Spectral Softening and the Structural Breakdown of Thermodynamic Equilibrium

Ilki Kim

arxiv: 2604.10216 · v1 · submitted 2026-04-11 · ❄️ cond-mat.stat-mech

Spectral Softening and the Structural Breakdown of Thermodynamic Equilibrium

Ilki Kim This is my paper

Pith reviewed 2026-05-10 15:23 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech

keywords spectral softeningthermodynamic equilibriumadiabatic breakdownquadratic Hamiltonianspartition function divergencedriven systemsphase-space geometrytimescale separation

0 comments

The pith

Spectral softening in driven quadratic systems makes the partition function diverge and breaks adiabaticity over a finite regime.

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

The paper shows that thermodynamic reversibility under slow driving fails near spectral degeneracy in quadratic Hamiltonian systems. As a soft-mode frequency collapses, the system's intrinsic dynamical timescale diverges, destroying the separation between driving and internal motion even for arbitrarily slow changes. This leads directly to a divergence in the canonical partition function, which removes any well-defined equilibrium ensemble. The effect is geometric, appearing in the phase-space structure of the bounded quadratic Hamiltonian and showing up identically in both quantum and classical descriptions.

Core claim

In driven quadratic Hamiltonian systems, adiabatic following is lost once the soft-mode frequency drops below a finite drive-dependent threshold rather than only at exact zero. The resulting divergence of the canonical partition function renders equilibrium ensembles ill-defined, so that quasistatic thermodynamic processes cannot be defined over a finite interval of parameters. This structural breakdown originates from the loss of an intrinsic frequency scale within a bounded quadratic system and is visible in the Wigner phase-space representation as well as its classical counterpart.

What carries the argument

The soft-mode frequency of the driven quadratic Hamiltonian, whose collapse removes the internal frequency scale, produces both a diverging dynamical timescale and a diverging canonical partition function.

If this is right

Adiabaticity fails across a finite band of soft-mode frequencies set by the drive strength rather than only at exact degeneracy.
Quasistatic thermodynamic cycles cannot pass through regions of spectral softening without losing their equilibrium foundation.
The same singular structure appears in the classical phase-space geometry, so the limitation is not quantum-specific.
Equilibrium statistical mechanics loses its footing whenever an intrinsic frequency scale is removed by softening in bounded quadratic systems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Realistic systems with weak anharmonic terms might still exhibit large deviations from equilibrium predictions in the vicinity of the softening threshold.
This geometric constraint could restrict the design of slowly driven devices or thermodynamic engines that must traverse near-degenerate points.
Non-quadratic or open-system extensions might restore a finite timescale through higher-order stabilization, offering a route to test the boundary of the quadratic assumption.

Load-bearing premise

The Hamiltonian stays strictly quadratic and bounded as the soft-mode frequency collapses, without higher-order terms or other physics becoming dominant near degeneracy.

What would settle it

Exact evaluation of the canonical partition function for a concrete time-dependent quadratic oscillator showing divergence precisely when the instantaneous frequency falls below the drive-rate threshold, together with numerical integration of the time-dependent Schrödinger equation revealing non-adiabatic excitations for arbitrarily slow ramps in that same regime.

Figures

Figures reproduced from arXiv: 2604.10216 by Ilki Kim.

read the original abstract

Under sufficiently slow driving, thermodynamics predicts reversible evolution through a sequence of equilibrium states. We show that this expectation fails near spectral degeneracy in driven quadratic Hamiltonian systems. As the soft-mode frequency collapses, the intrinsic dynamical timescale diverges and quadratic confinement is lost, leading to a breakdown of timescale separation and the failure of adiabatic following even under arbitrarily slow driving. More precisely, adiabaticity is lost once the soft-mode frequency falls below a finite, drive-dependent threshold, implying that the breakdown extends over a finite regime rather than being confined to a singular limit. Crucially, this dynamical instability is accompanied by a divergence of the canonical partition function, rendering equilibrium ensembles ill-defined and eliminating the foundation of quasistatic thermodynamic processes. This breakdown does not arise from unbounded Hamiltonians or critical slowing down, but emerges structurally from spectral softening within a bounded quadratic system. Analysis of the Wigner phase-space representation, together with its classical counterpart, reveals the same singular structure, demonstrating that this limitation is not uniquely quantum but originates from the underlying Hamiltonian phase-space geometry. These results show that thermodynamic reversibility is fundamentally constrained, as a direct consequence of the breakdown of equilibrium, whenever spectral softening removes an intrinsic frequency scale.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The finite-threshold loss of adiabaticity in softening quadratics is a concrete claim, but the partition-function divergence only occurs at exactly ω=0, so the link to ill-defined equilibria over a finite interval does not hold.

read the letter

The paper's core observation is that in driven quadratic systems, the soft-mode frequency dropping below a drive-dependent threshold causes adiabatic following to fail even for arbitrarily slow drives. This comes from the diverging intrinsic timescale and loss of quadratic confinement, analyzed through both classical phase space and the Wigner function. That part is new enough to be worth checking against existing adiabatic theorems for time-dependent oscillators. The work also correctly notes that the effect is structural and appears in bounded systems without needing unbounded potentials or external critical points. The phase-space geometry argument is a reasonable way to unify the classical and quantum sides. Credit for spelling out the finite-regime claim rather than defaulting to the singular ω→0 limit. The soft spot is the thermodynamic conclusion. The abstract states that the dynamical instability is accompanied by a divergence of the canonical partition function, rendering equilibrium ensembles ill-defined. For the instantaneous Hamiltonian ½mω(t)²q² + p²/2m, the partition function is proportional to 1/ω (or the quantum coth form), which remains finite for any ω>0 and only diverges at the singular point ω=0. That means the ensembles stay well-defined throughout the open interval where the paper says adiabaticity has already broken. The two claims are not coextensive, so the argument that quasistatic thermodynamics loses its foundation over a finite regime rests on a mismatch. If the full derivation introduces higher-order terms or a different ensemble definition that changes this, it needs to be shown explicitly; otherwise the link between dynamics and thermodynamics is overstated. The paper is aimed at people working on driven statistical mechanics and quantum control who care about the limits of quasistatic assumptions. A reader who already knows the standard harmonic-oscillator partition function will spot the gap quickly. It deserves a serious referee because the dynamical threshold claim is specific and falsifiable, even if the thermodynamic framing needs repair. I would send it to review but flag the partition-function step as the item that must be tightened or corrected.

Referee Report

1 major / 0 minor

Summary. The paper claims that in driven quadratic Hamiltonian systems, spectral softening causes adiabaticity to fail over a finite drive-dependent regime of soft-mode frequencies (below a threshold ω_c > 0) rather than only at the singular limit ω=0. This breakdown arises from diverging intrinsic timescales and loss of quadratic confinement, is accompanied by divergence of the canonical partition function (rendering equilibrium ensembles ill-defined), and originates from Hamiltonian phase-space geometry as revealed by Wigner and classical analyses. The result implies fundamental constraints on thermodynamic reversibility independent of unbounded Hamiltonians or critical slowing down.

Significance. If the central claims hold, the work would identify a structural limitation on quasistatic thermodynamics in bounded quadratic systems undergoing spectral softening, with potential implications for both classical and quantum driven systems. The phase-space approach offers a unified classical-quantum perspective and avoids reliance on fitted parameters or ad-hoc assumptions.

major comments (1)

[Abstract] Abstract (and central claim): The assertion that dynamical instability over the finite regime 0 < ω < ω_c 'is accompanied by a divergence of the canonical partition function, rendering equilibrium ensembles ill-defined' requires explicit support. For the instantaneous quadratic Hamiltonian H = p²/2m + ½ m ω(t)² q², the partition function Z ∝ 1/ω (classical) or Z ∝ 1/sinh(βħω/2) (quantum) remains finite for all ω > 0 and diverges only exactly at ω = 0. Clarify with the relevant derivation or equation how divergence occurs throughout the open interval rather than solely at the singular point; this link is load-bearing for the claim that equilibrium is eliminated over a finite regime.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the precise comment on the abstract and central claim. We address the point directly below, providing the requested clarification and derivation from the phase-space analysis. We have revised the manuscript to strengthen the exposition of this link while preserving the original results.

read point-by-point responses

Referee: [Abstract] Abstract (and central claim): The assertion that dynamical instability over the finite regime 0 < ω < ω_c 'is accompanied by a divergence of the canonical partition function, rendering equilibrium ensembles ill-defined' requires explicit support. For the instantaneous quadratic Hamiltonian H = p²/2m + ½ m ω(t)² q², the partition function Z ∝ 1/ω (classical) or Z ∝ 1/sinh(βħω/2) (quantum) remains finite for all ω > 0 and diverges only exactly at ω = 0. Clarify with the relevant derivation or equation how divergence occurs throughout the open interval rather than solely at the singular point; this link is load-bearing for the claim that equilibrium is eliminated over a finite regime.

Authors: We thank the referee for this observation. The instantaneous canonical partition function is indeed finite for any fixed ω > 0. However, the manuscript's claim concerns the driven system: the Wigner phase-space analysis (Section III and Appendix B) shows that below the drive-dependent threshold ω_c the soft-mode distribution spreads without bound on timescales set by the drive, because the adiabatic invariant is no longer conserved once the intrinsic frequency falls below ω_c. This renders the instantaneous equilibrium ensemble unreachable, even though the formal Z(ω) remains finite. The divergence of Z as ω → 0 is the static signature of the same loss of quadratic confinement that produces the dynamical breakdown at finite ω_c. We have added an explicit derivation (new Eq. (14) and revised text in Section II.C) demonstrating that the phase-space volume explored by the Wigner function diverges as 1/ω for ω < ω_c under the given drive protocol, making the canonical ensemble inapplicable over the open interval. The abstract has been rephrased to read: 'this dynamical instability is accompanied by the limiting divergence of the canonical partition function as the soft-mode frequency softens below a finite threshold, rendering equilibrium ensembles ill-defined over the finite regime.' This clarifies the connection without altering the central result. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives its central claims about the loss of adiabaticity over a finite regime and the accompanying divergence of the partition function directly from the phase-space geometry of the driven quadratic Hamiltonian, analyzed via the Wigner representation and its classical limit. No load-bearing steps reduce by construction to self-definitions, fitted parameters renamed as predictions, or chains of self-citations; the threshold for breakdown and the singular structure emerge from the explicit time-dependent frequency in the Hamiltonian without circular equivalence to the inputs. The derivation remains self-contained against the stated model assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the domain assumption that the system is governed by a quadratic Hamiltonian whose phase-space geometry directly determines the validity of equilibrium ensembles under slow driving.

axioms (2)

domain assumption The Hamiltonian is quadratic and remains bounded even as a mode frequency approaches zero.
Invoked throughout the abstract as the setting in which spectral softening produces the claimed breakdown.
domain assumption Thermodynamic equilibrium is defined via the canonical partition function for slow driving.
The standard premise whose failure is demonstrated when the partition function diverges.

pith-pipeline@v0.9.0 · 5501 in / 1316 out tokens · 37267 ms · 2026-05-10T15:23:21.341648+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

32 extracted references · 32 canonical work pages

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The instantaneous eigenvalues ofˆH1(t)are E(1) n+,n−(t) =E (0) n+,n− +c(t)(5a) with c(t) =− ω2 2 pc(t)2 m(ω2 1 −ω 2

, p 0(t) =− ω2 1 pc(t) ω2 1 −ω 2 2 .(4) These expressions are valid forω1 ̸=ω 2. The instantaneous eigenvalues ofˆH1(t)are E(1) n+,n−(t) =E (0) n+,n− +c(t)(5a) with c(t) =− ω2 2 pc(t)2 m(ω2 1 −ω 2

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A. Messiah,Quantum Mechanics, Vol. II (North-Holland, Amsterdam, 1962). Appendix A: Hamiltonian Structure and Instantaneous Eigenstates We start from the quadratic Hamiltonian atp c = 0, denoted ˆH0, and introduce the standard ladder operators ˆax = r mω1 2ℏ ˆx+ i√2mℏω1 ˆpx ,ˆa y = r mω1 2ℏ ˆy+ i√2mℏω1 ˆpy .(A1) Here bothˆax andˆay are defined with the sa...

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[31]

(2) of the main text)

The Hamiltonian then takes the diagonal formˆH0 =ℏ(ω 1+ω2) ˆn++ℏ(ω1−ω2) ˆn−+ℏω1, whereˆn± = ˆa† ±ˆa± (see Eq. (2) of the main text). The corresponding eigenstates are |n+(0), n−(0)⟩= (ˆa† +)n+(ˆa† −)n− p (n+!)(n−!) |0,0⟩.(A2) Equivalently, introducing the canonical variables ˆQ± = s ℏ 2M ω± (ˆa± + ˆa† ±), ˆP± =−i r Mℏω ± 2 (ˆa± −ˆa† ±),(A3) we obtain the ...

work page
[32]

Substituting these expressions into Eq

, σ v(t) =− √ 2ω 2 2 pc(t) ω2 1 −ω 2 2 (A6) forω 1 ̸=ω 2. Substituting these expressions into Eq. (A5), all linear terms cancel, yieldingˆH4(t) = ˆH0 +c(t) in the form of the undriven quadratic Hamiltonian plus a scalar shift, without modifying the spectrum, where c(t)is a time-dependent real number, as explicitly given in Eq. (5b) of the main text. The i...

work page

[1] [1]

The instantaneous eigenvalues ofˆH1(t)are E(1) n+,n−(t) =E (0) n+,n− +c(t)(5a) with c(t) =− ω2 2 pc(t)2 m(ω2 1 −ω 2

, p 0(t) =− ω2 1 pc(t) ω2 1 −ω 2 2 .(4) These expressions are valid forω1 ̸=ω 2. The instantaneous eigenvalues ofˆH1(t)are E(1) n+,n−(t) =E (0) n+,n− +c(t)(5a) with c(t) =− ω2 2 pc(t)2 m(ω2 1 −ω 2

work page

[2] [2]

This structural change has direct consequences for the canonical partition function, as discussed below

(5b) As the soft-mode point is approached, the quadratic confinement of the(−)normal mode is progressively weakened and vanishes in the limit, where the mode becomes marginal, corresponding to a flat direction in phase space (see Appendix A). This structural change has direct consequences for the canonical partition function, as discussed below. The diver...

work page

[3] [3]

L. D. Landau and E. M. Lifshitz,Statistical Physics, Part 1, 3rd ed. (Pergamon Press, Oxford, 1980)

work page 1980

[4] [4]

Born and V

M. Born and V. Fock, Z. Phys.51, 165 (1928)

work page 1928

[5] [5]

T. Kato, J. Phys. Soc. Jpn.5, 435 (1950)

work page 1950

[6] [6]

M. V. Berry, Proc. R. Soc. London A392, 45 (1984). 8

work page 1984

[7] [7]

Torrontegui, S

E. Torrontegui, S. Ibáñez, S. Martínez-Garaot, M. Modugno, A. del Campo, D. Guéry-Odelin, A. Ruschhaupt, X. Chen, and J. G. Muga, Adv. At. Mol. Opt. Phys.62, 117 (2013)

work page 2013

[8] [8]

Polkovnikov, K

A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalattore, Rev. Mod. Phys.83, 863 (2011)

work page 2011

[9] [9]

Albash and D

T. Albash and D. A. Lidar, Rev. Mod. Phys.90, 015002 (2018)

work page 2018

[10] [10]

Wigner and J

E. Wigner and J. von Neumann, Phys. Z.30, 467 (1929)

work page 1929

[11] [11]

L. D. Landau, Phys. Z. Sowjetunion2, 46 (1932)

work page 1932

[12] [12]

Zener, Proc

C. Zener, Proc. R. Soc. London A137, 696 (1932)

work page 1932

[13] [13]

S. N. Shevchenko, S. Ashhab, and F. Nori, Phys. Rep.492, 1 (2010)

work page 2010

[14] [14]

P. C. Hohenberg and B. I. Halperin, Rev. Mod. Phys.49, 435 (1977)

work page 1977

[15] [15]

Folk and G

R. Folk and G. Moser, J. Phys. A39, R207 (2006)

work page 2006

[16] [16]

U. C. Täuber,Critical Dynamics(Cambridge University Press, Cambridge, 2014)

work page 2014

[17] [17]

Goldenfeld,Lectures on Phase Transitions and the Renormalization Group(Addison-Wesley, 1992)

N. Goldenfeld,Lectures on Phase Transitions and the Renormalization Group(Addison-Wesley, 1992)

work page 1992

[18] [18]

Ruelle,Statistical Mechanics: Rigorous Results(W

D. Ruelle,Statistical Mechanics: Rigorous Results(W. A. Benjamin, New York, 1969)

work page 1969

[19] [19]

E. H. Lieb and R. Seiringer,The Stability of Matter in Quantum Mechanics(Cambridge University Press, Cambridge, 2010)

work page 2010

[20] [20]

Wigner, Phys

E. Wigner, Phys. Rev.40, 749 (1932)

work page 1932

[21] [21]

Hillery, R

M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, Phys. Rep.106, 121 (1984)

work page 1984

[22] [22]

W. H. Zurek, Nature412, 712 (2001)

work page 2001

[23] [23]

Breuer and F

H.-P. Breuer and F. Petruccione,The Theory of Open Quantum Systems(Oxford University Press, Oxford, 2002)

work page 2002

[24] [24]

Weiss,Quantum Dissipative Systems, 4th ed

U. Weiss,Quantum Dissipative Systems, 4th ed. (World Scientific, Singapore, 2012)

work page 2012

[25] [25]

Ingold, inQuantum Transport and Dissipation, edited by T

G.-L. Ingold, inQuantum Transport and Dissipation, edited by T. Dittrich et al. (Wiley-VCH, Weinheim, 1998) pp. 213–248

work page 1998

[26] [26]

Farhi, J

E. Farhi, J. Goldstone, S. Gutmann, and M. Sipser, Science292, 472 (2001)

work page 2001

[27] [27]

Jarzynski, Phys

C. Jarzynski, Phys. Rev. Lett.78, 2690 (1997)

work page 1997

[28] [28]

Talkner, E

P. Talkner, E. Lutz, and P. Hänggi, Phys. Rev. E75, 050102(R) (2007)

work page 2007

[29] [29]

Campisi, P

M. Campisi, P. Talkner, and P. Hänggi, Rev. Mod. Phys.83, 771 (2011)

work page 2011

[30] [30]

Messiah,Quantum Mechanics, Vol

A. Messiah,Quantum Mechanics, Vol. II (North-Holland, Amsterdam, 1962). Appendix A: Hamiltonian Structure and Instantaneous Eigenstates We start from the quadratic Hamiltonian atp c = 0, denoted ˆH0, and introduce the standard ladder operators ˆax = r mω1 2ℏ ˆx+ i√2mℏω1 ˆpx ,ˆa y = r mω1 2ℏ ˆy+ i√2mℏω1 ˆpy .(A1) Here bothˆax andˆay are defined with the sa...

work page 1962

[31] [31]

(2) of the main text)

The Hamiltonian then takes the diagonal formˆH0 =ℏ(ω 1+ω2) ˆn++ℏ(ω1−ω2) ˆn−+ℏω1, whereˆn± = ˆa† ±ˆa± (see Eq. (2) of the main text). The corresponding eigenstates are |n+(0), n−(0)⟩= (ˆa† +)n+(ˆa† −)n− p (n+!)(n−!) |0,0⟩.(A2) Equivalently, introducing the canonical variables ˆQ± = s ℏ 2M ω± (ˆa± + ˆa† ±), ˆP± =−i r Mℏω ± 2 (ˆa± −ˆa† ±),(A3) we obtain the ...

work page

[32] [32]

Substituting these expressions into Eq

, σ v(t) =− √ 2ω 2 2 pc(t) ω2 1 −ω 2 2 (A6) forω 1 ̸=ω 2. Substituting these expressions into Eq. (A5), all linear terms cancel, yieldingˆH4(t) = ˆH0 +c(t) in the form of the undriven quadratic Hamiltonian plus a scalar shift, without modifying the spectrum, where c(t)is a time-dependent real number, as explicitly given in Eq. (5b) of the main text. The i...

work page