Thermodynamics Basis

The Degeneracy Problem

Heat capacity and entropy share the same SI dimension:

Quantity	SI Unit	SI Dimension	Physical Meaning
Heat capacity (C)	J/K	ML²T⁻²Θ⁻¹	Energy required to raise temperature
Entropy (S)	J/K	ML²T⁻²Θ⁻¹	Microstate counting / disorder

Same dimension. Different physics.

• Heat capacity: dQ/dT — a process-dependent response function
• Entropy: ∫dQ_rev/T — a state function measuring disorder

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Conceptual Difference

Heat Capacity

• Question answered: "How much energy to raise temperature by 1 K?"
• Type: Response function (can depend on process: C_p vs C_v)
• Formula: C = dQ/dT
• Intensive form: Specific heat c = C/m or molar heat C_m = C/n

Entropy

• Question answered: "How many microstates correspond to this macrostate?"
• Type: State function (path-independent)
• Formula: S = k_B ln Ω (statistical), dS = dQ_rev/T (thermodynamic)
• Intensive form: Specific entropy s = S/m or molar entropy S_m = S/n

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Extended Basis

from ucon.basis import Basis, BasisComponent, Vector
from ucon.core import Dimension, Unit
from fractions import Fraction

THERMO = Basis("Thermodynamics", [
    BasisComponent("mass", "M"),              # 0
    BasisComponent("length", "L"),            # 1
    BasisComponent("time", "T"),              # 2
    BasisComponent("temperature", "Θ"),       # 3
    BasisComponent("statistical", "Σ"),       # 4 — microstate/entropy character
])

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Dimensional Vectors

Quantity	Vector	Interpretation
Energy	M¹L²T⁻²Θ⁰Σ⁰	Mechanical/thermal energy
Heat capacity	M¹L²T⁻²Θ⁻¹Σ⁰	Thermal response
Entropy	M¹L²T⁻²Θ⁻¹Σ¹	Statistical/microstate measure
Boltzmann constant	M¹L²T⁻²Θ⁻¹Σ¹	Entropy per microstate (k_B)
Temperature	M⁰L⁰T⁰Θ¹Σ⁰	Thermal intensity

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Implementation

# Dimensions
energy = Dimension(
    vector=Vector(THERMO, (1, 2, -2, 0, 0)),
    name="energy"
)

heat_capacity = Dimension(
    vector=Vector(THERMO, (1, 2, -2, -1, 0)),
    name="heat_capacity"
)  # ML²T⁻²Θ⁻¹Σ⁰

entropy = Dimension(
    vector=Vector(THERMO, (1, 2, -2, -1, 1)),
    name="entropy"
)  # ML²T⁻²Θ⁻¹Σ¹

statistical_factor = Dimension(
    vector=Vector(THERMO, (0, 0, 0, 0, 1)),
    name="statistical_factor"
)  # Σ¹

# Units
joule_per_kelvin_heat = Unit(
    name="joule_per_kelvin_C",
    shorthand="J/K",
    dimension=heat_capacity
)

joule_per_kelvin_entropy = Unit(
    name="joule_per_kelvin_S",
    shorthand="J/K",
    dimension=entropy
)

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The Boltzmann Constant

k_B connects temperature to energy via entropy:

S = k_B ln Ω

In the extended basis, k_B has dimension Σ¹ — it's the bridge between statistical mechanics (microstates) and thermodynamics (temperature).

When we set k_B = 1 (natural units for statistical mechanics), we're collapsing the statistical dimension:

# Forward: Σ¹ → dimensionless via k_B
# Inverse: dimensionless → Σ¹ by dividing by k_B

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Dimensional Algebra

Heat absorbed from capacity:

Q = C × ΔT
M¹L²T⁻²Θ⁰Σ⁰ = (M¹L²T⁻²Θ⁻¹Σ⁰) × (Θ¹)  ✓

Entropy from heat (reversible):

ΔS = Q_rev / T
M¹L²T⁻²Θ⁻¹Σ¹ = (M¹L²T⁻²Θ⁰Σ⁰) / (Θ¹) × (Σ¹)  ← needs statistical factor

Wait — this reveals something important. The formula dS = dQ/T doesn't automatically give you entropy's statistical character. The Σ¹ comes from the definition of entropy as a statistical quantity, not from the algebra.

This is analogous to torque vs energy: the angle factor comes from the physics (τ = dW/dθ), not from pure dimensional analysis.

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Related Degeneracies

Specific Heat vs Specific Entropy

Quantity	SI Dimension	Extended
Specific heat	L²T⁻²Θ⁻¹	L²T⁻²Θ⁻¹Σ⁰
Specific entropy	L²T⁻²Θ⁻¹	L²T⁻²Θ⁻¹Σ¹

Gas Constant vs Entropy

The gas constant R = 8.314 J/(mol·K) has the same dimension as molar entropy:

Quantity	SI Dimension	Extended
Gas constant R	ML²T⁻²Θ⁻¹N⁻¹	ML²T⁻²Θ⁻¹N⁻¹Σ⁰
Molar entropy	ML²T⁻²Θ⁻¹N⁻¹	ML²T⁻²Θ⁻¹N⁻¹Σ¹

R is a proportionality constant; molar entropy is a state function.

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Safety Guarantees

# These are now caught:

heat_capacity_water + entropy_water
# raises: incompatible dimensions (Σ⁰ vs Σ¹)

# Must be explicit about what you're computing:
total_entropy = entropy_system + entropy_surroundings  # ✓ (both Σ¹)
total_capacity = C_water + C_container                 # ✓ (both Σ⁰)

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Free Energy Functions

The statistical dimension clarifies thermodynamic potentials:

Potential	Definition	Entropy Term
Internal energy U	—	No explicit S
Helmholtz F	U - TS	Contains Σ¹ via S
Gibbs G	H - TS	Contains Σ¹ via S
Enthalpy H	U + PV	No explicit S

When computing G = H - TS: - H has dimension M¹L²T⁻²Σ⁰ (energy) - T has dimension Θ¹ - S has dimension M¹L²T⁻²Θ⁻¹Σ¹

TS: Θ¹ × M¹L²T⁻²Θ⁻¹Σ¹ = M¹L²T⁻²Σ¹

Problem: H (Σ⁰) and TS (Σ¹) have different dimensions!

Resolution: Gibbs energy inherits the statistical character:

G = H - TS  →  G has dimension M¹L²T⁻²Σ¹

Or we recognize that H implicitly carries Σ⁰ and the subtraction is valid only when we track that TS "converts" to pure energy via the temperature multiplication.

This is a deep point — the extended basis reveals that free energies are intrinsically statistical quantities, not pure mechanical energies.

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Practical Value

Textbook sanity check:

"The entropy of the system increased by 50 J/K." "The heat capacity is 50 J/K."

In SI, these are dimensionally indistinguishable. In the extended basis: - First statement: Σ¹ quantity changed - Second statement: Σ⁰ property measured

Simulation validation:

In molecular dynamics, you compute entropy from microstate sampling (statistical) and heat capacity from energy fluctuations (thermodynamic response). Conflating them is a methodological error the extended basis can catch.

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Summary

Degeneracy	Hidden Dimension	Resolution
Heat capacity vs Entropy	Statistical (Σ)	Entropy is ML²T⁻²Θ⁻¹Σ¹
Gas constant vs Molar entropy	Statistical (Σ)	Molar entropy carries Σ¹
Specific heat vs Specific entropy	Statistical (Σ)	Same pattern, intensive

The statistical dimension captures whether a quantity describes: - Σ⁰: Bulk thermal response (heat capacity, gas constant) - Σ¹: Microstate counting (entropy, Boltzmann constant)