Solitons and Gibbs Measures for Nonlinear Schrödinger Equations

Kirkpatrick, Kay

doi:10.1051/mmnp/20127209

Cited by 5 publications

(4 citation statements)

References 65 publications

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“…Finally, it would be worth clarifying to what extent the scenario described here survives when passing to continuous models: some preliminary simulations provide encouraging results, but computationally heavy simulations are required before drawing reliable conclusions. Finally, let us note, en passant, that a richer scenario would presumably arise in higher-dimensional setups, due to the existence of a finite activation energy threshold for the discrete breathers (see the recent review [32]).…”

Section: Discussionmentioning

confidence: 99%

Discrete breathers and negative-temperature states

Iubini¹,

Franzosi²,

Livi³

et al. 2013

New J. Phys.

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We explore the statistical behaviour of the discrete nonlinear Schrödinger equation as a test bed for the observation of negative-temperature (i.e. above infinite temperature) states in Bose-Einstein condensates in optical lattices and arrays of optical waveguides. By monitoring the microcanonical temperature, we show that there exists a parameter region where the system evolves towards a state characterized by a finite density of discrete breathers and a negative temperature. Such a state persists over very long (astronomical) times since the convergence to equilibrium becomes increasingly slower as a consequence of a coarsening process. We also discuss two possible mechanisms for the generation of negative-temperature states in experimental setups, namely, the introduction of boundary dissipations and the free expansion of wavepackets initially in equilibrium at a positive temperature. Gesellschaft optics [1]. For instance, BEC in optical lattices are ideal benchmarks to investigate the role of nonlinearity and spatial discreteness in quantum transport phenomena [2,3]. The refined experimental techniques now available [4-8] enable investigations and applications of BEC in quantum coherence, quantum control, quantum information processing and the quantumclassical correspondence [9]. In addition, BEC in optical lattices can be considered as the 'atomic analogues' of light propagating in waveguide arrays [10].The discrete nonlinear Schrödinger equation (DNLSE) is a basic semiclassical model for the study of BEC in optical lattices [11] and light propagating in arrays of optical waveguides [10]. Numerical studies have revealed the important role played by localized solutions, notably discrete breathers [11,12]. Many aspects of the relationship between breathers and transport properties in BEC have recently been reviewed in [13]. The DNLSE has also been found to exhibit unusual thermodynamic features. A first statistical-mechanics study of the DNLSE identified a region in the parameter space characterized by the spontaneous formation of breathers and conjectured it to correspond to negative-temperature (NT) states [14]. NT states have attracted the curiosity of researchers since the pioneering work in systems of quantum nuclear spins [15]. From a thermodynamic point of view, the presence of NT states implies that the system's entropy is a decreasing function of the internal energy. In a series of recent papers [16][17][18][19], Rumpf provided a convincing theoretical argument that excludes the physical occurrence of NT equilibrium states in the DNLSE. In particular, he showed that even in the region of parameter space where breathers form spontaneously via a modulational instability, the DNSLE eventually reaches a maximum entropy (equilibrium) state formed by a background at infinite temperature superposed on a single breather that collects the 'excess' energy. It was also observed that the convergence to the equilibrium state predicted by Rumpf would need transients lasting over astronomical times [20]. Therefore, ...

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Section: Discussionmentioning

confidence: 99%

Discrete breathers and negative-temperature states

Iubini¹,

Franzosi²,

Livi³

et al. 2013

New J. Phys.

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“…In mathematics community, we can mention most notably [9], and [8] (cf. [22], for a review). In [9], the Hamiltonian (2.2) is considered such that N h 2 → 0, as h → 0, and N → ∞, where N denotes the number of particles, and h is the interparticle distance.…”

Section: Introductionmentioning

confidence: 99%

A stochastic thermalization of the Discrete Nonlinear Schrödinger Equation

Hannani

Olla

2022

Stoch PDE: Anal Comp

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We introduce a mass conserving stochastic perturbation of the discrete nonlinear Schrödinger equation that models the action of a heat bath at a given temperature. We prove that the corresponding canonical Gibbs distribution is the unique invariant measure. In the onedimensional cubic focusing case on the torus, we prove that in the limit for large time, continuous approximation, and low temperature, the solution converges to the steady wave of the continuous equation that minimizes the energy for a given mass.

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“…However, for p > 6, so such Z exists. See also [13,16] for alternative constructions of the Gibbs measure.…”

Section: Introductionmentioning

confidence: 99%

Concentration of the invariant measures for the periodic Zakharov, KdV, NLS and Gross--Piatevskii equations in 1D and 2D

Blower¹

2015

Preprint

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This paper concerns Gibbs measures ν for some nonlinear PDE over the D-torus u) x∈T du(x) which is normalized and formally invariant under the flow generated by the PDE. The paper proves that (Ω N , • L 2 , ν) is a metric probability space of finite diameter that satisfies the logarithmic Sobolev inequalities for the periodic KdV , the focussing cubic nonlinear Schrödinger equation and the periodic Zakharov system. For suitable subset of Ω N , a logarithmic Sobolev inequality also holds in the critical case p = 6. For D = 2, the Gross-Piatevskii equation has H = T 2 ∇u 2 − T 2 (V * |u| 2 )|u| 2 , for a suitable bounded interaction potential V and the Gibbs measure ν lies on a metric probability space (Ω, • H −s , ν) which satisfies LSI. In the above cases, (Ω, d, ν) is the limit in L 2 transportation distance of finite-dimensional (Ω n , • , ν n ) given by Fourier sums.

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Solitons and Gibbs Measures for Nonlinear Schrödinger Equations

Cited by 5 publications

References 65 publications

Discrete breathers and negative-temperature states

Discrete breathers and negative-temperature states

A stochastic thermalization of the Discrete Nonlinear Schrödinger Equation

Concentration of the invariant measures for the periodic Zakharov, KdV, NLS and Gross--Piatevskii equations in 1D and 2D

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