8,956 research outputs found
Projection method for droplet dynamics on groove-textured surface with merging and splitting
The geometric motion of small droplets placed on an impermeable textured
substrate is mainly driven by the capillary effect, the competition among
surface tensions of three phases at the moving contact lines, and the
impermeable substrate obstacle. After introducing an infinite dimensional
manifold with an admissible tangent space on the boundary of the manifold, by
Onsager's principle for an obstacle problem, we derive the associated parabolic
variational inequalities. These variational inequalities can be used to
simulate the contact line dynamics with unavoidable merging and splitting of
droplets due to the impermeable obstacle. To efficiently solve the parabolic
variational inequality, we propose an unconditional stable explicit boundary
updating scheme coupled with a projection method. The explicit boundary
updating efficiently decouples the computation of the motion by mean curvature
of the capillary surface and the moving contact lines. Meanwhile, the
projection step efficiently splits the difficulties brought by the obstacle and
the motion by mean curvature of the capillary surface. Furthermore, we prove
the unconditional stability of the scheme and present an accuracy check. The
convergence of the proposed scheme is also proved using a nonlinear
Trotter-Kato's product formula under the pinning contact line assumption. After
incorporating the phase transition information at splitting points, several
challenging examples including splitting and merging of droplets are
demonstrated.Comment: 26 page
Continuum limit of a mesoscopic model with elasticity of step motion on vicinal surfaces
This work considers the rigorous derivation of continuum models of step
motion starting from a mesoscopic Burton-Cabrera-Frank (BCF) type model
following the work [Xiang, SIAM J. Appl. Math. 2002]. We prove that as the
lattice parameter goes to zero, for a finite time interval, a modified discrete
model converges to the strong solution of the limiting PDE with first order
convergence rate.Comment: 52 page
Data-driven Efficient Solvers and Predictions of Conformational Transitions for Langevin Dynamics on Manifold in High Dimensions
We work on dynamic problems with collected data that
distributed on a manifold . Through the
diffusion map, we first learn the reaction coordinates where is a manifold isometrically embedded into an
Euclidean space for . The reaction coordinates
enable us to obtain an efficient approximation for the dynamics described by a
Fokker-Planck equation on the manifold . By using the reaction
coordinates, we propose an implementable, unconditionally stable, data-driven
upwind scheme which automatically incorporates the manifold structure of
. Furthermore, we provide a weighted convergence analysis of
the upwind scheme to the Fokker-Planck equation. The proposed upwind scheme
leads to a Markov chain with transition probability between the nearest
neighbor points. We can benefit from such property to directly conduct
manifold-related computations such as finding the optimal coarse-grained
network and the minimal energy path that represents chemical reactions or
conformational changes. To establish the Fokker-Planck equation, we need to
acquire information about the equilibrium potential of the physical system on
. Hence, we apply a Gaussian Process regression algorithm to
generate equilibrium potential for a new physical system with new parameters.
Combining with the proposed upwind scheme, we can calculate the trajectory of
the Fokker-Planck equation on based on the generated equilibrium
potential. Finally, we develop an algorithm to pullback the trajectory to the
original high dimensional space as a generative data for the new physical
system.Comment: 59 pages, 16 figure
Gradient flow approach to an exponential thin film equation: global existence and latent singularity
In this work, we study a fourth order exponential equation, derived from thin film growth on crystal surface in multiple
space dimensions. We use the gradient flow method in metric space to
characterize the latent singularity in global strong solution, which is
intrinsic due to high degeneration. We define a suitable functional, which
reveals where the singularity happens, and then prove the variational
inequality solution under very weak assumptions for initial data. Moreover, the
existence of global strong solution is established with regular initial data.Comment: latent singularity, curve of maximal slope. arXiv admin note: text
overlap with arXiv:1711.07405 by other author
Quantum coherence of the molecular states and their corresponding currents in nanoscale Aharonov-Bohm interferometers
By considering a nanoscale Aharonov-Bohm (AB) interferometer containing a
parrallel-coupled double dot coupled to the source and drain electrodes, we
investigate the AB phase oscillations of transport current via the bonding and
antibonding state channels. The results we obtained justify the experimental
analysis given in [Phys. Rev. Lett. \textbf{106}, 076801 (2011)] that bonding
state currents in different energy configurations are almost the same. On the
other hand, we extend the analysis to the transient transport current
components flowing through different channels, to explore the effect of the
parity of bonding and antibonding states on the AB phase dependence of the
corresponding current components in the transient regime. The relations of the
AB phase dependence between the quantum states and the associated current
components are analyzed in details, which provides useful information for the
reconstruction of quantum states through the measurement of the transport
current in such systems. With the coherent properties in the quantum dot states
as well as in the transport currents, we also provide a way to manipulate the
bonding and antibonding states by the AB magnetic flux.Comment: 10 pages, 7 figure
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