Langevin dynamics simulation with dipole–dipole interactions: Massive performance improvements and advanced analytical integrator
Introduction
Soft condensed matter and, particularly, complex fluid are of crucial interest and topicality in modern physical, chemical, and life sciences [[1], [2]]. They have also found applications in nanotechnology and medicine [[3], [4]]. Models of such matter are naturally based on statistical physics [[5], [6], [7], [8]]. Many practical calculations in the frameworks of such models are performed by means of a computer simulation: implementations of molecular dynamics (MD) [9], Monte Carlo [10], and hybrid Monte Carlo [11] algorithms. The partly modelled implicit solvent transforms MD concept into Langevin dynamics (LD) [[9], [12]], or to Brownian dynamics (BD) in case of the viscous limit approximation and getting rid of the inertial term of the stochastic differential equation of motion (Langevin equation) [13]. However, it is important to mention here that whereas MD and LD are numerical models of -micro- and -canonical ensembles, respectively, the BD does not fully model a statistical ensemble, rather its distinct aspect on the certain time scale. An introduction of a special friction term implying a mutual nanoparticles motion provides a Stokesian dynamics method [14]. A scaling need of the LD simulation towards a “mesoscopic” hydrodynamic description without the microscopic kinetic details loss had been implemented by dissipative particle dynamics (DPD) [[12], [15]] or other methods [16]. Any improvement of LD method can be easily generalized to DPD methods and other canonical ensemble models as it was done for a rotational motion quaternion based algorithm significant optimization [15]. The LD and DPD methods often are implemented within same simulation package and share many similar routines like Velocity Verlet (VV) integrator [[17], [18], [19]], interactions calculations, etc. Example of such package is “Extensible Simulation Package for the Research on Soft Matter Systems” (ESPResSo) [[20], [21], [22]].
Unlike BD, the LD simulation requires that the finite-difference scheme time-step order of magnitude should be smaller than the temporal parameters of the Langevin equation. Real physical problem (like Brownian motion of interacting nanoparticles) often corresponds to incommensurable characteristic times of a translational and rotational motions. As far as the time-step is common, this issue leads to an over-precision in one of these motion simulation and, finally, to a significant computational overhead. Probably, due to this technical problem, there are the research works which claim a relation between translational and rotational friction coefficients as a numerical method parameter for the equilibration time speed up. As a numerical approach of equilibria location, this approach is correct. Contrarily, a dynamic stochastic process consistent simulation, probably, should be performed based on real (i.e. derived from physical experiments or material parameters) translational and rotational friction coefficients, e.g. ones determined by Stokes’ law assuming the limit of low Reynolds number and the hydrodynamic-originated Langevin parameters [9]. Otherwise, numerical results could not be fully connected to real physical experiments: especially in case of dynamic processes investigations, electromagnetic radiation propagation and scattering, noise related phenomena, Brownian motion investigations, correct separation of metastable and the real equilibria, etc.
The method limitation of the incommensurable characteristic times requires a solution which will be suggested in the present work for ball dipole nanoparticles only. An LD simulation of the large ferrofluid aggregate is selected as a demo for such a challenge resolution. Such system as any other one with long-range each-with-each nanoparticles’ interactions naturally corresponds to a calculation problem of a polynomial complexity of the type. The Barnes–Hut approximation [23] of the long-range dipole–dipole interaction implementation has been used in the present work.
Specific open-source MD package ESPResSo has been selected to publish a new method in a ready to be used way within a stable implementation with a large set of successfully passed automated validation tests scripted by other researchers from significantly different research programs within the computational soft matter science.
Section 2.1 is dedicated to the theoretical basis of a further derivation of known VV integrator and original one within Sections 2.2 Velocity Verlet fluctuative–dissipative integrator, 2.3 Analytical fluctuative–dissipative integrator , respectively. Section 3.1 describes parameters of real physical experiment applications: just to be as close to the physical reality as possible which is crucial in order to emphasize suggested analytical integrator capabilities for specific applied tasks. Section 3.2 reveals a technical implementation of the integrator in a way freely available for the open-source community. Finally, Section 4 is dedicated to a quantitative and illustrative demo of suggested integrator benefits and its critical comparison with other VV-like methods.
Section snippets
Theoretical foundation
The MD simulation method of the -microcanonical ensemble should be symplectic (at an absence of fluctuative–dissipative components), energy-conservative [24], strictly time-reversible [18], and numerically stable [25]. These requirements are more simply achievable in practice for numerical integrators which are canonical transformations. Let us define the and the conjugate momentum as th degrees of freedom canonical coordinates, whole set of which (system mechanical state) is
Materials and conditions
In order to test suggested AI, one is required to apply it to a real physical problem. One is also important to use material parameters and other experimental conditions which are similar to such reported by different researchers in the same applied science area. Let us consider a ferrofluid suspension [38] consisting of kerosene (dynamic viscosity Pa s) as a carrier liquid and magnetite (, a mass density ) ferromagnetic/superparamagnetic [39] nanoparticles
Results and discussion
The present simulation detailed setup parameters had been published in Tcl-script [53]. The original simplest cubic lattice structure of randomly switching 15.44 nm and 25.54 nm nanoparticles with the lattice parameter equal to the double large nanoparticle diameter have been selected in order to simulate the visual Stockmayer polymer-like structure quick enough formation [61]. The AI with/without current CUDA implementation of the BH method has been validated by the set of 111
Conclusions
Hence, a conjunction of Analytical fluctuative–dissipative integrator with GPU based Barnes–Hut long-range dipole–dipole interaction calculation simulation drastically increases Langevin simulation calculation capabilities: starting from 1 million nanoparticles a difference is minutes vs. years of Unix epoch time range. The Analytical fluctuative–dissipative integrator is a Velocity Verlet like modification of Brownian dynamics methods based on a formalism of the Liouville operator Trotter
Acknowledgements
We thank Mr. Alexander (Polyakov) Peletskyi and his scientific team for an approval of the Barnes–Hut GPU simulation source code publishing to open-source repositories. We thank Dr. Serhii Shulyma, Prof. Valeriy F. Kovalenko and Dr. Mykhaylo V. Petrychuk for experimental support, material parameters providing and discussions. We thank Mr. Nicola Tanygin for the simulation hardware appliance setup support. We thank ESPResSo and Ubuntu community for the open-source software maintenance which was
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