Discrete Values of Circulation and Quantized Vortex Lines in DeFi

# Discrete Values of Circulation and Quantized Vortex Lines in DeFi To express the gist of our discussion mathematically, with an emphasis on *discrete values of circulation*, *quantized vortex* lines in DeFi, and *trajectory statistics* of vortices in quantum turbulence (QT), we introduce a framework that models these phenomena using mathematical formalisms relevant to both quantum fluid dynamics and financial markets. ### Discrete Values of Circulation and Quantized Vortex Lines in DeFi In superfluid helium, the circulation $$\Gamma$$ around a quantized vortex is quantized and given by: $$\Gamma = n \cdot \kappa,$$ where $$n$$ is an integer, and $$\kappa = \frac{h}{m}$$ is the quantum of circulation, with $$h$$ being Planck's constant and $$m$$ the mass of the helium atom. --- ***In analogy to DeFi liquidity pools, we define the quantized movement of assets as:*** $$\Delta Q = n \cdot q,$$ where $$\Delta Q$$ represents the quantized movement (circulation) of assets within the pool, $$n$$ is an integer denoting the number of discrete transactions or interactions, and $$q$$ is the minimum quantizable unit of the asset, analogous to the quantum of circulation $$\kappa$$ in superfluids. This quantization reflects the discrete nature of transactions and the influence of smart contract rules on asset paths. ### Trajectory Statistics of Vortices in Quantum Turbulence The trajectory of a quantized vortex in superfluid helium, exhibiting quantum turbulence, can be described using the statistics of its position $$X(t)$$ over time. The superdiffusive behavior observed in QT is characterized by the mean square displacement (MSD) showing a power-law time dependence: $$\langle X^2(t) \rangle \sim t^{\alpha},$$ where $$\alpha > 1$$ indicates superdiffusion, typical for quantum turbulence *rather than simple Brownian motion* $$\alpha = 1$$. Translating this to DeFi asset movements within liquidity pools, we consider the asset's position or value $$V(t)$$ in the pool over time, with its MSD also following a power-law indicative of superdiffusive behavior in a turbulent market: $$\langle V^2(t) \rangle \sim t^{\beta},$$ where $$\beta > 1$$ denotes the superdiffusive movement of assets, influenced by trading actions, market dynamics, and the programmable settings of smart contracts. This model allows for analyzing asset interactions within the pool and offers insights into the market's underlying dynamics, aiding in the prediction of future asset movements. ### Conclusion By adopting mathematical formalisms from quantum fluid dynamics, we can model the discrete and superdiffusive behaviors of assets within DeFi liquidity pools, drawing parallels to the quantized vortices in superfluid helium and their trajectory statistics in quantum turbulence. This approach not only enhances our understanding of the complex dynamics governing DeFi markets but also opens up new avenues for research into the predictive modeling of asset movements within these novel financial ecosystems.