Here we present supplementary online video material concerning the following publications:
Philipp Heinen1, Aleksandr N. Mikheev1,2, Christian-Marcel Schmied1, and Thomas Gasenzer1,2
1Kirchhoff-Institut für Physik, Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer Feld 227, 69120 Heidelberg, Germany
2Institut für Theoretische Physik, Ruprecht-Karls-Universität Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany
We examine coarsening of field-excitation patterns of the sine-Gordon (SG) model, in two and three spatial dimensions, identifying it as universal dynamics near non-thermal fixed points. The SG model is relevant in many different contexts, from solitons in quantum fluids to structure formation in the universe. The coarsening process entails anomalously slow self-similar transport of the spectral distribution of excitations towards low energies, induced by the collisional interactions between the field modes. The focus is set on the non-relativistic limit exhibiting particle excitations only, governed by a Schrödinger-type equation with Bessel-function nonlinearity. The results of our classical statistical simulations suggest that, in contrast to wave turbulent cascades, in which the transport is local in momentum space, the coarsening is dominated by rather non-local processes corresponding to a spatial containment in position space. The scaling analysis of a kinetic equation obtained with path-integral techniques corroborates this numerical observation and suggests that the non-locality is directly related to the slowness of the scaling in space and time. Our methods, which we expect to be applicable to more general types of models, could open a long-sought path to analytically describing universality classes behind domain coarsening and phase-ordering kinetics from first principles, which are usually modelled in a near-equilibrium setting by a phenomenological diffusion-type equation in combination with conservation laws.
Philipp Heinen1, Aleksandr N. Mikheev1,2, and Thomas Gasenzer1,2
1Kirchhoff-Institut für Physik, Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer Feld 227, 69120 Heidelberg, Germany
2Institut für Theoretische Physik, Ruprecht-Karls-Universität Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany
We extend the theory of non-thermal fixed points to the case of anomalously slow universal scaling dynamics according to the sine-Gordon model. This entails the derivation of a kinetic equation for the momentum occupancy of the scalar field from a non-perturbative two-particle irreducible effective action, which re-sums a series of closed loop chains akin to a large-N expansion at next-to-leading order. The resulting kinetic equation is analysed for possible scaling solutions in space and time that are characterised by a set of universal scaling exponents and encode self-similar transport to low momenta. Assuming the momentum occupancy distribution to exhibit a scaling form we can determine the exponents by identifying the dominating contributions to the scattering integral and power counting. If the field exhibits strong variations across many wells of the cosine potential, the scattering integral is dominated by the scattering of many quasiparticles such that the momentum of each single participating mode is only weakly constrained. Remarkably, in this case, in contrast to wave turbulent cascades, which correspond to local transport in momentum space, our results suggest that kinetic scattering here is dominated by rather non-local processes corresponding to a spatial containment in position space. The corresponding universal correlation functions in momentum and position space corroborate this conclusion. Numerical simulations performed in accompanying work yield scaling properties close to the ones predicted here.
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All videos by Philipp Heinen
The videos show the approach of the sine-Gordon system towards a non-thermal fixed point. The evolution shown is computed in the non-relativistic limit, where the evolution is governed by a non-linear Schrödinger equation with Bessel function nonlinearity, cf. Eq. (7) in the first of the above papers.
Video 1: Sine-Gordon system evolving close to a non-thermal fixed point, for m/Q = 20, F0 = 103. The upper panels show the 2D spatial distribution of the field amplitude (left) and phase (right). The lower panels depict the momentum spectrum f(t,p) = ⟨ψ+(t,p)ψ(t,p)⟩ (left) and the density-density correlator D(t,p) = ⟨ρ(t,p)ρ(t,p)⟩ (right), defined as the Fourier transform of D(t,x,y) = ⟨ρ(t,x)ρ(t,y)⟩, with ρ(t,x) = |ψ(t,x)|2 , w.r.t. x – y, both averaged over the angular orientations of p, and each on a double-logarithmic scale.
The initial state, in each case, corresponds to a box-like even occupancy of all momentum modes up to a maximum cutoff, with random phases in each mode and random Gaussian noise added to all modes, including the empty modes at large momenta. The video corresponds to the evolution, 4 snapshots of which are shown in Fig. 1 of the first paper above. The time is indicated, in units of Q–1, at the top of each video.
Video 2: Sine-Gordon system evolving close to a non-thermal fixed point, for m/Q = 80, F0 = 103. The upper panels show the 2D spatial distribution of the field amplitude (left) and phase (right). The lower panels depict the momentum spectrum f(t,p) = ⟨ψ+(t,p)ψ(t,p)⟩ (left) and the density-density correlator D(t,p) = ⟨ρ(t,p)ρ(t,p)⟩ (right), defined as the Fourier transform of D(t,x,y) = ⟨ρ(t,x)ρ(t,y)⟩, with ρ(t,x) = |ψ(t,x)|2 , w.r.t. x – y, both averaged over the angular orientations of p, and each on a double-logarithmic scale.
The initial state, in each case, corresponds to a box-like even occupancy of all momentum modes up to a maximum cutoff, with random phases in each mode and random Gaussian noise added to all modes, including the empty modes at large momenta. The time is indicated, in units of Q–1, at the top of each video.
