Pith. sign in

REVIEW 2 major objections 4 minor 57 references

Low-temperature anomalies in one-dimensional exactly solvable fluids

T0 review · 2 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper proves that in exactly solvable one-dimensional fluids, negative thermal expansion arises whenever the interparticle potential's curvature increases with distance or the potential has a singularity inside its attraction well…

desk verdict Exact 1D results give the cleanest proof yet that phi''' > 0 drives low-T NTE; the non-analyticity claim is real but oversold as a general mechanism. read the letter →

arxiv 2507.01568 v2 pith:PYBYYMKD submitted 2025-07-02 cond-mat.soft cond-mat.stat-mech

Igor Travěnec , Ladislav Šamaj This is my paper · ORCID
classification cond-mat.softcond-mat.stat-mech MSC 82B0582B2182B26
keywords one-dimensionalfluidsexactlysolvablemodelsnegativethermalexpansionhardrodsnearest-neighborinteractionslow-temperatureequationofstatecompressibilityfactor
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

The paper studies one-dimensional fluids of hard rods with soft nearest-neighbor attractions that are exactly solvable, and asks when the fluid contracts on heating at fixed pressure — negative thermal expansion. It derives a low-temperature expansion for the mean spacing, $l(T,p)=x^*(p)-\frac{T}{2}\,\frac{\phi'''(x^*)}{[\phi''(x^*)]^2}+o(T)$, which shows that any smooth single-minimum potential whose curvature increases with distance, $\phi'''(x^*)>0$, gives the anomaly at low temperatures. It then shows, in three exactly solved models, that a singularity (a kink) inside the attraction basin also produces the anomaly, regardless of whether the kink lies to the left, at, or to the right of the minimum. The paper also proves that quantizing the classical pure hard-rod gas, which has no anomaly, does not induce one. If correct, these are model-independent sufficient conditions for low-temperature contraction in exactly solvable one-dimensional fluids, together with plateau and double-plateau isotherms of the compressibility factor.

What carries the argument

The central object is the saddle-point expansion of the exact isothermal-isobaric equation of state, $l(T,p)=-\partial_p\ln\hat\Omega(\beta p)$, where $\hat\Omega(s)$ is the Laplace transform of the pair Boltzmann factor. At low temperature the integral is dominated by the minimum of $f(x)=\phi(x)+px$; the ground-state spacing $x^*(p)$ solves $p=-\phi'(x^*)$, and the first correction follows from $\partial_p\ln\phi''(x^*)$, giving $-\phi'''(x^*)/[\phi''(x^*)]^2$ as the prefactor of $T/2$. This identity does the argument's work: its sign decides normal versus negative thermal expansion for analytic potentials. At the incompressibility pressure $p_i$ the analytic expansion changes to a $\sqrt{T}$ correction, while above $p_i$ the spacing grows as $a+T/(p-p_i)$; at the jump pressure $p_j$ of the cubic model the interior and boundary minima coexist, and for non-analytic potentials the exact Laplace transforms yield explicit low-temperature expansions with negative linear-$T$ terms.

What would settle it

For the cubic potential (5.23) at $p=1$, evaluating the exact equation of state (2.8) numerically at small $T$ should give $l(T,1)-x^*(1)\approx -T/5$; a positive slope at arbitrarily small $T$ would overturn the curvature condition. For the fused linear-quadratic potential (6.5) at $p=2$, the same evaluation should show $l(T,2)\approx 5/4-4\sqrt{\pi}\,T^{3/2}$; absence of this decrease would refute the singularity mechanism.

Watch

Extended reading notes

Core claim

On its own terms, the paper's central discovery is that negative thermal expansion in these one-dimensional fluids is controlled by two features of the soft interaction potential inside the attraction well. For analytic potentials with a single minimum, the exact low-temperature expansion gives the linear coefficient of the mean spacing as $-\frac{1}{2}\phi'''(x^*)/[\phi''(x^*)]^2$, so the spacing decreases with temperature precisely when the potential's curvature increases with separation. This occurs for the cubic potential that is sharp to the right of its minimum, and the anomaly is accompanied by a discontinuous jump in the ground-state spacing at a pressure $p_j$ where the interior minimum of $\phi(x)+px$ becomes degenerate with the hard-core boundary. For pressures just above $p_j$ the anomaly persists up to about $p\approx 1.75$, as shown numerically. For non-analytic potentials, three fused models (kink at, left of, and right of the minimum) all have exact low-temperature expansions with negative linear-$T$ coefficients, establishing singularity inside the basin of attraction as a second sufficient mechanism. Quantization of pure hard rods is shown not to produce the anomaly, indicating that quantum mechanics is not the origin of NTE in these models.

Load-bearing premise

The low-temperature expansion assumes $\phi(x)+px$ has a single non-degenerate interior minimum for the relevant pressures; this fails at the jump pressure, where the hard-core boundary and interior minima are degenerate, and the NTE interval just above the jump is therefore supported by numerical evaluation of the exact equation of state rather than by a complete asymptotic derivation.

Editorial extensions

If this is right

  • For a smooth single-minimum attractive nearest-neighbor potential, low-temperature NTE is decided by the sign of $\phi'''(x^*)$: a positive third derivative at the ground-state spacing gives contraction, while a negative one gives normal expansion on heating.
  • At the incompressibility pressure $p_i$, the linear-$T$ correction gives way to a $\sqrt{T}$ correction, and for $p>p_i$ the spacing grows as $a+T/(p-p_i)$, so the anomaly is confined to pressures where the interior minimum branch dominates (with a numerically verified extension up to $p\approx 1.75$ in the cubic model).
  • If the soft potential is non-analytic anywhere inside the attraction well, the exact low-temperature expansion has a negative linear-$T$ coefficient, producing NTE irrespective of whether the singularity sits at, left of, or right of the potential minimum.
  • Quantizing the pure hard-rod gas does not create NTE, indicating that quantum fluctuations are irrelevant to the anomaly in these one-dimensional fluids.
  • Low-temperature isotherms of the compressibility factor show a wide plateau for single-minimum potentials and a double plateau with a steep rise near $n\approx 1/a_m$ for models with a ground-state spacing jump, signalling a large density window in which the pressure response is very weak.

Reading between the lines

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

  • The two sufficient mechanisms suggest a design rule for NTE materials beyond the usual two-length-scale repulsive cores: use a smooth attraction whose curvature steepens with separation, or introduce a controlled kink inside the well; the one-dimensional exact solutions then guarantee low-temperature contraction without fitted parameters.
  • The persistence of NTE for pressures just above the jump pressure is verified numerically rather than by a complete asymptotic derivation; a full treatment would need to include the coexisting hard-core boundary branch alongside the interior branch in the saddle-point expansion.
  • The 'as soon as' claim for non-analytic potentials rests on three piecewise examples; proving it for the whole class of single-minimum non-analytic potentials, such as a cusp with a vanishing derivative at the minimum, remains an open problem.
  • The double-plateau compressibility factor implies an extended regime of almost pressure-independent density; a similar effect might be engineered in experimental one-dimensional quantum gases whose interactions approximate a kinked potential.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper studies exactly solvable one-dimensional fluids of hard rods with attractive nearest-neighbor soft interactions, using the exact isothermal-isobaric partition function. For smooth single-minimum potentials it derives the low-temperature expansion l(T,p)=x*(p)+(T/2)d/dp ln φ''(x*(p))+o(T), which gives negative thermal expansion (NTE) when φ'''>0, and it analyzes the square-well, quadratic, and cubic models. For piecewise non-analytic potentials it presents three models (two linear segments; linear-quadratic fusions with the non-analyticity left or right of the minimum) and finds NTE in certain parameter ranges. It also treats quantum hard rods via the Bethe-ansatz solution and concludes that quantization does not induce NTE, and it plots compressibility-factor isotherms showing plateau structures.

Significance. The paper's main strengths are its exact equation of state (2.8)–(2.9), the parameter-free sign criterion for smooth potentials (Eq. (2.23)), and the explicit low-T asymptotics for several non-analytic models, all checked against numerical integration of the exact formulas. The quantum hard-rod section uses the exact Bethe-ansatz solution and correctly identifies that the classical NTE-free behavior is not turned into NTE by quantum fluctuations. If the universal non-analyticity claim were true, the paper would establish a new mechanism for NTE; however, that claim is not supported by the presented arguments and is contradicted by the paper's own formulas. The paper remains valuable as a set of exact model calculations and a correct smooth-potential criterion, but the abstract and conclusion overstate the scope of the non-analytic result.

major comments (2)
  1. [§6 and §7] The universal non-analyticity criterion is not supported. The abstract states that NTE is present 'if the potential exhibits a singularity within the basin of attraction,' and §7 states that NTE occurs 'as soon as the soft potential is non-analytic at a point within the interval x∈(a,a′).' The three examples of §§6.1–6.3 do not imply this. In fact, Eq. (6.3) for the two-linear-segment model has a positive leading coefficient α(p) when the minimum is closer to the hard core; e.g., with a=1, a′=2, ε=1, ε_a=0, and a_m=1.3, α(p)>0 for 0<p<0.95. Similarly, in Eq. (6.7) the leading coefficient is positive for 2<p<2.5. The conclusion in §7 itself concedes that no precise mathematical criterion was identified. The authors should replace the universal claim with the weaker statement that NTE occurs for the piecewise models studied, or prove a condition on the potential that guarantees the negative sign of the leading low-temperature coefficient.
  2. [§5.2.2, Eq. (5.29)] At p=p_j=27/16 the function f(x)=φ(x)+px has two degenerate minima, the interior point x*=3/2 and the hard-core boundary x=1. Eq. (5.29) includes p_j in the branch '0<p≤27/16' and keeps only the interior saddle, yielding a linear-T coefficient −1/(9−4p_j). The boundary minimum, although subdominant in weight (its integrand contributes O(1/β) versus the interior O(β^{−1/2})), shifts the mean spacing by O(√T) with a negative coefficient because l_bnd−x*=−1/2. Hence the correct leading correction at p_j is a negative √T term, not the displayed finite O(T) term. The NTE conclusion at p_j survives, and the numerical evidence for p in (p_j,1.75) is convincing, but the asymptotic formula (5.29) at the endpoint should be stated separately with the boundary contribution included.
minor comments (4)
  1. [§6.3, Eq. (6.8)] In the definition of ϕ(x) for the fusion of quadratic and linear models, the second branch is also labeled 'if 1 < x < 7/4'; the second line should be 'if 7/4 < x < 2.'
  2. [§5.2.2] The sentence 'in spite of the positive prefactor 1/p to the to the linear T-correction to l(0)' contains a duplicated 'to the,' and a similar duplication appears in the caption of Fig. 9.
  3. [§2.2, Eq. (2.23)] The expression ∂/∂p ln φ''(x*(p)) should use a total derivative d/dp, since x*(p) itself depends on p; this removes ambiguity in the derivation.
  4. [§7] The opening of the conclusion, 'we did not identify a precise mathematical criterion,' is in tension with the deterministic phrasing in the abstract ('NTE anomaly is present if ... or if the potential exhibits a singularity'); please reconcile these statements after revising the claim.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: all NTE criteria are derived from the exact partition-function/EoS with stated model inputs, and the self-citations used are independent background results.

full rationale

The paper's central derivation flows from the exact isothermal-isobaric expression for the mean spacing, Eq. (2.8)-(2.9), to the saddle-point low-temperature expansion Eq. (2.23), l(T,p) = x*(p) - (T/2) phi'''(x*)/[phi''(x*)]^2 + o(T). The NTE criterion 'curvature of the soft potential increases with distance' is therefore an algebraic sign condition on the input potential, not an output of any fit or of a prior NTE claim. The non-analytic-potential results in Sections 6.1-6.3 are obtained from explicit Laplace transforms and low-T expansions with stated parameters; no fitted parameter is renamed as a prediction. The paper's self-citations are [33] (a textbook by one of the authors used for the standard Bethe-ansatz solution of quantum hard rods) and [45] (the authors' earlier repulsive-core NTE study used as background). Neither is load-bearing for the present conclusions: the quantum-hard-rod solution has stated assumptions that do not include the target result, and the present NTE analysis is carried out from the exact EoS rather than imported from [45]. The over-general claim in Section 7 that NTE occurs 'as soon as' the potential is non-analytic is supported only by three examples and the paper itself concedes that no precise mathematical criterion was identified; that is a rigor/correctness limitation, not circular reasoning. Similarly, the use of Eq. (2.23) at the jump pressure p_j in Eq. (5.29) is a validity concern about a degenerate saddle point, but the omitted boundary contribution is not a fitted input and the exact-integral numerics are independent checks. No step in the derivation reduces to its own input by construction.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The derivations use the exact 1D isothermal-isobaric partition function with no new degrees of freedom. The only hand-picked numbers are model parameters for the examples; the NTE criteria depend on the sign of the third derivative or the presence of a non-analyticity, not on the parameter values. Background theorems (Radin-Schulman periodicity, Bethe ansatz) are cited from the published literature.

free parameters (2)
  • Cubic shape parameter t = 1/3 (phi'''>0) and 3/5 (phi'''<0)
    Chosen by hand to illustrate the two signs of the third derivative; the NTE result holds for intervals of t (jump for t in [0,1/2)), so the numerical value is not fitted to any data.
  • Potential scales a, a', a_m, epsilon, epsilon_a = a=1, a'=2, a_m=3/2, epsilon=1, epsilon_a=0
    Set the length and energy units and the position of the well minimum for the figures. The qualitative NTE criteria do not depend on these specific values.
assumptions (4)
  • standard math Radin-Schulman theorem: the ground state of 1D hard-core particles with finite-range soft potentials is periodic
    Invoked in Section 2.2 (reference [52]) to justify the ground-state analysis via the periodic spacing x*(p).
  • standard math Bethe-ansatz exact solution of 1D quantum hard rods (cited from reference [33])
    Section 3.2 uses the analytic Bethe ansatz, Eqs. (3.7)-(3.8), as the starting point for the quantum no-NTE conclusion.
  • domain assumption Restriction a' <= 2a in the pair potential (1.3)
    Ensures each particle interacts only with nearest neighbors, making the Laplace-transform exact solution (2.8)-(2.9) valid; all analyzed potentials obey this.
  • domain assumption Soft-core potential is attractive with a single minimum, Eqs. (2.1)-(2.5)
    Defines the class of potentials studied; the square well with degenerate minimum is the sole exception, treated separately.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Low-temperature anomalies in one-dimensional exactly solvable fluids." pith.science (2026). https://pith.science/paper/PYBYYMKD

@misc{pith2026250701568,
  author       = {Pith},
  title        = {Pith review of: Low-temperature anomalies in one-dimensional exactly solvable fluids},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PYBYYMKD}},
  note         = {Machine review of arXiv:2507.01568}
}
read the original abstract

Previous experiments and numerical simulations have revealed that a limited number of two- and three-dimensional particle systems contract in volume upon heating isobarically. This anomalous phenomenon is known as negative thermal expansion (NTE). The present paper focuses on the possibility of NTE in exactly solvable one-dimensional fluids. Firstly, the quantization of classical pure hard rods (free of NTE) does not induce NTE which indicates an unimportant role of quantum mechanics in the topic. Secondly, the classical hard rods with various types of soft nearest-neighbor interactions that contain a basin of attraction with only one minimum are investigated. The ground-state analysis reveals that, for certain potentials, increasing the pressure can lead to a discontinuous jump in the mean spacing between particles. The low-temperature analysis of the exact equation of state indicates that the NTE anomaly is present if the curvature of the soft potential increases with the distance between particles or if the potential exhibits a singularity within the basin of attraction. Isotherms of the compressibility factor, which measures the deviation of the thermodynamic behavior of a real gas from that of an ideal gas, demonstrate typical plateau or double-plateau shapes in large intervals of particle density.

Figures

Figures reproduced from arXiv: 2507.01568 by the authors.

Figure 1
Figure 1. figure 1. Formula (5.5) tells us that the incompressibility pressure [PITH_FULL_IMAGE:figures/full_fig_p016_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

57 extracted references · 45 canonical work pages

  1. [1]

    Doyle, S., Kuh s, W.F.: Lat- tice constants and thermal expansion of H 2O and D 20 ice Ih in between 10 and 265 K

    R¨ ottger, K., Endriss, A., Ihringer, J., S. Doyle, S., Kuh s, W.F.: Lat- tice constants and thermal expansion of H 2O and D 20 ice Ih in between 10 and 265 K. Acta Crystallogr. B 50, 644–648 (1994), https://doi.org/10.1107/S0108768194004933

    work page doi:10.1107/s0108768194004933 1994

  2. [2]

    Evans, J.S.O.: Negative thermal expansion materials. J. Chem. Soc. Dalton Trans. 19, 3317–3326 (1999), https://doi.org/10.1039/A904297K

    work page doi:10.1039/a904297k 1999

  3. [3]

    Greenwood, N.N., Earnshaw, A.: Chemistry of the elements , 2nd edn., Butterworth-Heinemann, (1997), https://doi.org/10.101 6/C2009-0-30414-6

    work page 1997

  4. [4]

    Nature (London) 409, 318–321 (2001), https://doi.org/10.1038/35053024

    Errington, J.R., Debenedetti, P.G.: Relationship betwe en structural order and the anomalies of liquid water. Nature (London) 409, 318–321 (2001), https://doi.org/10.1038/35053024

    doi:10.1038/35053024 2001

  5. [5]

    Nano Lett

    Yoon, D., Son, Y-W., Cheong, H.: Negative thermal expansi on coefficient of graphene measured by Raman spectroscopy. Nano Lett. 11, 3227–3231 (2011), https://doi.org/10.1021/nl201488g

    doi:10.1021/nl201488g 2011

  6. [6]

    Science 272, 90–92 (1996), https://doi.org/10.1126/science.272.5258.90

    Mary, T.A., Evans, J.S.O., Vogt, T., Sleight, A.W.: Negat ive thermal ex- pansion from 0.3 to 1050 K in ZrW 2O8. Science 272, 90–92 (1996), https://doi.org/10.1126/science.272.5258.90

    work page doi:10.1126/science.272.5258.90 1996

  7. [7]

    Biernacki, S., Scheffler, M.: Negative thermal expansion o f diamond and zinc-blende semiconductors, Phys. Rev. Lett. 63, 290–293 (1989), https://doi.org/10.1103/PhysRevLett.63.290

    work page doi:10.1103/physrevlett.63.290 1989

  8. [8]

    Forster, P.M., Yokochi, A., Sleight, A.W.: Enhanced nega tive thermal expansion in Lu 2W3O12. J. Solid State Chem. 140, 157–158 (1998), https://doi.org/10.1006/jssc.1998.7967

    work page arXiv 1998

Show all 57 references
  1. [9]

    Rechtsman, M.C., Stillinger, F.H., Torquato, S.: Negati ve thermal expansion in single-component systems with isotropic interactions. J. Phys. Chem. A 111, 12816–12821 (2007), https://doi.org/10.1021/jp076859l

    2007 doi

  2. [10]

    Negative thermal expansion in single-component systems with isotropic interactions

    Kuzkin, V.A.: Comment on “Negative thermal expansion in single-component systems with isotropic interactions”. J. Phys. Chem. A 118, 9793–9794 (2014), https://doi.org/10.1021/jp509140n

    2014 doi

  3. [11]

    Batten, R.D., Stillinger, F.H., Torquato, S.: Novel low -temperature behav- ior in classical many-particle systems. Phys. Rev. Lett. 103, 050602 (2009), https://doi.org/10.1103/PhysRevLett.103.050602

    2009 doi

  4. [12]

    Batten, R.D., Stillinger, F.H., Torquato, S.: Interact ions leading to disordered ground states and unusual low-temperature behavior. Phys. Rev. E 80, 031105 (2009), https://doi.org/10.1103/PhysRevE.80.031105

    2009 doi

  5. [14]

    Velasco, E., Mederos, L., Navascues, G., Hemmer, P.C., S tell, G.: Complex phase behavior induced by repulsive interactions. Phys. Re v. Lett. 85, 122– 125 (2000), https://doi.org/10.1103/PhysRevLett.85.12 2

    2000 doi

  6. [15]

    Ryzhov, V.N., Stishov, S.M.: Repulsive step potential: A model for liquid-liquid phase transition. Phys. Rev. E 67, 010201(R) (2003), https://doi.org/10.1103/PhysRevE.67.010201

    2003 doi

  7. [16]

    Kumar, P., Buldyrev, S.V., Sciortino, F., Zaccarelli, E ., Stanley, H.E.: Static and dynamic anomalies in a repulsive spherical ramp liquid: Theory and simulation. Phys. Rev. E 72, 021501 (2005), https://doi.org/10.1103/PhysRevE.72.021501

    2005 doi

  8. [17]

    Xu, L., Buldyrev, S.V., Angell, C.A., Stanley, H.E.: The rmody- namics and dynamics of the two-scale spherically symmetric Jagla ramp model of anomalous liquids. Phys. Rev. E 74, 031108 (2006), https://doi.org/10.1103/PhysRevE.74.031108 31

    2006 doi

  9. [18]

    Yan, Z., Buldyrev, S.V., Giovambattista, N., Debenedet ti, P.G., Stanley, H.E.: Family of tunable spherically symmetric potentials t hat span the range from hard spheres to waterlike behavior. Phys. Rev. E 73, 051204 (2006), https://doi.org/10.1103/PhysRevE.73.051204

    2006 doi

  10. [19]

    Barros de Oliveira, A., Netz, P.A., Barbosa, M.C.: Which mechanism underlies the water-like anomalies in core-softened potentials?. Eu r. Phys. J. B 64, 481– 486 (2008), https://doi.org/10.1140/epjb/e2008-00101- 6

    2008 doi

  11. [20]

    Europhys

    Barros de Oliveira, A., Netz, P.A., Barbosa, M.C.: An ubi quitous mechanism for water-like anomalies. Europhys. Lett. 85, 36001 (2009), https://doi.org/10.1209/0295-5075/85/36001

    2009 doi

  12. [21]

    Buldyrev, S.V., Malescio, G., Angell, C.A., Giovambatt ista, N., Prestipino, S., Saija, F., Stanley, H.E., Xu, L.: Unusual phase behavior of o ne-component systems with two-scale isotropic interactions. J. Phys.: C ondens. Matter 21, 504106 (2009), https://doi.org/10.1088/0953...

    2009 doi

  13. [22]

    Zhou, S., Solana, J.R.: Inquiry into thermodynamic beha vior of hard spheres plus repulsive barrier of finite height. J. Chem. Phys. 131, 204503 (2009), https://doi.org/10.1063/1.3265984

    2009 doi

  14. [23]

    Coquand, O., Sperl, M.: Temperature expansions in the sq uare shoul- der fluid II: Thermodynamics. J. Chem. Phys. 152, 124113 (2020), https://doi.org/10.1063/1.5142662

    2020 doi

  15. [24]

    Hemmer, P.C., Stell, G.: Fluids with several phase trans itions. Phys. Rev. Lett. 24, 1284–1287 (1970), https://doi.org/10.1103/PhysRevLet t.24.1284

    1970 doi

  16. [25]

    Stell, G., Hemmer, P.C.: Phase transitions due to softne ss of the potential core. J. Chem. Phys. 56, 4274–4286 (1972), https://doi.org/10.1063/1.1677857

    1972 doi

  17. [26]

    Jagla, E.A.: Core-softened potentials and the anomalou s properties of water. J. Chem. Phys. 111, 8980–8986 (1999), https://doi.org/10.1063/1.480241

    1999 doi

  18. [27]

    , Frenkel, D.: Quasibinary amorphous phase in a three-dimensional system of particles with repulsive-shoulder interactions

    Fomin, Yu.D, Gribova, N.V., Ryzhov, V.N., Stishov, S.M. , Frenkel, D.: Quasibinary amorphous phase in a three-dimensional system of particles with repulsive-shoulder interactions. J. Chem. Phys. 129, 064512 (2008), https://doi.org/10.1063/1.2965880

    2008 doi

  19. [28]

    Gribova, N.V., Fomin, Yu.D., Frenkel, D., Ryzhov, V.N.: Waterlike thermo- dynamic anomalies in a repulsive-shoulder potential syste m. Phys. Rev. E 79, 051202 (2009), https://doi.org/10.1103/PhysRevE.79.05 1202

    2009 doi

  20. [29]

    Physica A 244, 358–369 (1997) https://doi.org/10.1016/S0378- 4371(97)00246-X

    Stillinger, F.H., Stillinger, D.K.: Negative thermal e xpansion in the Gaussian core model. Physica A 244, 358–369 (1997) https://doi.org/10.1016/S0378- 4371(97)00246-X

    1997 doi

  21. [30]

    Tonks, L.: The complete equation of state of one, two, and three di- mensional gases of hard elastic spheres. Phys. Rev. 50, 955–963 (1936), https://doi.org/10.1103/PhysRev.50.955

    1936 doi

  22. [31]

    Zernike, F., Prins, J.A.: Die beugung von rontgenstrahl en in flus- sigkeiten als effekt der molekulanordnung. Z. Physik 41, 184–194 (1927), https://doi.org/10.1007/BF01391926

    1927 doi

  23. [32]

    Chaos, Solitons & Fractals 14, 23–28 (2002), https://doi.org/10.1016/S0960-0779(01) 00178-3

    Wadati, M., Kato, G.: One-dimensional hard-core Bose ga s. Chaos, Solitons & Fractals 14, 23–28 (2002), https://doi.org/10.1016/S0960-0779(01) 00178-3

    2002 doi

  24. [33]

    ˇSamaj, L., Bajnok, Z.: Introduction to the statistical phys ics of integrable many-body systems, Cambridge University Press , (2013), https://doi.org/10.1017/CBO9781139343480

    2013 doi

  25. [34]

    Takahashi, H.: A simple method for treating the statisti cal mechanics of one-dimensional substances. Proc. Phys. Math. Soc. Japan 24, 60–62 (1942), https://doi.org/10.11429/ppmsj1919.24.0 60

    1942 doi

  26. [35]

    Bishop, M., Boonstra, M.: Exact partition functions for some one-dimensional models via the isobaric ensemble. Am. J. Phys. 51, 564–566 (1983), https://doi.org/10.1119/1.13204

    1983 doi

  27. [36]

    Salsburg, Z.W., Zwanzig, R.W., Kirkwood, J.G.: Molecul ar distribution func- tions in a one-dimensional fluid. J. Chem. Phys. 21, 1098–1107 (1953), https://doi.org/10.1063/1.1699116

    1953 doi

  28. [37]

    Lebowitz, J.L., Zomick, D.: Mixtures of hard spheres wit h nonadditive diam- eters: Some exact results and solution of PY equation. J. Che m. Phys. 54, 3335–3346 (1971), https://doi.org/10.1063/1.1675348 32

    1971 doi

  29. [38]

    Fluid Phase Equilib

    Heying, M., Corti, D.S., The one-dimensional fully non- additive binary hard rod mixture: exact thermophysical properties. Fluid Phase Equilib. 220, 85– 103 (2004), https://doi.org/10.1016/j.fluid.2004.02.01 8

    2004 doi

  30. [39]

    Santos, A.: Exact bulk correlation functions in one-dim ensional nonadditive hard-core mixtures. Phys. Rev. E 76, 062201 (2007), https://doi.org/10.1103/PhysRevE.76.062201

    2007 doi

  31. [40]

    Ben-Naim, A., Santos, A.: Local and global properties of mixtures in one- dimensional systems, Part II: Exact results for the Kirkwoo d-Buff Integrals. J. Chem. Phys. 131, 164512 (2009), https://doi.org/10.1063/1.3256234

    2009 doi

  32. [41]

    : One dimensional lattice fluid mixture with nearest neighbour interactions

    Sahnoun, A.Y., Djebbar, M., Benmessahib, T., Bakhti, B. : One dimensional lattice fluid mixture with nearest neighbour interactions. J. Phys. A: Math. Theor. 57, 325007 (2024), 10.1088/1751-8121/ad6538

    2024 doi

  33. [42]

    XIII Simple Models o f Equilibrium and Nonequilibrium Phenomena ed

    Percus, J.K.: Exactly solvable models of classical many -body systems, In Studies in Statistical Mechanics, vol. XIII Simple Models o f Equilibrium and Nonequilibrium Phenomena ed. J. L. Lebowitz, North Holland, (1987)

    1987

  34. [43]

    Santos, A.: A Concise Course on the Theory of Classical Fl uids, Chapter 5, Springer, (2016), https://doi.org/10.1007/978-3-319-2 9668-5

    2016 doi

  35. [44]

    Jones, G.L.: The low-temperature properties of the one- dimensional fluid. J. Stat. Phys. 44, 237–248 (1985), https://doi.org/10.1007/BF01010915

    1985 doi

  36. [45]

    Travˇ enec, I., ˇSamaj, L.: 1D fluids with repulsive nearest-neighbour inter ac- tions: Low-temperature anomalies. J. Phys. A: Math. Theor. 58, 195005 (2025), https://doi.org/10.1088/1751-8121/add22c

    2025 doi

  37. [46]

    Radin, C.: Low temperature and the origin of crystalline symmetry. Int. J. Mod. Phys. B 1, 1157–1191 (1987), https://doi.org/10.1142/S021797928 7001675

    1987 doi

  38. [47]

    Le Bris, C., Lions, P.-L.: From atoms to crystals: a mathe matical journey. Bull. Am. Math. Soc. 42, 291–363 (2005), https://doi.org/10.1090/S0273-0979-0 5- 01059-1

    2005 doi

  39. [48]

    EMS Surv

    Blanc, X., Lewin, M.: The crystallization conjecture: a review. EMS Surv. Math. Sci. 2, 255–306 (2015), https://doi.org/10.4171/EMSS/13

    2015 doi

  40. [49]

    Physica A 92, 343–361 (1978), https://doi.org/10.1016/0378-4371(78) 90136-X

    Ventev¨ ogel, W.J.: On the configuration of a one-dimensi onal system of inter- acting particles with minimum potential energy per particl e. Physica A 92, 343–361 (1978), https://doi.org/10.1016/0378-4371(78) 90136-X

    1978 doi

  41. [51]

    Physica A 99, 569–580 (1979), https://doi.org/10.1016/0378-4371(79) 90072-4

    Ventev¨ ogel, W.J., Nijboer, B.R.A.: On the configuratio n of systems of inter- acting particles with minimum potential energy per particl e. Physica A 99, 569–580 (1979), https://doi.org/10.1016/0378-4371(79) 90072-4

    1979 doi

  42. [52]

    Radin, C., Schulman, L.S.: Periodicity of classical gro und states. Phys. Rev. Lett. 51, 621–622 (1983), https://doi.org/10.1103/PhysRevLett. 51.621

    1983 doi

  43. [53]

    Montero, A.M., Santos, A.: Triangle-Well and Ramp Inter actions in One- Dimensional Fluids: A Fully Analytic Exact Solution. J. Sta t. Phys. 175, 269– 288 (2019), https://doi.org/10.1007/s10955-019-02255- x

    2019 doi

  44. [54]

    Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Ser ies, and Products, 6th edn., Academic Press, (2000), https://doi:10.1090/s0025 -5718-1972-0415970-6

    2000 doi

  45. [55]

    Evans, M.R., Majumdar, S.N., Pagonabarraga, I., Trizac , E.: Condensation transition in polydisperse hard rods. J. Chem. Phys. 132, 014102 (2010), https://doi.org/10.1063/1.3263913

    2010 doi

  46. [56]

    Science 305, 1125–1128 (2004), https://doi.org/10.1126/science.1100700

    Kinoshita, T., Wenger, T., Weiss, D.S.: Observation of a one- dimensional Tonks-Girardeau gas. Science 305, 1125–1128 (2004), https://doi.org/10.1126/science.1100700

    2004 doi

  47. [57]

    Nature 466, 597–600 (2010), https://doi.org/10.1038/nature09259

    Haller, E., Hart, R., Mark, M.J., Danzl, J.G., Reichsoll ner, L., Gustavsson, M., Dalmonte, M., Pupillo, G., Nagerl, H-C.: Pinning quantum ph ase transition for a Luttinger liquid of strongly interacting bosons. Nature 466, 597–600 (2010), https://doi.org/10.1038/nature09259

    2010 doi

  48. [58]

    Kr¨ uger, P., Hofferberth, S., Mazets, I.E., Lesanovsky, I., Schmiedmayer, J.: Weakly interacting Bose gas in the one-dimensional limit. P hys. Rev. Lett. 105, 265302 (2010), https://doi.org/10.1103/PhysRevLett.1 05.265302

    2010 doi

  49. [59]

    Langen, T.: Non-equilibrium dynamics of one-dimension al Bose gases, Chapter 2, Springer, (2015), https://doi.org/10.1007/978-3-319 -18564-4

    2015 doi

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.