Effective Thermodynamic Systems

• Effective thermodynamic systems are mathematical or physical constructions that capture macroscopic behavior by linking well-defined state functions with energy and entropy balances.
• They use state-space structures, port dynamics, and geometric frameworks like contact manifolds to enforce the laws of thermodynamics in both equilibrium and nonequilibrium settings.
• Applications range from engineered energy conversion and analog computation to insights in black hole and cosmological horizon thermodynamics, illustrating their versatility.

An effective thermodynamic system is a mathematical or physical construction whose state variables, flows, and governing relations are designed to capture the macroscopic thermodynamic behavior of an underlying complex, possibly non-equilibrium, stochastic, or composite system. Effective thermodynamic systems are characterized by the existence of well-defined macroscopic state functions (such as internal energy, entropy, and free energy), balance laws (including the first and second laws), and operationally meaningful conjugate variables, yet may arise in settings—quantum open systems, strongly coupled reservoirs, multicomponent media, nonlinear oscillators, nonclassical fluids—where the standard derivation from microcanonical equilibrium is either inapplicable or only approximate. The development, mathematical structure, and applications of such systems are central to modern nonequilibrium physics, stochastic process theory, applied mathematics, and the design of engineered energy-conversion or information-processing devices.

1. Foundational Principles and State-Space Structure

The effective thermodynamic system is formalized by specifying a finite set of extensive state variables $x$ (e.g., internal energy $U$, entropy $S$, volume $V$), inputs/outputs (ports) associated to flows of energy, work, and matter, and a geometric framework—usually based on contact manifolds or port-Hamiltonian systems—that encodes the constitutive relations and permissible dynamics. In the lumped-parameter formulation, the dynamics are captured by

$\dot{x} = f(x) + G(x)u,$

where $f(x)$ describes autonomous internal dissipation or reaction terms and $G(x)u$ injects heat, work, or mass flow through the system’s ports. The conjugate "outputs" $y$ are the corresponding intensive variables, such as temperature $1/T$, generalized chemical potential, or pressure–volume derivatives. The first law (energy balance) and second law (entropy production) are encoded as dissipativity inequalities:

$\dot{U} = q - w,\qquad \dot{S} \geq \frac{q}{T},$

with $q$ the input heat flux and $w$ the mechanical or chemical power, and the storage functions $U(x)$, $S(x)$ acting as Lyapunov and monotonicity functionals (Schaft, 2020).

The state manifold is often embedded into a contact manifold with local coordinates $(U, S, V; T, P)$ and canonical contact form $\alpha = dU - T\,dS + P\,dV$, constrained to a Legendre submanifold $\mathcal{L}$ defined by the constitutive relations (e.g., $U=U(S, V)$). This structure ensures the geometric unification of different energy and entropy representations and allows modular interconnection of subsystems at the level of ports, subject to the requirement of global entropy non-decrease and energy balance.

2. Effective Thermodynamic Laws in Composite and Open Systems

Effective thermodynamic systems can be constructed for hybrid or nonequilibrium setups provided appropriate invariances and symmetries are respected. In open quantum Markovian systems, the effective thermodynamic description emerges by enforcing strict energy conservation between system and reservoir, leading to generators $L$ that commute with the system Hamiltonian and possess the full GKLS (Lindblad) form. This ensures thermodynamic compatibility at the dynamical map level, guarantees steady states of Gibbs (or NESS) form, imposes detailed-balance on the couplings, and prohibits unphysical phenomena such as exceptional-point degeneracies (Dann et al., 2020).

For multipartite or strongly-coupled systems interacting with multiple reservoirs, a global master equation with jump operators diagonal in the complete system Hamiltonian is necessary to ensure the compatibility of the effective thermodynamic description with the conservation laws, entropy production, and non-negativity constraints of the second law (Dann et al., 2020). The effective entropy production decomposes into adiabatic (housekeeping) and nonadiabatic (relaxational) parts, directly linked to the monotonic decay of Kullback–Leibler divergence from nonequilibrium states to the reference steady-state, with sharp Lyapunov functionals governing the relaxation (Taye, 10 Sep 2025).

3. Parameter Tuning, Nonlinearity, and Macroscopic Control

Effective thermodynamic systems can exhibit strong tunability via internal parameters that modulate the spectrum, coupling, or response. In the context of quantum systems with position-dependent effective mass, such as two-dimensional nonlinear oscillators, explicit closed-form thermodynamic functions (partition function, energy, specific heat, free energy, entropy) reveal that tuning a nonlinearity parameter $k$ can dramatically alter the macroscopic heat capacity and approach to the classical Dulong–Petit law at high temperature. Negative $k$ ensures thermodynamic stability and well-defined saturation, with the crossover from quantum to classical thermodynamics controlled by $k$ (Bokpe et al., 16 Sep 2025). This mechanism is directly applicable to the design of quantum thermal machines and optoelectronic devices where thermal response and memory can be engineered.

Effective thermodynamic systems built around universal coupling rules—matching the ratio of energy and matter fluxes to a reservoir-defined constant $\Theta$—enable fine optimization of efficiency and entropy generation in far-from-equilibrium steady states. Adjusting system Hamiltonian, tunnel couplings, or interaction strengths can steer the system towards regimes of tight coupling, maximal reversible efficiency, or optimal performance at finite dissipation (Su et al., 2019). Notably, maximum efficiency is generally achieved at finite entropy production in non-equilibrium setups.

4. Effective Thermodynamics in Nonclassical and Multiphase Media

The effective thermodynamic system concept extends rigorously to systems with multiple coexisting phases or nontrivial interface effects. In partially saturated soils, a thermodynamically consistent treatment defines effective stress through a variational principle around a ‘suctionless’ reference, followed by the introduction of constraints encoding intrinsic capillary (suction) effects. The effective stress tensor,

$$\sigma'_{ij} = \sigma_{ij} - P_T\,\delta_{ij},\quad P_T = u_A - \chi(u_A - u_W),$$

is derived as a thermodynamic emergent, where $u_A$, $u_W$ are measured air and water pressures, and $\chi$—Bishop's parameter—encapsulates detailed information about the soil–water retention curve and the phase densities. Multiple limiting forms are recovered as special cases, and the theoretical framework guarantees explicit thermodynamic consistency between state variables, energy minimization, and observed macroscopic responses (Jiang et al., 2022).

5. Dynamical Effective Thermodynamic Systems: Matrix Computation and Information Processing

Effective thermodynamic systems can be implemented as physical or analog computational devices, exploiting fluctuations, dissipation, and stochastic evolution to perform complex operations with asymptotic computational advantages. Consider the realization of matrix exponentiation via a thermodynamic algorithm: encoding a matrix $A$ in an overdamped Langevin system

$dx = -A\,x\,dt + \mathcal{N}(0,\,2\beta^{-1}A\,dt)$

yields, by time-averaged two-time correlations, an empirical estimator for $e^{-A\tau}$. Hardware instantiations involve coupled resistor–capacitor networks where thermal noise, via the fluctuation–dissipation theorem, injects parallel fluctuations across $d$ units. The required estimator emerges as a continuous measurement of equilibrium fluctuations, and the protocol exhibits asymptotic $O(d^2)$ scaling, exploiting "thermodynamic parallelism"—the physical integration of all $d^2$ matrix elements in a single collective correlation measurement (Duffield et al., 2023). This noise-driven parallelization is essential for practical speedup and energy efficiency in thermodynamic computation.

The same principles extend to broader classes of AI-accelerator hardware (e.g., thermodynamic stochastic processing units, SPUs), where continuous-variable Langevin dynamics are mapped to inference or linear algebra tasks, enabling scalable performance for sampling, matrix inversion, and uncertainty quantification in probabilistic models (Melanson et al., 2023).

6. Effective Thermodynamics in Black Hole and Cosmological Horizon Physics

In gravitational physics, effective thermodynamic systems are constructed to characterize the macroscopic properties of two-horizon regions (e.g., de Sitter hairy black holes) by enforcing the universal thermodynamic first law,

$$dM = T_\mathrm{eff}\,dS - P_\mathrm{eff}\,dV + \phi_\mathrm{eff}\,dQ + k_\mathrm{eff}\,dH.$$

Here, global thermodynamic quantities are systematically derived as effective macroscopic observables (temperature, pressure, electric potential, etc.) of the composite system. The resulting heat capacities display Schottky-type anomalies—single sharp peaks as functions of effective temperature or geometric parameters—indicating behavior akin to finite quantum two-level systems. Matching the effective heat capacity profile to that of a two-level model provides estimates for the effective number of microstates localized between horizons, connecting horizon thermodynamics to underlying quantum microphysics and offering guidance for further investigation of horizon thermodynamics within quantum gravity frameworks (Bao et al., 11 Nov 2025).

7. Variational and Path-Integral Approaches: Large Deviations and Path Entropy

From a probabilistic and field-theoretic perspective, the effective thermodynamic system is intimately connected to large-deviation rate functions, maximum-caliber (path entropy) principles, and the Freidlin–Wentzell Hamiltonian structure. For stochastic Markov processes or reaction-diffusion systems, effective actions constructed by the Doi–Peliti formalism or Jaynes' Maximum Caliber derivations encode the duration-weighted path entropy and emerge as the controlling functionals of the macroscopic thermodynamic description in and far from equilibrium:

$$S_\mathrm{path} = -\sum_j p_j \log p_j, \quad \text{with mean constraint} \ \langle M^{xy} \rangle.$$

The path-integral formalism recovers all equilibrium and nonequilibrium potentials, fluctuation relations, and state-ensemble or path-ensemble thermodynamic functions (Smith, 2011).

8. Summary Table: Prototypical Features and Realizations

Physical Setting	Effective Thermodynamic Structure	Key Parameters/Quantities
Lumped-parameter system	$(U, S, V)$, port-Hamiltonian, contact	Heat/work ports, storage functions
Open quantum Markov system	Global master equation, GKLS structure	Jump ops, Bohr shells, detailed balance
Composite multicomponent fluid	Variational free-energy, effective stress	Densities, capillarity, $\chi$
Quantum oscillator w/ eff. mass	Canonical $Z(T, k)$, $U$, $C_v$	Nonlinearity $k$, spectrum
Analog matrix computation	OU process two-time correlation	$A$, $\beta$, correlation time
Black-hole spacetime	Global first law, effective $T_\mathrm{eff}$	$S$, $V$, Schottky anomaly

Effective thermodynamic systems offer a universal, modular, and rigorous framework for understanding and engineering macroscopic behavior in a wide spectrum of physical, chemical, biological, informational, and gravitational systems. Their construction depends crucially on symmetry constraints, variational principles, and the emergent coarse-grained variables, bridging the microscopic dynamical particulars with operationally testable macroscopic laws.