Pith. sign in

REVIEW 3 major objections 4 minor 1 cited by

Generalized Beth--Uhlenbeck entropy formula from the $\Phi-$derivable approach

T0 review · 3 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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

Pith's one-line read Dense-fermion entropy reduces exactly to a generalized Beth–Uhlenbeck phase-shift formula at two-loop order.

desk verdict A clean formal identity for the two-loop Φ-derivable bosonic entropy, but the leap to a generalized Beth-Uhlenbeck formula for fermions rests on an identification that is asserted, not proven. read the letter →

arxiv 2512.03876 v1 pith:TYPJR3SF submitted 2025-12-03 nucl-th hep-thphysics.plasm-ph

David Blaschke , Gerd Röpke , Gordon Baym This is my paper · ORCID
classification nucl-thhep-thphysics.plasm-ph
keywords generalizedBeth-UhlenbeckformulaΦ-derivableapproachentropyphaseshiftssquaredLorentzianMotteffecttwo-particlecorrelationsthermodynamicpotential
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 shows that in the Φ-derivable approach to dense fermion systems with strong two-particle correlations, the bosonic contribution to the entropy is exactly (at the two-loop level of the Φ-functional) the generalized Beth–Uhlenbeck formula: an integral over the Bose distribution times a unique spectral density, 2 sin²δ(ω,q) ∂δ(ω,q)/∂ω, where δ is the two-particle phase shift. This unifies two major approaches to quantum statistics, extending the classic Beth–Uhlenbeck virial expansion beyond the low-density limit. A surprising consequence is that near a resonance the entropy spectral weight is a squared Lorentzian, Γ³/[(ω−ω_R)²+Γ²]², not the simple Breit–Wigner shape one might expect. Because the derivation uses self-consistent quasiparticle energies, the formula includes Pauli blocking, Mott dissociation of bound states, and respects Levinson's theorem.

What carries the argument

The load-bearing object is the two-loop ('sunset') Φ-functional, the sum of two-particle-irreducible skeleton diagrams. Its stationarity gives the Dyson equations for the fermion and boson propagators. The proof uses a derivative optical theorem, Eq. (21): ReD′ ImΠ − ImD ReΠ′ = 2Im(Π_I D Π_I D^{*′}), which together with the polar representation D=|D|e^{iδ} converts the entropy spectral function into 2 sin²δ ∂δ/∂ω. The vanishing of S′ at two-loop order (shown for scalar φ³ theory in an appendix) ensures that the entropy has no explicit Φ-derivative contribution, leaving the phase-shift integral as the exact boson entropy.

What would settle it

Choose a model with a known exact solution, e.g., a one-dimensional Fermi gas with a λδ potential where the two-body phase shift is analytic. Compute the entropy from Eq. (24) using that phase shift and compare with the exact thermodynamic Bethe ansatz entropy; a discrepancy beyond the two-loop truncation error would falsify the claim of exactness at two-loop order.

Watch

Extended reading notes

Core claim

The central result is Eq. (24): S_b = 2V ∫ d³q/(2π)³ ∫ dω/π σ_b(ω) sin²δ(ω,q) ∂δ(ω,q)/∂ω. The authors derive it from the Φ-derivable thermodynamic potential by showing that, at two-loop order, the term S′ vanishes, so the entropy is carried entirely by the statistical factors and the propagator spectral densities. A 'derivative optical theorem' recasts the spectral density in terms of the phase shift δ of the bosonic propagator. Near the two-particle resonance, with δ = −arctan[Γ/(ω−ω_R)], the spectral weight becomes a squared Lorentzian rather than a simple Lorentzian, which sharpens the entropy contribution of resonant correlations. The relation between the Beth–Uhlenbeck formula and the Φ

Load-bearing premise

The derivation assumes that all relevant correlations in the dense fermion system are two-particle in nature, so a two-loop Φ-functional and a single bosonic propagator suffice; if three-body or higher clusters contribute significantly at the densities of interest, Eq. (24) misses those entropy contributions.

Editorial extensions

If this is right

  • The generalized Beth–Uhlenbeck formula provides a self-consistent quasiparticle picture valid beyond low density, including scattering continua and bound states with Pauli blocking.
  • Resonant correlations contribute a squared-Lorentzian entropy peak, sharper than a Breit–Wigner; models that simply broaden hadronic spectral functions without phase shifts can violate the in-medium Levinson theorem.
  • The exactness at two-loop order gives a rigorous bridge between Φ-derivable resummations and phase-shift representations, useful for nuclear matter, quark matter, and Coulomb plasmas.
  • Mott dissociation of bound states emerges naturally from the self-consistent mean-field shifts and the occupation factors in the spectral density.

Reading between the lines

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

  • If the squared-Lorentzian structure persists in higher orders for the entropy, then standard resonance-gas treatments that assign a simple Lorentzian width to hadron states should be re-examined for consistency with sum rules and conservation laws.
  • The derivative optical theorem may generalize to higher-order vertex functions, suggesting an entire hierarchy of generalized Beth–Uhlenbeck formulas for three- and four-particle clusters.
  • One could test the formula directly in ultracold Fermi gases across a Feshbach resonance, where two-body phase shifts are precisely known and the entropy can be extracted from measured density profiles; a mismatch would indicate multi-body correlations beyond the two-loop truncation.
  • The same machinery should yield a generalized Beth–Uhlenbeck formula for particle number and pressure, not just entropy, providing a fully consistent equation of state.
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

3 major / 4 minor

Summary. The paper claims to derive a generalized Beth-Uhlenbeck (BU) entropy formula from the Φ-derivable approach at two-loop order. Starting from the thermodynamic potential for a relativistic fermion-boson system (QED as concrete example), the entropy is written as S = S_f + S_b + S'. Using the identity (14) and partial integration, S_b is expressed as Eq. (15) with spectral function (17). Introducing the phase δ of the boson propagator D and using a 'derivative optical theorem' (21), the paper rewrites the spectral function as 2δ' sin²δ, leading to the central result Eq. (24): S_b = 2V ∫ d³q/(2π)³ ∫ dω/π σ_b(ω) sin²δ ∂δ/∂ω. Near a resonance ansatz (25), this gives a squared Lorentzian (27). The proof that S'=0 is given for scalar φ³ in Appendix A and cited for QED/QCD. The paper claims the result includes Mott dissociation and Levinson's theorem, and discusses applications to quark and nuclear matter.

Significance. The algebraic derivation from Eq. (15) to (24) is internally consistent, and Appendix B provides a clean derivative optical theorem. The demonstration in Appendix A that S'=0 for the two-loop sunset is a useful concrete check. If Eq. (24) were rigorously identified as the two-particle entropy contribution of a dense fermion system, it would be a valuable bridge between Φ-derivable thermodynamics and the generalized BU approach. The prediction of a squared-Lorentzian spectral weight near a resonance (Eq. (27)) is a concrete, falsifiable qualitative result. However, the significance is presently conditional on an identification that the paper does not prove.

major comments (3)
  1. [Section II.B, Eqs. (22)-(24)] The phase δ is defined as the phase of the boson propagator D, not as the two-particle scattering phase shift of the constituent fermions. In the QED example of Sec. II, D is the photon propagator; its phase encodes the collective plasmon mode and Landau damping, not the electron-positron scattering T-matrix phase that enters the standard Beth-Uhlenbeck formula. The 'derivative optical theorem' (21)/(B8) is a relation between D and Π, not between the two-fermion T-matrix and its derivative. No step maps δ to the T-matrix phase shift, nor is it shown that S_b is the correlation contribution to the fermion entropy rather than the entropy of the exchanged boson. The central claim of the paper is thus asserted rather than demonstrated.
  2. [Eqs. (10), (15), (24)] The paper's title and abstract promise a formula for the entropy of a dense fermion system, but the derivation only treats the bosonic contribution S_b. The fermionic contribution S_f in Eq. (11) is not rewritten in BU form, and the total entropy S is not shown to reduce to Eq. (24) (with S' = 0). The sentence 'we focus on the bosonic entropy' after Eq. (13) narrows the scope, but this qualification is absent from the abstract and conclusions, which overstate the result.
  3. [Abstract and Section III] The claims that the formalism includes Mott dissociation of bound states and satisfies Levinson's theorem are delegated to earlier work (Refs. [14,15,18]); they are not derived here. The only explicit spectral model in this paper is the resonance ansatz Eq. (25), which is a Breit-Wigner form, not a bound-state pole. If bound-state contributions are to be included in the phase δ, a demonstration of how they appear and how Levinson's theorem applies in the present framework is needed. As it stands, the bound-state content of Eq. (24) is not evidenced.
minor comments (4)
  1. [Throughout] The text contains numerous typographical errors, e.g., 'thermodyanamic', 'The quation', 'thequark-gluonplasma', and garbled equations in Appendix A (e.g., the integration measure in Eq. (A8)). The manuscript needs careful proofreading.
  2. [Eq. (8)] The slashed integral symbol is nonstandard; it is defined in parentheses, but a more conventional notation (e.g., PV ∫) would improve readability.
  3. [Appendix A, Eq. (A8)] The integration measure 'd4k′/2π' appears garbled; the prefactor '−g^2/(2·2)' should be checked against Eq. (A7).
  4. [Section III] The term 'self-consistent Hartree-Fock approximation' is not standard for the sunset two-loop Φ; please clarify the terminology.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Eq. (24) is an algebraic rewrite of the two-loop boson entropy, not a fitted prediction.

full rationale

The derivation chain is self-contained in the relevant sense. Starting from the Phi-derivable thermodynamic potential, the entropy is decomposed into fermionic, bosonic, and residual pieces, and the residual piece S' is shown to vanish at two-loop order, with an explicit proof for scalar phi^3 theory in Appendix A and references for QED/QCD. No parameter is fitted to reproduce Eq. (24). The key step introducing the polar phase delta of the boson propagator D=|D|exp(i delta) is a definition, and the subsequent optical-theorem identity is an algebraic relation for D and Pi, not an imposed physical input. Thus Eq. (24) is an exact rewrite of the two-loop boson entropy in terms of that phase, not a prediction forced by a prior commitment to the generalized Beth-Uhlenbeck formula. The squared-Lorentzian result, Eq. (27), follows from an explicitly stated Breit-Wigner ansatz for delta and is not used to define the general formula. The paper also acknowledges the limitation that higher-order clusters are left open, and the interpretive identification of delta with a two-particle scattering phase shift is a physical-association issue rather than circularity. Self-citations are present but are supplemented by derivations in the paper and are not the load-bearing justification for the central result.

Assumptions & free parameters 2 free parameters · 7 assumptions · 0 invented entities

The central derivation relies on the Φ-derivable stationarity, two-loop truncation with S'=0, spectral representations, and the generalized optical theorem; it also assumes that bosonic S_b captures the correlated entropy and that the Breit-Wigner ansatz illustrates the shape. No fitting parameters; the Γ and ω_R in the ansatz are illustrative.

free parameters (2)
  • Resonance width Γ = not fitted (ansatz, Eq. 25)
    Assumed Breit-Wigner width in the illustrative phase-shift δ = -arctan[Γ/(ω-ω_R)] used to obtain the squared-Lorentzian entropy spectral function (27). Not needed for the general formula (24).
  • Resonance pole energy ω_R(q) = not fitted (pole of Re D^{-1}=0)
    Position of the boson/plasmon pole in the same illustrative ansatz; could in principle be computed from the self-consistent Dyson equation, but is not calculated in this paper.
assumptions (7)
  • domain assumption The thermodynamic potential is of the Φ-derivable form, Eq. (1), and is stationary under variations of G and D.
    Invoked in Section II.A; stationarity is the basis for dropping spectral-function derivatives in the entropy (Eq. 10).
  • domain assumption Only the two-loop 'sunset' skeleton diagram for Φ is kept, for which S'=0.
    Section II.A and Fig. 1; the cancellation is proved for scalar φ^3 in Appendix A and cited from Refs. [5,6] for QED/QCD.
  • standard math Spectral representations of the propagators and Matsubara frequency sums/contour integration are valid.
    Used throughout Section II.A to pass from Eq. (1) to Eq. (8).
  • standard math The generalized optical theorem, Eq. (B8), and the phase-shift representation D=|D|e^{iδ} hold.
    Appendix B derives Eq. (21); assumes existence of the propagator inverse and an energy-independent free propagator in Coulomb gauge.
  • domain assumption The entropy of the correlated part is captured by the bosonic contribution S_b; the fermionic quasiparticle entropy S_f is not included in Eq. (24).
    The paper states 'we focus on the bosonic entropy S_b' and never assembles the full S=S_f+S_b; this limits the scope of the entropy formula.
  • domain assumption Generalized phase shifts in medium account for Mott dissociation and obey the in-medium Levinson theorem.
    Asserted in the abstract and Discussion; supported only by citations to [10,14,15,18] rather than a calculation in this paper.
  • ad hoc to paper The illustrative Breit-Wigner phase shift δ=-arctan[Γ/(ω-ω_R)] describes a resonance.
    Eq. (25)-(27); chosen by hand to exhibit the squared-Lorentzian shape, not derived from a specific microscopic interaction.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Generalized Beth--Uhlenbeck entropy formula from the $\Phi-$derivable approach." pith.science (2026). https://pith.science/paper/TYPJR3SF

@misc{pith2026251203876,
  author       = {Pith},
  title        = {Pith review of: Generalized Beth--Uhlenbeck entropy formula from the $\Phi-$derivable approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TYPJR3SF}},
  note         = {Machine review of arXiv:2512.03876}
}
abstract

We derive a generalized Beth-Uhlenbeck formula for the entropy of a dense fermion system with strong two-particle correlations, including scattering states and bound states. We work within the $\Phi-$derivable approach to the thermodynamic potential. The formula takes the form of an energy-momentum integral over a statistical distribution function times a unique spectral density. In the near mass-shell limit, the spectral density reduces, contrary to na\"{i}ve expectations, not to a Lorentzian but rather to a "squared Lorentzian" shape. The relation of the Beth-Uhlenbeck formula to the $\Phi$-derivable approach is exact at the two-loop level for $\Phi$. The formalism we develop, which extends the Beth-Uhlenbeck approach beyond the low-density limit, includes Mott dissociation of bound states, in accordance with Levinson's theorem, and the self-consistent back reaction of correlations in the fermion propagation. We discuss applications to further systems, such as quark matter and nuclear matter.

Figures

Figures reproduced from arXiv: 2512.03876 by the authors.

Figure 1
Figure 1. FIG. 1: The two-loop functional [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Self-energy diagrams derived from the [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Feynman diagram for the [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Generalized Beth-Uhlenbeck Approach to the 2+1D Gross-Neveu Model

    cond-mat.mes-hall 2026-04 conditional novelty 5.0 of 10

    Generalized Beth-Uhlenbeck entropy density suppresses low-energy Landau damping in the 2+1D Gross-Neveu model while preserving bound-state effects, yielding a sharper exciton-to-fermion Mott crossover.

Reference graph

Works this paper leans on

34 extracted references · 1 linked inside Pith · cited by 1 Pith paper

  1. [1]

    L. P. Kadanoff and G. Baym,Quantum Statistical Mechanics, 1962

    1962

  2. [2]

    Conservation laws and correlation functions,

    G. Baym and L. P. Kadanoff, “Conservation laws and correlation functions," Phys. Rev.124l 287 (1961)

    1961

  3. [3]

    Self consistent approximations in many body systems,

    G. Baym, “Self consistent approximations in many body systems,” Phys. Rev.127, 1391 (1962)

    1962

  4. [4]

    The entropy expression of a Fermi liquid in shielded potential approximation. Application to3He,

    E. Riedel, "The entropy expression of a Fermi liquid in shielded potential approximation. Application to3He," Zeitschr. Phys.210, 403 (1968)

    1968

  5. [5]

    Self-consistent approximations in relativistic plasmas: Quasiparticle analysis of the thermodynamic properties,

    B. Vanderheyden and G. Baym, “Self-consistent approximations in relativistic plasmas: Quasiparticle analysis of the thermodynamic properties,” J. Statist. Phys.93, 843 (1998)

    1998

  6. [6]

    J. P. Blaizot, E. Iancu and A. Rebhan, “Approximately self-consistent resummations for the thermodynamics of the quark gluon plasma

  7. [7]

    Entropy and density,” Phys. Rev. D63, 065003 (2001)

    2001

  8. [8]

    The quantum theory of the non-ideal gas. I. Deviations from the classical theory,

    G. Uhlenbeck, E. Beth, “The quantum theory of the non-ideal gas. I. Deviations from the classical theory,” Physica3, 729 (1936)

    1936

Show all 34 references
  1. [9]

    The quantum theory of the non-ideal gas. II. Behaviour at low temperatures,

    E. Beth, G. Uhlenbeck, “The quantum theory of the non-ideal gas. II. Behaviour at low temperatures,” Physica4, 915 (1937)

    1937

  2. [10]

    The Mass Action Law in Two-Component Fermi Systems Revisited. Excitons and Electron-Hole Pairs,

    R. Zimmermann, H. Stolz, “The Mass Action Law in Two-Component Fermi Systems Revisited. Excitons and Electron-Hole Pairs,” physica status solidi (b)131, 151 (1985)

    1985

  3. [11]

    Generalized Beth-Uhlenbeck approach for hot nuclear matter,

    M. Schmidt, G. Röpke, H. Schulz, “Generalized Beth-Uhlenbeck approach for hot nuclear matter,” Annals Phys.202, 57 (1990)

    1990

  4. [12]

    The Influence of Dynamical Effects on the Two-Particle States (Excitons) in the Electron-Hole Plasma,

    G. Röpke, K. Kilimann, W. D. Kraeft, D. Kremp and R. Zimmermann, "The Influence of Dynamical Effects on the Two-Particle States (Excitons) in the Electron-Hole Plasma," Phys. Stat. Sol. (b)88, K59 (1978)

    1978

  5. [13]

    Dynamical screening and self-energy of excitons in the electron- hole plasma,

    R. Zimmermann, K. Kilimann, W. D. Kraeft, D. Kremp and G. Röpke, "Dynamical screening and self-energy of excitons in the electron- hole plasma," Phys. Stat. Sol. (b)90, 175 (1978)

    1978

  6. [14]

    The influence of two-particle states (excitons) on the dielectric function of the electron—hole plasma,

    G. Röpke and R. Der, "The influence of two-particle states (excitons) on the dielectric function of the electron—hole plasma," phys. stat. sol. (b)92, 501 (1979)

    1979

  7. [15]

    GeneralizedBeth–Uhlenbeckapproachtomesonsanddiquarksinhot, dense quark matter,

    D.Blaschke,M.Buballa,A.Dubinin,G.RöpkeandD.Zablocki,“GeneralizedBeth–Uhlenbeckapproachtomesonsanddiquarksinhot, dense quark matter,” Annals Phys.348, 228 (2014)

    2014

  8. [16]

    Polyakov-loop suppression of colored states in a quark-meson-diquark plasma,

    D. Blaschke, A. Dubinin and M. Buballa, “Polyakov-loop suppression of colored states in a quark-meson-diquark plasma,” Phys. Rev. D 91, no. 12, 125040 (2015)

    2015

  9. [17]

    Thermodynamics of a quark plasma beyond the mean field: A generalized Beth- Uhlenbeck approach,

    J. Hüfner, S. P. Klevansky, P. Zhuang and H. Voss, “Thermodynamics of a quark plasma beyond the mean field: A generalized Beth- Uhlenbeck approach,” Annals Phys.234, 225 (1994)

    1994

  10. [18]

    Thermodynamics of a quark - meson plasma in the Nambu-Jona-Lasinio model,

    P. Zhuang, J. Hüfner and S. P. Klevansky, “Thermodynamics of a quark - meson plasma in the Nambu-Jona-Lasinio model,” Nucl. Phys. A576, 525 (1994)

    1994

  11. [19]

    PiondissociationandLevinson‘stheoreminhotPNJLquarkmatter,

    A.Wergieluk,D.Blaschke,Y.L.KalinovskyandA.Friesen,“PiondissociationandLevinson‘stheoreminhotPNJLquarkmatter,”Phys. Part. Nucl. Lett.10, 660 (2013)

    2013

  12. [20]

    Quark-Hadron Phase Transition in the PNJL model for interacting quarks,

    K. Yamazaki and T. Matsui, “Quark-Hadron Phase Transition in the PNJL model for interacting quarks,” Nucl. Phys. A913, 19 (2013)

    2013

  13. [21]

    Variationalpath-integralapproachtoback-reactionsofcompositemesonsintheNambu–Jona-Lasiniomodel,

    D.BlaschkeandD.Ebert,“Variationalpath-integralapproachtoback-reactionsofcompositemesonsintheNambu–Jona-Lasiniomodel,” Nucl. Phys. B921, 753 (2017)

    2017

  14. [22]

    S Matrix formulation of statistical mechanics,

    R. Dashen, S. K. Ma and H. J. Bernstein, “S Matrix formulation of statistical mechanics,” Phys. Rev.187, 345-370 (1969)

    1969

  15. [23]

    Thermodynamik mit Resonanzzuständen

    W. Weinhold, "Thermodynamik mit Resonanzzuständen", Dissertation (unpublished), TU Darmstadt (1998)

    1998

  16. [24]

    Kinetic theory of radiation in non-equilibrium relativistic plasmas,

    V. G. Morozov and G. Röpke, "Kinetic theory of radiation in non-equilibrium relativistic plasmas," Annals Phys.324, 1261 (2009)

    2009

  17. [25]

    Cluster virial expansion for nuclear matter within a quasi- particle statistical approach,

    G. Röpke, N.-U. Bastian, D. Blaschke, T. Klähn, S. Typel and H. H. Wolter, “Cluster virial expansion for nuclear matter within a quasi- particle statistical approach,” Nucl. Phys. A897, 70 (2013). 8

    2013

  18. [26]

    Beth-Uhlenbeck equation for the thermodynamics of fluctuations in a generalized (2+1)D Gross- Neveu model,

    B. Mahato, D. Blaschke and D. Ebert, “Beth-Uhlenbeck equation for the thermodynamics of fluctuations in a generalized (2+1)D Gross- Neveu model,” Phys. Rev. D112, 036013 (2025)

    2025

  19. [27]

    Calculating energy shifts in terms of phase shifts,

    Z. Yu, G. Baym, and C. J. Pethick, “Calculating energy shifts in terms of phase shifts," J. Phys. B: At. Mol. Opt. Phys. 44 195207-1- 6,(2011); arXiv:0911.2853

    2011 arXiv

  20. [28]

    Mott dissociation of pions and kaons in hot, dense quark matter,

    D. Blaschke, A. Dubinin, A. Radzhabov and A. Wergieluk, “Mott dissociation of pions and kaons in hot, dense quark matter,” Phys. Rev. D96, no.9, 094008 (2017)

    2017

  21. [29]

    Mott-hadron resonance gas and lattice QCD thermodynamics,

    D. Blaschke, A. Dubinin and L. Turko, “Mott-hadron resonance gas and lattice QCD thermodynamics,” Phys. Part. Nucl.46, no. 5, 732 (2015)

    2015

  22. [30]

    Generalized Beth-Uhlenbeck approach to the equation of state for quark-hadron matter,

    D. Blaschke, A. Dubinin and L. Turko, “Generalized Beth-Uhlenbeck approach to the equation of state for quark-hadron matter,” Acta Phys. Polon. Supp.10, 473-480 (2017)

    2017

  23. [31]

    Thermodynamics of quark matter with multiquark clusters in an effective Beth- Uhlenbeck type approach,

    D. Blaschke, M. Cierniak, O. Ivanytskyi and G. Röpke, “Thermodynamics of quark matter with multiquark clusters in an effective Beth- Uhlenbeck type approach,” Eur. Phys. J. A60, no.1, 14 (2024)

    2024

  24. [32]

    EquationofMotionMethodforstronglycorrelated Fermi systems and Extended RPA approaches,

    P.Schuck,D.S.Delion,J.Dukelsky,M.Jemai,E.Litvinova,G.Röpke,M.Tohyama,"EquationofMotionMethodforstronglycorrelated Fermi systems and Extended RPA approaches," Physics Reports929, 1-84 (2021)

    2021

  25. [33]

    Bound states and superfluidity in strongly coupled Fermion systems,

    G. Röpke, "Bound states and superfluidity in strongly coupled Fermion systems," Ann. Physik (Leipzig)3, 145 (1994)

    1994

  26. [34]

    Cluster Mean-Field Approach to the Steady-State Phase Diagram of Dissipative Spin Systems,

    Jiasen Jin et al., "Cluster Mean-Field Approach to the Steady-State Phase Diagram of Dissipative Spin Systems," Phys. Rev. X6, 031011 (2016). Appendix A: Proving𝑆′=0in scalar field theory with a𝜙3 self-interaction at two-loop order InthisAppendix,wedemonstratethecancellationex...

    2016

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.