First Law of Thermodynamics

Idea

We can deduce the First Law of Thermodynamics directly from the principle of maximum entropy in Classical Statistical Mechanics.
The usual thermodynamic statement,

emerges naturally when entropy is maximized subject to multiple constraints.
Related: entropy#Important relation.

Setup

• Consider a system with state space $X$.
• Assign functions:
  • Energy: $E:X\to \mathbb{R}$
  • Volume: $V:X\to \mathbb{R}$
• For a probability distribution $p:X\to [0,\mathrm{\infty })$, define:
  • Shannon entropy:

• Expected values:$$⟨E⟩={\int }_{X}E(x)p(x)dx,⟨V⟩={\int }_{X}V(x)p(x)dx$$

Maximizing Entropy with Two Constraints

• Suppose we only know $⟨E⟩=e$ and $⟨V⟩=v$.
• By the principle of maximum entropy, the “best” distribution $p$ maximizes $H$ under these constraints.
• Introducing Lagrange multipliers $\beta ,\gamma$, one finds:

From Information Theory to Thermodynamics

• Replace Shannon entropy by Gibbs entropy:

• Solve for $d⟨E⟩$:$$d⟨E⟩=\frac{1}{k\beta }dS-\frac{\gamma }{\beta }d⟨V⟩$$
• Define temperature and pressure as:$$T=\frac{1}{k\beta },P=\frac{\gamma }{\beta }$$
• The relation becomes:$$d⟨E⟩=TdS-Pd⟨V⟩$$

Interpretation

This equation decomposes changes in the system’s expected energy into two distinct parts:

• Controlled work: $-Pd⟨V⟩$, coming from macroscopic parameters we can manipulate directly. An increase in volume yields work in the surroundings, which translates into an energy reduction.
• Non-controlled contribution: $TdS$, the part of the energy change not associated with mechanical work. Traditionally, thermodynamics calls this non-controlled part heat, but more generally it represents any mechanism that increases entropy and contributes to the energy balance without being controlled work.

Relation to the Classical First Law

In classical thermodynamics, the First Law says that
energy is conserved: it can change form but is neither created nor destroyed.

• Here, $d⟨E⟩$ tracks the change in expected energy.
• The balance shows that such changes always decompose into:
  • work done through controllable variables,
  • plus non-controlled contributions tied to entropy.

Thus the statistical mechanics formulation recovers the First Law as a precise statement of conservation of energy.