Modular Thermodynamic Framework

• Modular Thermodynamic Framework is a methodology for constructing and analyzing thermodynamic models using independently defined, symmetry-driven modules.
• It enables hierarchical refinement and rapid adaptation across diverse fields, from quantum gravity to materials modeling.
• The framework rigorously connects microscopic statistical indices to macroscopic observables, incorporating dualities and quantum corrections through modular forms.

The modular thermodynamic framework is a principle and methodology for constructing, analyzing, and applying thermodynamic models and theories using standardized, independently defined components—“modules”—which may encode statistical, physical, or mathematical structure. Modular construction facilitates compositional extensibility, transparent physical bookkeeping, hierarchical refinement, and rapid adaptation to diverse physical scenarios. Across quantum resource theories, high-energy theory, materials modeling, and information processing, the modular strategy clarifies the relationship between microscopic degrees of freedom, macroscopic thermodynamic observables, and the symmetry or invariance properties that govern their transformations.

1. Modular Partition Functions and Thermodynamic Potentials

A foundational example of modular thermodynamic construction arises in string-theoretic black hole entropy computations, where the microscopic partition function $Z(\tau)$ is expressed as a $q$-series of a modular or mock modular form: $Z(\tau) = \sum_{n \ge \alpha} d(n) q^{n - \alpha}$ with $q = e^{2\pi i \tau}$ and $\tau = x + iy$ as the complexified temperature variable. The trace formulation

$$Z(\tau) = \operatorname{Tr}_{\mathcal{H}_{\text{micro}}}(q^{L_0 - c/24})$$

aggregates BPS or weighted quantum states in a Hilbert space. Thermodynamic potentials are then modularly defined:

• Inverse temperature as $\beta = 2\pi \tau_2$,
• Free energy $F(\tau) = -\frac{1}{\beta}\ln Z(\tau)$,
• Entropy $S(\tau) = (1 - \beta \partial_\beta) \ln Z(\tau)$,
• In holomorphic language, $$S = (1 - \tau\partial_{\tau}) \ln Z(\tau) - (1 - \bar\tau\partial_{\bar\tau}) \ln \overline{Z(\tau)}$$.

This formalism enables direct identification and computation of macroscopic thermodynamic observables from the modular structure of the microscopic index, rendering the framework inherently modular and symmetry-driven (Murthy, 2023).

2. Modular Bootstrap and Quantum Corrections

Modularity under $SL(2,\mathbb{Z})$ transformations generates high- and low-temperature dualities. The $S$ transformation $\tau \to -1/\tau$ interchanges thermal regimes:

• For a modular form of weight $k$,

$Z(-1/\tau) = (-i\tau)^k Z(\tau)$

ensuring that Cardy-like growth in the degeneracy $d(N)$ is controlled by the most polar term at low temperature: $d(N) \sim \exp\left(2\pi\sqrt{\alpha N}\right)$ and entropy exhibits Bekenstein–Hawking scaling $S \sim 4\pi\sqrt{\alpha N}$ ($A/4G$ law).

Further quantum corrections—logarithmic, power law, nonperturbative—are systematically encapsulated by the Rademacher expansion for modular forms: $$d(N) = 2\pi\left(\frac{\pi}{2}\right)^{7/2} \sum_{c=1}^{\infty} c^{-9/2} K_c(N) I_{7/2}\left(\frac{\pi\sqrt N}{c}\right)$$ where $K_c(N)$ are Kloosterman sums and $I_\nu(z)$ are modified Bessel functions. Subleading terms correspond to orbifold contributions in the gravitational path integral (Murthy, 2023).

3. Mock Modular Refinements and Sectoral Decomposition

In more general setups (e.g., $N=4$ black holes in Type II on $K3 \times T^2$), the index $Z_{\text{full}}(\tau)$ splits into sectoral components $Z_{r-\text{BH}}(\tau)$ enumerating $r$-center black hole microstates. The single-center sector $Z_{1-\text{BH}}(\tau)$ is generically a mock modular form, admitting a nonholomorphic completion: $$\widehat Z_{1-\text{BH}}(\tau) = Z_{1-\text{BH}}(\tau) + (i\tau_2)^{-k'} \int_{-\bar\tau}^{i\infty} \frac{d\tau'}{(\tau'+\tau)^{2 - k'}} \overline{S(\tau')}$$ with shadow $S(\tau)$. Thermodynamics proceeds as for genuine modular forms, with corrections extracted via Rademacher–Zagier expansions for mock forms. This sectoral modularity refines the quantum state-counting and correction structure in gravitational thermodynamics (Murthy, 2023).

4. Summary of Modular Thermodynamic Recipe

The modular thermodynamic framework is succinctly characterized by the following procedural elements:

Step	Modular Construction	Thermodynamic Mapping
1. Microscopic Index	$Z(\tau)$ as $q$-series of modular/mock modular form of weight $k$	$f((a\tau+b)/(c\tau+d)) = (c\tau+d)^k f(\tau)$ or completed version
2. Potentials	$\beta = 2\pi\tau_2$, $F = - (1/\beta)\ln Z$, $S = (1 - \beta\partial_\beta)\ln Z$	Modular theory provides direct computation pathways
3. High/Low-T Duality	$$Z(\tau_2 \to 0^+) \sim \tau_2^k Z(\tau_2 \to \infty)$$, Cardy: $S \sim 4\pi\sqrt{\alpha N}$	Exploits modular symmetry of $SL(2,\mathbb{Z})$
4. Quantum Corrections	Rademacher expansion of $d(N)$—all quantum corrections included	Full account of logarithmic, orbifold, non-perturbative contributions
5. Mock Modular Sectors	Single-center index is mock modular; corrections via shadow integral	Completions yield refined quantum corrections for sectoral microstate counting

This formalism elevates modular symmetry to the principle that both determines and resums the gravitational path integral for BPS black holes in string theory, capturing all corrections from leading area law down to exponentially suppressed effects (Murthy, 2023).

5. Broader Impact and Generalization

Although originally developed in the context of string-theoretic black hole microstate counting, the modular thermodynamic framework is applicable to any quantum-statistical system whose microscopic index is modular or mock modular in form. Its modular architecture unifies entropy bounds, state-counting methods, and correction hierarchies via algebraic and analytic properties of modular forms. This allows for systematic exploration of high-energy statistical mechanics, quantum gravity, and related symmetry-constrained thermodynamic theories.

A plausible implication is that similar modular thermodynamic recipes may be constructed in statistical mechanics, condensed matter systems (via conformal blocks and modular Hamiltonians), and beyond, whenever the partition function possesses modular invariance or quasi-invariance under relevant symmetry groups.

6. Connections to Quantum Information and Resource Theories

The modular approach resonates structurally with quantum thermodynamic resource theories, where systems are described modularly by (operator, intensive variable) pairs and composite systems are assembled by module addition; free states and free operations are characterized by tensor-product and unitary invariance constructions (Halpern, 2014). Both frameworks exploit symmetry and compositional extensibility for generalized thermodynamic modeling, including quantum statistical state-counting, entropy monotonicity, and sectoral correction hierarchies. These connections reinforce modularity as a unifying theme transcending traditional domain boundaries in thermodynamics.