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# System prepended metadata

title: Gibbs Free Energy in Automated Market Maker (AMM)

---

# Gibbs Free Energy in Automated Market Maker (AMM)

Gibbs Free Energy (G) is a thermodynamic potential that combines enthalpy (H) and entropy (S) to determine the maximum reversible work a system can perform at constant temperature (T) and pressure (P). The formula for Gibbs Free Energy is:

```
G = H - T * S
```

In the context of an Automated Market Maker (AMM), we can use the Gibbs Free Energy concept to analyze the system's behavior. Let's define the variables in terms of an AMM:

1. Enthalpy (H): Represents the total value locked in the AMM (e.g., token amounts).
2. Entropy (S): Represents the distribution of tokens in the liquidity pool (e.g., token ratio).
3. Temperature (T): Represents the market conditions or external factors influencing the AMM.

Consider a simple AMM with two tokens, A and B. The pool's state can be represented by the number of tokens in the pool, x and y, respectively. The invariant k is the product of the token amounts:

```
k = x * y
```

Assuming a constant product market maker (CPMM) model, the price (P) of token A in terms of token B can be calculated as follows:

```
P = y / x
```

In this context, we can consider the enthalpy (H) as the total value locked in the pool:

```
H = x * P_x + y * P_y
```

Where `P_x` and `P_y` are the prices of tokens A and B, respectively.

The entropy (S) can be represented as the distribution of tokens in the pool. We can use the Shannon entropy formula to measure the pool's token distribution, normalized by the invariant k:

```
S = - [(x / k) * ln(x / k) + (y / k) * ln(y / k)]
```

The temperature (T) can be considered as external market factors, like trading fees or slippage in the AMM.

Given these definitions, we can now calculate the Gibbs Free Energy (G) for the AMM as:

```
G = H - T * S
```

A lower value of G indicates a more favorable condition for trades and liquidity provision. Bots and liquidity providers can potentially compute this information to decide when to enter or exit the market. Additionally, it can help in designing AMMs with better stability, efficiency, and resistance to external market shocks.

*Note: this is an analogy only, an AMM as a mathematical object may be much more complex and function according to different dynamics.*

Not investment advice.

**Sources:**

J. Ferreira–Martins a et al. (2020) Ads5–schwarzschild deformed black branes and hydrodynamic transport coefficients, Nuclear Physics B. Available at: https://www.sciencedirect.com/science/article/pii/S0550321320301735 (Accessed: 09 May 2023). 