# Mathematical Framework for Liquidity Ranges in Uniswap: An Analogy to Quantum Physics

## 1. Energy Bands in Quantum Physics:
- **Valence Band Energy $$E_v$$** Represents the upper limit of the valence band.

- **Conduction Band Energy $$E_c$$** Represents the lower limit of the conduction band.

- **Band Gap $$E_g$$** Defined as $$E_g = E_c - E_v$$ It represents the energy difference between the valence and conduction bands.

## 2. Liquidity Ranges in Uniswap:
- **Minimum Price $$P_{min}$$** Lower boundary of the liquidity range.

- **Maximum Price $$P_{max}$$** Upper boundary of the liquidity range.

- **Liquidity Range Width $$L_r$$**: Defined as $$L_r = P_{max} - P_{min}$$ It represents the width of the price range within which liquidity is provided.

## 3. Market Dynamics:
- **Market Price $$P_m$$**: Current market price of the asset.
- **Active Liquidity Condition**: Liquidity is active if $$P_{min} \leq P_m \leq P_{max}$$. This is analogous to a conducting state in quantum physics.
- **Inactive Liquidity Condition**: Liquidity is inactive if $$P_m < P_{min}$$ or $$P_m > P_{max}$$. This is analogous to an insulating state in quantum physics.

## 4. Mathematical Formulation of Analogies:
- **Analogy for Active Liquidity (Conducting State)**
- Quantum Physics: Electrons can transition to the conduction band if the provided energy $$E$$ satisfies $$E \geq E_g$$.
- Uniswap: Liquidity is active if $$P_{min} \leq P_m \leq P_{max}$$. Mathematically, $$L_r$$ should be chosen considering market volatility $$\sigma$$ to maximize the probability of $$P_m$$ staying within $$L_r$$.
- Probability of Active Liquidity: $$P(A) = \int_{P_{min}}^{P_{max}} f(P_m, \sigma) \, dP_m$$, where $$f(P_m, \sigma)$$ is the probability density function of $$P_m$$ based on market volatility $$\sigma$$.

- **Analogy for Inactive Liquidity (Insulating State)**
- Quantum Physics: Electrons remain in the valence band if $$E < E_g$$.
- Uniswap: Liquidity is inactive if $$P_m$$ falls outside $$[P_{min}, P_{max}]$$. This state is more probable as $$L_r$$ becomes narrower relative to market volatility $$\sigma$$.

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