Linear Landau damping, Schr\"{o}dinger equation, and fluctuation theorem

Hideo Sugama

REVIEW 2 major objections 4 minor 16 references

Linear Landau damping, Schr\"{o}dinger equation, and fluctuation theorem

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper shows that linear Landau damping obeys the fluctuation theorem, so the apparent irreversibility of collisionless plasma damping coexists with exact time-reversal symmetry and detailed balance between energy-loss and gain events.

desk verdict The finite-N fluctuation theorem for linear Landau damping is correct and new; the claimed N→∞ limit is an unproved assumption, and the Gaussian ensemble used is not a probability measure on the Hilbert space. read the letter →

arxiv 2505.23110 v2 pith:7CFOQNQE submitted 2025-05-29 physics.plasm-ph

Hideo Sugama This is my paper

classification physics.plasm-ph

keywords LandaudampingVlasov-PoissonsystemSchrödingerequationrepresentationfluctuationtheoremCase-VanKampenmodesstochasticrelativeentropycollisionlessplasmairreversibility

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 tries to show that Landau damping, the decay of electric-field oscillations in a collisionless plasma, is not a paradox but a statistical-mechanical process with a built-in arrow of time. It rewrites the linearized Vlasov-Poisson system as a Schrödinger equation, whose unitary evolution preserves an exact quadratic invariant built from the perturbed distribution function. From that representation it defines a stochastic relative entropy $\Delta S$ for random initial perturbations and proves the fluctuation theorem $P(\Delta S)/P(-\Delta S)=\exp(\Delta S)$, together with its integral version and a non-negative ensemble-averaged entropy production. In the specific Gaussian ensemble studied, $\Delta S=Q(1/T_{\rm res}-1/T_0)$, where $Q$ is the decrease in electric-field energy and $T_0$, $T_{\rm res}$ are effective temperatures of the collective mode and the higher-order velocity-space modes that absorb the energy. A sympathetic reader would care because this puts the irreversible decay of plasma waves into the same nonequilibrium-statistical-mechanics family as other reversible-dynamics fluctuation theorems.

What carries the argument

The carrying object is the Schrödinger representation of linear Landau damping: the state vector $|\psi(\tau)\rangle=\hat A|\tilde f(\tau)\rangle$, the Hermitian Hamiltonian $\hat H=\hat A\hat\Xi\hat A$ with its ladder structure in the Hermite basis, and the unitary evolution $\hat U(\tau)=\exp(-i\tau\hat H)$. This makes the conservation law and time-reversal symmetry explicit. The second ingredient is the finite $N_{\rm cvk}$-dimensional subspace spanned by Case-Van Kampen eigenvectors satisfying $\langle N_{\rm cvk}|{\rm CVK},\zeta\rangle=0$, which gives a one-to-one correspondence between the first $N_{\rm cvk}$ Hermite components and an invariant subspace of exact Hamiltonian eigenstates; that finite unitary dynamics is where the fluctuation theorem is formulated and numerically validated.

What would settle it

Compute the probability density $P(\Delta S)$ for the same Gaussian ensemble at substantially larger truncations, say $N_{\rm cvk}=50$ or $100$, and at late times beyond $\omega_{\rm p}t=5$; if the ratio $P(\Delta S)/P(-\Delta S)$ departs from $\exp(\Delta S)$ or the $N_{\rm cvk}$-dependence does not converge, the infinite-dimensional claim fails. An independent check would be a reversible Vlasov simulation of many small random perturbations whose field-energy changes are binned into $P(Q)$, with the prediction that $\ln[P(Q)/P(-Q)]=Q(1/T_{\rm res}-1/T_0)$.

Watch

Extended reading notes

Core claim

The central claim is that the linearized Vlasov-Poisson equation for a single Fourier mode is equivalent to a Schrödinger equation $i\,d|\psi\rangle/d\tau = \hat H|\psi\rangle$ with a Hermitian Hamiltonian $\hat H=\hat A\hat\Xi\hat A$, where $\hat A$ modifies the $n=0$ Hermite component and $\hat\Xi$ is the velocity operator. The squared norm $\langle\psi|\psi\rangle$ is conserved and equals four times the invariant $D[f_1]=E^{(2)}/T - S_f^{(2)}$, the difference between second-order energy and entropy perturbations. Using discrete Case-Van Kampen eigenstates selected by $\langle N_{\rm cvk}|{\rm CVK},\zeta\rangle=0$, the paper constructs a finite-dimensional unitary evolution and proves, following the standard fluctuation-theorem argument, that the stochastic relative entropy obeys $P(\Delta S)/P(-\Delta S)=\exp(\Delta S)$ and $\langle e^{-\Delta S}\rangle=1$. For a Gaussian initial ensemble with mode-dependent inverse temperatures, $\Delta S=Q(1/T_{\rm res}-1/T_0)$, so Landau damping is interpreted as transfer of electric-field energy from the $n=0$ state at effective temperature $T_0$ to the $n\ge1$ states acting as a cooler reservoir at $T_{\rm res}=\rho T_0$; the numerical test with $N_{\rm cvk}=20$ and $10^6$ samples verifies the ratio at moderate entropies.

Load-bearing premise

The argument rests on assuming that the truncated set of Case-Van Kampen modes used in the numerics really converges to the true infinite-dimensional plasma dynamics; if the truncated subspace does not match the full system, the fluctuation theorem is only a property of the toy model.

Editorial extensions

If this is right

The irreversibility of Landau damping is quantified by an exponentially asymmetric probability ratio: field-energy gain events are suppressed by $\exp(-\Delta S)$ relative to loss events.
The ensemble-averaged entropy production equals the Kullback-Leibler divergence between the evolved and initial probability distributions, so damping is literally information loss about the initial velocity-space perturbation.
Electric-field energy decay can be re-described as heat flow from a collective mode with effective temperature $T_0$ into higher-order velocity-space modes at lower temperature $T_{\rm res}=\rho T_0$.
Because the theorem is exact for every finite $N_{\rm cvk}$ and the $N_{\rm cvk}=10$ and $20$ results coincide for $\omega_{\rm p}t\le2$, the fluctuation relation is expected to hold for the infinite-dimensional system in that time range.
In the full nonlinear system, the same field-energy loss $Q$ reappears as kinetic-energy gain, linking the linear fluctuation theorem to quasilinear energy conservation.

Reading between the lines

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

If the construction extends to multiple Fourier modes and to other phase-mixing linear waves, fluctuation relations of this form may be generic for collisionless relaxation, not special to one-mode electron Landau damping.
The effective-reservoir picture suggests a practical diagnostic: measure the statistics of $Q$ in particle-in-cell or Vlasov simulations starting from random small-amplitude perturbations and check whether $\ln[P(Q)/P(-Q)]$ is linear with slope $1/T_{\rm res}-1/T_0$.
The paper's initial ensemble is an artificial Gaussian superposition of modes; whether real thermal or turbulent initial conditions sample such an ensemble, and hence whether the theorem describes actual spontaneous fluctuations, is not established here.
A clean numerical extension would push the truncation to much larger $N_{\rm cvk}$ and to late times to see whether the fluctuation relation survives where the $N_{\rm cvk}=10$ and $20$ results separate, which would locate the physical time limit of the theorem.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

Here's what you should know about this one. The finite-dimensional part is solid. The paper takes the linearized Vlasov-Poisson system, rewrites it as a Schrödinger equation via a Hermite expansion (which Ameri et al. already did), and then, for any finite truncation Ncvk, constructs a Gaussian ensemble on the initial state that is time-reversal symmetric. From that, the detailed fluctuation theorem P(ΔS)/P(−ΔS)=exp(ΔS) follows by standard arguments, and the Monte Carlo verifies it. That's a correct, citable result, and the discrete Case–Van Kampen spectral representation used to set up the truncation is a genuinely new addition.

The two-temperature reading of ΔS as Q(1/T_res−1/T0) is neat but essentially built into the choice β_n=β0/ρ; it's an interpretation of the result rather than an independent discovery. Not a flaw, just don't oversell it.

The problem is the jump to the infinite-dimensional system. The initial Gaussian (15) has ⟨||Ψ(0)||²⟩ growing like Ncvk, so on the Hilbert space ℓ² the ensemble is not a probability measure—it lives on the space of all sequences, where the unitary evolution exp(−iτH) is not defined on typical draws. The finite-N subspaces are also not nested: S_N imposes ψ_N=0 while S_{N+1} imposes ψ_{N+1}=0, so there is no single limiting stochastic process. The numerical agreement between Ncvk=10 and 20 at short times covers low-order moments only; the fluctuation theorem constrains the full distribution, including the tails, which the data do not resolve. So the statement that the theorem 'is considered to hold' for Ncvk→∞ is an assumption, not a consequence of the finite-N identities. The stress-test note makes this precise, and I think it's right.

Minor issues: the invariance of D[f1] is asserted without proof, though it's a standard quadratic invariant and easily checked; the fluctuation theorem derivation is sketched, though for finite N it's a textbook application; and there are no error bars or code, but the Monte Carlo is simple enough.

Who should read this? People working on fluctuation theorems in kinetic theory and on quantum formulations of the Vlasov equation. The finite-N theorem is worth having. The infinite-N claim should not be taken as established.

My recommendation: accept for peer review, but the referee should insist on either a proof of a well-defined limit (e.g., with a trace-class covariance that keeps samples in the Hilbert space) or an explicit restriction of the claim to finite N. As it stands, the paper conflates a proved theorem with a plausible extrapolation.

Referee Report

2 major / 4 minor

Summary. The manuscript reformulates the linearized Vlasov-Poisson system as a time-dependent Schrödinger equation in a Hermite basis, with a Hamiltonian H = A Ξ A. It introduces a quadratic invariant D[f1] that is the difference of normalized field energy and second-order entropy, interprets its conservation as unitarity, and constructs finite-dimensional invariant subspaces from discrete Case-Van Kampen eigenvectors satisfying ⟨Ncvk|CVK,ζ⟩=0. For a Gaussian initial ensemble with inverse temperatures β0 and β0/ρ, the stochastic relative entropy ΔS is shown to equal Q(1/T_res−1/T0), where Q is the electric-field energy loss. The fluctuation theorem P(ΔS)/P(−ΔS)=e^{ΔS} is stated and verified numerically with 10^6 samples for Ncvk=20.

Significance. The finite-N construction is internally consistent, and the numerical verification is a useful demonstration that fluctuation-theorem relations can be exhibited in a collisionless plasma model. The mapping to a Schrödinger equation and the existence of the quadratic invariant are valuable connections to known spectral theory of the linearized Vlasov-Poisson system. However, the physical claim as advertised extends to the infinite-dimensional Vlasov-Poisson system, and that extension is not established by the manuscript. As a finite-dimensional model, the result is a sound and interesting illustration; as an infinite-dimensional statement, it requires a rigorous limit argument that is currently absent.

major comments (2)

[Paragraph after Eq. (16)] The statement that the fluctuation theorem 'is considered to hold for the infinite-dimensional system as the Ncvk→∞ limit' is not supported by the argument given. With the initial Gaussian of Eq. (15), ⟨||Ψ(0)||²⟩=2β0^{-1}[1+ρ(Ncvk−1)] diverges as Ncvk→∞, so the limiting ensemble is not a probability measure on the finite-energy Hilbert space on which e^{−iτH} acts. The invariant subspaces S_N defined by ψ_N=0 are not nested: imposing ψ_N=0 for different N selects different boundary conditions, so the truncated processes do not converge to a single stochastic process on a fixed state space. Agreement of Ncvk=10 and Ncvk=20 for ω_pt≤2 concerns low-order moments and does not control the full distribution P(ΔS), especially its tails, which are exactly what Eq. (13) constrains. The theorem should be stated for finite Ncvk, or the limit claim should be replaced with a proof.
[Eqs. (2) and (13)] The two central derivations are only sketched. The conservation of D[f1] is asserted with 'It can be shown', and the fluctuation theorem is introduced with 'following a procedure similar to Ref. 13'. For a self-contained Letter, provide the derivation of Eq. (2) (or a precise reference with equation numbers) and spell out the proof of Eq. (13), including how the time-reversal operator T (with T H T = −H) and the symmetry P[Tψ;0]=P[ψ;0] imply the detailed balance, and state exactly which classes of initial densities the theorem covers.

minor comments (4)

[Before Eq. (16)] The equality defining the normalized mean squared components is inconsistent with Eq. (15): from Eq. (15), ⟨|Ψ_n|²⟩=1/β_n, hence β0⟨|Ψ_n|²⟩/2 = (1/2)⟨|Ψ_n|²⟩/⟨|Ψ_0|²⟩ and ⟨||Ψ||²⟩=β0^{-1}[1+ρ(N−1)], not 2β0^{-1}[...]. Please correct the factor 2 and the figure normalization.
[Fig. 2] Please clarify whether the plotted ratio is the empirical P(ΔS)/P(−ΔS) with the theoretical exp(ΔS) as a reference line, and state the number of samples per ΔS bin; the statement 'better accuracy for smaller ΔS' would be more informative with error bars.
[After Eq. (16)] The interpretation of the n≥1 modes as a 'thermal reservoir' with temperature T_res is an ensemble choice rather than a physical bath; this should be stated explicitly to avoid overclaiming that the reservoir is part of the Vlasov-Poisson dynamics.
[Finite CVK subspace construction] The paper would benefit from one sentence explaining why the selected CVK states with ⟨N|CVK,ζ⟩=0 are exactly the eigenvectors of the N×N submatrix H in Eq. (10), since the tridiagonal structure is what validates the one-to-one correspondence claimed in the text.

Circularity Check

1 steps flagged · score 3.0 of 10

Fluctuation theorem is a general unitary-dynamics identity; the specific entropy-energy relation is fixed by the Gaussian ansatz and effective-temperature definitions.

self definitional [Eqs. (15)-(16) and surrounding text]
"Here, we assume βn = β0/ρ for n = 1, 2, · · ·, Ncvk − 1, where β0 > 0 and 0 < ρ < 1. Then, we obtain ... ∆S[ψ(0);τ ] = Q(1/Tres − 1/T0), where ... 1/T0 ≡ 4β0(1+κ2)/T and 1/Tres ≡ 1/(T0ρ)."

ΔS is defined in Eq. (11) as the log-ratio of the initial Gaussian probability densities. Inserting the chosen βn values and using conservation of the total norm, this log-ratio is exactly a constant times the change in |ψ0|², which is then identified with the electric-field-energy loss Q. Since T0 and Tres are defined in terms of β0 and ρ, the claimed entropy-production formula is an algebraic identity of the chosen ensemble rather than an independent consequence of the Vlasov-Poisson dynamics. The 'thermal reservoir' temperatures are names for the Gaussian parameters, so this part of the result is constructed by the ansatz.

full rationale

The Schrödinger mapping and the unitary evolution are legitimate reformulations of the linearized Vlasov-Poisson system. The fluctuation theorem P(ΔS)/P(−ΔS)=exp(ΔS) follows from the definition of the stochastic relative entropy plus time-reversal symmetry, so it is a general mathematical identity and not circular. The main constructed element is Eq. (16): once βn=β0/ρ is chosen and T0 and Tres are defined through β0 and ρ, the relation ΔS=Q(1/Tres−1/T0) is algebraically forced. This is an explicit modeling ansatz rather than a prediction from first principles; it gives physical interpretation but does not undermine the fluctuation theorem itself. The Ncvk→∞ extrapolation is an unproved assumption about the continuum limit, which is a correctness gap rather than a circularity. No load-bearing self-citation chain is present.

Assumptions & free parameters 4 free parameters · 6 assumptions · 1 invented entities

The paper rests on one physical linearization assumption, one single-mode assumption, standard completeness of Case-Van Kampen modes, a finite-truncation correspondence that is asserted rather than proved, a chosen Gaussian initial ensemble with two free temperatures, and the standard time-reversal symmetry needed for fluctuation theorems. The two-temperature reservoir is an interpretive construction without independent evidence.

free parameters (4)

beta0 = not specified (positive)
Inverse temperature of the n=0 (electric-field) mode in the initial Gaussian ensemble; T0 is defined through it and the value affects ΔS but not the form of the fluctuation theorem.
rho = 1/20 in numerics
Ratio of reservoir to field-mode temperature; chosen to be less than 1 so that ΔS is proportional to Q with the advertised positive sign; a free modeling parameter.
Ncvk = 20 in numerics, 10 for convergence check
Truncation dimension of the Hamiltonian; numerical convergence parameter, not a physical constant.
kappa = 1/2 in numerics
Normalized wavenumber chosen for the numerical example; the fluctuation theorem is intended for general κ but only one value is simulated.

assumptions (6)

domain assumption The nonlinear Vlasov term is neglected; the linearized Vlasov-Poisson system is used throughout.
Explicitly stated near Eq. (2): 'Hereafter, we neglect the nonlinear term.' The Schrödinger mapping requires linearity.
domain assumption The system is periodic with a single Fourier mode k=2π/L; higher harmonics are not included.
Stated after Eq. (2): 'only the mode with k=2π/L is considered, and higher-order harmonics are not included.'
standard math The Case-Van Kampen modes form a complete orthonormal continuum basis for the present Hilbert space.
Invoked to construct eigenvectors of H; completeness is asserted without proof in this Letter and relies on standard plasma kinetic theory.
ad hoc to paper The Ncvk discrete Case-Van Kampen vectors selected by ⟨Ncvk|CVK,ζ⟩=0 span an invariant subspace in one-to-one correspondence with the first Ncvk Hermite components.
This is the key finite-truncation assumption; the one-to-one correspondence and the sufficiency of the boundary condition are stated but not proved.
ad hoc to paper The initial probability density is the Gaussian P[ψ(0);0] = Z^-1 exp(-Σβ_n|ψ_n(0)|^2) with β_n=β0/ρ for n≥1.
This ensemble is chosen so that ΔS equals Q(1/T_res−1/T0); it is not derived from the Vlasov-Poisson dynamics.
standard math The time-reversal operator T anti-commutes with H and the initial distribution is invariant under T.
Used in the standard fluctuation theorem proof; T is the diagonal matrix with (−1)^n entries and corresponds to velocity reversal.

invented entities (1)

Thermal reservoir of Hermite modes n≥1 with effective temperature T_res=T0ρ
purpose: Gives a thermodynamic interpretation of ΔS as entropy generated when electric-field energy (mode n=0 at T0) flows to velocity-space structure (modes n≥1 at lower T_res).
The reservoir is not a separate physical system; its temperature is fixed by the free parameter ρ and has no measurable handle independent of the model.

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Cite this review

Pith. "Pith review of Linear Landau damping, Schr\"{o}dinger equation, and fluctuation theorem." pith.science (2026). https://pith.science/paper/7CFOQNQE

@misc{pith2026250523110,
  author       = {Pith},
  title        = {Pith review of: Linear Landau damping, Schr\"odinger equation, and fluctuation theorem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7CFOQNQE}},
  note         = {Machine review of arXiv:2505.23110}
}

read the original abstract

A linearized Vlasov-Poisson system of equations is transformed into a Schr\"{o}dinger equation, which is used to demonstrate that the fluctuation theorem holds for the relative stochastic entropy, defined in terms of the probability density functional of the particle velocity distribution function in the Landau damping process. The difference between the energy perturbation, normalized by the equilibrium temperature, and the entropy perturbation constitutes a time-independent invariant of the system. This invariant takes the quadratic form of the perturbed velocity distribution function and corresponds to the squared amplitude of the state vector that satisfies the Schr\"{o}dinger equation. Exact solutions, constructed from a discrete set of Hamiltonian eigenvectors, are employed to formulate and numerically validate the fluctuation theorem for the Landau damping process. The results offer new insights into the formulations of collisionless plasma processes within the framework of nonequilibrium statistical mechanics.

Figures

Figures reproduced from arXiv: 2505.23110 by the authors.

**Figure 2.** FIG. 2. (a) Probability density function [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

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Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

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Pith tools

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

[1] [1]

L. D. Landau, J. Exp.\ Theor.\ Phys.\ 16, 574 (1946)

work page 1946

[2] [2]

K. M. Case, Ann.\ Phys. 7 , 349 (1959)

work page 1959

[3] [3]

N. G. Van Kampen, Physica 21 , 949 (1955)

work page 1955

[4] [4]

D. R. Nicholson, Introduction to Plasma Theory (John Wiley & Sons, New York, 1983), Chap. 6

work page 1983

[5] [5]

G. W. Hammett and F. W. Perkins, Phys.\ Rev.\ Lett.\ 64 , 3019 (1990)

work page 1990

[6] [6]

Sugama, Phys.\ Plasmas 6 , 3527 (1999)

H. Sugama, Phys.\ Plasmas 6 , 3527 (1999)

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Sugama and T.-H

H. Sugama and T.-H. Watanabe, J. Plasma Phys.\ 72 , 825 (2006)

work page 2006

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A. Biancalani, F. Palermo, C. Angioni, A. Bottino, F. Zonca, Phys. Plasmas 23 , 112115 (2016)

work page 2016