Self-organization of vortex-length distribution in quantum turbulence: An approach based on the Barabási-Albert model

The energy spectrum of decaying quantum turbulence at $T=0$ obeys Kolmogorov’s law. In addition to this, recent studies revealed that the vortex-length distribution (VLD), meaning the size distribution of the vortices, in decaying Kolmogorov quantum turbulence also obeys a power law. This power-law VLD suggests that the decaying turbulence has scale-free structure in real space. Unfortunately, however, there has been no practical study that answers the following important question: why can quantum turbulence acquire a scale-free VLD? We propose here a model to study the origin of the power law of the VLD from a generic point of view. The nature of quantized vortices allows one to describe the decay of quantum turbulence with a simple model that is similar to the Barabási-Albert model, which explains the scale-invariance structure of large networks. We show here that such a model can reproduce the power law of the VLD well.

Figure 1

Schematic of the model. At every time step one, and only one, loop is randomly chosen to divide into two daughter loops with the probability given by Eq. (3). If the chosen loop has length $l_i$, $\mathcal{R}{l}_i$ and $(1-\mathcal{R})l_i$ are assigned as the lengths of these two loops.

Figure 2

Solid and slashed lines represent time dependences of total line length assuming $\Pi \left(l\right)$ as Eqs. (3, 4), respectively.

Figure 3

(Color online) Simulated $⟨n\left(l\right)⟩$ at $t=100000$ (blue open squares) and 1 000 000 (red closed circles). The simulations used $\Pi \left(l\right)$ from Eq. (3). The slope of the solid line is given by the exponent $\alpha =1$.

Figure 4

(Color online) Simulated $⟨n\left(l\right)⟩$ at $t=100000$ (blue open squares) starting from random $⟨n\left(l\right)⟩$ (red closed circles) at $t=0$. The simulations used $\Pi \left(l\right)$ from Eq. (3). The slope of the solid line is given by the exponent $\alpha =1$.

Figure 5

(Color online) Simulated $⟨n\left(l\right)⟩$ at $t=100000$ (blue open squares) and 1 000 000 (red closed circles) for the case when $\Pi \left(l\right)$ came from Eq. (4) (i.e., constant preferential splitting).