Symplectic integration methods in molecular and spin dynamics simulations

Tsai, Shan-Ho; Krech, M.; Landau, D. P.

doi:10.1590/s0103-97332004000300009

Cited by 26 publications

(24 citation statements)

References 25 publications

(27 reference statements)

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“…They are called "symplectic integrators". The splitting of a non-integrable Heisenberg Hamiltonian into integrable parts gives rise to symplectic integrators by means of Suzuki-Trotter decompositions, which exist of various orders, see [15]. Such a splitting is always possible, just consider the decomposition into dimer Hamiltonians.…”

Section: Discussionmentioning

confidence: 99%

Heisenberg-Integrable Spin Systems

Steinigeweg

Schmidt

2008

Math Phys Anal Geom

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Abstract. We investigate certain classes of integrable classical or quantum spin systems. The first class is characterized by the recursively defined property P saying that the spin system consists of a single spin or can be decomposed into two uniformly coupled or disjoint subsystems with property P . For these systems the time evolution can be explicitely calculated. The second class consists of spin systems where all non-zero coupling constants have the same strength (spin graphs) possessing N − 1 independent, commuting constants of motion of Heisenberg type. These systems are shown to have the above property P and can be characterized as spin graphs not containing chains of length four. We completely enumerate and characterize all spin graphs up to N = 5 spins. Applications to the construction of symplectic numerical integrators for non-integrable spin systems are briefly discussed.

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Section: Discussionmentioning

confidence: 99%

Heisenberg-Integrable Spin Systems

Steinigeweg

Schmidt

2008

Math Phys Anal Geom

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“…To tackle this problem, Bulgac and Kusnezov (BK) introduced a deterministic constant-temperature dynamics [22][23][24] which can be applied to spins. A number of numerical approaches to integration of spin dynamics can be found in the literature [25][26][27][28]. However, BK dynamics, as any other deterministic canonical phase space flow, is able to correctly sample the canonical distribution only if the motion in phase space is ergodic on the timescale of the simulation.…”

Section: Introductionmentioning

confidence: 99%

Bulgac-Kusnezov-Nosé-Hoover thermostats

Sergi

Ezra

2010

Phys. Rev. E

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In this paper we formulate Bulgac-Kusnezov constant temperature dynamics in phase space by means of non-Hamiltonian brackets. Two generalized versions of the dynamics are similarly defined: one where the Bulgac-Kusnezov demons are globally controlled by means of a single additional Nosé variable, and another where each demon is coupled to an independent Nosé-Hoover thermostat. Numerically stable and efficient measure-preserving time-reversible algorithms are derived in a systematic way for each case. The chaotic properties of the different phase space flows are numerically illustrated through the paradigmatic example of the one-dimensional harmonic oscillator. It is found that, while the simple Bulgac-Kusnezov thermostat is apparently not ergodic, both of the Nosé-Hoover controlled dynamics sample the canonical distribution correctly. * Electronic address: sergi@ukzn.ac.za † Electronic address: gse1@cornell.edu

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“…Hence the decomposition X H = X 1 + X 2 does not necessarily lead to symplectic integrators. Nevertheless, we will illustrate this by an example which is connected with a numerical integrator used for bi-partite spin systems, see [6,9,10]. Such spin systems can be divided into two disjoint subsets of spins A and B, such that the interaction is only non-zero between spins of different subsets.…”

Section: A Counter-examplementioning

confidence: 99%

“…Unfortunately, symplectic integrators for spin systems have only rarely been considered in the literature, see [6,7,8]. The method of the independent time evolution of sublattices, proposed in [6,9,10], is volume-preserving but not symplectic, see 2.2.1 and [6]. Inspired by [10], we suggest to construct symplectic integrators based on a splitting of the spin Hamiltonian into two completely integrable Hamiltonians belonging to a special kind of systems [11,12].…”

Section: Introductionmentioning

confidence: 99%

“…The method of the independent time evolution of sublattices, proposed in [6,9,10], is volume-preserving but not symplectic, see 2.2.1 and [6]. Inspired by [10], we suggest to construct symplectic integrators based on a splitting of the spin Hamiltonian into two completely integrable Hamiltonians belonging to a special kind of systems [11,12]. These systems are called "B-partitioned systems" and their time evolution can be calculated as a sequence of rotations about fixed axes [12].…”

Section: Introductionmentioning

confidence: 99%

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Symplectic integrators for classical spin systems

Steinigeweg

Schmidt

2006

Computer Physics Communications

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We suggest a numerical integration procedure for solving the equations of motion of certain classical spin systems which preserves the underlying symplectic structure of the phase space. Such symplectic integrators have been successfully utilized for other Hamiltonian systems, e. g. for molecular dynamics or non-linear wave equations. Our procedure rests on a decomposition of the spin Hamiltonian into a sum of two completely integrable Hamiltonians and on the corresponding Lie-Trotter decomposition of the time evolution operator. In order to make this method widely applicable we provide a large class of integrable spin systems whose time evolution consists of a sequence of rotations about fixed axes. We test the proposed symplectic integrator for small spin systems, including the model of a recently synthesized magnetic molecule, and compare the results for variants of different order.

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Symplectic integration methods in molecular and spin dynamics simulations

Cited by 26 publications

References 25 publications

Heisenberg-Integrable Spin Systems

Heisenberg-Integrable Spin Systems

Bulgac-Kusnezov-Nosé-Hoover thermostats

Symplectic integrators for classical spin systems

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