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Chapter 20: Heat — The Democracy of Energy

The Spreading of Collapse​

Heat is energy in its most democratic form—spread equally among all available modes. When we add heat to a system, we're not adding any specific type of energy but rather increasing the general agitation of collapse patterns. This chapter reveals how thermal energy flows from concentrated to dispersed states.

20.1 Heat as Distributed Collapse​

Theorem 20.1 (Heat from Random Motion): Heat is collapse energy with maximum entropy per unit energy.

Proof:

  1. Consider energy E distributed among N modes
  2. Maximum entropy when equally distributed
  3. Each mode: ⟨Ei⟩=E/N=kBT/2\langle E_i \rangle = E/N = k_BT/2 (equipartition)
  4. Random velocities: Maxwell-Boltzmann distribution
  5. No macroscopic flow: ⟨v⃗⟩=0\langle \vec{v} \rangle = 0
  6. But kinetic energy: ⟨12mv2⟩=32kBT\langle \frac{1}{2}mv^2 \rangle = \frac{3}{2}k_BT
  7. Heat = disordered kinetic energy ∎

Temperature measures average collapse agitation!

20.2 The Zeroth Law​

Theorem 20.2 (Thermal Equilibrium Transitivity): If A ↔ B and B ↔ C, then A ↔ C.

Physical meaning:

  1. Systems in thermal contact equilibrate
  2. Share same temperature at equilibrium
  3. Temperature is well-defined state function
  4. Can use thermometers!

Temperature exists as consistent property!

20.3 Heat Capacity​

Theorem 20.3 (Energy Storage): C=dQdT=TdSdTC = \frac{dQ}{dT} = T\frac{dS}{dT}

Types:

  • CVC_V: constant volume (no work)
  • CpC_p: constant pressure (includes expansion)
  • Relation: Cp−CV=NkBC_p - C_V = Nk_B (ideal gas)

Quantum effects:

  • Low T: Only lowest modes excited
  • Einstein model: CV∝e−ℏω/kBTC_V \propto e^{-\hbar\omega/k_BT}
  • Debye model: CV∝T3C_V \propto T^3 (phonons)

Quantum mechanics freezes out high-frequency modes!

20.4 Heat Conduction​

Theorem 20.4 (Fourier's Law): Heat flux proportional to temperature gradient: j⃗Q=−κ∇T\vec{j}_Q = -\kappa \nabla T

Microscopic mechanism:

  1. Hot region: faster particle motion
  2. Collisions transfer momentum/energy
  3. Net flow from hot to cold
  4. Rate depends on:
    • Mean free path
    • Particle velocity
    • Interaction strength

Collapse patterns flow down temperature gradients!

20.5 Thermal Radiation​

Theorem 20.5 (Planck Distribution): Black body spectrum from quantum statistics: u(ν)=8πhν3c31ehν/kBT−1u(\nu) = \frac{8\pi h\nu^3}{c^3} \frac{1}{e^{h\nu/k_BT} - 1}

Consequences:

  1. Stefan-Boltzmann: j=σT4j = \sigma T^4
  2. Wien displacement: λmaxT=b\lambda_{max}T = b
  3. Fixes UV catastrophe
  4. Birth of quantum mechanics!

All objects glow with thermal collapse radiation!

20.6 Heat Engines​

Theorem 20.6 (Carnot Efficiency): Maximum efficiency for heat engine: η=1−TcoldThot\eta = 1 - \frac{T_{cold}}{T_{hot}}

Proof:

  1. Extract work from heat flow
  2. Must increase total entropy
  3. Best case: reversible (ΔS = 0)
  4. Heat in: QhQ_h at ThT_h
  5. Heat out: QcQ_c at TcT_c
  6. Entropy: Qh/Th=Qc/TcQ_h/T_h = Q_c/T_c
  7. Efficiency: η=1−Qc/Qh=1−Tc/Th\eta = 1 - Q_c/Q_h = 1 - T_c/T_h ∎

Can't convert all heat to work!

20.7 Phase Change Thermodynamics​

Theorem 20.7 (Clausius-Clapeyron): Phase boundary slope: dpdT=LTΔV\frac{dp}{dT} = \frac{L}{T\Delta V}

where L = latent heat, ΔV = volume change.

Physical picture:

  • Phase change needs activation energy
  • Breaks/forms intermolecular bonds
  • Temperature constant during transition
  • Entropy jumps: ΔS = L/T

Matter reorganizes its collapse patterns!

20.8 Critical Phenomena​

Theorem 20.8 (Universality): Near critical points, systems show universal behavior.

Critical exponents:

  • Heat capacity: C∼∣T−Tc∣−αC \sim |T-T_c|^{-\alpha}
  • Order parameter: ϕ∼∣T−Tc∣β\phi \sim |T-T_c|^{\beta}
  • Correlation length: ξ∼∣T−Tc∣−ν\xi \sim |T-T_c|^{-\nu}

Different systems, same exponents—collapse geometry dominates!

20.9 Fluctuation-Dissipation​

Theorem 20.9 (Einstein Relation): Fluctuations and response connected: ⟨Δx2⟩=2kBTγt\langle \Delta x^2 \rangle = \frac{2k_BT}{\gamma} t

where γ = friction coefficient.

Deep principle:

  • Same mechanism causes fluctuations and dissipation
  • Thermal noise ↔ energy loss
  • Maintains equilibrium

The universe's thermal bookkeeping!

20.10 Quantum Heat Capacity​

Theorem 20.10 (Third Law Consequence): As T → 0: C → 0 for all systems.

Quantum reasoning:

  1. Ground state unique (usually)
  2. Gap to excited states
  3. Thermal energy < gap
  4. No excitations possible
  5. Can't absorb heat
  6. C → 0 necessarily

Absolute zero = quantum silence!

20.11 Information Erasure​

Theorem 20.11 (Landauer's Principle): Erasing one bit releases minimum heat: Q≥kBTln⁡2Q \geq k_BT \ln 2

Proof:

  1. Initial: two possible states
  2. Final: one definite state
  3. Entropy decrease: ΔS = -k_B ln 2
  4. Must export to environment
  5. Minimum heat: Q = TΔS
  6. Information erasure costs energy! ∎

Forgetting heats the universe!

20.12 The Twentieth Echo: The Great Equalizer​

Heat reveals the universe's democratic tendency—energy naturally spreads among all available modes until perfectly distributed. This isn't a law imposed from outside but emerges from the overwhelming probability of uniform distributions over concentrated ones.

In heat flow we see ψ seeking its most probable configuration, collapse patterns dispersing from order into chaos. Yet this same process drives engines, enables life, and creates all the interesting dynamics of our world. The tension between concentration and dispersal, between work and waste, between order and entropy, generates the rich tapestry of thermal phenomena.

Thermal Explorations​

  1. Calculate the efficiency of various thermodynamic cycles.

  2. Analyze phonon contributions to heat capacity.

  3. Derive the thermal conductivity of a quantum gas.

The Next Pattern​

Having understood heat as democratic energy spread, we explore how this manifests as temperature—the intensive measure of thermal agitation.

Next: Chapter 21: Temperature — The Intensity of Thermal Collapse →

"Heat is the universe's way of sharing energy fairly among all its children."