# Automated Market Makers (AMMs) as Autochemotactic Processes: **Abstract** *An Exploration of Constant Function-based Market Makers (CFMM) and Constant Product Market Makers (CPMM) in Scalable Finance.* Automated Market Makers (AMMs) in decentralized finance systems function similarly to autochemotactic processes observed in certain biological species. The liquidity imbalances within these pools, akin to 'chemical stimuli' in biological contexts, drive algorithmic price adjustments in pursuit of market equilibrium. This study rigorously investigates this analogy using mathematical models, drawing parallels between liquidity tokens and autochemotactic particles. Let's denote: ```math A(t) = concentration of Token A at time t B(t) = concentration of Token B at time t R(t) = ratio of concentrations, A(t)/B(t) ``` Drawing from the Keller-Segel model for chemotaxis: ```math ∇·J(A) = -D_A ∇A + χ_A A ∇c ∇·J(B) = -D_B ∇B + χ_B B ∇c ``` Where: - \(J(A)\) and \(J(B)\) are fluxes of Token A and B. - \(D_A\) and \(D_B\) are the respective diffusion constants for Token A and B. - \(χ_A\) and \(χ_B\) denote chemotactic sensitivities. - \(c\) is the chemotactic signal, proportional to the liquidity imbalance in our analogy. Using a lattice-based model inspired by the self-avoiding walk principle, which reduces to the finance-adapted Keller-Segel model in continuous form, we interpret the alignment of trading behaviors within AMMs. These spontaneous market reactions, driven by concentration imbalances, can be mathematically framed as: ```math δA = -k_A * (A(t) - B(t)) + α * rand(-1, 1) δB = -k_B * (B(t) - A(t)) + α * rand(-1, 1) ``` Where: - \(k_A\) and \(k_B\) represent the strength of the 'repulsion' or the incentive for a token to balance its over-representation. - \(α\) is a scaling factor, capturing the random trading/swapping impact. For CFMM and CPMM mechanisms, the invariant nature ensures that: ```math A(t) * B(t) = constant ``` These mathematical constructs, while inspired by biological models, align with AMM mechanics, giving rise to stable liquidity shifts, resembling the transverse navigation in chemotactic systems. In conclusion, through the amalgamation of decentralized finance and autochemotactic modeling, we offer a pioneering perspective, potentially opening avenues for further optimization in AMM architectures. --- Please note: While this presents a mathematical framework, further empirical validation is crucial to establish these models' accuracy and applicability in real-world financial systems. The pseudo-code provided is a representation and should be adapted to a suitable mathematical environment for rigorous computations. **Sources:** Alignment interaction and band formation in assemblies of autochemorepulsive walkers - https://journals.aps.org/pre/abstract/10.1103/PhysRevE.108.034604 Kim, Z.Y. and Park, J.-H. (2023) Distinguishable cash, Bosonic Bitcoin, and fermionic non-fungible token, Frontiers. Available at: https://www.frontiersin.org/articles/10.3389/fphy.2023.1113714/full (Accessed: 16 September 2023).