pith. machine review for the scientific record. sign in
module module high

IndisputableMonolith.Thermodynamics.HeatCapacity

show as:
view Lean formalization →

This module implements the classical equipartition theorem in Recognition Science by defining energy per quadratic Hamiltonian term and computing heat capacities for monatomic and diatomic gases from the 8-tick cycle. Researchers in discrete foundations of statistical mechanics cite it to link the fundamental clock to macroscopic specific heats. The structure enumerates modes via the eight-tick phases and scales energies by the RS time quantum.

claimThe classical equipartition theorem states that each quadratic term in the Hamiltonian contributes average energy $k_B T/2$, so the internal energy is $U = (f/2) k_B T$ and the heat capacity at constant volume is $C_V = (f/2) k_B$ per particle, where $f$ is the number of modes fixed by the 8-tick structure.

background

The module resides in the Thermodynamics domain and imports the discrete 8-tick cycle (phases at 0, π/4, π/2, ..., 7π/4) from Foundation.EightTick, the base time quantum τ₀ = 1 tick from Constants, and empirical calibration data from ExternalAnchors. It applies the equipartition rule quoted in the module doc-comment: each quadratic term contributes kT/2, with examples for kinetic and harmonic terms. The 8-tick structure determines the available translational and rotational modes for ideal gases.

proof idea

This is a definition module with supporting lemmas. It defines classicalEnergy as the sum of (1/2)kT contributions per mode, then enumerates monatomicModes = 3 and diatomicModesRoomTemp = 5 directly from the eight-tick phases. Specific values such as monatomic_cv_value and diatomicCvRoom follow by substitution and scaling with the RS time quantum.

why it matters in Recognition Science

The module supplies the classical baseline for heat capacities that connects the T7 eight-tick octave to thermodynamic observables. It feeds into broader RS thermodynamic derivations and permits direct comparison with experiment via ExternalAnchors. The doc-comment anchors the development to the standard equipartition theorem for calibration against measured specific heats.

scope and limits

depends on (3)

Lean names referenced from this declaration's body.

declarations in this module (21)