Stochastic Schrödinger-Lohe model

doi:10.1016/j.jfa.2021.109224

Journal of Functional Analysis

Volume 281, Issue 10, 15 November 2021, 109224

https://doi.org/10.1016/j.jfa.2021.109224 Get rights and content

Abstract

The Schrödinger-Lohe model consists of wave functions interacting with each other, according to a system of Schrödinger equations with a specific coupling such that all wave functions evolve on the $L^{2}$ unit ball. This model has been extensively studied over the last decade and it was shown that under suitable assumptions on the initial state, if one waits long enough all the wave functions become arbitrarily close to each other, which we call a synchronization. In this paper, we consider a stochastic perturbation of the Schrödinger-Lohe model and show a weak version of synchronization for this perturbed model in the case of two oscillators.

Introduction

Synchronization is the emergence of a collective behavior inside a group of independent agents. Examples include crowds of people clapping at the same time, the collective flashing of some species of fireflies and the simultaneous electric activity of pacemaker cells in our hearts. This kind of phenomena is captured by the Kuramoto model that has been extensively studied. In [19], Lohe proposed a non-abelian generalization of that model. In [18], Lohe further developed the ideas of [19] with the goal of using synchronization in a quantum setting as a way to possibly duplicate quantum information.

A special case of the model proposed in [18], the Schrödinger-Lohe model, has been studied recently (see [1], [5], [6], [11] and the references therein). For the emergent dynamics, the analysis of the pairwise correlation functions between wave oscillators is essential, because we can reduce the analysis of the $L^{2}$ distance between wave oscillators to the analysis of their correlation using $L^{2}$ norm conservation. It seems that the first study on the (complete, practical) synchronization was achieved in [6] under somewhat restricted initial conditions. This restriction was relaxed by [1] in a natural manner, and a more refined complete synchronization in terms of initial condition was derived in [11] with a generalization of the model in term of different couplings. As one of generalizations of the model, the authors in [5] considered the Schrödinger-Lohe model adding a potential term.

In this paper, we investigate the Schrödinger-Lohe model when it is perturbed by a multiplicative noise. The use of a multiplicative noise is natural since it allows us to have conservation of mass in our model. We show that in this setting as well, a weak version of synchronization, in the case of two oscillators.

The result in [5] mentioned above is of particular interest to us because it tells us that a synchronization occurs in some weak sense provided the maximum difference among the values of potentials is small enough compared to the positive coupling strength of wave oscillators. If we interpret the potential terms as a deterministic perturbation of multiplicative type, and the maximum difference of potentials as the strength of noise, we may prove the same kind of synchronization concerning the balance between the coupling strength and the noise strength. This will be done indeed by an application of the large deviation principle in this paper.

We are interested in the large time behavior of the correlation functions as in the deterministic case. In the case of two oscillators, there is only one correlation function and it is possible to study the correlation function using explicit computations; we will consider that case for the remainder of the introduction. It turns out that our complex-valued correlation function satisfies a stochastic differential equation which is degenerate, in the sense that the dimension of the noise is strictly less than the dimension of the space where it lives. We will see in more detail about this stochastic differential equation that if the initial data is in the interior of the complex closed unit ball, the solution stays in the interior, and if the initial data is on the boundary of the ball, the solution stays on the boundary. We can thus restrict the stochastic differential equation either to the interior or the boundary, and we find that the diffusion restricted to the boundary is non degenerate and that there exists a unique invariant (thus ergodic) probability measure on the boundary. By this ergodicity of the invariant probability measure, we prove that the modulus of solutions which start in the interior of the ball converges to one almost surely, and this implies that there is no invariant probability measure in the interior. Since Doeblin's condition holds for the non-degenerate diffusion on the boundary, the law of solution whose initial data is on the boundary converges to the invariant measure exponentially in the total variation distance, thus it is recurrent.

These results imply a synchronization in the sense of distributions i.e. the $L^{2}$ distance of any two wave oscillators converges to $δ_{0}$ in distribution.

Two important questions regarding the stochastic model have not been answered in this work and will be pursued in the future. The first one is a modeling question: What happens for a different choice of noise and which noise makes the most sense from a physical point of view? The second one is on a synchronization result for more than two wave oscillators.

Section snippets

Preliminaries and main results

In this section, we precisely explain the results in this paper. We first give some notation. We denote by $L^{2} (R^{d}, C)$ the Lebesgue space of complex valued, square-integrable functions, and the inner product in the complex Hilbert space $L^{2} (R^{d}, C)$ is denoted by, $〈 ψ, φ 〉 = \int_{R^{d}} ψ (x) \overline{φ} (x) d x, ψ, φ \in L^{2} (R^{d}, C) .$ The norm in $L^{2} (R^{d}, C)$ is denoted by $‖ \cdot ‖$ . We define for an integer $k \geq 0$ the space $H^{k} (R^{d}, C)$ to be the set of all functions on $R^{d}$ whose derivative up to k-times exist in the weak sense and is in $L^{2} (R^{d}, C)$ . For any

Existence of solution in the stochastic case

In this section, we establish the global existence, uniqueness and $L^{2}$ -norm conservation of solution for (7). As mentioned in the introduction, it suffices to prove the following Proposition for Theorem 1 on the existence of the solution to (8). Let $k \geq 0$ be a fixed integer in what follows.

Proposition 5

Fix $K > 0$ and $ε > 0$ . Let $ψ_{j, 0} \in H^{k} (R^{d}, C)$ with $‖ ψ_{j, 0} ‖ = 1$ for $j = 1, 2, . . ., N$ . There exists a unique, ${F_{t}}_{t \geq 0}$ -adapted solution of (8), $(ϕ_{1}, ϕ_{2}, . . ., ϕ_{N}) \in C ([0, + \infty), H^{k} (R^{d}, C) \oplus \dots \oplus H^{k} (R^{d}, C))$ , a.s. with $(ϕ_{1} (0), ϕ_{2} (0), . . ., ϕ_{N} (0)) = (ψ_{1, 0}, ψ_{2, 0}$

Invariant probability measure

In the previous section, we saw that $h (t) = 〈 ψ_{1} (t), ψ_{2} (t) 〉$ satisfies (9). We shall thus study the stochastic differential equation (9) in this section. Here we recall that our interest is in the case where $h (0) = 〈 ψ_{1, 0}, ψ_{2, 0} 〉$ , therefore we need only consider initial conditions satisfying $| h (0) | \leq 1$ since $| h (0) | = | 〈 ψ_{1, 0}, ψ_{2, 0} 〉 | \leq ‖ ψ_{1, 0} ‖ \cdot ‖ ψ_{2, 0} ‖ = 1 .$ Moreover the $L^{2}$ conservation implies that $| h (t) | = | 〈 ψ_{1} (t), ψ_{2} (t) 〉 | \leq 1$ .

We shall establish in the following proposition the existence and uniqueness of solution of (9)

Convergence with initial condition in B

In this subsection we investigate the asymptotic behavior of the law of solution with the initial data $x \in B$ . To prove Theorem 2, it suffices to compare two solutions established in Proposition 6, i.e. the solution of (9) with initial data $x \in B$ and the stationary solution of (9) whose initial distribution is $μ_{K, ε}$ , unique invariant ergodic measure on ∂B.

Proposition 9

Let $h^{x} (t)$ and $g (t)$ be two solutions of (9) such that $h^{x} (0) \sim δ_{x}$ and $g (0) \sim μ_{K, ε}$ for a fixed $x \in B$ . Then, we have $| h^{x} (t) - g (t) | \overset{t \to + \infty}{\to} 0$ almost surely.

Proof

Set $D_{t} = h^{x}$

Acknowledgement

This work was partly supported by JSPS KAKENHI Grant Numbers JP19KK0066, JP20K03669. The authors are grateful to have useful discussions with D. Kim. The authors would also like to express their gratitude to K. Funano, Y. Hariya, A. de Bouard and D. Chafaï for their encouragements and discussions. The authors thank the anonymous reviewer for letting us know about the paper [14].

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Stochastic Schrödinger-Lohe model

Abstract

Introduction

Section snippets

Preliminaries and main results

Existence of solution in the stochastic case

Invariant probability measure

Convergence with initial condition in B

Acknowledgement

Stoch. Process. Appl.

A model of synchronization over quantum networks

J. Phys. A

On unique ergodicity for degenerate diffusions

Stochastics

Representation formula for stochastic Schrödinger evolution equations and applications

Nonlinearity

Semilinear Schrödinger Equations

Practical quantum synchronization for the Schrödinger-Lohe system

J. Phys. A, Math. Theor.

Quantum synchronization of the Schrödinger-Lohe model

J. Phys. A, Math. Theor.

Ergodicity for Infinite Dimensional Systems

Random Perturbations of Dynamical Systems

Synchronization and desynchronization of self-sustained oscillators by common noise

Phys. Rev. E

Convergence of Markov processes

Asymptotic behavior and stability for the Schrödinger-Lohe model

J. Math. Phys.