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Chapter 21: Temperature — The Intensity of Thermal Collapse

The Measure of Agitation​

Temperature is more than just "hotness"—it's the fundamental measure of how vigorously collapse patterns vibrate. Unlike extensive quantities like heat, temperature is intensive, describing the local intensity of thermal motion. This chapter reveals temperature as the energy density of collapse fluctuations.

21.1 Temperature from Kinetic Theory​

Theorem 21.1 (Temperature-Kinetic Energy Relation): For ideal gas: 32kBT=12m⟨v2⟩\frac{3}{2}k_BT = \frac{1}{2}m\langle v^2\rangle

Proof:

  1. Maxwell-Boltzmann distribution: f(v)∝e−mv2/2kBTf(v) \propto e^{-mv^2/2k_BT}
  2. Average kinetic energy: ⟨KE⟩=∫12mv2f(v)d3v\langle KE\rangle = \int \frac{1}{2}mv^2 f(v) d^3v
  3. Using spherical coordinates and integrating
  4. Result: ⟨KE⟩=32kBT\langle KE\rangle = \frac{3}{2}k_BT
  5. Temperature measures mean kinetic energy
  6. Extends to all quadratic degrees of freedom
  7. Equipartition theorem! ∎

Temperature = average energy per mode!

21.2 Statistical Definition​

Theorem 21.2 (Temperature from Entropy): 1T=(∂S∂E)V,N\frac{1}{T} = \left(\frac{\partial S}{\partial E}\right)_{V,N}

Interpretation:

  • T measures how entropy responds to energy
  • High T: Adding energy barely increases S
  • Low T: Small energy greatly increases S
  • Zero T: No entropy increase possible

Temperature controls entropy's energy appetite!

21.3 Negative Temperature​

Theorem 21.3 (Population Inversion): Systems with bounded energy can have T < 0.

Example - Two-level system:

  1. Ground state: E = 0, excited: E = ε
  2. All ground: S = 0, T = 0
  3. Half excited: S = max, T = ∞
  4. All excited: S = 0 again!
  5. Beyond half: ∂S/∂E < 0
  6. Therefore: T < 0
  7. Hotter than any positive T! ∎

Negative temperature = overpopulated high states!

21.4 Temperature in Different Ensembles​

Canonical Ensemble (fixed T, V, N):

  • Probability: Pi=e−Ei/kBT/ZP_i = e^{-E_i/k_BT}/Z
  • Partition function: Z=∑ie−Ei/kBTZ = \sum_i e^{-E_i/k_BT}
  • Free energy: F=−kBTln⁡ZF = -k_BT \ln Z

Microcanonical (fixed E, V, N):

  • All accessible states equally likely
  • Temperature emerges from ∂S/∂E

Grand Canonical (fixed T, V, μ):

  • Particle number fluctuates
  • Pi,N=e−(Ei−μN)/kBT/ΞP_{i,N} = e^{-(E_i-\mu N)/k_BT}/\Xi

Temperature mediates all thermal exchanges!

21.5 Quantum Temperature Effects​

Theorem 21.4 (Thermal de Broglie Wavelength): λth=h2πmkBT\lambda_{th} = \frac{h}{\sqrt{2\pi mk_BT}}

Significance:

  • Quantum effects when λ_th ~ interparticle spacing
  • Bose-Einstein condensation when overlap
  • Fermi degeneracy pressure
  • Classical limit: λ_th → 0 as T → ∞

Temperature sets quantum-classical boundary!

21.6 Temperature and Radiation​

Theorem 21.5 (Wien's Law): Peak wavelength: λmaxT=2.898×10−3\lambda_{max}T = 2.898 \times 10^{-3} m·K

Proof from Planck's law:

  1. Planck distribution: u(λ,T)=8πhcλ51ehc/λkBT−1u(\lambda,T) = \frac{8\pi hc}{\lambda^5}\frac{1}{e^{hc/\lambda k_BT}-1}
  2. Maximize: ∂u∂λ=0\frac{\partial u}{\partial \lambda} = 0
  3. Transcendental equation: x=5(1−e−x)x = 5(1-e^{-x})
  4. Solution: x=4.965...x = 4.965...
  5. Therefore: λmax=hcxkBT\lambda_{max} = \frac{hc}{xk_BT}

Objects glow with color of their temperature!

21.7 Thermodynamic Temperature Scale​

Theorem 21.6 (Absolute Temperature): Carnot efficiency defines absolute scale.

Construction:

  1. Efficiency: η=1−Qc/Qh\eta = 1 - Q_c/Q_h
  2. For reversible: η=1−Tc/Th\eta = 1 - T_c/T_h
  3. Measure Q ratios → determine T ratios
  4. Fix one point (triple point of water = 273.16 K)
  5. Entire scale determined
  6. Independent of working substance!

Temperature has absolute meaning via entropy!

21.8 Temperature in Relativity​

Theorem 21.7 (Relativistic Transformations): Moving thermometer reads: T′=T/γT' = T/\gamma?

Debate continues:

  1. Planck-Einstein: T′=T/γT' = T/\gamma (colder)
  2. Ott-Arzelies: T′=γTT' = \gamma T (hotter)
  3. Landsberg: T invariant
  4. Resolution: Depends on definition!

Temperature transforms non-trivially!

21.9 Hawking Temperature​

Theorem 21.8 (Black Hole Temperature): TH=ℏc38πGMkBT_H = \frac{\hbar c^3}{8\pi GMk_B}

Remarkable facts:

  • Smaller holes are hotter!
  • Solar mass BH: T ~ 10^-7 K
  • Evaporation accelerates
  • Information paradox emerges

Gravity creates thermal radiation!

21.10 Temperature Limits​

Lower Bound: T ≥ 0 (Third Law)

  • Cannot reach in finite steps
  • Quantum ground state

Upper Bound: Planck temperature?

  • TP=ℏc5GkB2≈1032T_P = \sqrt{\frac{\hbar c^5}{Gk_B^2}} \approx 10^{32} K
  • Quantum gravity regime
  • May not be meaningful

Temperature has fundamental bounds!

21.11 Effective Temperatures​

Non-equilibrium Systems:

  • Kinetic temperature (velocities)
  • Configurational temperature (positions)
  • Spin temperature (magnetic systems)
  • Different modes can have different T!

Example: Laser cooling

  • Momentum space: μK
  • Internal states: room temperature

Temperature becomes mode-specific!

21.12 The Twenty-First Echo: The Universal Thermometer​

Temperature emerges as nature's universal measure of thermal agitation—from the cosmic microwave background at 2.7 K to quasar accretion disks at billions of degrees. It's not just a number but a fundamental property controlling how energy flows between systems.

The deep connection between temperature, entropy, and information reveals that T measures more than molecular motion—it quantifies the density of possible states, the availability of energy, the direction of time itself. In temperature we find a bridge between the microscopic and macroscopic, between quantum discreteness and classical continuity.

Temperature Studies​

  1. Derive the Saha equation for ionization equilibrium.

  2. Calculate Debye temperature for various crystals.

  3. Analyze temperature in accelerated frames.

The Next Rhythm​

Understanding temperature as collapse intensity, we now explore how this drives mechanical work—the directed extraction of energy from thermal chaos.

Next: Chapter 22: Work — The Harvest of Organized Collapse →

"Temperature is the universe's heartbeat—measuring how vigorously existence vibrates."