Cooling Efficacy Across Scales

• Cooling efficacy is a quantitative measure of a system’s ability to remove heat or lower entropy using application-specific metrics such as heat transfer coefficient and final phonon population.
• Different regimes apply unique indicators—from convective heat transfer in electronics to phonon occupancy in quantum cooling—to assess and optimize cooling performance.
• Effective cooling design balances material properties, system constraints, and environmental conditions to achieve targeted thermal management and operational safety.

Cooling efficacy is the quantitatively assessed effectiveness with which a physical process, device, or protocol removes heat, suppresses motional or mechanical excitation, lowers entropy-like measures, or drives a system toward a colder target state under specified constraints. In the cited literature, the term does not denote a single universal observable. Instead, it is instantiated through application-dependent quantities such as heat transfer coefficient, thermal resistance, cooling power, sub-ambient temperature reduction, coefficient of performance, final phonon occupation, asymptotic qubit polarization, and the astrophysical ratio of star formation rate to X-ray-inferred cooling rate (Nguyen et al., 25 Jun 2026, Mukhopadhyay et al., 2024, Tomes et al., 2011, Lin et al., 2024, Haghighi et al., 2018).

1. Definitions and performance measures

Cooling efficacy is defined operationally by the observable that is most closely tied to the task being optimized. In convective electronics cooling, the central quantity is often the heat transfer coefficient, $h = \frac{q}{A\Delta T}$, or its dimensionless counterpart, the Nusselt number $Nu$ (Nguyen et al., 25 Jun 2026). In vapor-chamber and package-level studies, efficacy is instead reduced to an overall resistance, $$R_{\text{tot}} = \frac{T_{\text{evap}}-T_{\text{cond}}}{Q}$$, so that lower values indicate more effective heat spreading and rejection (Mukhopadhyay et al., 2024). In radiative cooling, a weather-normalized figure of merit is used, $R_C = E_{\text{sky}} - r(1-R_{\text{solar}})$, with $R_C>0$ indicating theoretical sub-ambient capability (Li et al., 2020).

Context	Metric	Expression
Jet impingement cooling	Heat transfer coefficient	$h=\frac{q}{A\Delta T}$
Wick-free vapor chamber	Total thermal resistance	$$R_{\text{tot}}=\frac{T_{\text{evap}}-T_{\text{cond}}}{Q}$$
Electroaerodynamic jet arrays	Electrical cooling efficiency	$COP_{eff}=\frac{Q_{fc}}{P_{elec}}$
Caloric micro-cooling	Coefficient of performance	$\text{COP}=\frac{q_{\text{gen}}}{W_{\text{in}}}$
Photothermal optomechanics	Photothermal cooling efficiency	$\eta_{ph}=\frac{d\Gamma_{eff}}{dP}$
Galaxy-cluster cooling flows	Cooling efficiency	$Nu$0

In quantum and atomic settings, efficacy is frequently measured by the target-state trajectory rather than by a heat-transfer coefficient. Brillouin cooling uses the final phonon population $Nu$1 as the principal indicator (Tomes et al., 2011). Optimal-control laser cooling evaluates the final average phonon occupation $Nu$2 at a prescribed terminal time (Li et al., 2021). Algorithmic cooling unifies long-time target polarization through

$Nu$3

and supplements this cooling-limit description with the coefficient of performance $Nu$4 and the Landauer Ratio $Nu$5 (Lin et al., 2024).

A further class of studies evaluates efficacy by whether a system remains within an application-defined operating band. For personal cooling garments, the relevant criterion is maintaining back-skin temperature in the thermal comfort zone $Nu$6 (Feng et al., 2024). For power electronics, it is the ability to keep hotspot temperatures below safe device limits such as $Nu$7 for MOSFETs or within the “normal operating range of ICs” (Mukhopadhyay et al., 2024, Mandalapu et al., 2019).

2. Governing constraints and trade-offs

Across disparate cooling platforms, efficacy is bounded by coupled trade-offs rather than by a single material or actuator parameter. In feedback cooling of ultracold atomic gases, the central limitation is the trade-off between spatial resolution, signal-to-noise ratio, and measurement-induced heating. Optical imaging is bounded by

$Nu$8

while sufficient visibility of density fluctuations requires a minimum measurement strength and therefore a minimum amount of destructiveness. Even nominally non-destructive measurements heat the gas through spontaneous emission and measurement backaction. The same analysis identifies rapid rethermalization as essential because feedback removes center-of-mass energy only from spatially resolved cells; without efficient collisions, repeated cooling steps become ineffective, as in single-component Fermi gases or $Nu$9D integrable systems (Mehdi et al., 2023).

This theme recurs in markedly different systems. In photothermal cooling of a micro-cantilever, efficacy is set by dynamical matching between thermal relaxation and mechanical motion. The response function

$$R_{\text{tot}} = \frac{T_{\text{evap}}-T_{\text{cond}}}{Q}$$0

is maximized near the condition $$R_{\text{tot}} = \frac{T_{\text{evap}}-T_{\text{cond}}}{Q}$$1, and the measured temperature dependence follows this optimum: the system is off-optimal at $$R_{\text{tot}} = \frac{T_{\text{evap}}-T_{\text{cond}}}{Q}$$2 with $$R_{\text{tot}} = \frac{T_{\text{evap}}-T_{\text{cond}}}{Q}$$3, but near-optimal at $$R_{\text{tot}} = \frac{T_{\text{evap}}-T_{\text{cond}}}{Q}$$4 with $$R_{\text{tot}} = \frac{T_{\text{evap}}-T_{\text{cond}}}{Q}$$5 (Fu et al., 2014). In multi-stage electroaerodynamic jet arrays, increasing stage count raises jet velocity and $$R_{\text{tot}} = \frac{T_{\text{evap}}-T_{\text{cond}}}{Q}$$6, but reduces efficiency through greater electrical-to-mechanical losses per added stage, reported as approximately $$R_{\text{tot}} = \frac{T_{\text{evap}}-T_{\text{cond}}}{Q}$$7 per stage (Nguyen et al., 25 Jun 2026).

A common misconception in engineering heat transfer is that maximizing thermal conductivity always improves cooling. The convective-meta thermal dispersion study explicitly rejects this as a general rule: under limited heat-capacity flow rates, a uniformly high-conductivity package can promote tangential spreading that impedes effective heat removal from an internal heat source. The reported remedy is the deliberate integration of low- and high-conductivity regions to suppress tangential transport while retaining radial transport (Zhou et al., 2024). A comparable misconception appears in primordial-gas cooling: an extremely large per-molecule local thermodynamic equilibrium cooling rate does not imply macroscopic relevance. Although $$R_{\text{tot}} = \frac{T_{\text{evap}}-T_{\text{cond}}}{Q}$$8 has an LTE cooling rate per molecule roughly a billion times larger than $$R_{\text{tot}} = \frac{T_{\text{evap}}-T_{\text{cond}}}{Q}$$9, its abundance is so low in standard primordial collapse that it contributes no more than a few percent of the total cooling (0809.0780).

3. Electronics, heat-flux management, and device-scale cooling

In compact electronics, cooling efficacy is often governed by how efficiently momentum and heat are delivered to small hotspots. Multi-stage ducted electroaerodynamic jet arrays provide a representative example. Single-stage annular $R_C = E_{\text{sky}} - r(1-R_{\text{solar}})$0 actuators reached maximum local $R_C = E_{\text{sky}} - r(1-R_{\text{solar}})$1, while two-stage devices reached up to $R_C = E_{\text{sky}} - r(1-R_{\text{solar}})$2 at the jet center. At fixed power input, the two-stage configuration yielded about $R_C = E_{\text{sky}} - r(1-R_{\text{solar}})$3 higher $R_C = E_{\text{sky}} - r(1-R_{\text{solar}})$4 than the single-stage configuration. In a direct comparison with a conventional $R_C = E_{\text{sky}} - r(1-R_{\text{solar}})$5 fan, a four-actuator array achieved equal or greater peak $R_C = E_{\text{sky}} - r(1-R_{\text{solar}})$6 at similar input power, avoided the central dead zone characteristic of rotary fans, and reduced system mass from $R_C = E_{\text{sky}} - r(1-R_{\text{solar}})$7 to $R_C = E_{\text{sky}} - r(1-R_{\text{solar}})$8. When integrated on an NVIDIA Jetson Nano, its thermal regulation during extended inference workloads was nearly identical to that of the stock fan (Nguyen et al., 25 Jun 2026).

At higher heat fluxes, phase-change cooling dominates. A microchannel two-phase R22 platform for $R_C = E_{\text{sky}} - r(1-R_{\text{solar}})$9D-ICs maintained temperatures of $R_C>0$0 while dissipating $R_C>0$1 per chip, whereas hotspots without direct liquid cooling exceeded $R_C>0$2 (Mandalapu et al., 2019). A wick-free vapor chamber for parallel $R_C>0$3 MOSFET operation reduced total thermal resistance from $R_C>0$4 in an unpatterned wick-free control to $R_C>0$5 with a wettability-patterned condenser, both evaluated at $R_C>0$6, while keeping MOSFETs below $R_C>0$7 (Mukhopadhyay et al., 2024).

For severe thermal environments, the relevant measure becomes the directly removed heat flux. Thermionic surface cooling in thermionic discharge produced an experimentally inferred cooling capacity of $R_C>0$8, with time-resolved measurements showing that the temperature drop begins essentially at plasma ignition. Segmented-cathode measurements established that more than $R_C>0$9 of the current was thermionic electron emission, supporting the interpretation that the cooling is genuinely emission-driven rather than an artifact of plasma heating (Bak et al., 2023).

Micro-scale solid-state cooling introduces yet another efficacy regime. In caloric micro-cooling, the achievable sub-ambient load depends strongly on heat-sink conditions. The reported system can cool an electronic component below room temperature at heat-flux densities up to $h=\frac{q}{A\Delta T}$0 with air-cooled heat sinks and up to $h=\frac{q}{A\Delta T}$1 with water-cooled heat sinks, with COP values exceeding $h=\frac{q}{A\Delta T}$2. PMN-10PT electrocaloric ceramic gives the highest COP, reported as well above $h=\frac{q}{A\Delta T}$3 and up to or exceeding $h=\frac{q}{A\Delta T}$4 in optimal low-flux conditions, whereas Ni-Ti at $h=\frac{q}{A\Delta T}$5 strain offers larger cooling power but lower COP because of hysteresis losses (Kalizan et al., 2021).

4. Personal cooling, radiative cooling, and environmental dependence

For wearable cooling, efficacy is typically defined by the ability to maintain skin temperature within a comfort band under realistic ambient and metabolic loads. A flexible thermoelectric active cooling garment using $h=\frac{q}{A\Delta T}$6 small thermoelectric devices on Dyneema composite fabric maintained back-skin temperature within the $h=\frac{q}{A\Delta T}$7 comfort zone up to $h=\frac{q}{A\Delta T}$8 ambient under forced convection at $h=\frac{q}{A\Delta T}$9 for a $$R_{\text{tot}}=\frac{T_{\text{evap}}-T_{\text{cond}}}{Q}$$0 metabolic rate. Under natural convection, the corresponding limit was $$R_{\text{tot}}=\frac{T_{\text{evap}}-T_{\text{cond}}}{Q}$$1. Forced convection increased the total heat transfer coefficient to approximately $$R_{\text{tot}}=\frac{T_{\text{evap}}-T_{\text{cond}}}{Q}$$2, compared with approximately $$R_{\text{tot}}=\frac{T_{\text{evap}}-T_{\text{cond}}}{Q}$$3 for natural convection plus radiation. At maximum power of about $$R_{\text{tot}}=\frac{T_{\text{evap}}-T_{\text{cond}}}{Q}$$4 at $$R_{\text{tot}}=\frac{T_{\text{evap}}-T_{\text{cond}}}{Q}$$5, a $$R_{\text{tot}}=\frac{T_{\text{evap}}-T_{\text{cond}}}{Q}$$6 battery of $$R_{\text{tot}}=\frac{T_{\text{evap}}-T_{\text{cond}}}{Q}$$7 supports at least $$R_{\text{tot}}=\frac{T_{\text{evap}}-T_{\text{cond}}}{Q}$$8 of operation (Feng et al., 2024).

In more extreme heat, a hybrid thermoelectric-hydrogel architecture achieves a different form of efficacy by separating the skin-side and evaporation-side thermal roles. At $$R_{\text{tot}}=\frac{T_{\text{evap}}-T_{\text{cond}}}{Q}$$9, the reported skin-temperature reduction was $COP_{eff}=\frac{Q_{fc}}{P_{elec}}$0 for TED-only cooling, $COP_{eff}=\frac{Q_{fc}}{P_{elec}}$1 for hydrogel-only cooling, and $COP_{eff}=\frac{Q_{fc}}{P_{elec}}$2 for the integrated TED-hydrogel system. The hybrid remained operable up to $COP_{eff}=\frac{Q_{fc}}{P_{elec}}$3, produced evaporative heat fluxes of approximately $COP_{eff}=\frac{Q_{fc}}{P_{elec}}$4 at $COP_{eff}=\frac{Q_{fc}}{P_{elec}}$5 and $COP_{eff}=\frac{Q_{fc}}{P_{elec}}$6 relative humidity, and sustained more than six hours of continuous operation with a $COP_{eff}=\frac{Q_{fc}}{P_{elec}}$7 hydrogel layer (Pei et al., 8 Jan 2025).

Passive radiative cooling studies emphasize that efficacy depends jointly on spectral selectivity and environmental loading. A CaCO$COP_{eff}=\frac{Q_{fc}}{P_{elec}}$8-acrylic paint with solar reflectance $COP_{eff}=\frac{Q_{fc}}{P_{elec}}$9 and sky-window emissivity $\text{COP}=\frac{q_{\text{gen}}}{W_{\text{in}}}$0 achieved daytime cooling power exceeding $\text{COP}=\frac{q_{\text{gen}}}{W_{\text{in}}}$1 and more than $\text{COP}=\frac{q_{\text{gen}}}{W_{\text{in}}}$2 sub-ambient cooling at noon, with $\text{COP}=\frac{q_{\text{gen}}}{W_{\text{in}}}$3 (Li et al., 2020). Under the humid, cloudy, and rapidly changing sky of Singapore, a radiative cooler with strong thermal insulation still achieved up to $\text{COP}=\frac{q_{\text{gen}}}{W_{\text{in}}}$4 daytime sub-ambient cooling and daytime cooling power up to $\text{COP}=\frac{q_{\text{gen}}}{W_{\text{in}}}$5. The study further showed that the cloud base is not a complete blackbody and can function as a heat sink for radiative cooling (Hwang, 2023). In photovoltaic modules, optics-based selective-spectral and radiative cooling reduce operating temperature by up to $\text{COP}=\frac{q_{\text{gen}}}{W_{\text{in}}}$6 for one-sun terrestrial modules and up to $\text{COP}=\frac{q_{\text{gen}}}{W_{\text{in}}}$7 for low-concentrated modules (Sun et al., 2017).

These results directly address the widespread assumption that radiative cooling is ineffective in humid or cloudy climates. The Singapore measurements indicate instead that the dominant design variable can be suppression of parasitic heat gains, with vacuum insulation reducing the effective heat transfer coefficient from $\text{COP}=\frac{q_{\text{gen}}}{W_{\text{in}}}$8 to approximately $\text{COP}=\frac{q_{\text{gen}}}{W_{\text{in}}}$9 (Hwang, 2023).

5. Atomic, optomechanical, and laser-cooling regimes

In quantum and atomic systems, cooling efficacy is often synonymous with occupation-number suppression or phase-space compression. Brillouin cooling provides a fully quantized example. For a $\eta_{ph}=\frac{d\Gamma_{eff}}{dP}$0 diameter $\eta_{ph}=\frac{d\Gamma_{eff}}{dP}$1 sphere pumped near $\eta_{ph}=\frac{d\Gamma_{eff}}{dP}$2, moderate input powers of a few mW reduce phonon occupation by a factor of approximately $\eta_{ph}=\frac{d\Gamma_{eff}}{dP}$3, while lowering the quality factor of the anti-Stokes optical mode can in theory enable cooling ratios above $\eta_{ph}=\frac{d\Gamma_{eff}}{dP}$4 (Tomes et al., 2011). The underlying relation

$\eta_{ph}=\frac{d\Gamma_{eff}}{dP}$5

makes explicit that stronger anti-Stokes dissipation improves cooling by accelerating removal of up-converted photons (Tomes et al., 2011).

Photothermal optomechanical cooling achieves a different efficacy optimum through temperature tuning. The effective damping increase per unit laser power rises from $\eta_{ph}=\frac{d\Gamma_{eff}}{dP}$6 at $\eta_{ph}=\frac{d\Gamma_{eff}}{dP}$7 to $\eta_{ph}=\frac{d\Gamma_{eff}}{dP}$8 at $\eta_{ph}=\frac{d\Gamma_{eff}}{dP}$9, a factor of $Nu$00 improvement, and then decreases slightly below $Nu$01 (Fu et al., 2014).

Laser cooling of narrow-line transitions can also be assessed by force and phase-space density. Sawtooth frequency sweeps on the $Nu$02 $Nu$03 transition in $Nu$04 yielded accelerations of approximately $Nu$05, compared with the standard Doppler limit of approximately $Nu$06, and cooled an ensemble from $Nu$07 to $Nu$08 in $Nu$09 with negligible atom loss (Norcia et al., 2017). In trapped-ion cooling, automatic-differentiation-based quantum control identifies parameter regimes beyond weak sideband coupling: for sideband cooling, $Nu$10 is reached at $Nu$11, and for EIT cooling $Nu$12 at $Nu$13, with cooling rates up to $Nu$14 and order-of-magnitude speedups relative to perturbative optima (Li et al., 2021).

Feedback cooling of ultracold gases extends these ideas to many-body matter. In a high-density, highly oblate quasi-$Nu$15D $Nu$16Rb cloud, a gas starting with $Nu$17 atoms at $Nu$18 can be cooled to the critical temperature of approximately $Nu$19 in $Nu$20, retaining over $Nu$21 atoms and incurring less atom loss than evaporation (Mehdi et al., 2023). The same work sharpens the boundary of applicability: visible density fluctuations, rapid rethermalization, and densities below the three-body loss threshold are all necessary for strong efficacy (Mehdi et al., 2023).

6. Thermodynamic and astrophysical generalizations

Cooling efficacy also appears in explicitly thermodynamic and astrophysical forms. In algorithmic cooling, the central question is not merely how cold the target qubit becomes, but at what energetic and entropic cost. The coefficient of performance is

$Nu$22

and the Landauer Ratio is

$Nu$23

Within this framework, PPA3 has $Nu$24 at every round, NOE2 has $Nu$25 after the first round, and larger PPA protocols lose efficiency as more rounds are performed or colder temperatures are approached. Improved variants reduce work input for a fixed target temperature, including an “energetically efficient PPA” and an “efficient xHBAC” with bounded $Nu$26 (Lin et al., 2024).

A related but bath-free notion of efficacy is developed for the Sachdev-Ye-Kitaev model. By coupling two SYK copies through

$Nu$27

the protocol creates a gapped adiabatic path from an EPR-product state to a low-temperature thermofield double state. High many-body fidelity requires times $Nu$28, while local observables can be cooled on times $Nu$29, independent of system size $Nu$30. Exact large-$Nu$31 numerics show cooling by a factor of $Nu$32 to the low-temperature regime (Schuster et al., 12 Nov 2025).

In galaxy clusters, cooling efficacy is tied to the cooling-flow problem. The CpH model defines

$Nu$33

and infers $Nu$34, compared with the classical cooling-flow value of approximately $Nu$35. The same spectral modeling gives $Nu$36 with intrinsic scatter $Nu$37, consistent with a picture in which MHD turbulent viscous heating balances radiative cooling (Haghighi et al., 2018).

Primordial-gas chemistry provides an instructive limiting case. $Nu$38 can be the third most important coolant after $Nu$39 and HD, and its LTE cooling rate per molecule is roughly $Nu$40 times that of $Nu$41. Yet, in standard non-irradiated collapse it contributes no more than about $Nu$42 of the total cooling rate, peaking near $Nu$43, and becomes dominant only under ionization rates regarded as unlikely in the early universe (0809.0780). This suggests that cooling efficacy is fundamentally system-level: microscopic cooling strength alone is insufficient unless supported by the appropriate kinetics, abundances, transport pathways, and dissipation channels.