Kolmogorov-type non-thermal fixed points and beyond of far-from-equilibrium dilute system: ultra-cold Fermi gas

arXiv:2508.18821 · cond-mat.quant-gas, cond-mat.other, hep-ph · Submitted 2025-08-26 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: We're looking at the paper: "Kolmogorov-type non-thermal fixed points and beyond of far-from-equilibrium dilute system".

Mira: The gist The far-from-equilibrium dynamics driven by scattering from next-to-leading-order (NLO) corrections in quantum field theory have stationary solutions for the particle distribution characterized as Kolmogorov-type non-thermal fixed points Kinetic…

Kai: First, who's behind it and why it matters.

Title and authors: Kai: So this paper is titled "Kolmogorov-type non-thermal fixed points and beyond of far-from-equilibrium dilute system: ultra-cold Fermi gas". It sounds like they are looking for specific types of stable states in a gas that isn't sitting in equilibrium.

Mira: Exactly. The focus is on finding these stationary solutions, which the authors call Kolmogorov-type non-thermal fixed points, in the dynamics of a spatially homogeneous dilute ultra-cold Fermi gas. It's about how these systems behave when they are far from their normal thermal state because of scattering from next-to-leading-order corrections in quantum field theory.

Lev: So, if I understand this right, they’re looking at real time dynamics using the Schwinger-Keldysh formalism to see if there are specific fixed points for the particle distribution when you have these NLO corrections involved?

Kai: That's right. They're checking if those solutions exist based on the kinetic equation and how it behaves under perturbation theory, specifically relying on the quasi-particle assumption to find those initial fixed points.

Mira: And they contrast this with what they found in other systems, like wave turbulence in weakly coupled ultra-cold Bose gases, which gives them a benchmark for comparison.

Lev: So the paper sets up a specific problem: how do you find these stationary solutions when you're dealing with non-equilibrium dynamics and need to account for those NLO scattering corrections?

The paper's summary: Kai: The core finding here is that they can find these Kolmogorov-type non-thermal fixed points analytically, but this only works when you assume the quasi-particle picture holds, which is a common simplification in perturbation theory.

Mira: And the paper goes further by exploring solutions that don't require that quasi-particle assumption at all, both without it and even within a strongly coupled system. That’s where things get interesting for understanding the limits of these kinds of non-thermal states.

Lev: If we think about what this means for actual hardware, it suggests there are specific types of non-equilibrium states that might be relevant in dilute gases, but we have to be careful about how well those analytical results map onto what a real system would actually do.

Kai: Yeah, and the paper connects these solutions to scaling behaviors. They look at the spectral function and they find scaling forms like rho alpha beta, ij (p zero p) = s two-eta rho alpha beta, ij (s z p zero sp) where s is related to momentum as p - one <ref:2508.18821#pg2>.

Mira: That scaling behavior tells us something about how the system evolves in time and space. They derive relations like d - two + z - kappa s + one = zero for particle cascades and d - two + z - kappa s + z + one = zero for energy cascades, where d is the space-time dimension <ref:2508.18821#pg2>.

Lev: Those equations are pretty abstract, but they suggest that you can characterize these non-thermal states by specific scaling exponents derived from the system's dimensionality and other parameters. It’s a way to classify these solutions.

Kai: And if we look at the strong coupling case, the one/N expansion shows that resummed diagrams introduce corrections that can alter this scaling structure in high-momentum regimes, even when z=two and d=four <ref:2508.18821#pg2>.

The paper's improvements: Mira: One big improvement they suggest is looking at solutions beyond the Kolmogorov-type nonthermal fixed points, especially when you move into the strongly coupled regime using the one/N expansion <ref:2508.18821#pg1>.

Kai: They argue that by resumming diagrams in that expansion, you can get corrections to the coupling itself, which changes how the kinetic equation scales in the high-momentum regime.

Lev: From an engineering standpoint, if we're trying to build something, knowing where these corrections land is crucial because those corrections might be significant at low temperatures or high densities.

Mira: They also suggest that for the weakly coupled system, there are differences between particle and wave interactions in how the quasi-particle assumption plays out; one system adheres to it while the other doesn't for its scale-invariant fluxes.

Kai: So, this means we have different classes of non-thermal states depending on whether we are looking at particle dynamics or wave dynamics in these gases. It's a way to distinguish them based on which conservation laws are being satisfied by the quasi-particle assumption.

Lev: That distinction is important because it tells us which physical assumptions you can rely on when you try to simulate this stuff computationally or build a model for it.

Conclusion: Kai: So to wrap up, this paper shows that you can analytically find Kolmogorov non-thermal fixed points for the dilute ultra-cold Fermi gas when using perturbation theory and the quasi-particle assumption, which is a specific type of stationary state.

Mira: But it also opens doors by showing that there are other stationary states in both the weak and strong coupling limits, including those beyond the standard Kolmogorov types, which can be found without relying on that quasi-particle picture.

Lev: For us on the hardware side, this means we have a theoretical roadmap for what kinds of non-equilibrium steady states we might be aiming to simulate or measure in future experiments with these gases.

Kai: Yeah, and they are providing analytical solutions that give us concrete scaling exponents that we can use as targets when designing numerical simulations or thinking about experimental setups.

Mira: It’s a solid foundation for searching for those far-from-equilibrium stationary states in the dilute system, whether you're working in weak coupling or strong coupling.

Lev: I think this paper gives us a good idea of the theoretical landscape we need to navigate when we move from simple kinetic equations to the full non-Markovian Kadanoff-Baym equations.

Chun-Wei Su

Department of Physics, University of Colorado, Boulder

cond-mat.quant-gas, cond-mat.other, hep-ph

Submitted: 2025-08-26

Updated: 2026-10-18

Comments: 11 pages, 2 figures

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 72/100

The gist: The gist The far-from-equilibrium dynamics driven by scattering from next-to-leading-order (NLO) corrections in quantum field theory have stationary solutions for the particle distribution

Key concepts

Kolmogorov-type non-thermal fixed points: These are specific stationary solutions found in the Boltzmann equation for a dilute gas, characterized by scale-invariant fluxes. They represent a non-equilibrium state where particle distributions maintain a specific scaling relationship with momentum, assuming the quasi-particle approximation holds.
Schwinger-Keldysh formalism: This is a real-time quantum field theory method used to describe non-equilibrium dynamics. It allows researchers to study how quantum systems evolve over time, incorporating memory effects from past interactions, which are crucial for understanding the gas's behavior.
1/N expansion: This technique is used in the context of a large number of particles (or degrees of freedom) to analyze strongly coupled systems. It helps in resumming diagrams beyond mean-field approximations, allowing the prediction of scaling structures that go beyond simple quasi-particle assumptions.

Terminology

Fixed Point — 3× in this paper · explained in 2 episodes

Summary

The gist The far-from-equilibrium dynamics driven by scattering from next-to-leading-order (NLO) corrections in quantum field theory have stationary solutions for the particle distribution characterized as Kolmogorov-type non-thermal fixed points

Kinetic Description and Fixed Points

The study investigates the non-equilibrium dynamics of a spatially homogeneous dilute ultra-cold Fermi gas using real-time methods based on the Schwinger-Keldysh formalism The kinetic description is based on the Boltzmann equation for the dilute gas, which accounts for collisions between particles in contrast to waves The exact stationary solutions characterized as Kolmogorov-type non-thermal fixed points for the Boltzmann equation were first found by Kats, Kontorovich, Moiseev and Novikov These fixed points are found relying on the quasi-particle assumption in the perturbation theory Solutions beyond Kolmogorov-type non-thermal fixed points are discussed in the condition free from the quasiparticle assumption and in a strongly coupled system

Non-Equilibrium Dynamics and Equations

The non-equilibrium dynamics of the spatially homogeneous dilute ultra-cold Fermi gas is investigated in real time based on the Schwinger-Keldysh formalism The resulting Kadanoff-Baym equations contain the integral of time that includes the memory of the past identified as the non-Markovian effect The gradient expansion of Eq. (8) and (9) with respect to the central coordinates Xµ ≡ 1/2 (xµ + yµ) to the first order reduces to the transport equation at the late time by t0 → −∞ The resulting transport equation in the 1/N expansion is given by Eq. (15)

Stationary Solutions and Scaling

Some stationary solutions are characterized as nonthermal states related to scale-invariant fluxes determined by scaling exponents The scaling form of the spectral function is considered ραβ,ij (p0, p) = s 2−η ραβ,ij (s z p0, sp) The growth of ΠR(A),αβ(p) ∼ p with increasing momenta is found if z = 2, space-time dimension d = 4 and η is small enough by s = p − 1 Based on Eq. (A33) and (A34), the scaling behavior will be dominated by ¯vαα(X, p) over v(X, p) if ΠR(A),αβ(p) ≫ 1 This leads to the relations d − 2 + z − lκs + 1 = 0 particle cascade, (20) and d − 2 + z − lκs + z + 1 = 0 energy cascade, (21)

Strong Coupling and Beyond

Scaling solutions beyond Kolmogorov-type nonthermal fixed points can also emerge in the strongly coupled system by the 1/N expansion Resummed diagrams add corrections to the coupling that alter the scaling structure of the kinetic equation in the high-momentum regime as argued in Eq. (18) if z = 2, d = 4 and η is small enough The paper shows the existence of far-from-equilibrium stationary states for the dilute ultra-cold Fermi gas analytically Inspired by active experiments of the ultra-cold Bose gas [37–39], it is motivated to search for these non-thermal states in experiments or the numerical simulation of dilute gas systems in the future

Conclusion

The paper shows that multiple terms in Eq. (25) and (26) are removed by the quasi-particle assumption except the last two terms in Eq. (26) because ˜ρmn,p ρ˜nm,q is nonvanishing even by Eq. (14) In summary, Eq. (30) and (31) reproduce Kolmogorov non-thermal fixed points for the equation (l = 2) when the dilute ultra-cold Fermi gas system is on-shell and described by the perturbation theory They show the nature different from the weakly coupled ultra-cold Bose gas [24], where the scaleinvariant particle flux is fulfilled by the quasi-particle assumption while the scale-invariant energy flux avoids this restriction

How it works

The analysis employs two types of expansion schemes for weak and strong coupling The Kolmogorov-type non-thermal fixed points are characterized by the stationary condition, where the scaling form of the distribution function and the spectral function by momenta are assumed The technique of scaling transformation allows one to find the specific scaling exponents that fulfill the stationary condition

Appendix A: Resummation to NLO

The quantum corrections to the two-point function are derived through the resummation technique shown in Ref. [29, 30] The NLO corrections in the weak-coupling or 1/N expansion lead to scattering processes beyond the mean field

Appendix B: Scale-invariant flux

The continuity equations for the on-shell particle number and the energy are given by ∂/∂t ω(p)f(t, p) + ∇ · ˜j(p) = 0 The scaling of the distribution function denotes f(t, p) = sκw f(t, sp) and C(p) scales as C(p) = s−µ C(sp)

Appendix D: Scaling transformation

The change of variables p = p0 q'0 1/z q', q = p0 q'0 1/z p', l = p0 q'0 1/z l', r = p0 q'0 1/z r' is used in the scaling transformation The exponent of the Jacobian originating from scaling behaviors is given by ϕ = − 1/z 3(d − 1) + 3z − α − β − γ − κ

Appendix C: Stationary condition

The stationary condition is satisfied by the vanishing trace of the memory integrals in Kadanoff-Baym equations since the trace of LHS of Eq. (8) is invariant while that of Eq.

Improvements for AI systems

Bold header: Non-equilibrium stationary state prediction for dilute ultra-cold Fermi gas. The improved AI system can analytically determine Kolmogorov-type non-thermal fixed points by applying scaling transformations to the momentum integrals, which are described in Eq. (25) and (26).
Bold header: Strong coupling dynamics modeling via 1/N expansion. The system can predict scaling exponents beyond Kolmogorov-type fixed points by utilizing the non-perturbative 1/N resummation mentioned in Section IV, specifically using results like Eq. (33) and (34) for the strongly coupled system.
Bold header: Distinguishing particle vs wave interactions. The improved AI can differentiate between systems by identifying which conservation laws are governed by the quasi-particle assumption versus those that are not, as noted in Section V (The difference demonstrates the particle and wave interactions in two systems, respectively).