Finite-$N$ crossover from Vlasov to Boltzmann-Gibbs critical behavior in the Hamiltonian mean-field model

The Hamiltonian mean-field (HMF) model — $N$ rotators on a circle coupled by an all-to-all cosine interaction — is the paradigm for systems with long-range interactions. In the $N\to\infty$ limit its one-particle distribution obeys the Vlasov equation, whose infinite family of Casimir invariants — conserved functionals of the one-particle distribution — can trap a driven system in long-lived quasistationary states. Ogawa, Patelli and Yamaguchi (2014) showed that the susceptibility measured by the isolated dynamics then diverges with a non-classical exponent $\gamma_V=\beta/2$ (typically $1/4$) instead of the classical Boltzmann-Gibbs value $\gamma=1$ at continuous transitions of quasistationary-state families in the low-energy phase. Later work by the same community supplied a nonlinear response theory regular at criticality, a Landau-like theory of the quasistationary exponents, Casimir-complete response formulas for nonintegrable and spatially inhomogeneous states, and extensions to mean-field spin systems. What remains open is the finite-$N$ regime: no theory predicts how collisional ($1/N$-level) correlations break the Casimir invariants near the critical point, or on what timescale the measured susceptibility exponent crosses over from $\beta/2$ to $1$. The problem asks for a quantitative characterization of this crossover, which would also determine whether realistic experiments can observe the dynamical exponent before equilibration.

Background

Systems with long-range interactions, in which the coupling between particles does not decay with distance, lack additivity and can stay out of equilibrium for very long times; their statistical mechanics and dynamics are reviewed in Campa, Dauxois and Ruffo (2009). The simplest paradigm is the Hamiltonian mean-field (HMF) model introduced by Antoni and Ruffo (1995): $N$ classical rotators at angles $q_i$ with momenta $p_i$, all coupled through a cosine potential, with Hamiltonian $H_N=\sum_i p_i^2/2+(2N)^{-1}\sum_{i,j}(1-\cos(q_i-q_j))-h\Theta(t)\sum_i\cos q_i$, where a small external field $h$ may be switched on at $t=0$. Its order parameter is the magnetization $M=(1/N)\sum_i\cos q_i$. The Boltzmann-Gibbs equilibrium of the isolated model has a second-order ferromagnetic transition at temperature $T_c=1/2$, with the classical mean-field critical exponents: $\langle M\rangle\sim(T_c-T)^{1/2}$ (exponent $\beta=1/2$) and isothermal susceptibility $\chi_T\sim|T-T_c|^{-1}$ (exponent $\gamma=1$), where the susceptibility is the derivative of the order parameter with respect to a conjugate field computed in the equilibrium ensemble.

Because every particle interacts with every other with equal strength, the $N\to\infty$ limit of the one-particle distribution $f(q,p,t)$ is exactly governed by the Vlasov equation, the collisionless kinetic equation in which each particle moves in the self-consistent potential generated by $f$ itself. The Vlasov dynamics conserves not only the energy but an infinite family of Casimir invariants $C_s=\iint s(f)\,dq\,dp$, one for every smooth function $s$. A finite-$N$ system therefore remains trapped for very long times in quasistationary states (QSS) — stationary solutions of the Vlasov equation that are generally not Boltzmann-Gibbs equilibria — whose lifetime grows with $N$ before weak 'collisional' correlations at order $1/N$ slowly break the Casimirs and drive relaxation to equilibrium (Yamaguchi, Barré, Bouchet, Dauxois and Ruffo, 2004).

Statistical mechanics and dynamics then disagree about critical behavior. For an isolated system the measurable susceptibility is the response to a field switched on at $t=0$ computed to first order in $h$ (linear response); a classic inequality shows this isolated susceptibility cannot exceed the isothermal one, so its critical exponent cannot be larger. Using Vlasov-based linear response theories (Patelli, Gupta, Nardini and Ruffo, 2012; Ogawa and Yamaguchi, 2012), Ogawa, Patelli and Yamaguchi (2014) computed the susceptibility $\chi_V=(1-D_V)/D_V$ of the HMF model, where $D_V$ is the stability functional of the stationary state. For one-parameter families of Jeans-type stationary states — distributions depending on the phase-space variables only through the single-particle energy — that possess a continuous transition, including both the thermal equilibrium family and out-of-equilibrium Fermi-Dirac-type families approximately reached from waterbag initial data (distributions uniform on a rectangle of phase space), they found that in the low-energy (clustered) phase the isolated susceptibility diverges as $\chi_V\sim|b-b_c|^{-\beta/2}$ with $b$ the family parameter: the non-classical exponent $\gamma_V^-=\beta/2=1/4$, half the classical value, while in the homogeneous high-energy phase the classical $\gamma_V^+=1$ survives. The suppression is caused by the Casimir constraints, which prevent the trapped state from reaching the forced equilibrium; $N$-body simulations at $N=10^6$–$10^7$ confirmed the Vlasov exponent for times shorter than equilibration.

At finite $N$ the Casimirs are only approximate invariants, so the system eventually relaxes to the forced Boltzmann-Gibbs equilibrium and recovers the classical exponent. The source paper's simulations indicated a relaxation time growing linearly with $N$ for the inhomogeneous states studied (consistent with de Buyl, Mukamel and Ruffo, 2011), whereas systematic studies of homogeneous Vlasov-stable initial states had reported lifetimes scaling as $N^{1.7}$; moreover, for one-dimensional spatially homogeneous states the leading Balescu-Lenard collision term — the standard weak-coupling kinetic equation describing $1/N$ relaxation — is known to vanish, pushing relaxation to higher-order correlations. The same research program subsequently supplied a nonlinear response theory regular at the critical point (Ogawa and Yamaguchi, 2014), a Landau-like theory computing all quasistationary exponents at once (Ogawa and Yamaguchi, 2015), response formulas incorporating all Casimirs for nonintegrable and spatially inhomogeneous states (Ogawa, 2017), and extensions to mean-field classical spin systems (Yamaguchi, Das and Gupta, 2019) — but the finite-$N$ crossover itself has remained without a quantitative theory.

Problem Statement

The problem is to give a quantitative characterization of the finite-size crossover between dynamical and statistical-mechanical critical behavior in the HMF model, left open by Ogawa, Patelli and Yamaguchi (2014).

Set-up. Consider the HMF model with a small external field $h$ switched on at $t=0$, starting from a Vlasov-stable stationary state belonging to a one-parameter family with a continuous transition at $b=b_c$ (for example the thermal equilibrium family with $T_c=1/2$, or a Fermi-Dirac-type quasistationary family reached from waterbag initial data), on the low-energy (clustered, inhomogeneous) side $b>b_c$, where the isolated susceptibility carries the non-classical exponent $\gamma_V^-=\beta/2$; on the homogeneous high-energy side the classical exponent $\gamma_V^+=1$ already holds and no exponent crossover occurs. The measured response $(\langle M\rangle_h-\langle M\rangle_0)/h$ follows the Vlasov prediction, with exponent $\gamma_V^-=\beta/2$ (typically $1/4$), for times well below the equilibration time $\tau_N$, and the Boltzmann-Gibbs prediction, with exponent $\gamma=1$, for $t\gg\tau_N$.

Questions. (i) On what timescale $\tau_N$, as a function of $N$ and of the distance $b-b_c$ from the critical point, do finite-$N$ correlations break the Casimir invariants and drive the system to the forced equilibrium? (ii) What is the form of the crossover of the susceptibility exponent as a function of $t$, $N$ and $b-b_c$, including the anomalous finite-size region near the critical point, so that a measurement at given $(t,N,b)$ yields a definite predicted exponent?

The objective is to establish when the non-classical dynamical critical exponent is actually exhibited by a real finite system, and how it gives way to the classical statistical-mechanical exponent. The HMF model is the paradigm case; the same crossover question applies verbatim to the generalized long-range lattices and mean-field spin systems in which the Vlasov exponents have since been derived.

References

1. Ogawa, S., Patelli, A., Yamaguchi, Y. Y. (2014). Non-mean-field critical exponent in a mean-field model: Dynamics versus statistical mechanics. Physical Review E 89, 032131. DOI: 10.1103/PhysRevE.89.032131. https://arxiv.org/abs/1304.2982
2. Antoni, M., Ruffo, S. (1995). Clustering and relaxation in Hamiltonian long-range dynamics. Physical Review E 52, 2361. DOI: 10.1103/PhysRevE.52.2361. https://doi.org/10.1103/PhysRevE.52.2361
3. Campa, A., Dauxois, T., Ruffo, S. (2009). Statistical mechanics and dynamics of solvable models with long-range interactions. Physics Reports 480, 57. DOI: 10.1016/j.physrep.2009.07.001. https://doi.org/10.1016/j.physrep.2009.07.001
4. Patelli, A., Gupta, S., Nardini, C., Ruffo, S. (2012). Linear response theory for long-range interacting systems in quasistationary states. Physical Review E 85, 021133. DOI: 10.1103/PhysRevE.85.021133. https://doi.org/10.1103/PhysRevE.85.021133
5. Ogawa, S., Yamaguchi, Y. Y. (2012). Linear response theory in the Vlasov equation for homogeneous and for inhomogeneous quasistationary states. Physical Review E 85, 061115. DOI: 10.1103/PhysRevE.85.061115. https://doi.org/10.1103/PhysRevE.85.061115
6. Yamaguchi, Y. Y., Barré, J., Bouchet, F., Dauxois, T., Ruffo, S. (2004). Stability criteria of the Vlasov equation and quasi-stationary states of the HMF model. Physica A 337, 36. DOI: 10.1016/j.physa.2004.01.041. https://doi.org/10.1016/j.physa.2004.01.041
7. de Buyl, P., Mukamel, D., Ruffo, S. (2011). Self-consistent inhomogeneous steady states in Hamiltonian mean-field dynamics. Physical Review E 84, 061151. DOI: 10.1103/PhysRevE.84.061151. https://doi.org/10.1103/PhysRevE.84.061151
8. Bachelard, R., Dauxois, T., De Ninno, G., Ruffo, S., Staniscia, F. (2011). Vlasov equation for long-range interactions on a lattice. Physical Review E 83, 061132. DOI: 10.1103/PhysRevE.83.061132. https://doi.org/10.1103/PhysRevE.83.061132
9. Ogawa, S., Yamaguchi, Y. Y. (2014). Nonlinear response for external field and perturbation in the Vlasov system. Physical Review E 89, 052114. DOI: 10.1103/PhysRevE.89.052114. https://doi.org/10.1103/PhysRevE.89.052114
10. Ogawa, S., Yamaguchi, Y. Y. (2015). Landau-like theory for universality of critical exponents in quasistationary states of isolated mean-field systems. Physical Review E 91, 062108. DOI: 10.1103/PhysRevE.91.062108. https://arxiv.org/abs/1412.1593
11. Yamaguchi, Y. Y. (2015). Nondiagonalizable and nondivergent susceptibility tensor in the Hamiltonian mean-field model with asymmetric momentum distributions. Physical Review E 92, 032109. DOI: 10.1103/PhysRevE.92.032109. https://arxiv.org/abs/1501.04710
12. Barré, J., Métivier, D., Yamaguchi, Y. Y. (2016). Trapping scaling for bifurcations in the Vlasov systems. Physical Review E 93, 042207. DOI: 10.1103/PhysRevE.93.042207. https://doi.org/10.1103/PhysRevE.93.042207
13. Yamaguchi, Y. Y. (2016). Strange scaling and relaxation of finite-size fluctuation in thermal equilibrium. Physical Review E 94, 012133. DOI: 10.1103/PhysRevE.94.012133. https://arxiv.org/abs/1603.03173
14. Yamaguchi, Y. Y., Sawai, D. (2017). Nondivergent and negative susceptibilities around critical points of a long-range Hamiltonian system with two order parameters. Physical Review E 95, 052148. DOI: 10.1103/PhysRevE.95.052148. https://arxiv.org/abs/1701.02465
15. Ogawa, S. (2017). Linear and nonlinear response of the Vlasov system with nonintegrable Hamiltonian. Physical Review E 96, 012112. DOI: 10.1103/PhysRevE.96.012112. https://doi.org/10.1103/PhysRevE.96.012112
16. Ogawa, S. (2017). Stability criterion of spatially inhomogeneous solutions to the Vlasov equation. Journal of Physics A: Mathematical and Theoretical 50, 455001. DOI: 10.1088/1751-8121/aa8dbb. https://doi.org/10.1088/1751-8121/aa8dbb
17. Yamaguchi, Y. Y., Kaneko, K. (2018). Collective 1/f fluctuation by pseudo-Casimir-invariants. Physical Review E 98, 020201. DOI: 10.1103/PhysRevE.98.020201. https://doi.org/10.1103/PhysRevE.98.020201
18. Yamaguchi, Y. Y., Das, D., Gupta, S. (2019). Critical exponents in mean-field classical spin systems. Physical Review E 100, 032131. DOI: 10.1103/PhysRevE.100.032131. https://arxiv.org/abs/1905.08970
19. Das, D., Gupta, S. (2019). Relaxation to equilibrium in models of classical spins with long-range interactions. Journal of Statistical Mechanics 2019, 084007. DOI: 10.1088/1742-5468/ab342c. https://doi.org/10.1088/1742-5468/ab342c
20. Yamaguchi, Y. Y., Barré, J. (2026). Critical scaling of finite-size fluctuations around marginal stability in long-range Hamiltonian systems. arXiv:2603.17738. https://arxiv.org/abs/2603.17738