Optical Thermodynamic Framework

Updated 21 November 2025

Optical thermodynamics is a framework that maps thermodynamic principles onto optical systems by conserving quantities like energy, power, and mode number.
It defines conjugate intensive variables such as optical temperature, chemical potential, and pressure, establishing state equations analogous to classical thermodynamics.
The framework enables quantitative analysis of entropy production, steady-state distributions, and radiation-pressure effects in complex photonic devices.

An optical thermodynamic framework is a theoretical construct that rigorously maps the macroscopic, statistical-mechanical principles of thermodynamics onto optical systems—including driven single-mode cavities, highly multimoded nonlinear waveguides, photonic networks, and quantum optical devices—by identifying a set of physically conserved invariants (e.g., energy, optical power, mode number), defining their conjugate intensive variables (e.g., optical temperature, chemical potential, pressure), and establishing state equations, fluctuation relations, and nonequilibrium extensions that parallel established thermodynamic laws. This framework enables a quantitative and universal description of energetics, entropy production, steady-state distributions, and collective radiation-pressure effects in photonic devices with arbitrary complexity.

1. Fundamental Thermodynamic Structure in Optical Systems

The construction of optical thermodynamics begins by identifying conserved extensive variables in the optical system under consideration. For generic multimode, weakly nonlinear photonic systems—such as fibers, waveguide lattices, or cavities—these are:

Internal energy $U$ , typically related to the weighted occupation of photonic eigenmodes (e.g., $U = -\sum_k \beta_k n_k$ where $\beta_k$ is mode propagation constant).
Total optical power $\mathcal{P}$ or photon number $N$ , given as $N = \sum_k n_k$ .
Number of modes $M$ , corresponding to system size or geometry.

The equilibrium modal occupancies are governed by a Rayleigh–Jeans (RJ) distribution:

$n_k = \frac{T}{\beta_k - \mu}$

where $T$ is the optical temperature and $\mu$ is the optical chemical potential, identified through maximization of Boltzmann entropy under constraints ( $U$ , $N$ ). The macroscopic thermodynamic entropy for such a photon gas is:

$S = \sum_k \ln n_k$

This structure yields a fundamental equation of state:

$U - \mu N = M T$

which directly parallels the standard thermodynamic relation $U - \mu N = pV$ but with the number of modes as an extensive variable (Wu et al., 2019, Ross et al., 4 Dec 2024).

2. Intensive Variables and Equations of State

By analogy to classical thermodynamics, all thermodynamic conjugate variables are defined as: $\frac{1}{T} = \frac{\partial S}{\partial U}\bigg|_{N, M}, \qquad -\frac{\mu}{T} = \frac{\partial S}{\partial N}\bigg|_{U, M}, \qquad \frac{\hat{p}}{T} = \frac{\partial S}{\partial M}\bigg|_{U, N}$ where $\hat{p}$ is the optical thermodynamic pressure, the variable conjugate to $M$ (Ross et al., 4 Dec 2024, Ren et al., 10 Apr 2024). The full Euler relation in this context reads:

$S\,T = (U - \mu N) + \hat{p} M$

In photonic networks with more parameters (e.g., coupling constants, strain), further generalized forces arise, leading to analogues of stress and strain terms in 2D waveguide lattices (Efremidis et al., 2021).

3. Optical Thermodynamic Pressure and Its Physical Significance

Optical thermodynamic pressure is rigorously defined as the derivative of the free energy with respect to $M$ at constant $T$ and $\mu$ ; microscopically, it is associated with the mechanical force exerted by the photon gas on the confining boundaries (e.g., waveguide walls):

$\hat{p} = Q p_{EM} + T \ln\left[\frac{-T}{\beta_M + \mu}\right] - T$

where $p_{EM}$ is the classical electromagnetic (radiation) pressure, and the second term is a purely entropic correction (Ren et al., 10 Apr 2024). Notably, for systems supporting multiple field species (e.g., polarizations), the optical thermodynamic pressures obey an "optical Dalton’s law": the total pressure is the sum of partial pressures from each species, and each term can be linked directly to corresponding electromagnetic stress-tensor components (Ren et al., 5 Apr 2024).

This formalism provides analytic, closed-form expressions for radiation pressure in complex, heavily multimoded nonlinear structures without recourse to detailed Maxwell stress-tensor calculations—thereby enabling efficient modeling of optomechanics and photonic phase transitions in regimes where standard field-based calculations are intractable.

4. Stochastic and Non-Equilibrium Optical Thermodynamics

In driven, open optical systems (e.g., a linear cavity with input drive and bath), a stochastic thermodynamic description is established by treating the cavity field as an overdamped Brownian variable subjected to input noise and dissipation. The intra-cavity field quadratures obey independent Langevin equations, with the steady-state distribution given by a Boltzmann law at an effective temperature set by the noise variance:

$P(\alpha_{R,I}) = Z_{R,I}^{-1} \exp[-\beta U_{R,I}(\alpha_{R,I})]$

The first and second laws are maintained both at the level of individual stochastic trajectories and as ensemble averages: $dU_R = dW - dQ \quad \Rightarrow \quad \langle dU_R \rangle = \langle dW \rangle - \langle dQ \rangle,\qquad \langle W \rangle \geq \Delta F$ This formalism supports universal fluctuation relations, such as the Jarzynski equality and Crooks fluctuation theorem, with additional insight into finite-time corrections and entropy production during nonadiabatic protocols (Ramesh et al., 26 Jul 2024). These stochastic relations provide rigorous tools for diagnosing entropy generation, free energy changes, and work fluctuations in nano-optical devices.

5. Applications: Transport, Coherent Control, and Quantum Effects

The framework generalizes to diverse settings:

Transport in Photonic Networks: The identification of optical temperature and chemical potential allows for analogues of thermal and electrical transport in photonic circuits, including ballistic and diffusive scaling, a photonic Wiedemann–Franz law, and the design of all-optical cooling and energy-harvesting protocols (Kurnosov et al., 2023).
Quantum Coherent Thermodynamics: In low-power or quantum regimes, the statistical machinery persists, with modal populations described by weighted Bose–Einstein (wBE) distributions, unifying classical and quantum optics under a single thermodynamic paradigm (Zitelli, 27 Sep 2024). The approach extends to the design and analysis of quantum engines, thermal machines, and the quantum thermodynamics of driven optical cavities (Schrauwen et al., 13 May 2025, 2001.08033, Passos et al., 2018).
Nonlinear Dynamical Phenomena: The theory predicts and experimentally confirms phenomena such as all-optical Joule–Thomson cooling, abrupt condensation of photon gases (temperature quenching to $T \to 0$ and ground-state funneling), and entropy-driven beam self-cleaning and modal purification (Kirsch et al., 15 Nov 2025, Ferraro et al., 2022, Wu et al., 2020, Selim et al., 2022).

6. Experimental Validation and Practical Implications

The predictions of the optical thermodynamic framework have been quantitatively validated in nonlinear multimode fibers, photonic lattices, and waveguide arrays. Key achievements include:

Calorimetric measurements of photon gases demonstrating thermalization and heat flow consistent with the second law (Ferraro et al., 2022).
Direct observation of mode occupations thermalizing to RJ or wBE distributions under random mode coupling or nonlinear mixing (Zitelli, 27 Sep 2024, Kirsch et al., 15 Nov 2025).
Analytic predictions of near- and far-field intensity patterns, validated to high accuracy against fully nonlinear simulations (Selim et al., 2022).

The framework enables efficient design strategies for integrated photonic devices, optomechanical sensors, and quantum information platforms, by embedding thermodynamically constrained modal distributions and coupling processes directly into engineering protocols (Sadeghi, 10 Nov 2025, Ren et al., 10 Apr 2024).

7. Extensions: Optical Thermodynamic Geometry and Quantum Metamaterials

Recent advances generalize optical thermodynamic theory through geometric and quantum extensions. For instance, in quantum thermodynamic transformation optics (QTTO), the geometry of the optical environment manipulates not only field distributions but also the local densities of energy and entropy, governed by scaling with the Jacobian determinant of the spatial mapping. This dictates quantitative rules for the interplay between geometry, temperature, and quantum pressure, and enables the continuous interpolation between quantum and classical Casimir forces in metamaterials—establishing design principles for managing field fluctuations and entropy in tailored photonic environments (Sadeghi, 10 Nov 2025).

This body of theory thus establishes optical thermodynamics as a unified, rigorous platform for analyzing and optimizing the energetics, statistical fluctuations, and information-processing capability of complex photonic systems across scales and regimes (Wu et al., 2019, Ferraro et al., 2022, Ramesh et al., 26 Jul 2024, Ren et al., 10 Apr 2024, Kurnosov et al., 2023, Schrauwen et al., 13 May 2025, Sadeghi, 10 Nov 2025, Kirsch et al., 15 Nov 2025, Ross et al., 4 Dec 2024).