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REVIEW 3 major objections 4 minor 62 references

Crossover between Solid-like and Liquid-like Behavior in Supercooled Liquids

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A ratio of pair-relaxation times marks a liquid-like to solid-like crossover in supercooled liquids, at a temperature matching the mode-coupling and Stokes-Einstein breakdown temperatures.

desk verdict New gamma_tau diagnostic shows a reproducible minimum near T_c/T_b, but calling it a solid-like state crossover overreaches the evidence. read the letter →

arxiv 2506.06957 v2 pith:PO4K72GH submitted 2025-06-08 cond-mat.soft cond-mat.dis-nn

X. R. Tian , D. M. Zhang , B. Zhang , D. Y. Sun , X. G. Gong This is my paper
classification cond-mat.softcond-mat.dis-nn
keywords supercooledliquidsglasstransitionalpharelaxationnearest-neighboratomicpairscenter-of-massandrelativemotionliquid-likesolid-likecrossoverStokes-Einsteinbreakdownmode-couplingtemperature
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

Supercooled liquids slow down dramatically between their melting point and glass transition without any thermodynamic singularity, and this paper argues that the hidden change is a switch in the kind of dynamics. The switch is detected by $\gamma_\tau$, the ratio of the $\alpha$-relaxation times of the center-of-mass and relative motions of nearest-neighbour atomic pairs: on cooling, $\gamma_\tau$ first falls (liquid-like behavior), reaches a minimum at a temperature $T_x$, then rises again (solid-like behavior). The paper claims that $T_x$ lies between $T_g$ and $T_m$ and is close to both the mode-coupling critical temperature $T_c$ and the Stokes-Einstein breakdown temperature $T_b$, so a single dynamical crossover could be the common origin of those anomalies. It also proposes four possible paths from liquid to glass, and the molecular-dynamics results for Al, Cu$_{50}$Zr$_{50}$, and a binary Lennard-Jones mixture all follow the continuous-crossover path.

What carries the argument

The machinery is the ratio $\gamma_\tau=\tau_+/\tau_-$, built from the self-intermediate scattering functions $F_+(q_{\max},t)$ and $F_-(q_{\max},t)$ for the center-of-mass coordinate $\mathbf{r}_+=\tfrac12(\mathbf{r}_i+\mathbf{r}_j)$ and relative coordinate $\mathbf{r}_-=\tfrac12(\mathbf{r}_i-\mathbf{r}_j)$ of nearest-neighbor atomic pairs, with $\tau_+$ and $\tau_-$ their $\alpha$-relaxation times at decay to $e^{-1}$. The ratio carries the argument because it has opposite temperature trends in the two regimes: in the liquid state it decreases on cooling, while in the solid state it decreases on heating, so the non-monotonic curve and its minimum provide the marker of the crossover.

What would settle it

A decisive numerical test: compute $\gamma_\tau(T)$ for the same systems at cooling rates spanning at least two decades, and also compute it for a simple fluid that does not vitrify. If the minimum shifts with cooling rate, or if the non-vitrifying fluid also shows a non-monotonic $\gamma_\tau$, then the crossover is an artifact of the nearest-neighbour pair decomposition rather than a state change.

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Extended reading notes

Core claim

The paper's central claim is that the ratio $\gamma_\tau = \tau_+ / \tau_-$ of the $\alpha$-relaxation times of the center-of-mass and relative motions of nearest-neighbor atomic pairs has a minimum at a temperature $T_x$ between $T_g$ and $T_m$, and that this minimum marks a crossover from liquid-like (LLL) to solid-like (SLL) supercooled-liquid behavior. On the liquid side, decreasing temperature strengthens interatomic bonds and makes relative motion harder than center-of-mass motion, so $\gamma_\tau$ decreases; on the solid side, when one partner in a pair is essentially fixed, the two relaxation channels coincide and $\gamma_\tau$ returns toward one, so warming causes $\gamma_\tau$ to decrease. A non-monotonic curve in $\gamma_\tau(T)$ distinguishes the two regimes, and the crossover temperature $T_x$ is identified as the minimum. The simulations of Al, Cu$_{50}$Zr$_{50}$, and a Kob-Andersen-type binary Lennard-Jones mixture all follow path $P_{IV}$, a smooth crossover rather than a jump, and $T_x$ is found to be close to the mode-coupling critical temperature $T_c$ and the Stokes-Einstein breakdown temperature $T_b$.

Load-bearing premise

The load-bearing premise is that a change in the temperature trend of a kinetic quantity marks a genuine state transition, because $\gamma_\tau$ approaches one at both high and low temperatures and only the criterion upgrades this non-monotonicity into a physical crossover at $T_x$.

Editorial extensions

If this is right

  • If $T_x$ is a genuine crossover, the Stokes-Einstein breakdown and the mode-coupling critical temperature share one microscopic origin: the switch from liquid-like to solid-like pair relaxation.
  • Because the glass transition at $T_g$ occurs entirely within the solid-like regime, no thermodynamic singularity is expected there, consistent with the picture of a glass as a frozen supercooled liquid.
  • Soft spots and liquid-like atoms in glasses acquire a logical basis: they are defects defined against a background that is already solid-like.
  • The four-path classification predicts that a glass former whose $\gamma_\tau$ jumps instead of passing through a smooth minimum would show a discontinuous dynamic event at $T_x$, distinguishing path $P_{III}$ from $P_{IV}$.
  • $\gamma_\tau$ is an operational diagnostic: computing it from pair trajectories locates the crossover in any simulated glass former without any model of the relaxation mechanism.

Reading between the lines

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

  • A natural extension is to test whether the minimum of $\gamma_\tau$ is cooling-rate independent; if $T_x$ is a true state change, its position should not drift with quench rate as long as crystallization is avoided.
  • The pair decomposition could be exported to experiments that track individual particles, such as colloidal suspensions, where $\tau_+$ and $\tau_-$ are extracted directly from trajectories and a dip in $\gamma_\tau$ could be observed without fitting a power law.
  • The table in the paper shows that the agreement among $T_x$, $T_c$, and $T_b$ is tighter for aluminum than for the binary mixture; a systematic study across more glass formers is the obvious next step to decide whether the three temperatures are literally the same quantity.
  • Extending the solid-side argument, deeply supercooled liquids below $T_x$ should show pair dynamics in which the relative coordinate relaxes while the center-of-mass stays nearly arrested; that asymmetry could be tested by decomposing the van Hove function into its plus and minus displacements.
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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 / 4 minor

Summary. The paper introduces a ratio gamma_tau = tau_plus/tau_minus of alpha-relaxation times obtained from the center-of-mass and relative motions of nearest-neighbor atomic pairs. On the basis of a thermodynamic criterion that a change in trend signals a state change, the authors argue that gamma_tau decreases with decreasing temperature in a liquid-like supercooled liquid (LLL) and decreases with increasing temperature in a solid-like supercooled liquid (SLL), so that its minimum at T_x marks a LLL-to-SLL crossover. Molecular dynamics simulations of Al, Cu50Zr50, and a Kob-Andersen binary Lennard-Jones mixture show a reproducible nonmonotonic gamma_tau(T) with a minimum at T_x between T_g and T_m. The authors compare T_x with the Stokes-Einstein breakdown temperature T_b and the mode-coupling theory critical temperature T_c, and conclude that all studied systems follow one of four proposed paths, P_IV, with dynamics and thermodynamics asynchronous.

Significance. If the LLL/SLL crossover were established as a genuine state change, the observable gamma_tau would provide a microscopic, trajectory-based marker for the well-known dynamic anomalies near the mode-coupling crossover, and would connect the Stokes-Einstein breakdown and MCT dynamical arrest to a single microscopic origin. The main strength is that gamma_tau is computed directly from MD trajectories, not fitted to T_b or T_c, and the minimum appears consistently in three different glass formers. The main weakness is that the theoretical inference from a nonmonotonic ratio to a state crossover relies on an unproven thermodynamic criterion applied to a kinetic quantity that is designed to approach unity in both the high- and low-temperature limits.

major comments (3)
  1. [Theoretical Insight, first paragraph and solid-side discussion] The load-bearing claim that the minimum of gamma_tau marks a LLL/SLL state crossover is not established. The authors themselves argue that gamma_tau tends to 1 at high temperature (uncorrelated motions) and also tends to 1 at low temperature (one atom fixed makes F_+ = F_-). Any ratio of two relaxation times with these boundary conditions and different intermediate temperature sensitivities is expected to be nonmonotonic, so the minimum by itself is not evidence of a phase or state transition. The thermodynamic criterion 'a change in trends indicates a phase transition or state transition' is asserted, not derived, and its application to a kinetic ratio is questionable. I recommend adding a control study, for example gamma_tau computed from non-neighbor pairs or from a system without a glass transition, to show that the minimum is specific to nearest-neighbor pair dynamics, and providing an independent structural, thermodynamic, or dynamical marker that changes at T_x.
  2. [Table I and Numerical Results] The claim that 'T_c and T_x are indeed very close' is not supported by the reported data for Cu50Zr50, where T_x = 900 ± 20 K and T_c = 792 ± 9 K differ by about 108 K, more than five combined error bars. For BLJ, T_x = 502 ± 23 K differs from T_b = 650 ± 23 K by about 148 K. Since the association of T_x with the known dynamic anomaly temperatures is central to the paper's significance, these discrepancies need to be explained or the claim must be qualified.
  3. [Solid-side analysis in Theoretical Insight] The derivation of the SLL branch assumes that if one atom in a nearest-neighbor pair is fixed, then F_i = 1 and hence F_+ = F_- and gamma_tau ~ 1. This ignores thermal vibrations, and the authors state that they do not actually heat a glass. The SLL behavior is thus justified by analogy rather than by calculation or simulation. I recommend testing the predicted increase of gamma_tau with temperature in a solid-like regime, for example by computing gamma_tau in a crystal or a deeply supercooled glass at low temperatures, to verify the assumed mechanism.
minor comments (4)
  1. [Eq. (3)] There appears to be a sign error in Eq. (3): the exponential factors inside the inequality are written as exp(t/tau_i) and exp(t/tau_j), but for a decaying SISF they should be exp(-t/tau_i) and exp(-t/tau_j). Please correct the signs.
  2. [Figures 2 and 3] The figures do not show error bars, although the reported T_x values include uncertainties. Adding error bars or at least stating the statistical uncertainty in the text would help assess the significance of the minimum in gamma_tau.
  3. [Cooling-rate discussion] The paper states that the possible paths may change with cooling rate, but no analysis of the cooling-rate dependence of T_x is presented. A brief sensitivity statement would clarify whether T_x is robust for the chosen rates.
  4. [Generalization to 'all systems'] The conclusion that 'all supercooled liquids studied in this work follow the path P_IV' is based on only three systems. The wording in the abstract and conclusion could be more cautious to avoid overgeneralization from this limited set.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: T_x is read from an independently simulated gamma_tau(T), and the comparison to T_b/T_c is post hoc rather than fitted.

full rationale

The central quantity gamma_tau is computed directly from MD trajectories via Eq. (1), and T_x is simply the minimum of the resulting gamma_tau(T) curve; it is not obtained by matching T_b or T_c. T_b is determined from a linear-fit breakdown of D*tau_alpha and T_c from the MCT power-law fit of tau_alpha, both using independent fitting procedures, so the reported proximity of T_x to T_b/T_c is an empirical comparison, not a constructed identity. The LLL/SLL labels are indeed defined by the slope of gamma_tau ('when gamma_tau decreases with decreasing temperature ... referred to as liquid-like'), which makes the phrase 'crossover' a definitional consequence of the minimum; however, this is an explicit classification rather than a hidden reduction, and the nonmonotonic shape itself is not forced by the two boundary limits alone because paths P_I, P_II, and P_III are admitted as alternatives. The paper's own caveats ('we do not and need not actually heat the solid glass here' and 'Mathematically, the occurrence of other paths cannot be ruled out') weaken the physical interpretation of the 'solid-like' branch, but they are scope statements, not evidence that a fitted parameter has been renamed as a prediction. No load-bearing self-citation or imported uniqueness theorem appears in the derivation. The main weakness is the thermodynamic criterion ('a change in trends indicates a phase transition or state transition') applied to a kinetic ratio; that is a validity assumption for the physical interpretation, which is a correctness risk rather than a circular step.

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

The observable gamma_tau is not fitted, but the analysis relies on several user-chosen parameters and stated physical assumptions. The free parameters are mostly analysis choices and the T_c fit used only for comparison.

free parameters (4)
  • Nearest-neighbor cutoff distance per system = 3.71 Å (Cu50Zr50), 3.92 Å (Al), 1.39 Å (BLJ)
    Defines which atomic pairs contribute to F± and gamma_tau; chosen from the first minimum of the pair distribution function, affects all computed tau± values.
  • NVT analysis segment length = 10 ns, 5 segments per 50 ns run
    Chosen by hand to be long enough for alpha relaxation but short enough to avoid re-combination of separated nearest-neighbor pairs; affects statistical quality of tau±.
  • Cooling rate per system = 1e12 K/s (Cu50Zr50), 1e10 K/s (Al), 9e8 K/s (BLJ)
    Chosen above the critical cooling rate; affects the glass states sampled and therefore the gamma_tau curves.
  • MCT fit parameters in Eq. (2) = T_c = 645±14 (Al), 491±8 (BLJ), 792±9 (Cu50Zr50); exponent and prefactor not reported
    Used only for comparison with T_x; T_c is obtained by fitting in a selected window [T_i, 1.15 T_i] with selection criteria, so the comparison is not parameter-free.
assumptions (4)
  • domain assumption The system will be in a well-defined liquid state at sufficiently high temperatures and in a solid state at sufficiently low temperatures.
    Invoked in Theoretical Insight as fundamental fact 1; not derived.
  • domain assumption The binding strength between atoms always decreases with increasing temperature.
    Invoked as fundamental fact 2; used to argue gamma_tau decreases with decreasing temperature on the liquid side and with increasing temperature on the solid side.
  • ad hoc to paper In the same phase or state, thermodynamic quantities change continuously with a definite trend; a change in trend indicates a phase transition or state transition.
    The central criterion used to identify LLL versus SLL and to infer a crossover; not a standard theorem.
  • domain assumption For a fixed atom in a pair, its self-intermediate scattering function equals 1, ignoring thermal vibrations, which implies F_+ = F_- and gamma_tau ~ 1 on the solid side.
    Used in the solid-side argument; ignores vibrational decorrelation and collective motion.

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Cite this review

Pith. "Pith review of Crossover between Solid-like and Liquid-like Behavior in Supercooled Liquids." pith.science (2026). https://pith.science/paper/PO4K72GH

@misc{pith2026250606957,
  author       = {Pith},
  title        = {Pith review of: Crossover between Solid-like and Liquid-like Behavior in Supercooled Liquids},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PO4K72GH}},
  note         = {Machine review of arXiv:2506.06957}
}
read the original abstract

In supercooled liquids, at a temperature between the glass transition temperature Tg and the melting point Tm, thermodynamic properties remain continuous, while dynamic behavior exhibits anomalies. The origin of such thermodynamics-dynamic decoupling has long been a puzzle in the field of glass researches. In this study, we show that the ratio of the alpha-relaxation time associated with the relative and center-of-mass coordinate of nearest-neighbor atomic pairs can effectively characterize the dynamic features of supercooled liquids. With this approach, supercooled liquids can be categorized into two distinct 'states' based on their dynamics: solid-like and liquid-like behaviors. We further propose four possible paths from the liquid to the final glass state, each exhibiting unique thermodynamic and dynamic behaviors. Two of these paths predict a characteristic temperature Tx between Tm and Tg, where a crossover between solid-like and liquid-like behaviors occurs in supercooled liquids. The molecular dynamics simulations of several supercooled liquids reveal that the actual path followed by all these systems undergo the crossover between solid-like and liquid-like behaviors. Tx is found to reside in a similar temperature range as the critical temperature Tc in the mode-coupling theory and the breakdown temperature Tb of the Stokes-Einstein relation. This crossover provides a new microscopic perspective for explaining macroscopic dynamic anomalies, and the absence of a typical thermodynamic phase transition at Tg.

Figures

Figures reproduced from arXiv: 2506.06957 by the authors.

Figure 1
Figure 1. Schematic diagram of the possible paths according to the change in [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. 𝜏+ and 𝜏− as a function of temperature for BLJ (a), Cu50Zr50 (b), and Al (c). The red and blue circles represent 𝜏+ and 𝜏−, respectively. For comparison, 𝜏𝛼 is also plotted (black circles). Based on the variation of 𝛾𝜏 with temperature, we found that all supercooled liquids studied in this work follow the path 𝑃𝐼𝑉 during the glass transition, as shown in [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. 𝛾𝜏 as a function of temperature for BLJ (a), Cu50Zr50 (b), and Al (c). The LLL-SLL crossover will definitely cause dynamic anomalies, as the two supercooled liquids are different from the kinetic perspective. The fact, 𝑇𝑥 lying between 𝑇𝑔 and 𝑇𝑚 , makes us believe that 𝑇𝑥 corresponds the characteristic temperature mentioned in the introduction part. To confirm this issue, we compared 𝑇𝑥 with the Stokes-Einstein brea… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Variation of the product of the diffusion coefficient and relaxation time [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.