# On universally optimal lattice phase transitions and energy minimizers of completely monotone potentials

@inproceedings{Luo2021OnUO, title={On universally optimal lattice phase transitions and energy minimizers of completely monotone potentials}, author={Senping Luo and Juncheng Wei and Wenming Zou}, year={2021} }

. We consider the minimizing problem for energy functionals with two types of competing particles and completely monotone potential on a lattice. We prove that the minima of sum of two completely monotone functions among lattices is located exactly on a special curve which is part of the boundary of the fundamental region. We also establish a universal result for square lattice being the optimal in certain interval, which is surprising. Our result establishes the hexagonal-rhombic-square…

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On minima of difference of theta functions and application to hexagonal crystallization

- Mathematics
- 2022

. Let z = x + iy ∈ H := { z = x + iy ∈ C : y > 0 } and θ ( α ; z ) = (cid:80) ( m,n ) ∈ Z 2 e − α πy | mz + n | 2 be the theta function associated with the lattice L = Z ⊕ z Z . In this paper we…

## References

SHOWING 1-10 OF 21 REFERENCES

Two-Dimensional Theta Functions and Crystallization among Bravais Lattices

- MathematicsSIAM J. Math. Anal.
- 2016

It is proved that if a function is completely monotonic, then the triangular lattice minimizes its energy per particle among Bravais lattices for any given density, and the global minimality is deduced, i.e., without a density constraint, of a triangular lattICE for some Lennard-Jones-type potentials and attractive-repulsive Yukawa potentials.

From the Ginzburg-Landau Model to Vortex Lattice Problems

- Mathematics, Physics
- 2010

It is shown that the vortices of minimizer of Ginzburg-Landau, blown-up at a suitable scale, converge to minimizers of W, thus providing a first rigorous hint at the Abrikosov lattices, which is a next order effect compared to the mean-field type results.

Universally optimal distribution of points on spheres

- Mathematics
- 2006

We study configurations of points on the unit sphere that minimize potential energy for a broad class of potential functions (viewed as functions of the squared Euclidean distance between points).…

Nonhexagonal Lattices From a Two Species Interacting System

- Materials ScienceSIAM J. Math. Anal.
- 2020

A two species interacting system motivated by the density functional theory for triblock copolymers contains long range interaction that affects the two species differently. In a two species periodic…

Two-component Bose-Einstein condensates with a large number of vortices.

- PhysicsPhysical review letters
- 2002

This work considers the condensate wave function of a rapidly rotating two-component Bose gas with an equal number of particles in each component and finds that the two components contain identical rectangular vortex lattices.

Completely monotonic functions

- Mathematics
- 2001

In this expository article we survey some properties of completely monotonic functions and give various examples, including some famous special functions. Such function are useful, for example, in…

Some classes of completely monotonic functions, II

- Mathematics
- 2006

A function $$f\!:(0,\infty)\rightarrow \mathbf{R}$$ is said to be completely monotonic if $$(-1)^n f^{(n)}(x)\geq 0$$ for all x > 0 and n = 0,1,2,.... In this paper we present several new classes of…

Modular Functions and Dirichlet Series in Number Theory

- Mathematics
- 1976

This is the second volume of a 2-volume textbook which evolved from a course (Mathematics 160) offered at the California Institute of Technology du ring the last 25 years. The second volume…

Some Absolutely Monotonic and Completely Monotonic Functions

- Mathematics
- 1974

The functions $(1 - r)^{ - 2|\lambda |} (1 - 2xr + r^2 )^{ - \lambda } $ are shown to be absolutely monotonic, or equivalently, that their power series have nonnegative coefficients for $ - 1 \leqq x…