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REVIEW 3 major objections 6 minor 85 references

A simplified unified wave-particle method for diatomic gases with rotational and vibrational non-equilibrium

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper extends the simplified unified wave-particle (SUWP) method to diatomic gases with rotational and vibrational non-equilibrium, deriving a quantified model-competition (QMC) mechanism that splits molecules into free-transport…

desk verdict A useful extension of the SUWP wave-particle framework to vibrational nonequilibrium, with a real but unquantified question about how free-transport particles are reconstructed. read the letter →

arxiv 2507.00720 v1 pith:KPGAFLUP submitted 2025-07-01 physics.flu-dyn cs.NAmath.NA

Sirui Yang , Chengwen Zhong , Ningchao Ding , Junzhe Cao , He Zhang , Congshan Zhuo , Sha Liu This is my paper · ORCID
classification physics.flu-dyncs.NAmath.NA
keywords multi-scaleflowsrarefiedgasdynamicswave-particlemethodrotationalnon-equilibriumvibrationalquantifiedmodel-competitionhypersonicflowDSMC
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper extends the simplified unified wave-particle (SUWP) method to diatomic gases in which rotational and vibrational internal energy modes are out of equilibrium, the regime encountered in hypersonic near-space flight. It splits molecules into free-transport particles and colliding molecules, with the split weighted by a newly derived quantified model-competition (QMC) mechanism built from the time integral of a vibrational kinetic model. The two groups are simulated by a collisionless particle solver and a three-temperature Navier-Stokes solver, respectively. Across shock tube, shock structure, cylinder, blunt wedge, sphere, Apollo 6 command module, and space station Mir cases, the authors report agreement with DUGKS and DSMC reference solutions while using substantially fewer CPU-hours and far less memory. If correct, the method gives an accurate and low-cost tool for multiscale diatomic gas flows with internal energy non-equilibrium.

What carries the argument

The quantified model-competition (QMC) mechanism is the central machinery. Starting from the integral solution $f(t) = e^{-\Delta t/\tau} f(\mathbf{x}_0, \mathbf{u}, t_0) + \tau^{-1} \int_0^{\Delta t} g(\mathbf{x}', \mathbf{u}, t') e^{t'/\tau} dt'$, the fraction $e^{-\Delta t/\tau}$ of molecules that never collide is treated as free transport and sampled as particles, while the complement is treated as hydrodynamic. A second-order Taylor expansion of the equilibrium state $g$ turns the hydrodynamic part into the N-S flux with the scale-dependent viscous coefficient $c_{\text{vis}}$ multiplying the viscous flux. The reconstruction of Type-F particles draws from the three equilibrium distributions $f^{\text{tr}}$, $f^{\text{rot}}$, $f^{\text{vib}}$ with probabilities set by the relaxation numbers $Z_{\text{rot}}$ and $Z_{\text{vib}}$, and the macroscopic flux is closed by a three-temperature kinetic inviscid flux (KIF) scheme.

What would settle it

A refinement study of the Mach-15 nitrogen shock structure: increase cell resolution and particle count while holding the physical model fixed, and compare the normal translational-temperature profile with DSMC. If the early rise upstream of the shock maximum persists or grows under refinement, it is an artifact of the equilibrium-sampling reconstruction; if it shrinks, it was a discretization effect. A second check is the leeward-side cylinder flow at Mach 5, where the paper reports faster temperature rise than DUGKS; quantifying that gap against DSMC would reveal the same mechanism.

Watch

Extended reading notes

Core claim

The central claim is that a second-order Chapman-Enskog expansion of the time-integral solution of the vibrational kinetic model collapses the colliding-molecule fraction onto the three-temperature Navier-Stokes equations, leaving a complementary free-transport fraction that can be handled by a collisionless particle solver. This produces an explicit weight $w_{\text{free}} = e^{-\Delta t/\tau}$ for particles and $w_{\text{hydro}} = 1 - e^{-\Delta t/\tau}$ for hydrodynamic molecules, together with a scale-dependent viscous coefficient $c_{\text{vis}} = 1 - (\Delta t/\tau) e^{-\Delta t/\tau}/(1 - e^{-\Delta t/\tau})$. The paper argues that the resulting SUWP-vib method reproduces reference solutions across Knudsen numbers from $10$ to $10^{-3}$ and Mach numbers up to $25$, and that because it reuses an N-S solver and a DSMC component, existing results for both can be directly adopted.

Load-bearing premise

The accuracy in strong non-equilibrium regions rests on the assumption that newly created free-transport particles can be sampled from the equilibrium distributions $f^{\text{tr}}$, $f^{\text{rot}}$, and $f^{\text{vib}}$ instead of from the actual non-equilibrium distribution, because those particles are supposed to carry the non-equilibrium part of the flow.

Editorial extensions

If this is right

  • The SUWP-vib method adapts cell by cell between continuum and rarefied descriptions, so one solver covers flows that cross the entire Knudsen range.
  • Because it is built from standard N-S and DSMC components, improvements to either component (relaxation models, boundary conditions, collision sampling) carry over directly into the coupled method.
  • The Apollo 6 case shows drag and lift within 0.42% and 3.06% of the DSMC benchmark while using about 57% fewer CPU-hours than the implicit DUGKS solver on the same mesh.
  • The space station Mir case shows the method runs on complex three-dimensional geometries with millions of cells, keeping memory demand well below that of deterministic velocity-space solvers.
  • The paper identifies an 'early rise' of translational temperature in shock structures and notes that modifying the relaxation time, as done in prior wave-particle work, is a possible future correction.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • (Editorial inference) Because Type-F particles are reconstructed from equilibrium distributions $f^{\text{tr}}, f^{\text{rot}}, f^{\text{vib}}$, the non-equilibrium deviation they are meant to carry may be under-sampled in strong non-equilibrium zones, which would worsen exactly where the method is most needed.
  • (Editorial inference) The QMC mechanism depends only on the form of the integral solution, so the same weighting could extend to dissociating and ionizing air by adding chemical or electronic energy modes to the relaxation targets.
  • (Editorial inference) The reported speedup compares an implicit DUGKS run on one hardware setup with SUWP-vib on another; a same-machine comparison with matched particle counts would make the efficiency claim more direct.
  • (Editorial inference) Mapping $w_{\text{free}}$ and $c_{\text{vis}}$ onto local continuum-breakdown parameters could give a quantitative criterion for when the particle solver must take over, usable in other hybrid methods.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. This paper extends the simplified unified wave-particle (SUWP) method to diatomic gases with rotational and vibrational non-equilibrium. Starting from the time-integral solution of a vibrational kinetic model, the authors derive a quantified model-competition (QMC) mechanism that splits molecules into colliding (Type-C) and free-transport (Type-F) parts, weights them by w_hydro = 1 - exp(-Δt/τ) and w_free = exp(-Δt/τ), and couples a three-temperature Navier-Stokes solver with a collisionless DSMC-type particle solver. The N-S solver uses a kinetic inviscid flux (KIF) and explicit relaxation source terms for rotational and vibrational energies; the particle solver handles particle removal, reconstruction, transport, and boundary conditions. The method is applied to shock tube, shock structure, cylinder, blunt wedge, sphere, Apollo 6, and Space Station Mir cases, with a reported efficiency comparison against implicit DUGKS.

Significance. If the central reconstruction approximation is sound, the paper offers a practical low-memory multiscale solver for high-speed aerospace flows, with an original derivation of the QMC weights for the three-temperature diatomic model and an explicit Chapman-Enskog reduction to three-temperature N-S equations. The manuscript ships a substantial set of benchmark computations, including a 2.6-million-cell Mir reentry, and reports concrete CPU-hour and memory advantages. No new free parameters are introduced beyond standard rotational/vibrational collision numbers and VHS gas data. The main strengths are the algorithmic construction and the breadth of demonstration; the main weaknesses are that most validation is against another discretization of the same kinetic model and that the particle-reconstruction approximation is not quantitatively bounded.

major comments (3)
  1. [Section 3.2, Eqs. (13)-(15), (26)] In the reconstruction step, Type-F particles are sampled from the equilibrium target distributions f_tr, f_rot, and f_vib rather than from the actual non-equilibrium distribution f(x0,u,ηrot,ηvib,0) that appears in the free-streaming term of Eq. (26). Since Type-F particles are precisely the fraction that should carry non-equilibrium information, replacing f by these equilibrium forms discards at least the deviatoric stress and higher-order moments; the Shakhov-like heat-flux terms in Eqs. (13)-(15) restore only part of the heat flux. In the intermediate-Knudsen regime w_free is not small, so this approximation can bias the particle flux and therefore the claimed accuracy. The reported early rise of Ttr,n (Sec. 4.2) and the leeward-side temperature discrepancy (Sec. 4.3) are consistent with such a bias, but they are not quantified and are partly shared with DUGKS, so they do not isolate the reconstruction error. Please add a quantitative test, for example comparing the statistical moments of reconstructed Type-F particles with the pre-removal particle distribution, or a benchmark in which the reconstruction is replaced by direct particle statistics, to bound the error in the target flow regimes.
  2. [Sections 4.1-4.7] Most validation is against the authors' own DUGKS solution [27] of the same vibrational kinetic model. This confirms consistency between two discretizations, but it cannot validate the physical model or the particle-reconstruction approximation independently. Independent DSMC comparisons appear in shock structures and cylinder/blunt-wedge cases, but the known discrepancies (early rise in Ttr,n, leeward-side temperature increase in cylinder flow, post-shock temperature decay) are described qualitatively and no error metrics are reported. For the paper's central claim of 'maintaining accuracy' in strong non-equilibrium, please provide quantitative errors against DSMC (for example L1 or L2 norms of profile deviations) for rotational and vibrational temperatures and for wall heat flux, and state the range of Knudsen and Mach numbers over which the stated accuracy holds.
  3. [Section 4.6, Table 10] The efficiency comparison is not controlled: the implicit DUGKS calculation used a different processor generation (Xeon Gold 6258, 2.70 GHz, 640 cores) than the SUWP-vib calculation (EPYC 7763, 2.45 GHz, 108 cores), and no memory usage is measured even though the abstract and conclusion claim one-to-two-orders-of-magnitude memory reduction. The reported speedup is based on CPU-hours (2092.8 vs 887.86 GHz·h), while the wall-clock time is actually larger for SUWP-vib (8.22 h vs 3.27 h); with different architectures and core counts this does not establish a robust efficiency advantage. Please report measured memory, provide a controlled or at least well-characterized comparison, and clarify whether the 60,000 KIF initialization steps are included in the SUWP total cost.
minor comments (6)
  1. [Throughout] There are numerous typographical errors, including 'desity' in figure captions, 'Fig. 3, Fig. 4 和Fig. 5' in Sec. 4.1, 'The the first term' in Sec. 3.1, and a stray 'equilibrium' in Eq. (A.7); a careful copyedit is needed.
  2. [Eq. (43)] The fixed-point iteration as written uses T_eq^{n+1,i} on the left-hand side and K_vib(T_eq^{n+1,i+1}) on the right-hand side, which is index-inconsistent; please state the intended iteration and the initial guess.
  3. [Section 4.6] The sentence 'The total computation takes 25000 time steps' after describing a 60,000-step KIF initialization is ambiguous; please clarify whether the KIF warm-up is included in the reported runtime and CPU-hours.
  4. [Section 3.2, Eqs. (13)-(15)] The heat-flux correction terms in f_tr, f_rot, and f_vib can make these distributions locally negative, but the paper does not describe how the sampling algorithm handles this; a brief reproducibility-oriented description is needed.
  5. [Section 4.4] The blunt-wedge case specifies ω1=0.75, ω2=0.25, ω3=0.75, and ω4=0.4, but the model in Section 2.2 defines only ω0 through ω3 and no ω4; please clarify the mapping of these coefficients to Eqs. (13)-(15) and whether δ also changes.
  6. [Section 3.3, Eq. (52)] The definition of the KIF weighting parameter β refers to ΔP and M_back from Ref. [11] without a local summary; for a self-contained presentation, please state the smooth-function arguments and typical parameter values.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the QMC weights and N-S flux are derived in-paper from the integral solution and Chapman-Enskog expansion, and the self-citations are contextual rather than load-bearing.

full rationale

The paper's central derivation is self-contained. The quantified model-competition weights w_free = exp(-Dt/tau) and w_hydro = 1 - exp(-Dt/tau) follow algebraically from the exact time integral solution of the stated BGK-type kinetic model (Eqs. 26-34), and the hydrodynamic flux is identified with the Navier-Stokes flux through the Chapman-Enskog expansion given in Appendix A, which recovers the Euler and N-S equations from the same model. No fitted constant is relabeled as a prediction, and no load-bearing result is imported solely from a self-citation: the QMC mechanism, the three-temperature N-S equations, and the KIF inviscid flux are all reproduced in the manuscript rather than merely cited. The paper cites prior work by the same group (Refs. 11, 27, 64, 71, 72, 80), but those citations provide context, model parameters, and benchmark data rather than the forced content of the derivation. The validation against DUGKS using the same physical model is a consistency check between two discretizations of the same kinetic model, but the paper also compares against DSMC in shock-structure, cylinder, blunt-wedge, and Apollo-6 cases, and against the independent DSMC benchmark of Moss et al. for Apollo-6, which supplies external support. The acknowledged Type-F particle reconstruction from the equilibrium-form distributions f_tr, f_rot, and f_vib is a modeling approximation that may limit accuracy in strong non-equilibrium, but it is not circular: the method's equations do not define the predicted result in terms of that approximation. Overall, no step in the claimed derivation reduces to its own inputs by construction, so the circularity score is low.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The method introduces no new physical entities. It depends on the vibrational kinetic model's parameters (Zrot, Zvib, heat-flux coupling constants) and on the standard VHS nitrogen parameters. The most significant assumption is the sampling of free-transport particles from equilibrium distributions, which is a known approximation in the SUWP framework.

free parameters (4)
  • Zvib (vibrational collision number) = 10, 30, 35, 50 per test case
    Set per case from prior literature (Tables 2-8); controls vibrational relaxation rate in the source terms.
  • Zrot (rotational collision number) = 3, 3.5, 4, 5 per case, or Parker formula
    Set per case; controls rotational relaxation. Parker's formula involves Z^∞_R=15.7 and T*=80 K.
  • Model heat flux coupling parameters = δ=1/1.55, ω0=0.2354, ω1=0.2354, ω2=0.3049, ω3=0.2354 (N2)
    From the Zhang kinetic model [27]; also ω4=0.4 in blunt wedge case; used in relaxation rates of heat fluxes.
  • VHS gas parameters = ω=0.74, μref=1.656e-5, Tref=273.15 K, Θvib=3371 K
    Standard nitrogen VHS parameters from Table 1; define viscosity and mean free path.
assumptions (4)
  • domain assumption The vibrational kinetic model (Eq. 12) with three equilibrium states f_tr, f_rot, f_vib accurately describes diatomic gas dynamics with translational, rotational, and vibrational nonequilibrium.
    The whole method is built on this phenomenological BGK-type model from Ref. [27]. The paper validates the discretization, not the model itself.
  • standard math Chapman-Enskog expansion truncated at first order (Appendix A) yields the three-temperature N-S equations used for the colliding-particle flux.
    Standard asymptotic expansion; assumes small Knudsen number for the hydrodynamic component, which is reasonable since the particle solver handles non-equilibrium.
  • ad hoc to paper Free-transport particles can be sampled from the equilibrium (or quasi-equilibrium) distributions f_tr, f_rot, f_vib rather than from the actual non-equilibrium distribution f.
    Section 3.2: reconstructed Type-F particles are sampled from the three equilibrium distributions. This discards the non-equilibrium deviation in the particle representation, a known source of the 'early rise' discrepancy.
  • domain assumption The time-integral solution (Eq. 26) assumes τ is constant during the time step, and the Taylor expansion of g to second order is valid.
    Standard UGKWP approximation; the paper does not discuss corrections for spatial/temporal variation of τ.

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Cite this review

Pith. "Pith review of A simplified unified wave-particle method for diatomic gases with rotational and vibrational non-equilibrium." pith.science (2026). https://pith.science/paper/KPGAFLUP

@misc{pith2026250700720,
  author       = {Pith},
  title        = {Pith review of: A simplified unified wave-particle method for diatomic gases with rotational and vibrational non-equilibrium},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KPGAFLUP}},
  note         = {Machine review of arXiv:2507.00720}
}
read the original abstract

The hypersonic flow around near-space vehicles constitutes a multi-scale flow problem. Due to insufficient molecular collisions to achieve equilibrium, rarefied gas effects are present in the flow field. Thus, numerical methods capable of accurately resolving multi-scale flows are required. Furthermore, high-temperature gas effects in hypersonic flows mean vibrational excitation of polyatomic molecules. Consequently, numerical methods accounting for non-equilibrium in rotational and vibrational internal energy modes are required. This study derives a quantified model-competition (QMC) mechanism for diatomic gases with rotational and vibrational non-equilibrium, starting from integral solutions of kinetic model equations with rotational and vibrational energy. The QMC mechanism categorize collisional and free-transport particles in cell, applying computational weighting based on their local scale regimes. We developed a simplified unified wave-particle (SUWP) method for diatomic gases based on QMC mechanism. For the macroscopic of the method, a three-temperature model accounting for rotational and vibrational energy is incorporated into both the kinetic inviscid flux scheme and {Navier-Stokes} solvers. For the microscopic of the method, a collisionless DSMC solver is employed to resolve non-equilibrium flow physics. This work validates the proposed SUWP method with rotational and vibrational non-equilibrium through benchmark cases, including shock tube, shock structures, flow past a cylinder, Apollo 6 command module and space station Mir. Compared to the DSMC and deterministic methods, the SUWP method exhibits favorable computational efficiency while maintaining accuracy.

Figures

Figures reproduced from arXiv: 2507.00720 by the authors.

Figure 1
Figure 1. The route of SUWP-vib method. (The blue region is the macroscopic solver, and the green region [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. The particle reconstruction of SUWP-vib method. [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. The (a) desity, (b) velocity and (c) temperature profiles of the Sod’s shock tube at Kn=10. [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figures from the paper (29 more)
Figure 4
Figure 4. Figure 4: The (a) desity, (b) velocity and (c) temperature profiles of the Sod’s shock tube at Kn=0.1. [PITH_FULL_IMAGE:figures/full_fig_p021_4.png]
Figure 5
Figure 5. Figure 5: The (a) desity, (b) velocity and (c) temperature profiles of the Sod’s shock tube at Kn=0.001. [PITH_FULL_IMAGE:figures/full_fig_p021_5.png]
Figure 6
Figure 6. Figure 6: The (a) desity, (b) translational temperature, (c) rotational temperature and vibrational temperature [PITH_FULL_IMAGE:figures/full_fig_p023_6.png]
Figure 7
Figure 7. Figure 7: The (a) desity, (b) translational temperature, (c) rotational temperature and vibrational temperature [PITH_FULL_IMAGE:figures/full_fig_p023_7.png]
Figure 8
Figure 8. Figure 8: The mesh of the flow past a cylinder. model are specified as follows: ω1 = 0.75, ω2 = 0.25, ω3 = 0.75, and ω4 = 0.4. This test case employs dimensional variables for computation and comparison. The particle number in cell Np = 2 × 102 [PITH_FULL_IMAGE:figures/full_fi…
Figure 9
Figure 9. Figure 9: The (a) pressure, (b) Mach number, (c) equilibrium temperature, (d) translational temperature, (e) [PITH_FULL_IMAGE:figures/full_fig_p034_9.png]
Figure 10
Figure 10. Figure 10: The (a) density and (b) temperature along the forward stagnation line of the cylinder at Ma=5. [PITH_FULL_IMAGE:figures/full_fig_p035_10.png]
Figure 11
Figure 11. Figure 11: The (a) pressure and (b) heat flux on the wall surface of the cylinder at Ma=5. [PITH_FULL_IMAGE:figures/full_fig_p035_11.png]
Figure 12
Figure 12. Figure 12: The (a) pressure, (b) Mach number, (c) equilibrium temperature, (d) translational temperature, [PITH_FULL_IMAGE:figures/full_fig_p036_12.png]
Figure 13
Figure 13. Figure 13: The (a) density and (b) temperature along the forward stagnation line of the cylinder at Ma=20. [PITH_FULL_IMAGE:figures/full_fig_p037_13.png]
Figure 14
Figure 14. Figure 14: The (a) pressure and (b) heat flux on the wall surface of the cylinder at Ma=20. [PITH_FULL_IMAGE:figures/full_fig_p037_14.png]
Figure 15
Figure 15. Figure 15: The mesh of the blunt wedge at Ma=15. 38 [PITH_FULL_IMAGE:figures/full_fig_p038_15.png]
Figure 16
Figure 16. Figure 16: The (a) pressure, (b) translational temperature, (c) rotational temperature, and (d) vibrational [PITH_FULL_IMAGE:figures/full_fig_p039_16.png]
Figure 17
Figure 17. Figure 17: The (a) pressure and (b) temperature along the forward stagnation line of the blunt wedge at [PITH_FULL_IMAGE:figures/full_fig_p040_17.png]
Figure 18
Figure 18. Figure 18: The (a) pressure, (b) shear stress and (c) heat flux on the wall surface of the blunt wedge at Ma=15, [PITH_FULL_IMAGE:figures/full_fig_p040_18.png]
Figure 19
Figure 19. Figure 19: The (a) pressure, (b) translational temperature, (c) rotational temperature, and (d) vibrational [PITH_FULL_IMAGE:figures/full_fig_p041_19.png]
Figure 20
Figure 20. Figure 20: The (a) pressure and (b) temperature along the forward stagnation line of the blunt wedge at [PITH_FULL_IMAGE:figures/full_fig_p042_20.png]
Figure 21
Figure 21. Figure 21: The (a) pressure, (b) shear stress and (c) heat flux on the wall surface of the blunt wedge at Ma=15, [PITH_FULL_IMAGE:figures/full_fig_p042_21.png]
Figure 22
Figure 22. Figure 22: The mesh of the sphere at Ma=10. 43 [PITH_FULL_IMAGE:figures/full_fig_p043_22.png]
Figure 23
Figure 23. Figure 23: The (a) pressure, (b) translational temperature, (c) rotational temperature, and (d) vibrational [PITH_FULL_IMAGE:figures/full_fig_p044_23.png]
Figure 24
Figure 24. Figure 24: The (a) density and (b) temperature along the forward stagnation line of the sphere at Ma=10. [PITH_FULL_IMAGE:figures/full_fig_p045_24.png]
Figure 25
Figure 25. Figure 25: The (a) pressure and (b) heat flux on the wall surface of the sphere at Ma=10. [PITH_FULL_IMAGE:figures/full_fig_p045_25.png]
Figure 26
Figure 26. Figure 26: The mesh of the Apollo 6 command module. [PITH_FULL_IMAGE:figures/full_fig_p046_26.png]
Figure 27
Figure 27. Figure 27: The (a) pressure, (b) translational temperature, (c) rotational temperature, and (d) vibrational [PITH_FULL_IMAGE:figures/full_fig_p047_27.png]
Figure 28
Figure 28. Figure 28: The (a) pressure and (b) heat flux on the wall surface of the Apollo 6 command module. [PITH_FULL_IMAGE:figures/full_fig_p048_28.png]
Figure 29
Figure 29. Figure 29: The module composition of the SS Mir: (1) the progress spacecraft; (2) Kvant; (3) the base block; [PITH_FULL_IMAGE:figures/full_fig_p048_29.png]
Figure 30
Figure 30. Figure 30: The mesh of the SS Mir. (a) (b) [PITH_FULL_IMAGE:figures/full_fig_p049_30.png]
Figure 31
Figure 31. Figure 31: The (a) density contours and (b) 3D iso-surface [PITH_FULL_IMAGE:figures/full_fig_p049_31.png]
Figure 32
Figure 32. Figure 32: The (a) pressure, (b) Mach number, (c) equilibrium temperature, (d) translational temperature, [PITH_FULL_IMAGE:figures/full_fig_p050_32.png]

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