Thermal Non-Equilibrium (TNE) Dynamics

• Thermal Non-Equilibrium (TNE) is defined as the failure of steady thermal equilibrium under heating, cooling, or transport conditions, seen in solar loops and kinetic systems.
• TNE studies use models like discrete Boltzmann formulations and hydrodynamic simulations to capture non-conserved kinetic moments and thermal runaways.
• Implications of TNE include solar-coronal rain, modified nucleation rates, and extended state spaces in discrete thermodynamics, offering practical insights and diagnostics.

Thermal non-equilibrium (TNE) denotes a class of nonequilibrium thermal states for which steady thermal closure fails under the imposed heating, cooling, transport, or collisional conditions. In solar-coronal physics, TNE refers to the absence of any steady fluid equilibrium in a coronal loop or flux tube despite quasi-steady heating, so the plasma undergoes repeated evaporation–condensation cycles rather than settling into a static or steady-flow state (Klimchuk et al., 2019). In kinetic and discrete Boltzmann formulations, TNE denotes the departure of the distribution function from local equilibrium, quantified through non-conserved kinetic moments such as non-organized momentum fluxes and non-organized energy fluxes (Zhang et al., 2022). A broader thermodynamic literature extends the term to discrete systems, nucleation, multiphase flows, and open coronal outflows, where temperature variables, internal variables, or mesoscale gradients cannot be reduced to a single equilibrium description (Muschik, 2020, Kiefer et al., 2023, Scott et al., 2024).

1. Terminological scope and operative definitions

The modern literature uses TNE in several technically distinct but structurally related ways. In all of them, the common feature is that local or global thermal balance cannot be represented by a steady equilibrium state or by low-order constitutive closure.

Domain	Operative definition of TNE	Representative manifestations
Solar coronal loops	No steady fluid equilibrium exists under quasi-steady, highly stratified heating	Evaporation–condensation cycles, coronal rain, prominences, long-period EUV pulsations
Discrete Boltzmann and compressible-flow modeling	Deviation of $f$ from $f^{eq}$ measured by non-conserved moments of $f-f^{eq}$	Viscous stress, heat flux, higher-order fluxes, shock-layer nonequilibrium
Discrete non-equilibrium thermodynamics	Enlarged state space with contact temperature $\Theta$ and internal variables $\xi$	Accompanying reversible processes, embedding theorem, generalized cyclic efficiency
Kinetic nucleation	Different temperatures for gas, cluster translational motion, and cluster internal modes	Modified forward and backward nucleation rates, order-of-magnitude changes in cluster abundances

In the solar literature, TNE is explicitly distinguished from thermal instability in the strict sense. Instability presupposes an equilibrium that can be perturbed, whereas TNE means that no equilibrium exists for the chosen heating and geometry; the local cooling episode is therefore better described as thermal runaway within a global limit cycle (Klimchuk et al., 2019). In discrete Boltzmann work, by contrast, TNE is the kinetic departure from local equilibrium itself, often contrasted with hydrodynamic non-equilibrium and organized through central moments of $f-f^{eq}$ (Gan et al., 2015, Zhang et al., 2022).

This suggests a broad unifying interpretation: across fields, TNE marks the failure of a reduced steady or near-equilibrium thermal description, but the mathematically relevant object differs by discipline—field-aligned fluid balance in coronal loops, kinetic moments in compressible flows, or non-equilibrium state variables in discrete thermodynamics.

2. Solar-coronal TNE as a limit-cycle process

In coronal loops, TNE occurs when heating is both highly stratified and quasi-constant or quasi-steady, usually concentrated toward the loop footpoints (Froment et al., 2019, Froment et al., 2017). Under those conditions, footpoint heating drives chromospheric evaporation, the corona fills with denser plasma, radiative losses strengthen, the upper loop cools, and condensations form. The cycle can repeat as long as the heating remains sufficiently stable compared with the cooling time (Froment et al., 2018).

A standard heating prescription used in 1D loop studies is

$$H(s) = H_0 + H_1\left(e^{-g(s)/\lambda_1}+e^{-g(L-s)/\lambda_2}\right),$$

with

$$g(s)=\max(s-\Delta,0), \qquad \Delta=5~\mathrm{Mm},$$

where $H_0$ is a weak background term, $H_1$ sets footpoint heating strength, and $f^{eq}$0 are the heating scale lengths in the two legs (Froment et al., 2018). In this formulation, small $f^{eq}$1 concentrates heating low in the atmosphere, driving evaporation without adequately heating the apex. The loop then fills until radiative losses exceed available coronal heating and runaway cooling begins.

The global TNE cycle may terminate in either incomplete or complete condensations. In the parameter survey of Froment and collaborators, complete condensations were identified when temperature somewhere in the coronal part of the loop dropped below $f^{eq}$2 MK, whereas incomplete condensations remained at coronal temperatures (Froment et al., 2018). This distinction is observationally consequential: complete condensations are associated with coronal rain, while incomplete condensations can still generate long-period EUV pulsations without obvious chromospheric condensates (Froment et al., 2017).

Klimchuk and Luna derived two practical criteria that summarize the coronal-loop regime. As a rule of thumb, the ratio of apex to footpoint heating rates must be less than about $f^{eq}$3, and asymmetries must be less than about a factor of $f^{eq}$4, although the precise values are case dependent (Klimchuk et al., 2019). Their formulation emphasizes that strongly bottom-heavy heating is necessary, but large enough asymmetry in heating and/or cross-sectional area can replace TNE with a steady end-to-end flow whose enthalpy transport restores energy balance.

3. Observational manifestations in the corona

Long-period EUV pulsations and coronal rain are treated in the recent solar literature as two observational expressions of the same TNE mechanism (Froment et al., 2018). Froment and collaborators argued that the observed pulsations are not wave signatures: for periods of $f^{eq}$5–$f^{eq}$6 hr, standard MHD coronal loop modes are implausible, and the power spectra are better matched by periodic trains of pulses with random amplitudes superimposed on a power-law background (Auchère et al., 2016). Their model,

$f^{eq}$7

predicts a power spectral density with both a continuum term from amplitude randomness and a harmonic comb from periodic repetition, consistent with observed Fourier and wavelet signatures (Auchère et al., 2016).

A multi-scale observational demonstration was given with SDO/AIA and the Swedish 1-m Solar Telescope. Within the same loop bundle, a dominant coronal periodicity of about $f^{eq}$8 hr was detected, with close peaks at $f^{eq}$9 and $f-f^{eq}$0, corresponding to $f-f^{eq}$1 and $f-f^{eq}$2 hr (Froment et al., 2019). Time-lag analysis over the pulsating region showed positive lags for the coronal sequence $f-f^{eq}$3, $f-f^{eq}$4, $f-f^{eq}$5, and $f-f^{eq}$6, with median lags of $f-f^{eq}$7, $f-f^{eq}$8, $f-f^{eq}$9, and $\Theta$0 minutes, respectively, consistent with cooling from roughly $\Theta$1 MK through $\Theta$2, $\Theta$3, and $\Theta$4 MK (Froment et al., 2019). DEM analysis found the DEM-weighted temperature varying between about $\Theta$5 and $\Theta$6 MK and the total emission measure varying from $\Theta$7 to $\Theta$8, with a relative amplitude of about $\Theta$9 (Froment et al., 2019).

The same event linked the coronal cycle to chromospheric-scale rain. SST/CRISP and SST/CHROMIS resolved rain into narrow strands with typical widths of about $\xi$0, or roughly $\xi$1 km, while overall rain blob widths ranged from $\xi$2 to $\xi$3 km with an average around $\xi$4 km (Froment et al., 2019). Doppler velocities of $\xi$5–$\xi$6 and plane-of-sky velocities up to about $\xi$7 were measured; average total velocities along pulsating-loop paths were around $\xi$8–$\xi$9 (Froment et al., 2019). Combined H$f-f^{eq}$0 and Ca II K diagnostics yielded inferred average temperatures of roughly $f-f^{eq}$1–$f-f^{eq}$2 kK for pulsating-loop paths, with inferred densities typically $f-f^{eq}$3–$f-f^{eq}$4 and average values around $f-f^{eq}$5–$f-f^{eq}$6 (Froment et al., 2019). Using blob densities around $f-f^{eq}$7–$f-f^{eq}$8, the observed blob number, and widths, the derived mass fluxes were roughly $f-f^{eq}$9–$$H(s) = H_0 + H_1\left(e^{-g(s)/\lambda_1}+e^{-g(L-s)/\lambda_2}\right),$$0, which the authors interpreted as significant mass circulation through the corona–chromosphere system (Froment et al., 2019).

Spectroscopic confirmation of the predicted coronal-temperature flows is more difficult. In a Hinode/EIS search across $$H(s) = H_0 + H_1\left(e^{-g(s)/\lambda_1}+e^{-g(L-s)/\lambda_2}\right),$$1 raster datasets cross-matched with known AIA pulsation events, only $$H(s) = H_0 + H_1\left(e^{-g(s)/\lambda_1}+e^{-g(L-s)/\lambda_2}\right),$$2 datasets showed periodic Doppler variations compatible with TNE downflows (Pelouze et al., 2019). The clearest cases showed loop-leg velocity fluctuations of $$H(s) = H_0 + H_1\left(e^{-g(s)/\lambda_1}+e^{-g(L-s)/\lambda_2}\right),$$3 and $$H(s) = H_0 + H_1\left(e^{-g(s)/\lambda_1}+e^{-g(L-s)/\lambda_2}\right),$$4, much smaller than the $$H(s) = H_0 + H_1\left(e^{-g(s)/\lambda_1}+e^{-g(L-s)/\lambda_2}\right),$$5 projected velocities expected from simulation (Pelouze et al., 2019). The paper attributed the low detection rate to line-of-sight dilution, projection effects, low signal-to-noise ratio, limited cadence, and residual Hinode/EIS orbital drift. It explicitly concluded that the absence of a Doppler signature does not rule out TNE (Pelouze et al., 2019).

4. Geometry, asymmetry, and numerical control parameters

Large simulation surveys show that TNE occupies restricted regions of parameter space rather than arising generically in all loops. In a study of $$H(s) = H_0 + H_1\left(e^{-g(s)/\lambda_1}+e^{-g(L-s)/\lambda_2}\right),$$6 1D hydrodynamic simulations over three loop geometries, TNE occurred only for specific combinations of loop geometry, heating strength, and heating stratification (Froment et al., 2018). In the semi-circular loop, TNE appeared mainly near symmetric heating; in one skewed observed loop, it shifted toward asymmetric heating with $$H(s) = H_0 + H_1\left(e^{-g(s)/\lambda_1}+e^{-g(L-s)/\lambda_2}\right),$$7; and in another observed loop the preferred TNE region changed with heating strength (Froment et al., 2018). The simulated periods ranged from about $$H(s) = H_0 + H_1\left(e^{-g(s)/\lambda_1}+e^{-g(L-s)/\lambda_2}\right),$$8 to $$H(s) = H_0 + H_1\left(e^{-g(s)/\lambda_1}+e^{-g(L-s)/\lambda_2}\right),$$9 hr, with longer loops generally showing longer periods, and the lag between temperature and density was typically about $$g(s)=\max(s-\Delta,0), \qquad \Delta=5~\mathrm{Mm},$$0–$$g(s)=\max(s-\Delta,0), \qquad \Delta=5~\mathrm{Mm},$$1 of the cycle period (Froment et al., 2018).

A realistic loop study based on linear force-free field extrapolations showed that, for a loop of length $$g(s)=\max(s-\Delta,0), \qquad \Delta=5~\mathrm{Mm},$$2 Mm and apex altitude about $$g(s)=\max(s-\Delta,0), \qquad \Delta=5~\mathrm{Mm},$$3 Mm, TNE only appeared when the heating was asymmetric, with the eastern leg more strongly heated than the western one (Froment et al., 2017). The successful simulation used

$$g(s)=\max(s-\Delta,0), \qquad \Delta=5~\mathrm{Mm},$$4

$$g(s)=\max(s-\Delta,0), \qquad \Delta=5~\mathrm{Mm},$$5

and reproduced a $$g(s)=\max(s-\Delta,0), \qquad \Delta=5~\mathrm{Mm},$$6-hour periodic TNE regime with incomplete condensations (Froment et al., 2017). The cross-correlation delay between temperature and density was about $$g(s)=\max(s-\Delta,0), \qquad \Delta=5~\mathrm{Mm},$$7 minutes in the simulation, compared with $$g(s)=\max(s-\Delta,0), \qquad \Delta=5~\mathrm{Mm},$$8 minutes from the observations (Froment et al., 2017). The same paper emphasized that EUV cooling signatures and time lags are not uniquely diagnostic of impulsive heating, because a constant heating model can produce the same ordering across AIA channels (Froment et al., 2017).

The role of asymmetry in rain formation was addressed with a much larger survey of $$g(s)=\max(s-\Delta,0), \qquad \Delta=5~\mathrm{Mm},$$9 simulations. Using the reconstructed geometry of the “rain bow” event and comparison cases from Froment and collaborators, the study found that both prominences and TNE cycles, with or without coronal rain, can form within the same magnetic structure (Pelouze et al., 2021). Asymmetric loops were overall less likely to produce coronal rain regardless of the heating. In symmetric loops, coronal rain formed when the heating was also symmetric; in asymmetric loops, rain formed only when the heating compensated the asymmetry (Pelouze et al., 2021). For the six stereoscopically based geometries denoted S10–S98, TNE occurred in $H_0$0 of the explored parameter space; among TNE cases, $H_0$1 produced coronal rain, $H_0$2 produced prominence-like structures, and $H_0$3 produced TNE without rain (Pelouze et al., 2021).

Numerical treatment of the transition region is also decisive. For a $H_0$4 Mm loop, inadequate transition-region resolution can suppress TNE entirely or distort its periodicity (Johnston et al., 2019). In HYDRAD, runs with refinement levels RL $H_0$5 relaxed to static equilibrium rather than entering TNE, RL $H_0$6–$H_0$7 produced TNE but with substantially different cycle periods, and convergence required RL $H_0$8, corresponding to spatial resolution better than $H_0$9 km (Johnston et al., 2019). The converged reference period was about $H_1$0 hr. The paper further showed that TNE is suppressed unless the background heating is sufficiently small: for $H_1$1, the loop relaxed to static equilibrium (Johnston et al., 2019). An unresolved-transition-region jump-condition method implemented in LareJ reproduced the converged behavior at far lower cost, with average coronal density and temperature errors of about $H_1$2 and $H_1$3, respectively (Johnston et al., 2019).

5. Kinetic-moment formulations in compressible and multiphase flows

In discrete Boltzmann modeling, TNE is defined by non-conserved kinetic moments of the deviation from local equilibrium. A representative set of central-moment measures is

$H_1$4

where $H_1$5 is the non-organized momentum flux (generalized viscous stress), $H_1$6 is the non-organized energy flux (generalized heat flux), $H_1$7 is the flux of $H_1$8, and $H_1$9 is the flux of $f^{eq}$00 (Zhang et al., 2023). At first order, these quantities connect directly to macroscopic gradients through transport coefficients: $f^{eq}$01

$f^{eq}$02

so non-organized momentum flux depends on velocity gradients and viscosity, while non-organized energy flux depends on temperature gradients and thermal conductivity (Zhang et al., 2023).

Several papers use scalar or vector summaries of TNE intensity. One recurring choice is

$f^{eq}$03

or its density-weighted spatial average $f^{eq}$04, which acts as a compact measure of departure from equilibrium (Gan et al., 2015, Sun et al., 2022). Another formulation introduces

$f^{eq}$05

to emphasize that no single scalar fully characterizes nonequilibrium depth (Zhang et al., 2022). This literature explicitly distinguishes hydrodynamic non-equilibrium from thermodynamic non-equilibrium: non-central moments contain both organized flow and thermal deviations, whereas central moments isolate the thermal contribution (Gan et al., 2015, Guo et al., 3 Feb 2025).

As the degree of nonequilibrium deepens, higher-order kinetic moments become necessary. A family of ES-BGK-based discrete Boltzmann models was constructed to retain first- through sixth-order TNE effects, with D2V16, D2V25, D2V36, D2V49, D2V64, and D2V81 discrete velocity sets (Zhang et al., 2022). The study concluded that to obtain satisfying hydrodynamic quantities in a two-dimensional free jet, the model should include at least up to the third-order TNE effects (Zhang et al., 2022). In three-dimensional high-speed compressible flows, a second-order Burnett-level D3V55 model preserved the equilibrium moment set

$f^{eq}$06

allowing recovery of nonlinear constitutive relations for viscous stress and heat flux beyond the Navier–Stokes level (Guo et al., 3 Feb 2025). A later super-Burnett D3V91 construction extended this to third-order TNE in 3D and showed that odd-order central moments, including heat flux and viscous-stress flux, are primarily governed by temperature gradients, whereas even-order central moments, including viscous stress and heat-flux-related flux, are dominated by velocity gradients; these leading dependences are substantially modified by density gradients, secondary gradients, higher derivatives, and cross-couplings (Lai et al., 4 Jun 2026).

In multiphase systems, TNE diagnostics are used not only to describe local kinetics but also to segment process stages. During phase separation, the TNE strength $f^{eq}$07 attains its maximum at the end of the spinodal decomposition stage and then decreases during domain growth (Gan et al., 2015). In shock–bubble interaction, viscosity has only limited influence on macroscopic parameters during the rapid shock-compression stage but significantly affects the TNE features, strengthening both non-organized momentum flux and non-organized energy flux and increasing entropy production from both channels (Zhang et al., 2023).

6. Extensions to coalescence, open plasmas, nucleation, and discrete thermodynamics

The same mesoscopic language has been extended to droplet and bubble coalescence. In bubble coalescence, the averaged $f^{eq}$08-component of the non-organized momentum flux, $f^{eq}$09, identifies two characteristic instants: the time when the mean coalescing speed reaches its maximum, at which the ratio of minor to major axes is about $f^{eq}$10, and the time when the ratio $f^{eq}$11 reaches $f^{eq}$12 for the first time (Sun et al., 2022). The study reported that TNE intensity, coalescence acceleration, and the negative slope of boundary length show a high degree of correlation and attain their maxima simultaneously (Sun et al., 2022). A related droplet-coalescence analysis found that the temporal evolutions of $f^{eq}$13 and the mean entropy production rate $f^{eq}$14 provide concise and consistent stage criteria: maxima correspond to the fastest decrease of liquid–vapor interface length, and valleys correspond to the most elongated elliptical state of the merged droplet (Sun et al., 2023).

Under non-isothermal droplet coalescence, latent heat introduces a competition between temperature-gradient-driven and velocity-gradient-driven nonequilibrium effects. Before contact, $f^{eq}$15 and $f^{eq}$16 dominate; after contact, $f^{eq}$17 and $f^{eq}$18 become dominant (Sun et al., 24 Feb 2025). The paper concluded that latent heat-induced temperature rise significantly refrains the TNE intensity in the thermal case, so the non-isothermal system has a lower peak TNE intensity, longer duration, and more complex spatiotemporal evolution than the isothermal case (Sun et al., 24 Feb 2025). The same work reported a much longer coalescence initiation time in the non-isothermal case, with $f^{eq}$19 for the isothermal case and $f^{eq}$20 for the non-isothermal case (Sun et al., 24 Feb 2025).

Solar-plasma applications are no longer limited to closed loops. A one-dimensional HYDRAD study showed that TNE can also occur on open field lines in a transonic solar wind when sufficiently concentrated foot-point heating creates a radiation-dominated thermal sink in the low corona (Scott et al., 2024). The imposed heating profile was

$f^{eq}$21

with $f^{eq}$22, $f^{eq}$23, and $f^{eq}$24 Mm (Scott et al., 2024). In the simulation sequence, steady solutions existed up to about $f^{eq}$25, while a tiny increase to $f^{eq}$26 triggered TNE cycles (Scott et al., 2024). The marginal case had a cycle period of about $f^{eq}$27 hr; as $f^{eq}$28 increased, the period dropped to about $f^{eq}$29 hr by $f^{eq}$30 and approached about $f^{eq}$31 hr for $f^{eq}$32 (Scott et al., 2024). This result directly generalizes the loop-centered view of TNE by showing that appreciable mass and energy fluxes do not preclude nonequilibrium cycling on open structures.

In kinetic nucleation, TNE means that the gas temperature, cluster kinetic temperature, and cluster internal temperature are not equal: $f^{eq}$33 The forward rate coefficient is modified through the temperature-weighted reduced mass $f^{eq}$34, and the backward rate is derived from the law of mass action under thermal non-equilibrium (Kiefer et al., 2023). For homogeneous TiO$f^{eq}$35 nucleation at a gas temperature of $f^{eq}$36 K, differences in kinetic cluster temperatures as small as $f^{eq}$37 K increased the formation of larger TiO$f^{eq}$38 clusters by over an order of magnitude; at $f^{eq}$39 K, an increase in cluster temperature of around $f^{eq}$40 K reduced the formation of a larger TiO$f^{eq}$41 cluster by over an order of magnitude (Kiefer et al., 2023). These results show that small thermal offsets can materially alter the stability of large clusters.

A more axiomatic formulation appears in the thermodynamics of discrete Schottky systems. There the non-equilibrium state space is

$f^{eq}$42

where $f^{eq}$43 is the contact temperature and $f^{eq}$44 denotes internal variables (Muschik, 2020). Projection onto the equilibrium subspace generates a reversible accompanying process,

$f^{eq}$45

and the embedding theorem ensures compatibility between non-equilibrium entropy and equilibrium entropy along this projection (Muschik, 2020). The contact temperature is defined operationally by the inequality

$f^{eq}$46

and the paper shows that if entropy production is independent of internal energy, the contact temperature reduces to the thermostatic equilibrium temperature even though non-equilibrium is still under consideration (Muschik, 2020). This is a conceptually different use of TNE from the loop or DBM literatures, but it shares the same central theme: equilibrium temperature alone is not always an adequate state descriptor.

Across these literatures, TNE is therefore not a single model but a family of precise nonequilibrium constructs. In solar loops it constrains where and how heating is deposited; in discrete Boltzmann modeling it organizes the hierarchy of non-conserved kinetic moments; in multiphase and coalescence problems it tracks stage transitions and entropy production; in nucleation it controls forward and backward rates through small temperature offsets; and in discrete-system thermodynamics it motivates an enlarged state space with measurable non-equilibrium temperature. This suggests that the strongest common denominator of TNE is not a particular equation set, but the requirement to treat thermal evolution with variables or closures beyond steady equilibrium.