pith. sign in

arxiv: 2409.05645 · v2 · pith:BJ3FXYURnew · submitted 2024-09-09 · 🧮 math.PR

Trend to equilibrium and Newtonian limit for the relativistic Langevin equation with singular potentials

Manh Hong Duong , Hung Dang Nguyen This is my paper

Pith reviewed 2026-05-23 21:15 UTC · model grok-4.3

classification 🧮 math.PR MSC 60H1082C3135Q83
keywords relativistic Langevin equationalgebraic mixing ratesNewtonian limitsingular potentialsLyapunov functionsinteracting particle systemsstochastic differential equations
0
0 comments X

The pith

Relativistic Langevin equations with singular potentials converge algebraically to equilibrium and approximate classical Langevin dynamics on finite time intervals.

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

The paper examines systems of interacting particles governed by relativistic kinetic energy, external confining potentials, singular repulsive forces, and additive white noise. It shows that these systems approach their unique steady states at algebraic rates of any order, in contrast to the exponential mixing known for classical Langevin equations. The algebraic rates follow from specially constructed Lyapunov functions adapted to irregular potentials. The work further proves that solutions of the relativistic equations converge to those of the classical Langevin equations on any fixed time window as the speed of light tends to infinity.

Core claim

The relativistic systems satisfy algebraic mixing rates of any order toward their unique statistically steady states. This relies on the construction of Lyapunov functions that adapt techniques for irregular potentials. The Newtonian limit holds: solutions of the relativistic equation approximate those of the classical Langevin equation on any finite time window as the speed of light tends to infinity.

What carries the argument

Lyapunov functions adapted to singular repulsive forces and confining potentials, used to derive algebraic decay rates and control the relativistic-to-classical limit.

If this is right

  • The unique invariant measure is reached at any algebraic rate by tuning the potential parameters.
  • On finite time intervals the relativistic trajectories stay close to classical ones when the speed of light is large.
  • The same Lyapunov construction applies to other relativistic kinetic energies with similar singularity structure.
  • The algebraic rates persist under small perturbations of the interaction forces.

Where Pith is reading between the lines

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

  • The result suggests that relativistic corrections slow down equilibration in a controllable, power-law manner rather than destroying it.
  • One could test whether the same Lyapunov technique yields rates for relativistic mean-field limits or for systems with multiplicative noise.
  • The finite-time Newtonian approximation may extend to expectations of observables that are continuous in the particle paths.

Load-bearing premise

The external confining potentials and singular repulsive forces allow construction of Lyapunov functions that yield algebraic mixing.

What would settle it

Numerical simulation of the relativistic particle system showing whether the decay of the distance to equilibrium is algebraic of arbitrary order or slower.

read the original abstract

We study a system of interacting particles in the presence of the relativistic kinetic energy, external confining potentials, singular repulsive forces as well as a random perturbation through an additive white noise. In comparison with the classical Langevin equations that are known to be exponentially attractive toward the unique statistically steady states, we find that the relativistic systems satisfy algebraic mixing rates of any order. This relies on the construction of Lyapunov functions adapting to previous literature developed for irregular potentials. We then explore the Newtonian limit as the speed of light tends to infinity and establish the validity of the approximation of the solutions by the Langevin equations on any finite time window.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 2 minor

Summary. The manuscript studies a system of interacting particles governed by a relativistic Langevin equation that includes external confining potentials, singular repulsive forces, and additive white noise. It claims that, in contrast to the classical (non-relativistic) case which yields exponential convergence to equilibrium, the relativistic dynamics admit algebraic mixing rates of arbitrary order. This is obtained by constructing suitable Lyapunov functions that adapt techniques previously developed for irregular or singular potentials. The paper further establishes a Newtonian limit: as the speed of light tends to infinity, solutions of the relativistic system converge to those of the classical Langevin equation on any finite time window.

Significance. If the Lyapunov constructions succeed for the relativistic kinetic term without hidden regularity assumptions, the work would extend polynomial mixing results to relativistic stochastic particle systems, a setting of interest in mathematical physics and kinetic theory. The finite-time Newtonian limit supplies a rigorous justification for using classical approximations over bounded intervals. The adaptation of existing Lyapunov techniques for singular forces is a methodological strength provided the adaptation is carried through explicitly.

major comments (2)
  1. [Section detailing the Lyapunov-function construction for the relativistic generator] The central claim of algebraic mixing rates of arbitrary order rests on the successful adaptation of Lyapunov functions to the relativistic generator. The manuscript must explicitly verify that the relativistic kinetic energy term preserves the coercivity and moment-growth conditions used in the referenced literature on irregular potentials; otherwise the arbitrary-order rates do not follow.
  2. [Section on the Newtonian limit (finite-time approximation)] In the Newtonian-limit argument, uniform-in-c a-priori estimates are required to pass to the limit on finite time windows while controlling the singular repulsive forces. The manuscript should confirm that these estimates remain valid independently of the speed-of-light parameter.
minor comments (2)
  1. [Abstract] The abstract states that rates are 'of any order' but does not indicate whether this means polynomial decay of every degree or a specific family of rates; a brief clarification would improve readability.
  2. [Notation and preliminaries] Notation for the relativistic energy/momentum should be introduced once and used consistently; minor inconsistencies in the definition of the kinetic term appear in early sections.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and will revise the manuscript to incorporate the requested explicit verifications.

read point-by-point responses

  1. Referee: [Section detailing the Lyapunov-function construction for the relativistic generator] The central claim of algebraic mixing rates of arbitrary order rests on the successful adaptation of Lyapunov functions to the relativistic generator. The manuscript must explicitly verify that the relativistic kinetic energy term preserves the coercivity and moment-growth conditions used in the referenced literature on irregular potentials; otherwise the arbitrary-order rates do not follow.

    Authors: We thank the referee for this observation. In the revised manuscript we will add an explicit lemma verifying that the relativistic kinetic energy satisfies the required coercivity and moment-growth conditions, confirming that the Lyapunov constructions from the literature on irregular potentials apply directly without additional assumptions. This will make the adaptation fully rigorous and justify the arbitrary-order algebraic rates. revision: yes

  2. Referee: [Section on the Newtonian limit (finite-time approximation)] In the Newtonian-limit argument, uniform-in-c a-priori estimates are required to pass to the limit on finite time windows while controlling the singular repulsive forces. The manuscript should confirm that these estimates remain valid independently of the speed-of-light parameter.

    Authors: We agree that uniformity in the speed-of-light parameter is essential. The a-priori estimates in our Newtonian-limit proof are already derived independently of this parameter. In the revision we will add an explicit remark or short lemma stating this c-independence, including the control on singular repulsive forces, to clarify the passage to the limit on finite time windows. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation constructs Lyapunov functions for the relativistic generator to obtain algebraic mixing rates of arbitrary order, adapting techniques from prior literature on irregular potentials, then passes to the Newtonian limit on finite-time windows. No load-bearing step reduces by definition, by fitting a parameter to the target quantity, or by a self-citation chain whose cited result itself depends on the present claims. The argument is presented as a self-contained mathematical construction from the stated assumptions on potentials and noise.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the existence of suitable Lyapunov functions for the relativistic generator and on standard well-posedness assumptions for the singular potentials; no new entities are introduced and no numerical parameters are fitted.

axioms (2)
  • domain assumption The confining and repulsive potentials admit Lyapunov functions that control the relativistic dynamics at algebraic rates, adapting constructions from prior literature on irregular potentials.
    Invoked to obtain the mixing rates; location implicit in the abstract's description of the proof strategy.
  • domain assumption The relativistic Langevin equation is well-posed under the stated potential assumptions.
    Required before any convergence or limit statement can be formulated.

pith-pipeline@v0.9.0 · 5625 in / 1391 out tokens · 20135 ms · 2026-05-23T21:15:49.817313+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

55 extracted references · 55 canonical work pages

  1. [1]

    J. A. Alc´ antara and S. Calogero. On a relativistic Fokke r-Planck equation in kinetic theory. Kinet. Relat. Mod. , 4(2):401–426, 2011

    work page 2011

  2. [2]

    J. Angst. Trends to equilibrium for a class of relativist ic diffusions. J. Math. Phys. , 52(11), 2011. 37

    work page 2011

  3. [3]

    Arnold and G

    A. Arnold and G. Toshpulatov. Trend to equilibrium and hy poelliptic regularity for the relativistic Fokker-Planck equation. arXiv preprint arXiv:2406.15139 , 2024

    work page arXiv 2024

  4. [4]

    Bakry, P

    D. Bakry, P. Cattiaux, and A. Guillin. Rate of convergenc e for ergodic continuous markov processes: Lyapunov versus poincar´ e.J. Funct. Anal. , 254(3):727–759, 2008

    work page 2008

  5. [5]

    Barbachoux, F

    C. Barbachoux, F. Debbasch, and J. Rivet. Covariant Kolm ogorov equation and entropy current for the rela- tivistic Ornstein-Uhlenbeck process. Eur. Phys. J. B , 23:487–496, 2001

    work page 2001

  6. [6]

    Bolley, D

    F. Bolley, D. Chafa ¨ ı, and J. Fontbona. Dynamics of a plan ar Coulomb gas. Ann. Appl. Probab. , 28(5):3152–3183, 2018

    work page 2018

  7. [7]

    Calogero

    S. Calogero. The Newtonian limit of the relativistic Bol tzmann equation. J. Math. Phys. , 45(11):4042–4052, 2004

    work page 2004

  8. [8]

    Calogero

    S. Calogero. Exponential convergence to equilibrium fo r kinetic Fokker-Planck equations. Commun. Partial Differ. Equ. , 37(8):1357–1390, 2012

    work page 2012

  9. [9]

    L. Chen, X. Li, P. Pickl, and Q. Yin. Combined mean field lim it and non-relativistic limit of Vlasov–Maxwell particle system to Vlasov–Poisson system. J. Math. Phys. , 61(6), 2020

    work page 2020

  10. [10]

    Chevalier and F

    C. Chevalier and F. Debbasch. Relativistic diffusions: a unifying approach. J. Math. Phys. , 49(4), 2008

    work page 2008

  11. [11]

    Conrad and M

    F. Conrad and M. Grothaus. Construction, ergodicity an d rate of convergence of N-particle Langevin dynamics with singular potentials. J. Evol. Equ. , 10(3):623–662, 2010

    work page 2010

  12. [12]

    Cooke, D

    B. Cooke, D. P. Herzog, J. C. Mattingly, S. A. McKinley, a nd S. C. Schmidler. Geometric ergodicity of two-dimensional Hamiltonian systems with a Lennard–Jones -like repulsive potential. Commun. Math. Sci. , 15(7):1987–2025, 2017

    work page 1987

  13. [13]

    Debbasch

    F. Debbasch. A diffusion process in curved space–time. J. Math. Phys. , 45(7):2744–2760, 2004

    work page 2004

  14. [14]

    Debbasch and C

    F. Debbasch and C. Chevalier. Relativistic stochastic processes. In Nonequilibrium Statistical Mechanics and Nonlinear Physics(AIP Conference Proceedings Volume 913) , volume 913, pages 42–48, 2007

    work page 2007

  15. [15]

    Debbasch, K

    F. Debbasch, K. Mallick, and J.-P. Rivet. Relativistic Ornstein–Uhlenbeck process. J. Stat. Phys. , 88:945–966, 1997

    work page 1997

  16. [16]

    Debbasch and J

    F. Debbasch and J. Rivet. A diffusion equation from the re lativistic Ornstein–Uhlenbeck process. J. Stat. Phys. , 90:1179–1199, 1998

    work page 1998

  17. [17]

    R. Douc, G. Fort, and A. Guillin. Subgeometric rates of c onvergence of f-ergodic strong Markov processes. Stoch. Process. Their Appl. , 119(3):897–923, 2009

    work page 2009

  18. [18]

    Dunkel and P

    J. Dunkel and P. H¨ anggi. Theory of relativistic browni an motion: The (1+ 1)-dimensional case. Phys. Rev. E , 71(1):016124, 2005

    work page 2005

  19. [19]

    Dunkel and P

    J. Dunkel and P. H¨ anggi. Theory of relativistic Browni an motion: The (1+3)-dimensional case. Phys. Rev. E , 72(3):036106, 2005

    work page 2005

  20. [20]

    Dunkel and P

    J. Dunkel and P. H¨ anggi. Relativistic Brownian motion . Phys. Rep. , 471(1):1–73, 2009

    work page 2009

  21. [21]

    M. H. Duong. Formulation of the relativistic heat equat ion and the relativistic kinetic Fokker–Planck equations using GENERIC. Physica A , 439:34–43, 2015

    work page 2015

  22. [22]

    M. H. Duong and H. D. Nguyen. Asymptotic analysis for the generalized Langevin equation with singular potentials. J. Nonlinear Sci. , 34(4):1–63, 2024

    work page 2024

  23. [23]

    Einstein

    A. Einstein. Does the enertia of a body depend on its ener gy content? Ann. Phys. , 18:639–641, 1905

    work page 1905

  24. [24]

    Einstein

    A. Einstein. On the electrodynamics of moving bodies. Ann. Phys. , 17:891, 1905

    work page 1905

  25. [25]

    Elskens, M

    Y. Elskens, M. K.-H. Kiessling, and V. Ricci. The Vlasov limit for a system of particles which interact with a wave field. Commun. Math. Phys. , 285:673–712, 2009

    work page 2009

  26. [26]

    J. A. A. F´ elix and S. Calogero. Newtonian limit and tren d to equilibrium for the relativistic Fokker-Planck equation. J. Math. Phys. , 54(3), 2013

    work page 2013

  27. [27]

    Fort and G

    G. Fort and G. Roberts. Subgeometric ergodicity of stro ng Markov processes. Ann. Appl. Probab. , 15(1A):1565– 1589, 2005

    work page 2005

  28. [28]

    N. E. Glatt-Holtz, D. P. Herzog, S. A. McKinley, and H. D. Nguyen. The generalized Langevin equation with power-law memory in a nonlinear potential well. Nonlinearity, 33(6):2820, 2020

    work page 2020

  29. [29]

    F. Golse. The mean-field limit for a regularized Vlasov- Maxwell dynamics. Commun. Math. Phys. , 310:789–816, 2012

    work page 2012

  30. [30]

    Grothaus and P

    M. Grothaus and P. Stilgenbauer. A hypocoercivity rela ted ergodicity method with rate of convergence for singularly distorted degenerate Kolmogorov equations and applications. Integr. Equ. Oper. Theory , 83(3):331— -379, 2015

    work page 2015

  31. [31]

    Z. Haba. Relativistic diffusion. Phys. Rev. E , 79(2):021128, 2009

    work page 2009

  32. [32]

    Z. Haba. Relativistic diffusion of elementary particle s with spin. J. Phys. A , 42(44):445401, 2009

    work page 2009

  33. [33]

    Z. Haba. Energy and entropy of relativistic diffusing pa rticles. Mod. Phys. Lett. A , 25(31):2683–2695, 2010

    work page 2010

  34. [34]

    M. Hairer. Exponential mixing for a stochastic partial differential equation driven by degenerate noise. Nonlin- earity, 15(2):271, 2002. 38

    work page 2002

  35. [35]

    M. Hairer. Exponential mixing properties of stochasti c PDEs through asymptotic coupling. Probab. Theory Relat. Fields, 124(3):345–380, 2002

    work page 2002

  36. [36]

    M. Hairer. How hot can a heat bath get? Commun. Math. Phys. , 292:131–177, 2009

    work page 2009

  37. [37]

    Hairer and J

    M. Hairer and J. C. Mattingly. Yet another look at Harris ’ ergodic theorem for Markov chains. Seminar on Stochastic Analysis, Random Fields and Applications VI , 63:109–117, 2011

    work page 2011

  38. [38]

    D. P. Herzog, S. Hottovy, and G. Volpe. The small-mass li mit for Langevin dynamics with unbounded coefficients and positive friction. J. Stat. Phys. , 163(3):659–673, 2016

    work page 2016

  39. [39]

    D. P. Herzog and J. C. Mattingly. Ergodicity and Lyapuno v functions for Langevin dynamics with singular potentials. Commun. Pure Appl. Math. , 72(10):2231–2255, 2019

    work page 2019

  40. [40]

    D. J. Higham, X. Mao, and A. M. Stuart. Strong convergenc e of Euler-type methods for nonlinear stochastic differential equations. SIAM J. Numer. Anal. , 40(3):1041–1063, 2002

    work page 2002

  41. [41]

    H¨ ormander

    L. H¨ ormander. Hypoelliptic second order differential equations. Acta Math. , 119(1):147–171, 1967

    work page 1967

  42. [42]

    Karatzas and S

    I. Karatzas and S. Shreve. Brownian Motion and Stochastic Calculus , volume 113. Springer Science & Business Media, 2012

    work page 2012

  43. [43]

    Khasminskii

    R. Khasminskii. Stochastic stability of differential equations , volume 66. Springer Science & Business Media, 2011

    work page 2011

  44. [44]

    Kiessling and A

    M.-H. Kiessling and A. S. Tahvildar-Zadeh. On the relat ivistic Vlasov-Poisson system. Indiana Univ. Math. J. , pages 3177–3207, 2008

    work page 2008

  45. [45]

    E. H. Lieb and H.-T. Yau. The stability and instability o f relativistic matter. Commun. Math. Phys. , 118:177–213, 1988

    work page 1988

  46. [46]

    Lu and J

    Y. Lu and J. C. Mattingly. Geometric ergodicity of Lange vin dynamics with Coulomb interactions. Nonlinearity, 33(2):675, 2019

    work page 2019

  47. [47]

    Mai and M

    L.-S. Mai and M. Mei. Newtonian limit for the relativist ic Euler-Poisson equations with vacuum. J. Diff. Equ. , 313:336–381, 2022

    work page 2022

  48. [48]

    J. C. Mattingly, A. M. Stuart, and D. J. Higham. Ergodici ty for SDEs and approximations: locally Lipschitz vector fields and degenerate noise. Stoch. Process. Their Appl. , 101(2):185–232, 2002

    work page 2002

  49. [49]

    Ottobre and G

    M. Ottobre and G. A. Pavliotis. Asymptotic analysis for the generalized Langevin equation. Nonlinearity, 24(5):1629, 2011

    work page 2011

  50. [50]

    Pal and S

    P. Pal and S. Deffner. Stochastic thermodynamics of rela tivistic Brownian motion. New J. Phys. , 22(7):073054, 2020

    work page 2020

  51. [51]

    G. A. Pavliotis. Stochastic Processes and Applications: Diffusion Processes , the Fokker-Planck and Langevin Equations, volume 60. Springer, 2014

    work page 2014

  52. [52]

    Rey-Bellet

    L. Rey-Bellet. Ergodic properties of Markov processes . In Open Quantum Systems II: The Markovian Approach , pages 1–39. Springer, 2006

    work page 2006

  53. [53]

    R. M. Strain. Global Newtonian limit for the relativist ic Boltzmann equation near vacuum. SIAM J. Math. Anal., 42(4):1568–1601, 2010

    work page 2010

  54. [54]

    Stroock and S

    D. Stroock and S. Varadhan. On degenerate elliptic-par abolic operators of second order and their associated diffusions. Commun. Pure Appl. Math. , 25(6):651–713, 1972

    work page 1972

  55. [55]

    C. Villani. Hypocoercivity, volume 202. American Mathematical Society, 2009. 1 School of Mathematics, University of Birmingham, Birmingh am, UK 2 Department of Mathematics, University of Tennessee, Knoxv ille, Tennessee, USA 39

    work page 2009