Violations of conservation laws in viscous liquid dynamics

Jeppe C. Dyre

REVIEW

Violations of conservation laws in viscous liquid dynamics

Jeppe C. Dyre

Soft Condensed Matter · Submitted 1 Mar 2006 · 25 references

Pith's verdict: The open-system point about momentum and energy is real, but the density non-conservation argument is mathematically inconsistent and should not carry the paper's central claim.

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arXiv IDcond-mat/0603023 (version 4)

Pith ID5NV74RFW, Pith's permanent record of this paper and its review

CitationJeppe C. Dyre. "Violations of conservation laws in viscous liquid dynamics." arXiv:cond-mat/0603023 (2006). https://pith.science/paper/5NV74RFW

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Editor's note

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Hi X,

The short version: this is a clearly written opinion piece from an established voice, and the open-system observations about momentum and energy leakage are legitimate. But the new-seeming part—the claim that coarse-grained density in a viscous liquid is a non-conserved field—doesn't survive contact with the continuity equation. It should not be accepted as a theoretical result.

What the paper does well: the order-of-magnitude estimates are illuminating. The momentum diffusion length over an alpha relaxation time is a nice physical illustration of why mode-coupling theory's conservation-based starting point may be inapplicable in the ultraviscous regime. The framing in terms of characteristic times τ_A is clean, and the distinction between adiabatic and isothermal-like response functions is worth keeping. The review of the inherent-state/flow-event picture is concise and well cited.

The soft spots: the momentum and energy 'violations' are really statements about a subsystem coupled to its environment. Conservation laws are not violated; the sample simply isn't isolated. That is a fair point but shouldn't be phrased as a violation. The load-bearing problem is Eq. (2). For any physical density, ∂ρ/∂t = -∇·J exactly, and coarse-graining preserves this. You cannot write ∂ρ/∂t as a sum of point sources and sinks without a compensating divergence; otherwise the k=0 mode is not conserved, contradicting the paper's own premise. Also the elastic displacement field argument misreads the character of a shear transformation: the far field of a localized internal rearrangement has zero net divergence, so a coarse-grained density over a volume larger than the event core is transported, not created or destroyed. Thus the 'apparent violation' of particle number conservation collapses, and the k-independent decay rates have no basis in this argument.

I'd send this to a referee, but with the expectation that the density part needs a serious rewrite or removal. The paper is better as a commentary on open-system effects than as a claim about modified conservation laws. Worth discussing in a reading group precisely because the error is instructive.

Best,

[Your name]

Referee report

Significance

The paper draws attention to a real and often overlooked timescale separation: for highly viscous liquids the momentum and heat diffusion lengths over an alpha-relaxation time can exceed the sample dimensions, so the sample cannot be treated as isolated. This is a useful caveat for interpreting experiments and simulations. The order-of-magnitude estimates (transverse momentum diffusing more than a kilometre in one second; heat diffusion through a 100 µm layer) are plausible. However, the paper's central particle-number claim is not merely underived but inconsistent with the exact continuity equation, and the momentum/energy statements are overframed as violations of conservation laws rather than as open-system effects. The paper would need substantial revision to make its claims precise and defensible.

Major objections (3)

[§3, Eq. (2)] The derivation of Eq. (2) is inconsistent with particle-number conservation, which the paper explicitly grants. For point particles, ρ(r,t)=Σ_i m δ(r−r_i(t)) obeys the continuity equation ∂ρ/∂t=−∇·J exactly, and any time-independent coarse-graining kernel yields ∂ρ_K/∂t=−∇·J_K; hence the k=0 mode is exactly conserved and the time derivative of a coarse-grained density cannot contain the local source term Σ b_μ δ(r−r_μ)δ(t−t_μ). The elastic argument does not repair this: the field u∝r/r³ has ∇·u=4πδ(r), a monopole source, whereas an internal shear transformation produces zero net volume change and a far field whose integrated divergence vanishes. Consequently, the statements that density 'has the appearance of a non-conserved field' and that decay rates are k-independent are unsupported; this part of the abstract and conclusion must be revised or removed.
[§1 and Conclusion] The discussion of momentum and energy conflates non-conservation with openness. The estimates following Eq. (1) show that τ_mom and τ_en can be shorter than the alpha relaxation time, which is correctly interpreted as the sample exchanging momentum and heat with its surroundings. That is a statement about boundary conditions, not a violation of the conservation laws; for the combined system of sample plus holder the laws hold exactly. The stronger conclusions that 'the basic conservation laws cannot be used for arriving at a proper description' and that 'Newton's laws do not provide a useful starting point' do not follow; one can equally describe the open sample by Newton's laws with explicit reservoir forces or boundary terms. The framing should be changed to 'the sample is not an isolated system on the alpha-relaxation timescale,' which is a valid and useful point.
[§3, k-space claim] The assertion that Eq. (2) 'corresponds to decay rates that are k-independent' is not derived; no equation of motion or correlation function is analyzed to show this. Even if one accepted the source-term representation, the k-dependence of density correlations would depend on the spatial correlations of the events b_μ and r_μ, which Eq. (2) does not specify. Since the comparison with conserved variables' k² decay is a central argument for the non-conserved character of density, this step should either be proved or removed.

Minor objections (5)

[Abstract] The abstract contains a duplicated word: 'violated for for highly viscous liquids' should read 'violated for highly viscous liquids.'
[§1] The phrase 'the kinematic diffusion constant increases' is imprecise; the text earlier defines the kinematic viscosity ν, not a 'diffusion constant,' and the terminology is confusing.
[§3] The statement 'Like the Coulomb electrical field u(r) has zero divergence' is misleading, since the Coulomb field has a δ-function divergence at its source; the intended statement is presumably that the divergence vanishes away from the event.
[References] Ref. [16] lists three publications used together, but the text cites it for multiple distinct claims (solidity, Eq. (2), k-independent decay rates); the text should identify which of the three supports each claim.
[Eq. (1)] The characteristic time τ_A is defined by an integral over an infinite upper limit; for slowly decaying autocorrelation functions this definition needs a convergence caveat, though this does not affect the main argument.

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The paper

The paper claims

For highly viscous liquids, momentum and energy are not conserved over laboratory time scales.

The reading

This paper asks what principles should underpin a theory of viscous liquids near the glass transition. It argues that in the laboratory, highly viscous liquids exchange momentum and energy with the measuring equipment on the same time scale as the alpha relaxation, so these quantities are not conserved within the sample. It further argues that because a viscous liquid is a 'solid that flows,' relaxing through rare, localized flow events, the coarse-grained density field behaves as a non-conserved field, with no long-range density change from individual events. If correct, the standard conservation-law-based equations of liquid dynamics do not apply to the highly viscous regime, and stochastic descriptions become the natural starting point.

What carries the argument

The argument runs on two objects. The first is the characteristic time $\tau_A$ defined in Eq. (1) for any fluctuating quantity $A$; comparing the momentum and energy diffusion times with the $\alpha$ relaxation time shows that momentum and energy leave the sample before the liquid relaxes. The second is Eq. (2), the coarse-grained time derivative of density written as a sum of instantaneous, localized flow events $b_\mu \delta(r-r_\mu)\delta(t-t_\mu)$; the displacement field of each event goes as $u \propto r/r^3$ to leading order, which is divergence-free and produces no long-range density change. This makes density appear non-conserved in coarse-grained descriptions, giving $k$-independent decay rates rather than the $k^2$ behavior of conserved fields.

Extended reading notes

Core claim

The paper's central claim is that for highly viscous liquids under realistic laboratory conditions, the fundamental conservation laws fail within the sample: momentum and energy diffuse to and from the measuring apparatus within one alpha relaxation time, so they are not conserved, while particle number, though strictly conserved, appears to be violated in coarse-grained descriptions because density changes arrive as localized point events. The consequence is that descriptions based on Newton's second law and conservation laws, including mode-coupling theory, cannot provide a proper foundation for viscous liquid dynamics. Instead, the appropriate tools are stochastic descriptions such as Langevin or Smoluchowski equations for density, dipole density, and stress.

Load-bearing premise

The conclusion that particle number is apparently violated rests on assuming relaxation happens through rare, spatially localized flow events whose elastic displacement field is divergence-free at long range; if rearrangements are extended or long-range correlated, the non-conserved density picture fails.

Editorial extensions

If this is right

Momentum-based hydrodynamics and mode-coupling theory, both built on Newton's second law and conservation laws, cannot be expected to describe the highly viscous liquid phase.
The relevant linear response functions for viscous liquids are those of a system continuously exchanging momentum and energy with its surroundings, not the constant-momentum or isolated-system response functions.
Coarse-grained density fluctuations in viscous liquids decay with wavevector-independent rates, so hydrodynamic treatments that assume particle-number conservation misrepresent density dynamics.
A stochastic description of viscous liquid dynamics is the natural starting point, for example Langevin or Smoluchowski equations for density, dipole density, and stress tensor.
Whether momentum and energy appear conserved depends on sample size and thermal contact; once the alpha relaxation time exceeds the relevant diffusion times, laboratory measurements necessarily exchange these quantities with the apparatus.

Reading between the lines

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

A direct test of the density claim would be to measure the low-wavevector decay rate of density fluctuations in a deeply supercooled liquid: conserved-density hydrodynamics predicts a $k^2$ dependence, while Eq. (2) predicts a plateau, and this can be checked with existing simulation data.
If momentum and energy exchange with the apparatus is real, then dielectric or mechanical spectra near the glass transition should depend on sample size and electrode or thermal coupling; such finite-size dependence would be a physical signature rather than a numerical artifact.
The two kinds of frequency-dependent response functions distinguished in the paper could be compared in the same material by varying sample thickness, and a measurable difference would mark the onset of the viscous regime.
The flow-event picture implies that spatial correlations of relaxation should be localized, so measuring four-point density correlations would probe the key assumption behind Eq. (2) directly.

Original abstract

The laws expressing conservation of momentum and energy apply to any isolated system, but these laws are violated for for highly viscous liquids under laboratory conditions because of the unavoidable interactions with the measuring equipment over the long times needed to study the dynamics. Although particle number conservation applies strictly for any liquid, the solidity of viscous liquids implies that even this conservation law is apparently violated in coarse-grained descriptions of density fluctuations.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,324 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.29 · human participants 0.03 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.05 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v4. Results and the main claim are underlined; show every role.

The laws expressing conservation of momentum and energy apply to any isolated system, but these laws are violated for for highly viscous liquids under laboratory conditions because of the unavoidable interactions with the measuring equipment over the long times needed to study the dynamics. Although particle number conservation applies strictly for any liquid, the solidity of viscous liquids implies that even this conservation law is apparently violated in coarse-grained descriptions of density fluctuations.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The laws expressing conservation of momentum and energy apply to any isolated system, but these laws are violated for for highly viscous liquids under laboratory conditions because of the unavoidable interactions with the measuring equipment over the long times needed to study the dynamics.	result 0.77	0.21	0.69	asserted 1.00
s1	Although particle number conservation applies strictly for any liquid, the solidity of viscous liquids implies that even this conservation law is apparently violated in coarse-grained descriptions of density fluctuations.	result 0.79	0.25	0.19	asserted 0.63

What kind of paper the classifier reports this is: position or commentary 0.74 · evidence from argument 0.95.

What was examined, and what was not

sources read

arXiv listing abstract v4

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

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S sections
T body sentences
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (25)

[1]

W. Kauzmann, Chem. Rev. 43 219 (1948)

View on Pith 1948
[2]

G. Harrison, The dynamic properties of supercooled liquids (Academic Press, New York, 1976)

View on Pith 1976
[3]

S. Brawer, Relaxation in viscous liquids and glasses (American Ceramic Society, Columbus, Ohio, 1985)

View on Pith 1985
[4]

S. R. Elliott, Physics of amorphous materials (Longman Scientiﬁc, London, 1990)

View on Pith 1990
[5]

C. A. Angell, Science 267 1924 (1995)

View on Pith 1995
[6]

P. G. Debenedetti, Metastable liquids: Concepts and principles (Princeton University Press, Princeton, 1996)

View on Pith 1996
[7]

C. A Angell, K. L. Ngai, G. B. McKenna, P. F. McMillan, and S . W. Martin, J. Appl. Phys. 88 3113 (2000)

View on Pith 2000
[8]

C. Alba-Simionesco, C. R. Acad. Sci. Paris (Ser. IV) 2 203 (2001)

View on Pith 2001
[9]

G. Tammann, J. Soc. Glass Technol. 9 166 (1925)

View on Pith 1925
[10]

J. P. Boon and S. Yip, Molecular hydrodynamics (McGraw-Hill, New York, 1980)

View on Pith 1980
[11]

J. P. Hansen and I. R. McDonald, Theory of simple liquids 2nd Ed. (Academic Press, New York, 1986)

View on Pith 1986
[12]

J.–L. Barrat and J.–P. Hansen, Basic concepts for simple and complex liquids (Cambridge University Press, Cambridge, 2003)

View on Pith 2003
[13]

W. G¨ otze and L. Sj¨ ogren, Rep. Progr. Phys. 55 241 (1992)

View on Pith 1992
[14]

S. P. Das, Rev. Mod. Phys. 76 785 (2004)

View on Pith 2004
[15]

The Debye-Stokes-Einstein relation does not always ap ply, but the occasional 1-3 order of magnitude deviations do not aﬀect the argument presented here. See, e.g.: F. Fujara, B. G eil, H. Sillescu, and G. Fleischer, Z. Phys. B 88 195 (1992); F. H. Stillinger and J. A. Hodgdon, Phys. Rev. E 50 2064 (1994); G. Diezemann, H. Sillescu, G. Hinze, and R. B¨ oh...

View on Pith 1992
[16]

J. C. Dyre, Phys. Rev. E 72 Art. No. 011501 (2005); Europhys. Lett. 71 646 (2005); e-print cond-mat/0602362

View on Pith Pith review arXiv 2005
[17]

F. H. Stillinger and T. A. Weber, Phys. Rev. A 28 2408 (1983)

View on Pith 1983
[18]

M. Goldstein, J. Chem. Phys. 51 3728 (1969)

View on Pith 1969
[19]

A. Heuer, Phys. Rev. Lett. 78 4051 (1997)

View on Pith 1997
[20]

S. Sastry, P. G. Debenedetti, and F. H. Stillinger, Natu re 393 554 (1998). 4

View on Pith 1998
[21]

L. Angelani, R. Di Leonardo, G. Ruocco, A. Scala, and F. S ciortino, Phys. Rev. Lett. 85 5356 (2000)

View on Pith 2000
[22]

T. B. Schrøder, S. Sastry, J. C. Dyre, and S. C. Glotzer, J . Chem. Phys. 112 9834 (2000)

View on Pith 2000
[23]

Phenomenologically one often deﬁnes a solid as a system which does not ﬂow, implying that a “solid that ﬂows” is a contradiction in terms. Whether a system ﬂows or not, howeve r, is a matter of time scale; even a crystal ﬂows at ﬁnite temperature: In a gravitational ﬁeld, for instance, a verti cally ﬂat crystalline sample has lower free energy than, e.g ....

View on Pith
[24]

J. C. Dyre, Phys. Rev. E 59 2458 (1999)

View on Pith 1999
[25]

L. D. Landau and E. M. Lifshitz, Theory of elasticity , 2nd Ed. (Pergamon, London, 1970). “Ordinary” liquids Viscous liquids Relaxation time Picoseconds Seconds, years, ... Momentum conservation Crucially important Disobeyed Energy conservation Crucially important Disobeyed Particle number conservation Crucially important Apparently disobeyed Fig. 1 Basic...

View on Pith 1970

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

W. Kauzmann, Chem. Rev. 43 219 (1948)

View on Pith 1948

[2] [2]

G. Harrison, The dynamic properties of supercooled liquids (Academic Press, New York, 1976)

View on Pith 1976

[3] [3]

S. Brawer, Relaxation in viscous liquids and glasses (American Ceramic Society, Columbus, Ohio, 1985)

View on Pith 1985

[4] [4]

S. R. Elliott, Physics of amorphous materials (Longman Scientiﬁc, London, 1990)

View on Pith 1990

[5] [5]

C. A. Angell, Science 267 1924 (1995)

View on Pith 1995

[6] [6]

P. G. Debenedetti, Metastable liquids: Concepts and principles (Princeton University Press, Princeton, 1996)

View on Pith 1996

[7] [7]

C. A Angell, K. L. Ngai, G. B. McKenna, P. F. McMillan, and S . W. Martin, J. Appl. Phys. 88 3113 (2000)

View on Pith 2000

[8] [8]

C. Alba-Simionesco, C. R. Acad. Sci. Paris (Ser. IV) 2 203 (2001)

View on Pith 2001

[9] [9]

G. Tammann, J. Soc. Glass Technol. 9 166 (1925)

View on Pith 1925

[10] [10]

J. P. Boon and S. Yip, Molecular hydrodynamics (McGraw-Hill, New York, 1980)

View on Pith 1980

[11] [11]

J. P. Hansen and I. R. McDonald, Theory of simple liquids 2nd Ed. (Academic Press, New York, 1986)

View on Pith 1986

[12] [12]

J.–L. Barrat and J.–P. Hansen, Basic concepts for simple and complex liquids (Cambridge University Press, Cambridge, 2003)

View on Pith 2003

[13] [13]

W. G¨ otze and L. Sj¨ ogren, Rep. Progr. Phys. 55 241 (1992)

View on Pith 1992

[14] [14]

S. P. Das, Rev. Mod. Phys. 76 785 (2004)

View on Pith 2004

[15] [15]

The Debye-Stokes-Einstein relation does not always ap ply, but the occasional 1-3 order of magnitude deviations do not aﬀect the argument presented here. See, e.g.: F. Fujara, B. G eil, H. Sillescu, and G. Fleischer, Z. Phys. B 88 195 (1992); F. H. Stillinger and J. A. Hodgdon, Phys. Rev. E 50 2064 (1994); G. Diezemann, H. Sillescu, G. Hinze, and R. B¨ oh...

View on Pith 1992

[16] [16]

J. C. Dyre, Phys. Rev. E 72 Art. No. 011501 (2005); Europhys. Lett. 71 646 (2005); e-print cond-mat/0602362

View on Pith Pith review arXiv 2005

[17] [17]

F. H. Stillinger and T. A. Weber, Phys. Rev. A 28 2408 (1983)

View on Pith 1983

[18] [18]

M. Goldstein, J. Chem. Phys. 51 3728 (1969)

View on Pith 1969

[19] [19]

A. Heuer, Phys. Rev. Lett. 78 4051 (1997)

View on Pith 1997

[20] [20]

S. Sastry, P. G. Debenedetti, and F. H. Stillinger, Natu re 393 554 (1998). 4

View on Pith 1998

[21] [21]

L. Angelani, R. Di Leonardo, G. Ruocco, A. Scala, and F. S ciortino, Phys. Rev. Lett. 85 5356 (2000)

View on Pith 2000

[22] [22]

T. B. Schrøder, S. Sastry, J. C. Dyre, and S. C. Glotzer, J . Chem. Phys. 112 9834 (2000)

View on Pith 2000

[23] [23]

Phenomenologically one often deﬁnes a solid as a system which does not ﬂow, implying that a “solid that ﬂows” is a contradiction in terms. Whether a system ﬂows or not, howeve r, is a matter of time scale; even a crystal ﬂows at ﬁnite temperature: In a gravitational ﬁeld, for instance, a verti cally ﬂat crystalline sample has lower free energy than, e.g ....

View on Pith

[24] [24]

J. C. Dyre, Phys. Rev. E 59 2458 (1999)

View on Pith 1999

[25] [25]

L. D. Landau and E. M. Lifshitz, Theory of elasticity , 2nd Ed. (Pergamon, London, 1970). “Ordinary” liquids Viscous liquids Relaxation time Picoseconds Seconds, years, ... Momentum conservation Crucially important Disobeyed Energy conservation Crucially important Disobeyed Particle number conservation Crucially important Apparently disobeyed Fig. 1 Basic...

View on Pith 1970