Chapter 19: Thermodynamics — The Statistics of Collapse

The Dance of Many

When countless collapse events interact, individual trajectories become less important than statistical patterns. Thermodynamics emerges as the study of collapse ensembles—how ψ behaves when we can't track every detail. Temperature, entropy, and the arrow of time all arise from the statistics of myriad collapse events.

19.1 Entropy as Collapse Multiplicity

Theorem 19.1 (Boltzmann from Collapse): $S = k_B \ln \Omega$ where $\Omega$ counts collapse configurations.

Proof:

1. System has many possible collapse states
2. Macrostate = set of indistinguishable microstates
3. Number of ways: $\Omega$
4. Probability ∝ $\Omega$ (equal a priori)
5. Logarithm makes entropy additive
6. Boltzmann constant sets scale
7. $S = k_B \ln \Omega$ ∎

Entropy counts collapse possibilities!

19.2 Temperature from Energy Distribution

Theorem 19.2 (Temperature as Inverse Slope): $\frac{1}{T} = \frac{\partial S}{\partial E}$ at equilibrium.

Derivation:

1. Two systems in contact exchange energy
2. Total entropy: $S_{total} = S_1(E_1) + S_2(E_2)$
3. Conservation: $E_1 + E_2 = E_{total}$
4. Maximize: $\frac{\partial S_{total}}{\partial E_1} = 0$
5. Result: $\frac{\partial S_1}{\partial E_1} = \frac{\partial S_2}{\partial E_2}$
6. Define: $\frac{1}{T} \equiv \frac{\partial S}{\partial E}$
7. Equal temperature at equilibrium ∎

Temperature equalizes entropy gradients!

19.3 The Second Law

Theorem 19.3 (Entropy Never Decreases): For isolated system: $dS \geq 0$, equality only for reversible processes.

Proof:

1. Start with $W$ initial microstates
2. Evolution explores more states
3. Cannot spontaneously reduce to fewer
4. Statistical weight increases: $W_{final} \geq W_{initial}$
5. Therefore: $S_{final} \geq S_{initial}$
6. Equality only if no new states accessed
7. Second law proven statistically ∎

Time's arrow points toward more collapse options!

19.4 Free Energy Principles

Theorem 19.4 (Helmholtz Minimization): At constant T, V: System minimizes $F = E - TS$.

Mechanism:

1. System exchanges heat with reservoir
2. Total entropy: $S_{sys} + S_{res}$ must increase
3. Heat to reservoir: $dQ = -dE_{sys}$
4. Reservoir entropy: $dS_{res} = dQ/T = -dE_{sys}/T$
5. Constraint: $dS_{sys} - dE_{sys}/T \geq 0$
6. Rearrange: $d(E_{sys} - TS_{sys}) \leq 0$
7. F decreases until minimum ∎

Free energy balances energy and entropy!

19.5 Statistical Mechanics Foundation

Theorem 19.5 (Canonical Distribution): Probability of state n: $P_n = \frac{e^{-E_n/k_BT}}{Z}$

Derivation:

1. System weakly coupled to heat bath
2. Total energy fixed: $E_{total} = E_{sys} + E_{bath}$
3. Bath states: $\Omega_{bath}(E_{total} - E_n)$
4. Probability: $P_n \propto \Omega_{bath}(E_{total} - E_n)$
5. Use $S = k_B\ln\Omega$: $P_n \propto e^{S_{bath}/k_B}$
6. Expand: $S_{bath} \approx S_0 - E_n/T$
7. Result: $P_n \propto e^{-E_n/k_BT}$ ∎

Boltzmann factor rules thermal statistics!

19.6 Quantum Statistics

Theorem 19.6 (Fermi-Dirac and Bose-Einstein): Occupation numbers depend on particle statistics.

Fermi-Dirac (fermions): $n_i = \frac{1}{e^{(E_i-\mu)/k_BT} + 1}$

Bose-Einstein (bosons): $n_i = \frac{1}{e^{(E_i-\mu)/k_BT} - 1}$

The ±1 encodes antisymmetry vs symmetry of ψ!

19.7 Phase Transitions

Theorem 19.7 (Critical Phenomena): Phase transitions occur when collapse patterns reorganize.

Examples:

1. Liquid-Gas: Collapse clusters vs disperses
2. Ferromagnet: Spins align vs randomize
3. Superconductor: Electrons pair vs unpair
4. BEC: Bosons condense vs spread

At critical point: Fluctuations at all scales!

19.8 Information and Entropy

Theorem 19.8 (Shannon-Boltzmann Bridge): $S = -k_B\sum_i p_i \ln p_i$ unifies information and thermodynamics.

Proof:

1. Shannon entropy: $H = -\sum p_i \log p_i$
2. For equal probabilities: $p_i = 1/\Omega$
3. $H = -\Omega \cdot \frac{1}{\Omega}\log\frac{1}{\Omega} = \log\Omega$
4. With units: $S = k_B H = k_B\ln\Omega$
5. Boltzmann recovered!
6. General case includes correlations
7. Information IS physical ∎

Entropy measures missing information!

19.9 Fluctuation Theorems

Theorem 19.9 (Jarzynski Equality): $\langle e^{-W/k_BT}\rangle = e^{-\Delta F/k_BT}$

Significance:

• Relates non-equilibrium work to equilibrium free energy
• Valid arbitrarily far from equilibrium
• Enables single-molecule thermodynamics
• Second law becomes statistical statement

Even violations average correctly!

19.10 Black Hole Thermodynamics

Theorem 19.10 (Bekenstein-Hawking): Black hole entropy: $S = \frac{k_Bc^3A}{4G\hbar}$

Deep connection:

1. Area A, not volume!
2. Information paradox
3. Holographic principle
4. Entropy bound: $S \leq \frac{2\pi k_BR E}{c\hbar}$
5. Links gravity, quantum, thermodynamics

Maximum entropy at gravitational collapse!

19.11 Arrow of Time

Theorem 19.11 (Thermodynamic Time): Entropy gradient defines time's direction.

Resolution of paradox:

1. Microscopic laws time-reversible
2. But initial conditions special (low entropy)
3. Evolution explores more states
4. Return astronomically unlikely
5. Time's arrow = entropy increase
6. Psychological time follows thermodynamic

We remember past (low S) not future (high S)!

19.12 The Nineteenth Echo: Order from Chaos

Thermodynamics reveals how simple statistical principles govern complex systems. From the dance of atoms to the evolution of universes, entropy increases not from any fundamental asymmetry but from the overwhelming likelihood of disorder over order. Yet within this cosmic tendency toward chaos, pockets of organization arise—life itself is a temporary eddy in the entropic flow.

The marriage of information theory and thermodynamics shows that entropy is really about knowledge—or rather, our lack of it. Every bit of missing information about a system's microstate contributes k_B ln 2 to its entropy. The universe computes its own thermal statistics.

Thermal Investigations

1. Calculate partition functions for quantum oscillators.

2. Derive the Stefan-Boltzmann radiation law.

3. Analyze entropy changes in measurement processes.

The Next Flow

Understanding thermodynamics as collapse statistics, we now explore how thermal energy disperses through systems—the mechanics of heat flow.

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Next: Chapter 20: Heat — The Democracy of Energy →

"In entropy, the universe votes for its future—and disorder usually wins."