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Positive-density ground states of the Gross-Pitaevskii equation

Mathieu Lewin, Phan Thành Nam

Abstract

We consider the nonlinear Gross-Pitaevskii equation at positive density, that is, for a bounded solution not tending to 0 at infinity. We focus on infinite ground states, which are by definition minimizers of the energy under local perturbations. When the Fourier transform of the interaction potential takes negative values we prove the existence of a phase transition at high density, where the constant solution ceases to be a ground state. The analysis requires mixing techniques from elliptic PDE theory and statistical mechanics, in order to deal with a large class of interaction potentials.

Positive-density ground states of the Gross-Pitaevskii equation

Abstract

We consider the nonlinear Gross-Pitaevskii equation at positive density, that is, for a bounded solution not tending to 0 at infinity. We focus on infinite ground states, which are by definition minimizers of the energy under local perturbations. When the Fourier transform of the interaction potential takes negative values we prove the existence of a phase transition at high density, where the constant solution ceases to be a ground state. The analysis requires mixing techniques from elliptic PDE theory and statistical mechanics, in order to deal with a large class of interaction potentials.
Paper Structure (43 sections, 38 theorems, 466 equations, 1 figure)

This paper contains 43 sections, 38 theorems, 466 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Definition of infinite ground states
  4. Boundedness of solutions to the GP equation
  5. Existence, sign and complex phase of infinite ground states
  6. Thermodynamic properties
  7. Canonical and grand-canonical energies
  8. Equivalence of ensembles
  9. Thermodynamic properties of infinite ground states
  10. Phase transition
  11. Low and high density limits
  12. Low density limit
  13. High density limit
  14. Local and global bounds
  15. Local bounds
  16. ...and 28 more sections

Key Result

Lemma 2.3

Let $d\geqslant1$ and $w$ satisfy Assumption ass:w. (i) Any infinite ground state $u$ as in Definition def:infinite_GP_GS solves the GP equation eq:GP_infinite in the sense of distributions: Moreover, we have for all $h\in H^1(\mathbb{R}^d)$ having compact support and finite interaction energy. If $u$ is real-valued, the latter is equivalent to the positivity of the two operators in the sense o

Figures (1)

  • Figure 1: Plot of $|u|^2$ for $u$ the solution to the Dirichlet problem in dimension $d=1$ with the interaction potential $w(x)=(1+x^6)^{-1}$ in the interval $\Omega=(0,40)$ and the mentioned values of $\mu$. Finite differences were used with 50 discretization points per unit length. The displayed horizontal line is the constant solution $u^2_{\rm cnst}=\mu/\int_\mathbb{R} w$ and the mentioned effective density is $\rho=\int_0^{40}|u|^2/40$. These computations confirm the occurrence of a phase transition to a periodic ground state somewhere between $\mu=80$ and $\mu=90$, with oscillations arising from the boundary. Of course, there is no well defined sharp transition in finite volume. Evaluating the right-hand side of \ref{['eq:estim_mu_c']} gives that the constant solution $u_{\rm cnst}$ becomes linearly unstable at $\mu\approx 86.5$, which might thus be the true value of $\mu_c$. This hints for a second-order phase transition, with small oscillations developing on top of the constant function KirNep-71. Note that the lower bound in \ref{['eq:mc-mu2']} gives the rather loose estimate $\mu_c\geqslant 4.6$ for this potential $w$. Observe also that the period of the ground state $u$ does not vary very much with $\mu$, as predicted from Theorem \ref{['thm:high_density']} below. There are respectively 25 and 26 peaks for $\mu=90$ and $\mu=150$, hence a period of about $1.6$. For comparison, the frequency at which the linear instability of $u_{\rm cnst}$ happens (the solution of the minimum in \ref{['eq:estim_mu_c']}) is $k_0\approx3.9$ which remarkably gives the same period $2\pi/k_0\approx 1.6$. Finally, for $\mu=150$ the effective density $\rho(\mu)\approx 77$ of the solid is higher than the density $\mu/\int_\mathbb{R} w\approx 71$ of the fluid, as predicted in \ref{['eq:critical_mu']}.

Theorems & Definitions (72)

  • Definition 2.2: Infinite ground state
  • Lemma 2.3: First and second order conditions for infinite ground states
  • proof
  • Corollary 2.4: Constant solution for positive-definite $w$
  • proof
  • Theorem 2.5: Uniform bounds
  • Corollary 2.6: Existence of real positive infinite ground states
  • Theorem 2.7: Positivity of infinite ground states in low dimensions
  • Theorem 2.8: Uniqueness in 1D for positive-definite interactions
  • Theorem 2.9: Existence of vortex-type infinite ground states in 2D
  • ...and 62 more