# 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 \( \sigma \).

---