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Gibbs measures as local equilibrium KMS states for focusing nonlinear Schr\"odinger equations

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Pith's one-line read For focusing NLS and Hartree equations on the torus in dimensions 1–3, every sufficiently regular local KMS equilibrium state coincides with a local Gibbs measure on the mass ball.

desk verdict A dense but genuinely new converse result: local KMS states for focusing NLS/Hartree are characterized as local Gibbs measures, modulo explicit regularity and connectedness hypotheses that need referee scrutiny. read the letter →

arxiv 2412.05354 v1 pith:V7QWOOSH submitted 2024-12-06 math.AP math-phmath.MPmath.PR

Zied Ammari , Andrew Rout , Vedran Sohinger This is my paper · ORCID
classification math.APmath-phmath.MPmath.PR MSC 35Q5537D3560H0728C2035L05
keywords KMSstateslocalGibbsmeasuresfocusingnonlinearSchrödingerequationHartreeGaussianMalliavincalculusDirichletformsconcentrationinequalities
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 statistical equilibria of focusing nonlinear Schrödinger and Hartree equations on the torus with $d=1,2,3$. Because the focusing nonlinearity makes the global Gibbs measure non-normalizable, the authors work with localized Gibbs measures obtained by cutting off the (renormalized) mass. They prove two directions: every local Gibbs measure is a local KMS state and a stationary solution of the Liouville equation, and conversely every local KMS state whose density is sufficiently regular is a local Gibbs measure on the mass ball $B_R$, up to a normalization constant and up to arbitrary mass outside $B_R$. This gives the first characterization of local thermal equilibria for focusing dispersive PDEs on the torus, and it recovers almost sure global well-posedness for these equations. The interest is that the KMS condition, a standard criterion for thermal equilibrium, singles out exactly the same measures that the Hamiltonian structure suggests.

What carries the argument

The argument runs through Malliavin calculus on the Gaussian measure $\mu_0$ with covariance $A^{-1-s}$, where $A=-\Delta+1$. The central objects are the renormalized mass $M(u)$ (Wick-ordered $\|u\|_{L^2}^2$ in $d=2,3$), the local Gibbs measure with sharp cutoff $\chi^{(1)}_R(M)$, and the local KMS condition, which integrates the Poisson bracket identity against test functions supported on the mass ball. A Gaussian integration by parts formula shifts derivatives from test functions onto the Hamiltonian, while Aida's irreducibility theorem for Dirichlet forms on infinite-dimensional domains turns the resulting differential equation $\nabla(e^{-h_I}\rho)=0$ into constancy of $e^{-h_I}\rho$. The delicate geometric step is proving that the sublevel set $B_R$ is $H^1$-connected, meaning connected along Cameron-Martin translation directions, even though it is not convex for $d=2,3$; the paper constructs explicit paths handling the parallel-component scaling cases $\lambda\in(-\infty,-2)$ and $(-2,-1)$.

What would settle it

Find a probability measure $\mu=\rho\,d\mu_0$ with $\rho\in D^{1,2}(\mu_0)\cap L^4(\mu_0)$ satisfying the local KMS condition whose density on $B_R$ is not proportional to $e^{h_I}$. Concretely, the proof would break if the path construction in Lemma 5.4 failed for some $R$: for a configuration $u\in B_R$ and a direction $w\in B_R(u)$ with $w=\lambda u_n + w^\perp$, the explicit path for $\lambda\in(-\infty,-2)$ or $(-2,-1)$ must stay inside $B_R$; checking those paths on $\mathbb{T}^3$ with a numerical or analytic computation would settle the claim.

Watch

Extended reading notes

Core claim

The central claim is an equivalence theorem. Theorem 2.22 shows that the truncated Gibbs measure $\mu^{(1)}$ defined by density proportional to $e^{h_I}\chi^{(1)}_R(M)$ with respect to the Gaussian measure $\mu_0$ satisfies the local KMS condition of Definition 2.17. Theorem 2.23 proves the converse: if $d\mu=\rho\,d\mu_0$ with $\rho\in D^{1,2}(\mu_0)\cap L^4(\mu_0)$ and $\mu$ is a local KMS state, then $\rho(u)=c_0 e^{h_I(u)}$ for $\mu_0$-almost every $u$ in the mass ball $B_R=\{u: |M(u)|<R\}$. Thus, inside the mass ball, local KMS equilibrium states coincide with local Gibbs measures; outside the ball the measure is arbitrary, which is why the constant $c_0$ is fixed by the total mass. Theorem 2.19 adds that these local Gibbs measures are stationary solutions of the Liouville equation, yielding almost sure global existence as a corollary.

Load-bearing premise

The load-bearing premise is that the region of field configurations with renormalized mass below R is connected along the Cameron-Martin directions, so that the differential equation $\nabla(e^{-h_I}\rho)=0$ forces one constant density, and that the density is smooth enough to lie in the Malliavin Sobolev space $D^{1,2}\cap L^4$.

Editorial extensions

If this is right

  • Every local Gibbs measure is stationary for the Liouville equation and satisfies the local KMS condition, so the KMS criterion is satisfied by the natural truncated equilibria.
  • Any sufficiently regular local KMS state has the exponential-of-interaction Gibbs density on the mass ball, so the local thermal equilibrium class is exactly the local Gibbs class in that region.
  • Focusing NLS and Hartree initial data are almost surely globally well-posed with respect to the local Gibbs measure, recovered here from the Liouville-equation method rather than by direct flow construction.
  • The characterization holds on the torus in dimensions one, two, and three for both the local and Hartree nonlinearities covered by Assumption 2.10.

Reading between the lines

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

  • If the equivalence extends to the full state space, then uniqueness of local equilibria holds only modulo the mass outside the ball; this suggests that ergodicity, if it holds, must be understood relative to the conserved mass, and that mixing may fail because different exterior masses coexist.
  • The $H^1$-connectedness of the nonconvex mass sublevel sets is a standalone geometric fact that may transfer to other constructions of truncated Gibbs measures, for example in stochastic quantization or in mean-field limits of Bose gases.
  • A natural testable extension is to replace the sharp mass cutoff by smooth cutoffs or by other conserved quantities; the same differential-equation-plus-irreducibility strategy would predict the same Gibbs form on each connected component of the resulting sublevel set.
  • The method invites a follow-up: proving ergodicity of the local Gibbs measures for the focusing flows, the second step the authors announce, using the now-complete description of the equilibrium states.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 5 minor

Summary. This paper studies local equilibrium statistical mechanics for focusing NLS and Hartree equations on the torus T^d, d=1,2,3. It introduces local Gibbs measures defined with a cutoff on the (renormalized) mass M(u) and proves three main results: Theorem 2.19 shows these measures solve the Liouville equation; Theorem 2.22 shows they satisfy a local KMS condition; and Theorem 2.23 shows that, under the explicit hypothesis μ=ρdμ0 with ρ∈D^{1,2}(μ0)∩L^4(μ0), every local KMS state agrees with the Gibbs weight e^{h_I} up to a constant on the mass ball B_R. The proof of Theorem 2.23 uses Malliavin calculus, Aida's irreducibility theorem for Dirichlet forms, and a connectedness analysis of the mass sublevel set. The paper also gives a self-contained revision of Bourgain's normalizability proof via concentration inequalities.

Significance. If correct, this is the first local KMS characterization for focusing dispersive PDEs and provides a rigorous link between stationary solutions of the Liouville equation and local Gibbs measures. The paper is carefully written: the finite-dimensional heuristic, the Gaussian integration by parts, and the concentration arguments in Appendix B are detailed, and the main characterization is honestly stated with its hypotheses. The primary weakness is the proof of the H^1-connectedness lemma (Lemma 5.4), specifically the general case in Case 5, where an existence statement for a continuous path is asserted rather than proved. Because this lemma is necessary for Theorem 2.23, the completeness of the proof is affected. Within the stated hypotheses, I found no circularity or demonstrated mathematical error.

major comments (1)
  1. [Section 5, Lemma 5.4] The proof of the H^1-path connectedness of O(u) in Case 5 (after (5.24)) is not complete. The authors reduce the problem to finding a continuous f on [0,1] with f(0)=0, f(1)=1, and F1(t)<f(t)^2<F2(t) for all t, and they verify only the endpoint and positivity conditions in (5.25). Since Lemma 5.4 is the key geometric input for Theorem 2.23, this is a load-bearing point. Please supply a rigorous construction of f or state a general lemma showing that the conditions (5.25) imply the existence of such a continuous f. For example, one can use the homeomorphism Φ(t,y)=(t,(y-F1(t))/(F2(t)-F1(t))) of the open region between F1 and F2 to a rectangle.
minor comments (5)
  1. [Abstract and Section 1] The phrase 'all possible local KMS equilibrium states' overstates the conditional, localized conclusion of Theorem 2.23; please add the assumptions ρ∈D^{1,2}(μ0)∩L^4(μ0) and 'on B_R' to the claim.
  2. [Section 4, proof of Theorem 2.22] In equations (4.24) and (4.25), the limit should be δ→1 (as stated in the surrounding text), not δ→0.
  3. [Equation (4.14)] The last term contains 'χ′_R(M)'; it should be (χ_R^{(δ)})'(M) to match the preceding terms.
  4. [Section 5, Lemma 5.4, Case 4] The displayed formula for f(t) on (t0,1] appears to be missing a division sign; it should read f(t)=((1−t)θ(t))/((1−t0)θ(t0)) f(t0) so that f is continuous at t0 and f(1)=0.
  5. [Throughout] Several symbols appear corrupted in the text (e.g., '/BD', '/greaterorsimilar'); these are likely typesetting artifacts and should be cleaned up in the final version.

Circularity Check

0 steps flagged · score 2.0 of 10

No circularity in the central equivalence: local KMS states are proved to be locally Gibbs via a genuine differential equation, Aida's external irreducibility theorem, and real geometric content in Lemma 5.4; self-citations to [11] are programmatic but constitute independent, verifiable evidence.

full rationale

The paper's central claim (Theorem 2.23) is not circular. The local KMS condition (Definition 2.17) is stated purely in terms of the Poisson bracket {F,G} = <∇F,-i∇G> and the Hamiltonian vector field X = -iA + i∇h_I, with no reference to the Gibbs density e^{h_I}. The converse is proved in two genuinely independent stages: Proposition 5.1 derives the differential equation ∇ρ = ρ∇h_I on B_R from the local KMS condition together with the free-measure KMS identity (Proposition 4.1, whose content for h_I ≡ 0 is exactly the Gaussian integration-by-parts formula of Proposition 3.6); then Aida's theorem (Proposition 5.2, an external result of S. Aida, not of the present authors) converts ∇(e^{-h_I}ρ) = 0 on the H^1-connected set B_R into constancy, yielding ρ = c0 e^{h_I} on B_R. The H^1-connectedness of the non-convex renormalized-mass sublevel set (Lemma 5.4) is real geometric content in d = 2,3, so the conclusion is not built into either the definition of local KMS or the choice of test functions. No parameter is fitted and renamed a prediction, and no uniqueness theorem from the authors' prior work is invoked to forbid alternatives. The self-citations to [7,8,11] are numerous but load-bearing only through facts that are parameter-free, externally verifiable, and do not contain the target result (the Gaussian IPB formula and the free-measure KMS property reduce to standard Gaussian calculus). Two flagged non-circular concerns are weighed here: (i) Theorem 2.23 assumes ρ ∈ D^{1,2}(μ0) ∩ L^4(μ0), so the characterization covers exactly that class; the hypothesis is plainly stated, but it does narrow the abstract's phrasing 'all possible local KMS equilibrium states'. (ii) Lemma 5.4, Case 5, contains an omitted construction: the paper says 'instead of giving an explicit construction of f as before, we provide a rough argument justifying of the existence of such a function', asserting the existence of f with F1 < f^2 < F2 from endpoint and positivity checks. This is a missing justification, not a demonstrated failure (a continuous section exists since {(t,y) : 0 ≤ t ≤ 1, F1(t) < y < F2(t)} is an open strip with the endpoint conditions placing (0,0) and (1,1) in the correct fibers), and it does not amount to circularity. Overall the derivation chain is self-contained against external anchors; the moderate score reflects the programmatic reliance on the authors' own KMS/Malliavin framework from [11] without there being any circular reduction.

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

The central claims rest on the Malliavin calculus framework (integration by parts from [11]), on Aida's irreducibility theorem, on Bourgain's d=1 normalizability, and on the Hamiltonian reformulation of the PDEs. The parameters R, s, delta, and epsilon shape the statements but are not fitted values. No new physical entities are introduced; the renormalized mass and the local KMS condition are constructions within existing frameworks.

free parameters (4)
  • R (mass cutoff radius)
    R > 0 truncates the (renormalized) mass M(u) in the local Gibbs measure (2.34) and defines the local KMS domain B_R. The theorems hold for R as in Proposition 2.13(ii), which requires R sufficiently small in the d=1 quintic case. It is a domain parameter, not a fitted value.
  • s (Sobolev index)
    Assumption 2.3 fixes s in (d/2 - 1, 1], which determines the base space H^{-s}, the Gaussian mu0 with covariance A^{-1-s}, and the regularity of Cameron-Martin translations. The choice is forced by the trace condition (2.5), not fitted.
  • delta (cutoff shape parameter)
    Definition 2.12 introduces smooth cutoffs chi_R^(delta) with a smoothing parameter delta in (0,1); the theorems are obtained in the limit delta -> 1. The specific value of delta is technical.
  • epsilon (interaction decay exponent)
    Assumption 2.10(ii) requires |Vhat(k)| ≤ C(1+|k|)^{-epsilon} for d=2 and index -2-epsilon for d=3. This decay enters the concentration estimates in Appendix B and excludes the delta-potential in d=2, consistent with Brydges-Slade [24]. It is an input assumption, not a fit.
assumptions (4)
  • standard math Gaussian integration by parts on H^{-s} (Proposition 3.6)
    The Malliavin derivative and the integration by parts identity are quoted from [11, Proposition A.1] and used throughout Sections 3-5. This is background from Malliavin calculus.
  • standard math Aida's irreducibility theorem (Proposition 5.2)
    The theorem from Aida [2, Corollary 4] (and Kusuoka [49,50]) that a D^{1,2} function with vanishing gradient on an H^1-connected domain of positive measure is constant there. It is the engine of the converse characterization and is used unproved.
  • standard math Normalizability of the d=1 local Gibbs measure (Proposition 2.13(ii), d=1 cases)
    For d=1 the bound (2.32) is quoted from Bourgain [17, Lemma 3.10]. The paper uses it as an input for the cubic, quintic, and Hartree cases.
  • domain assumption Identification of the NLS/Hartree flow with the Hamiltonian vector field X = -iA + i∇h_I (2.42)-(2.45)
    The PDE is treated as an infinite-dimensional Hamiltonian system on H^{-s}; as noted in Remark 2.21, the vector field maps H^{-s} to itself only in the interaction representation, a structure imported from [7,11].

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Pith. "Pith review of Gibbs measures as local equilibrium KMS states for focusing nonlinear Schr\"odinger equations." pith.science (2026). https://pith.science/paper/V7QWOOSH

@misc{pith2026241205354,
  author       = {Pith},
  title        = {Pith review of: Gibbs measures as local equilibrium KMS states for focusing nonlinear Schr\"odinger equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/V7QWOOSH}},
  note         = {Machine review of arXiv:2412.05354}
}
read the original abstract

In this paper, we are concerned with the study of statistical equilibria for focusing nonlinear Schr\"odinger and Hartree equations on the d-dimensional torus when d=1,2,3. Due to the focusing nature of the nonlinearity in these PDEs, Gibbs measures have to be appropriately localized. First, we show that these local Gibbs measures are stationary solutions for the Liouville probability density equation and that they satisfy a local equilibrium Kubo-Martin-Schwinger (KMS) condition. Secondly, under some natural assumptions, we characterize all possible local KMS equilibrium states for these PDEs as local Gibbs measures. Our methods are based on Malliavin calculus in Gross-Stroock Sobolev spaces and on a suitable Gaussian integration by parts formula. To handle the technical problems due to localization, we rely on the works of Aida and Kusuoka on irreducibility of Dirichlet forms over infinite-dimensional domains. This leads us to the study of sublevel sets of the renormalized mass and their connectedness properties. In this paper, we also revisit Bourgain's proof of the normalizability of the local Gibbs measure for the focusing Hartree equation on the d-dimensional torus with d=2,3 by using concentration inequalities.

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Reference graph

Works this paper leans on

84 extracted references · 69 canonical work pages

  1. [11]

    Ammari, V

    Z. Ammari, V. Sohinger, Gibbs measures as unique KMS equilibrium states of nonlinea r Hamitonian PDEs, Rev. Mat. Iberoam. 39 (2023), no. 1, 29–90

    work page 2023

  2. [1]

    Aida, Differential calculus on path and loop spaces

    S. Aida, Differential calculus on path and loop spaces. II. Irreducib ility of Dirichlet forms on loop spaces, Bull. Sci. Math. 122 (1998), 635–666

    1998

  3. [2]

    Aida, On the irreducibility of Dirichlet forms on domains in infini te-dimensional spaces , Osaka Journal of Mathematics 37 (2000), no

    S. Aida, On the irreducibility of Dirichlet forms on domains in infini te-dimensional spaces , Osaka Journal of Mathematics 37 (2000), no. 4, 953–966

    2000

  4. [3]

    Aizenman, Geometric analysis of ϕ4 fields and Ising models

    M. Aizenman, Geometric analysis of ϕ4 fields and Ising models. Parts I and II , Comm. Math. Phys. 86 (1982), no.1, 1–48

    1982

  5. [4]

    Aizenman and H

    M. Aizenman and H. Duminil-Copin, Marginal triviality of the scaling limits of critical 4D Isi ng and ϕ4 4 models, Ann. Math. 194 (2021), no.1, 163–235

    2021

  6. [5]

    Aizenman, S

    M. Aizenman, S. Goldstein, C. Gruber, J.L. Lebowitz, P.A . Martin, On the equivalence between KMS- states and equilibrium states for classical systems , Comm. Math. Phys. 53 (1977), no. 3, 209–220

    work page 1977

  7. [6]

    Expansion of the Many-body Quantum Gibbs State of the Bose-Hubbard Model on a Finite Graph

    Z. Ammari, S. Farhat, S. Petrat, Expansion of the Many-body Quantum Gibbs State of the Bose- Hubbard Model on a Finite Graph , Preprint arXiv: 2405.04055(2024)

    work page Pith review arXiv 2024

  8. [7]

    Ammari, S

    Z. Ammari, S. Farhat, V. Sohinger, Almost sure existence of global solutions for general initi al value problems, Adv. Math., 453 (2024), No. 109805, 61

    work page 2024

Show all 84 references
  1. [8]

    Ammari, S

    Z. Ammari, S. Farhat, V. Sohinger, Invariant measures as probabilistic tools in the analysis o f nonlinear ODEs and PDEs , Quantum mathematics I, Springer INdAM Ser., 57 (2023), 301-317

    2023

  2. [9]

    Ammari, Q

    Z. Ammari, Q. Liard, On uniqueness of measure-valued solutions to Liouville’s e quation of Hamiltonian PDEs, Discrete Contin. Dyn. Syst. 38 (2018), no. 2, 723–748

    2018

  3. [10]

    Ammari, A

    Z. Ammari, A. Ratsimanetrimanana, High temperature convergence of the KMS boundary condition s: The Bose–Hubbard model on a finite graph . Commun. Contemp. Math. 23 (2021), no. 5, article no. 2050035, 18 pp

    2021

  4. [12]

    Arsen’ev, Invariant measures for classical dynamical systems with in finite phase space , Mat

    A.A. Arsen’ev, Invariant measures for classical dynamical systems with in finite phase space , Mat. Sb. (N.S.) 121 (163) (1983), no. 3, 297–309

    1983

  5. [13]

    Barashkov, M

    N. Barashkov, M. Gubinelli, A variational method for Φ 4 3, Duke Math. J. 169 (2020), no. 17, 3339–3415

    2020

  6. [14]

    Barashkov, P

    N. Barashkov, P. Laarne, Invariance of ϕ4 measure under nonlinear wave and Schr¨ odinger equations on the plane , Preprint arXiv: 2211.16111 (2022)

    2022 arXiv

  7. [15]

    Bogachev, Gaussian measures, Mathematical Surveys and Monographs 62, American Mathema t- ical Society, Providence, RI, 1998

    I.V. Bogachev, Gaussian measures, Mathematical Surveys and Monographs 62, American Mathema t- ical Society, Providence, RI, 1998

    1998

  8. [16]

    Boucheron, G

    S. Boucheron, G. Lugosi, P. Massart, Concentration Inequalities: A Nonasymptotic Theory of Ind e- pendence, Oxford University Press (2013)

    2013

  9. [17]

    Bourgain, Periodic nonlinear Schr¨ odinger equation and invariant me asures, Comm

    J. Bourgain, Periodic nonlinear Schr¨ odinger equation and invariant me asures, Comm. Math. Phys. 166 (1994), no. 1, 1–26

    1994

  10. [18]

    Bourgain, Invariant measures for the 2D-defocusing nonlinear Schr¨ odinger equation, Comm

    J. Bourgain, Invariant measures for the 2D-defocusing nonlinear Schr¨ odinger equation, Comm. Math. Phys. 176 (1996), no. 8, 421–445

    1996

  11. [19]

    Bourgain, Invariant measures for the Gross-Pitaevskii equation , J

    J. Bourgain, Invariant measures for the Gross-Pitaevskii equation , J. Math. Pures Appl. (9) 76 (1997), no. 8, 649–702

    1997

  12. [20]

    Bourgain, Global Solutions of Nonlinear Schr¨ odinger Equations, AMS Colloquium Publications, Vol

    J. Bourgain, Global Solutions of Nonlinear Schr¨ odinger Equations, AMS Colloquium Publications, Vol. 46 (1999)

    1999

  13. [21]

    Bourgain, Invariant meausures for NLS in infinite volume , Comm

    J. Bourgain, Invariant meausures for NLS in infinite volume , Comm. Math. Phys. 210 (2000), no. 3, 605–620

    2000

  14. [22]

    Bratelli, D.W

    O. Bratelli, D.W. Robinson, Operator algebras and quantum statistical mechanics. 2. Eq uilibrium states. Models in quantum statistical mechanics , Second edition, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1997

    1997

  15. [23]

    Brydges, J

    D. Brydges, J. Fr¨ ohlich, A.D. Sokal, A new proof of the existence and nontriviality of the continu um ϕ4 2 and ϕ4 3 theories, Comm. Math. Phys. 91 (1983), no. 2, 141–186

    1983

  16. [24]

    Brydges, G

    D. Brydges, G. Slade, Statistical mechanics of the 2-dimensional focusing nonli near Schr¨ odinger equa- tion, Comm. Math. Phys. 182 (1996), no. 2, 485–504

    1996

  17. [25]

    N. Burq, L. Thomann, N. Tzvetkov, Long time dynamics for the one dimensional non linear Schr¨ odinger equation, Annales de l’Institut Fourier 63 (2013), no. 6, 2137–2198

    2013

  18. [26]

    N. Burq, L. Thomann, N. Tzvetkov, Remarks on the Gibbs measures for nonlinear dispersive equa tions, Ann. Fac. Sci. Toulouse Math. 6 (2018), 27(3): 527–597

    2018

  19. [27]

    Cacciafesta, A.-S

    F. Cacciafesta, A.-S. de Suzzoni, Invariant measure for the Schr¨ odinger equation on the real line, J. Funct. Anal. 269 (2015), no. 1, 271–324

    2015

  20. [28]

    Cacciafesta, A.-S

    F. Cacciafesta, A.-S. de Suzzoni, Invariance of Gibbs measures under the flows of Hamiltonian e qua- tions on the real line , Commun. Contemp. Math. 22 (2020), no. 2, 1950012, 39 pp

    2020

  21. [29]

    Carlen, J

    E. Carlen, J. Fr¨ ohlich, J. Lebowitz, Exponential relaxation to equilibrium for a one-dimension al focus- ing non-linear Schr¨ odinger equation with noise , Comm. Math. Phys. 342 (2016), no. 1, 303–332

    2016

  22. [30]

    Chueshov, Equilibrium statistical solutions for dynamical systems w ith an infinite number of degrees of freedom, Mat

    I.D. Chueshov, Equilibrium statistical solutions for dynamical systems w ith an infinite number of degrees of freedom, Mat. Sb. (N.S.) 130 (172) (1986), no. 3, 394–403, 432

    1986

  23. [31]

    Deng, Two dimensional nonlinear Schr¨ odinger equation with rand om initial data , ANAL PDE 5 (2012), 913–960

    Y. Deng, Two dimensional nonlinear Schr¨ odinger equation with rand om initial data , ANAL PDE 5 (2012), 913–960. GIBBS MEASURES AND LOCAL KMS STATES FOR THE FOCUSING NLS 69

    2012

  24. [32]

    V.D. Dinh, N. Rougerie, Invariant Gibbs measures for 1D NLS in a trap , Preprint arXiv: 2301.02544, (2023)

    2023 arXiv

  25. [33]

    V.D. Dinh, N. Rougerie, L. Tolomeo, Y. W ang, Statistical mechanics of the radial focusing nonlinear Schr¨ odinger equation in general traps, Preprint arXiv: 2312.06232, (2023)

    2023

  26. [34]

    Fannes, J.V

    M. Fannes, J.V. Pul´ e, A.F. Verbeure, Integral representations of the classical KMS-states for q uasi-free evolutions, Rep. Mathematical Phys. 11 (1977), no. 3, 383–388

    1977

  27. [35]

    Forlano, R

    J. Forlano, R. Killip, M. Visan, Invariant measures for mKdV and KdV in infinite volume , Preprint arXiv: 2401.04292 (2024)

    2024 arXiv

  28. [36]

    P. K. Friz, M. Hairer, A course on rough paths: With an introduction to regularity s tructures, Univer- sitext, Springer, Cham (2020), xvi+346

    2020

  29. [37]

    Fr¨ ohlich,On the triviality of λφ4 d theories and the approach to the critical point in d > (−) 4 dimensions, Nucl

    J. Fr¨ ohlich,On the triviality of λφ4 d theories and the approach to the critical point in d > (−) 4 dimensions, Nucl. Phys. B 200 (1982), no.2, 281–296

    1982

  30. [38]

    Fr¨ ohlich, A

    J. Fr¨ ohlich, A. Knowles, B. Schlein, V. Sohinger, Gibbs Measures of Nonlinear Schr¨ odinger Equations as Limits of Many-Body Quantum States in Dimensions d ≤ 3, Comm. Math. Phys., 356 (2017), no. 3, 883–980

    2017

  31. [39]

    Fr¨ ohlich, A

    J. Fr¨ ohlich, A. Knowles, B. Schlein, V. Sohinger, A microscopic derivation of time-dependent correla- tion functions of the 1D cubic nonlinear Schr¨ odinger equat ion, Advances in Mathematics (2019), no. 353, 67–115

    2019

  32. [40]

    Fr¨ ohlich, A

    J. Fr¨ ohlich, A. Knowles, B. Schlein, and V. Sohinger. The mean-field limit of quantum Bose gases at positive temperature, J. Amer. Math. Soc. 35 (2022), no. 4, 955–1030

    2022

  33. [41]

    Fr¨ ohlich, A

    J. Fr¨ ohlich, A. Knowles, B. Schlein, and V. Sohinger. A path-integral analysis of interacting Bose gases and loop gases , J. Stat. Phys. 180 (2020), no. 1–6, 810–831

    2020

  34. [42]

    Fr¨ ohlich, A

    J. Fr¨ ohlich, A. Knowles, B. Schlein, and V. Sohinger. Interacting loop ensembles and Bose gases , Ann. Henri Poincar´ e24 (2023), no. 5, 1439–1503

    2023

  35. [43]

    Fr¨ ohlich, A

    J. Fr¨ ohlich, A. Knowles, B. Schlein, and V. Sohinger. The Euclidean Φ 4 2 theory as a limit of an interacting Bose gas , Preprint arXiv: 2201.07632 (2022). To appear in J. Eur. Mat h. Soc. (JEMS)

    2022 arXiv

  36. [44]

    Gallavotti, E.J

    G. Gallavotti, E.J. Verboven, On the classical KMS boundary condition . Nuovo Cimento B (11) 28 (1975), no. 1, 274–286

    1975

  37. [45]

    Glimm, A

    J. Glimm, A. Jaffe, Quantum Physics. A Functional Integral Point of View , Springer-Verlag, Second edition, 1987

    1987

  38. [46]

    Gubinelli, M

    M. Gubinelli, M. Hofmanov´ a, A PDE construction of the Euclidean φ4 3 quantum field theory , Comm. Math. Phys. 384 (2021), no. 1, 1–75

    2021

  39. [47]

    Gubinelli, N

    M. Gubinelli, N. Perkowski, An introduction to singular SPDEs , Stochastic partial differential equations and related fields, Springer Proc. Math. Stat., 229 (2018), 69–99

    2018

  40. [48]

    Haag, N.M

    R. Haag, N.M. Hugenholtz, M. Winnink, On the equilibrium states in quantum statistical mechanics , Comm. Math. Phys. 5 (1967), 215–236

    1967

  41. [49]

    Kusuoka, Analysis on Wiener spaces

    S. Kusuoka, Analysis on Wiener spaces. I. Nonlinear maps , J. Funct. Anal., 98 (1991), 122-168

    1991

  42. [50]

    Kusuoka, Analysis on Wiener spaces

    S. Kusuoka, Analysis on Wiener spaces. II. Differential forms , J. Funct. Anal., 103 (1992), 229-274

    1992

  43. [51]

    Lebowitz, H

    J. Lebowitz, H. Rose, E. Speer, Statistical mechanics of the nonlinear Schr¨ odinger equat ion, J. Stat. Phys. 50 (1988), 657–687

    1988

  44. [52]

    Lewin, P.-T

    M. Lewin, P.-T. Nam, N. Rougerie, Derivation of nonlinear Gibbs measures from many-body quan tum mechanics, Journal de l’ ´Ecole Polytechnique-Math´ ematiques2 (2015), 65–115

    2015

  45. [53]

    Lewin, P.-T

    M. Lewin, P.-T. Nam, N. Rougerie, Gibbs measures based on 1D (an)harmonic oscillators as mean -field limits, J. Math. Phys. 59 (2018), no. 4, 041901

    2018

  46. [54]

    Lewin, P.-T

    M. Lewin, P.-T. Nam, N. Rougerie, Classical field theory limit of 2D many-body quantum Gibbs st ates, Preprint arXiv: 1810.08370v1 (2018)

    2018 arXiv

  47. [55]

    Lewin, P.-T

    M. Lewin, P.-T. Nam, N. Rougerie, Derivation of renormalized Gibbs measures from equilibriu m many- body quantum Bose gases , J. Math. Phys, 60 (2019), no.6, 061901, 11 pp

    2019

  48. [56]

    Lewin, P.-T

    M. Lewin, P.-T. Nam, N. Rougerie, Classical field theory limit of many-body quantum Gibbs stat es in 2D and 3D , Invent. Math. 224 (2021), no. 2, 315–444

    2021

  49. [57]

    Liang, Y

    R. Liang, Y. W ang, Gibbs measure for the focusing fractional NLS on the torus , SIAM J. Math. Anal. 54 (2022), no. 6, 6096–6118

    2022

  50. [58]

    G. Li, R. Liang, Y. W ang Optimal divergence rate of the focusing Gibbs measure , Preprint arXiv: 2310.08783 (2023)

    2023 arXiv

  51. [59]

    Malliavin, Stochastic analysis , Grundlehren der mathematischen Wissenschaften, textbf3 13, (1997) Springer-Verlag, Berlin

    P. Malliavin, Stochastic analysis , Grundlehren der mathematischen Wissenschaften, textbf3 13, (1997) Springer-Verlag, Berlin

    1997

  52. [60]

    H. P. McKean, K. L. Vaninsky, Action-angle variables for the cubic Schr¨ odinger equation, Comm. Pure Appl. Math. 50 (1997), no. 6, 489–562

    1997

  53. [61]

    H. P. McKean, K. L. Vaninsky, Cubic Schr¨ odinger: the petit canonical ensemble in action -angle vari- ables, Comm. Pure Appl. Math. 50 (1997), no. 7, 593–622

    1997

  54. [62]

    Nahmod, G

    A. Nahmod, G. Staffilani, Randomness and nonlinear evolution equations , Acta Math. Sin. (Engl. Ser.) 35 (2019), no. 6, 903–932

    2019

  55. [63]

    Nelson The free Markoff field , J

    E. Nelson The free Markoff field , J. Functional Anal. 12 (1973), no. 2, 211–227. 70 ZIED AMMARI, ANDREW ROUT, AND VEDRAN SOHINGER

    1973

  56. [64]

    Nelson, Probability theory and Euclidean field theory , Constructive quantum field theory, Springer, 1973, pp

    E. Nelson, Probability theory and Euclidean field theory , Constructive quantum field theory, Springer, 1973, pp. 94–124

    1973

  57. [65]

    Nualart, The Malliavin calculus and related topics , Second edition, Probability and its Applications (New York), Springer-Verlag, Berlin, 2006

    D. Nualart, The Malliavin calculus and related topics , Second edition, Probability and its Applications (New York), Springer-Verlag, Berlin, 2006

    2006

  58. [66]

    T. Oh, M. Okamoto, L. Tolomeo, Focusing Φ 3 4 model with a Hartree-type nonlinearity , Preprint arXiv: 2009.03251 (2020), to appear in Mem. Amer. Math. Soc

    2020 arXiv

  59. [67]

    T. Oh, P. Sosoe, L. Tolomeo, Optimal integrability threshold for Gibbs measures associ ated with focusing NLS on the torus , Invent. Math. 227 (2022), no. 3, 1323–1429

    2022

  60. [68]

    T. Oh, L. Thomann, A pedestrian approach to the invariant Gibbs measures for th e 2-d defocusing nonlinear Schr¨ odinger equations, Stoch. Partial Differ. Equ. Anal. Comput. 6 (2018), no. 3, 397–445

    2018

  61. [69]

    T. Oh, L. Thomann. Invariant Gibbs measures for the 2-d defocusing nonlinear w ave equations , An- nales de la Facult´ e des sciences de Toulouse : Math´ ematiques, 29(1): 1–26, 2020

    2020

  62. [70]

    T. Oh, L. Tolomeo, Y. W ang, G. Zheng. Hyperbolic P (Φ) 2-model on the plane . Preprint arXiv: 2211.03735v2, (2022)

    2022

  63. [71]

    Peskov, The KMS state of a sine-Gordon system , Teoret

    N.V. Peskov, The KMS state of a sine-Gordon system , Teoret. Mat. Fiz. 64 (1985), no. 1, 32–40

    1985

  64. [72]

    Pulvirenti, G

    M. Pulvirenti, G. Riela, KMS condition for stable states of infinite classical system s, J. Math. Phys. 18 (1977), no. 12, 2364–2367

    1977

  65. [73]

    Rider, On the ∞-volume limit of the focusing cubic Schr¨ odinger equation, Comm

    B. Rider, On the ∞-volume limit of the focusing cubic Schr¨ odinger equation, Comm. Pure Appl. Math. 55 (2002), no. 10, 1231–1248

    2002

  66. [74]

    Robert, K

    T. Robert, K. Seong, L. Tolomeo, Y. W ang, Focusing Gibbs measures with harmonic potential , Preprint arXiv: 2212.11386 (2022), to appear in Ann. Inst. Henri Poin car´ e Probab. Stat

    2022 arXiv

  67. [75]

    A. Rout, V. Sohinger, A microscopic derivation of Gibbs measures for the 1D focusi ng cubic nonlinear Schr¨ odinger equation, Commun. Partial Differ. Equ. 48 (2023), no. 7–8, 1008–1055

    2023

  68. [76]

    A. Rout, V. Sohinger, A microscopic derivation of Gibbs measures for the 1D focusi ng quintic nonlinear Schr¨ odinger equation, Preprint arXiv: 2308.06569 (2023)

    2023 arXiv

  69. [77]

    Rudelson, R

    M. Rudelson, R. Vershynin, Hanson-Wright inequality and sub-gaussian concentration , Electron. J. Probab. 18 (2013), 1–9

    2013

  70. [78]

    Simon, The P (Φ) 2 Euclidean (Quantum) Field Theory , Princeton Univ

    B. Simon, The P (Φ) 2 Euclidean (Quantum) Field Theory , Princeton Univ. Press, 1974

    1974

  71. [79]

    Sohinger, A microscopic derivation of Gibbs measures for nonlinear Sc hr¨ odinger equations with unbounded interaction potentials , Int

    V. Sohinger, A microscopic derivation of Gibbs measures for nonlinear Sc hr¨ odinger equations with unbounded interaction potentials , Int. Math. Res. Not. IMRN (2022), no. 19, 14964–15063

    2022

  72. [80]

    Tolomeo, H

    L. Tolomeo, H. W eber, Phase transition for invariant measures of the focusing Sch r¨ odinger equation, Preprint arXiv: 2306.07697 (2023)

    2023

  73. [81]

    Vershynin, Introduction to the non-asymptotic analysis of random matr ices, In: Y.C

    R. Vershynin, Introduction to the non-asymptotic analysis of random matr ices, In: Y.C. Eldar, G. Kutyniok, eds, Compressed Sensing: Theory and Applications, Cambridge: Cambridge University Press, 2012

    2012

  74. [82]

    S. W atanabe, Lectures on stochastic differential equations and Malliavi n calculus , Tata Institute of Fundamental Research Lectures on Mathematics and Physics, 73 (1984), Tata Institute of Fundamental Research, Bombay; by Springer-Verlag, Berlin

    1984

  75. [83]

    Xian, Optimal mass normalizability for Gibbs measure associated with NLS on the 2D disc , Preprint arXiv: 2204.09561 (2022)

    T. Xian, Optimal mass normalizability for Gibbs measure associated with NLS on the 2D disc , Preprint arXiv: 2204.09561 (2022)

    2022 arXiv

  76. [84]

    Zhidkov, An invariant measure for the nonlinear Schr¨ odinger equati on (Russian), Dokl

    P.E. Zhidkov, An invariant measure for the nonlinear Schr¨ odinger equati on (Russian), Dokl. Akad. Nauk. SSSR 317 (1991), 543–546; translation in Soviet Math. Dokl. 43, 431–434. Univ Rennes, [UR1], CNRS, IRMAR - UMR 6625, F-35000 Rennes, France . Email address : zied.ammari@u...

    1991

Pith tools

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