Shunyaya Symbolic Mathematics — Symbolic Thermodynamics (2.29)

Abstract
Thermodynamics studies energy, entropy, and equilibrium. Shunyaya Symbolic Mathematics extends these concepts by embedding an alignment channel a, so stability drift is tracked alongside classical energy flows. Under collapse phi(m,a) = m, all statements reduce to the classical laws.

Symbolic entropy (macro/state variable)

For a symbolic state X = (m, a), define the symbolic entropy as the pair

S_s(X) = ( S(m) , a_S )     where  S(m) = k_B * log( Omega(m) )  and  a_S := a
  • Magnitude channel reproduces the classical Boltzmann entropy.
  • Alignment channel carries the stability of microscopic organization.
    Collapse: if a = +1, then S_s → ( k_B * log Omega , +1 ) and the alignment channel is inert.

Worked example.
If Omega(100) = 10^6, then S(100) = 6 * k_B * log(10).
S_s( (100, 0.7) ) = ( 6 * k_B * log(10) , 0.7 ) — same disorder measure, flagged as partially unstable.

Symbolic temperature (two sensitivities)

Classical inverse temperature (alignment held fixed):

(1/T) = (partial S / partial U)|_a

Use alignment rapidity for stability:

u_E = atanh( clamp(a_E, -1+eps, +1-eps) ),   eps = 1e-6

Define an alignment inverse-temperature (sensitivity of entropy to alignment drift at fixed energy):

(1/T_a) = (partial S / partial u_E)|_U

Interpretation

  • 1/T measures how entropy changes with energy (classical).
  • 1/T_a measures how entropy changes with stability drift; even at fixed energy, moving away from centre (changing u_E) can raise disorder.
    Reporting pair (recommended): Temperature summary = ( T , T_a ).

Symbolic first law (pair form, M2 default)

Classical:

dU = delta Q - delta W

Symbolic (componentwise on the pair with ominus for subtraction):

( dU, a_U ) = ( delta Q, a_Q )  ominus  ( delta W, a_W )
  • Internal energy, heat, and work each carry an alignment tag.
  • Magnitude channel is exactly classical under collapse.
  • Alignment tags propagate with the same bookkeeping used for core operations (Section 2.4, M2 default).

Symbolic second law (production in two channels)

Let S_s = (S, a_S) and define u_S = atanh(a_S). Decompose the differential of S_s into exchange and production:

Magnitude (classical Clausius):      dS >= delta Q / T
Alignment (nonnegative production):  du_S = delta u_S,exch + sigma_a    with  sigma_a >= 0
  • delta u_S,exch models alignment transfer with a reservoir (e.g., contact with a stabilizing bath).
  • sigma_a captures irreversible drift (fragility generation).
    Collapse: if a_S ≡ +1 (constant u_S), then sigma_a = 0 and we recover the classical inequality.

Symbolic free energies (M2 multiplication)

Helmholtz free energy (pair):

F_s = U_s  ominus  ( T_s  otimes  S_s ),   where  U_s = (U, a_U),  T_s = (T, a_T),  S_s = (S, a_S)
  • Magnitude channel: F = U - T * S (classical).
  • Alignment channel (M2):
a_F = tanh( atanh(a_U) - [ atanh(a_T) + atanh(a_S) ] )

This keeps |a_F| <= 1 without ad hoc clipping.

Gibbs free energy (pair):

G_s = H_s  ominus  ( T_s  otimes  S_s ),    with   H_s = U_s  oplus  ( P_s  otimes  V_s )

Worked mini-example (illustrative numbers, with clamp).
U_s = (100, 0.99), S_s = (6*k_B, 0.7), T_s = (300, 0.9).
Magnitude: F = 100 - 300 * (6*k_B) (classical).
Alignment:

atanh(0.99) ≈ 2.647
atanh(0.9)  ≈ 1.472
atanh(0.7)  ≈ 0.867
a_F = tanh( 2.647 - (1.472 + 0.867) ) ≈ tanh(0.307) ≈ 0.298

Interpretation: usable free energy exists, but its stability is diminished by misalignment in T and S relative to U.

Symbolic equilibrium (dual stationarity)

At equilibrium (closed, fixed V, N), require both:

  1. Classical condition: (partial S / partial U)|_constraints = 1/T (entropy maximized at fixed U).
  2. Alignment condition: (partial S / partial u_E)|_constraints = 0 (no net drift pressure), equivalently T_a -> infinity or u_E stationary.

For two systems A and B in weak contact:

  • Thermal equilibrium: T_A = T_B.
  • Alignment equilibrium: T_{a,A} = T_{a,B} (no net alignment flow).
    Collapse recovers the single classical criterion.

Process accounting (recommended manifest fields)

  • Heat/work entries: (delta Q, a_Q), (delta W, a_W) with declared provenance of a_*.
  • Temperature pair: (T, T_a) and how T_a was estimated (finite-difference on u_E or model-based).
  • Free-energy choice: M2 (default).
  • Edge handling: use u = atanh(a) internally; report a = tanh(u).

Worked mini-example (two-cell relaxation)

Two identical cells separated by a diathermal wall:

Cell A: U_A = 100, a_{U,A} = +0.9   (u_{U,A} ≈ 1.472)
Cell B: U_B = 100, a_{U,B} = +0.3   (u_{U,B} ≈ 0.309)

Assume equal classical temperature initially (T_A = T_B) but alignment contact allows u_U to exchange with a small relaxation rate r:

du_{U,A}/dt = -r * (u_{U,A} - u_{U,B})
du_{U,B}/dt = +r * (u_{U,A} - u_{U,B})

Solution: both u values exponentially converge to (u_{U,A} + u_{U,B})/2 ≈ 0.891, hence
a_U ≈ tanh(0.891) ≈ 0.712.

  • Classical energies unchanged.
  • Alignment equalizes, reducing fragility in B while slightly reducing A’s stability — a drift-neutral equilibrium.

Takeaway

Symbolic Thermodynamics augments classical thermodynamics with a centre-aware axis:

  • Entropy is reported as (S, a_S) — disorder in size plus stability metadata.
  • Temperature splits into (T, T_a), capturing sensitivity to energy and to alignment drift.
  • First/second laws hold in the magnitude channel and gain alignment bookkeeping with nonnegative drift production sigma_a.
  • Free energies use M2 to keep alignment bounded, and equilibrium requires both thermal and alignment stationarity.
    Under collapse (a ≡ +1), every statement reduces exactly to the classical laws, while the symbolic layer exposes hidden instabilities that classical formulations miss.

Navigation
Previous → Symbolic Relativity (2.28)
Next → General Extensions (2.30)

Disclaimer
Observation only. Results reproduce mathematically; domain claims require independent peer review. Defaults: mult_mode = M2, clamp_eps = 1e-6, |a| < 1 with rapidity u = atanh(a). All formulas are presented in plain text. Collapse uses phi(m,a) = m.