the tight-binding approximation. Canonical Hamiltonian equations for coupled lattice-charge dynamics are derived, and a linear analysis of linearized equations with the derivation of the dispersion relations is performed. Structure-preserving splitting methods are constructed by splitting the total Hamiltonian into the sum of Hamiltonians, for which the individual dynamics can be solved exactly. Symmetric methods are obtained with the Strang splitting of exact, symplectic flow maps leading to explicit second-order numerical integrators. Splitting methods that are symplectic and conserve exactly the charge probability are also proposed. Conveniently, they require only one solution of a linear system of equations per time step. The developed methods are computationally efficient and preserve the structure; therefore, they provide new means for qualitative numerical analysis and long-time simulations for charge transfer by nonlinear lattice excitations. The properties of the developed methods are explored and demonstrated numerically considering charge transport by mobile discrete breathers in an example model previously proposed for a layered crystal.https://www.mdpi.com/2227-7390/10/19/3460semi-classical Hamiltonian dynamicssplitting methodssymplectic integratorslattice modelscharge transferintrinsic localized modes |
| spellingShingle | Jānis Bajārs Juan F. R. Archilla Splitting Methods for Semi-Classical Hamiltonian Dynamics of Charge Transfer in Nonlinear Lattices Mathematics semi-classical Hamiltonian dynamics splitting methods symplectic integrators lattice models charge transfer intrinsic localized modes |
| title | Splitting Methods for Semi-Classical Hamiltonian Dynamics of Charge Transfer in Nonlinear Lattices |
| title_full | Splitting Methods for Semi-Classical Hamiltonian Dynamics of Charge Transfer in Nonlinear Lattices |
| title_fullStr | Splitting Methods for Semi-Classical Hamiltonian Dynamics of Charge Transfer in Nonlinear Lattices |
| title_full_unstemmed | Splitting Methods for Semi-Classical Hamiltonian Dynamics of Charge Transfer in Nonlinear Lattices |
| title_short | Splitting Methods for Semi-Classical Hamiltonian Dynamics of Charge Transfer in Nonlinear Lattices |
| title_sort | splitting methods for semi classical hamiltonian dynamics of charge transfer in nonlinear lattices |
| topic | semi-classical Hamiltonian dynamics splitting methods symplectic integrators lattice models charge transfer intrinsic localized modes |
| url | https://www.mdpi.com/2227-7390/10/19/3460 |
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