On the dynamical and statistical properties of a quartic mean-field Hamiltonian model

TL;DR

The paper investigates how integrability emerges and how chaos behaves in a quartic mean-field Hamiltonian as the number of degrees of freedom $N$ grows. It derives the Vlasov-limit description, showing that time-independent macroscopic moments $\mu_k$ yield an autonomous, integrable one-degree-of-freedom single-particle dynamics governed by an effective potential $V_{\mathrm{eff}}(x)$. Finite-$N$ analysis reveals that fluctuations scale as $N^{-1/2}$ and drive chaos with a largest Lyapunov exponent $\lambda_1 \propto N^{-1/3}$, while non-extensive $q$-statistics indicate that $q(t)$ converges to unity as time increases, signaling strong chaos and Gaussian statistics at long times. The results clarify that previously reported weak-chaos signatures were transient finite-time effects and provide a coherent dynamical-statistical picture linking the thermodynamic limit to finite-size behavior in mean-field systems.

Abstract

Mean-field systems provide a natural framework in which collective effects persist as the number of degrees of freedom N increases, raising fundamental questions about the emergence of integrability and the nature of chaos in large but finite systems. We investigate the dynamical and statistical properties of a quartic mean-field Hamiltonian model, with particular emphasis on the relation between the thermodynamic limit and finite-size chaotic dynamics. We first analyze the thermodynamic limit of the model within the Vlasov collisionless framework and derive the corresponding self-consistent single-particle description. We identify the conditions under which the mean-field dynamics becomes effectively autonomous and show numerically that fluctuations of the relevant intensive quantities vanish algebraically with N, supporting the emergence of integrability as N goes to infinity. We then study the finite-N dynamics by computing the largest Lyapunov exponent over an exceptionally wide range of N, spanning several orders of magnitude. We find that the largest Lyapunov exponent decays algebraically with N, consistently with the suppression of chaos in the thermodynamic limit for mean-field Hamiltonian models. Using tools from non-extensive statistical mechanics, we further analyze the time evolution of the entropic index q and demonstrate that, although transient values q > 1 may appear at intermediate times, q systematically converges to unity as the observation time increases. This behavior indicates that the finite-N dynamics is strongly chaotic in the asymptotic regime and that previously reported q > 1 values for the present models originate from finite-time effects rather than from a persistent weakly chaotic phase.

Figures (5)

• Figure 1: (a)--(c) Intensive moments $M_k(t)$ as a function of time for different values of $N$, and (d) the standard deviation $\sigma_{M_k}$ as a function of $N$, at $\varepsilon = 100$.
• Figure 2: (a) Average standard deviation of the effective energies [Eq. \ref{['eq:stds']}] as a function of the number of degrees of freedom $N$ for four values of the specific energy $\varepsilon$. The data follow a power-law decay $\sigma(N)=bN^{a}$. (b) Dependence of the prefactor $b$ on $\varepsilon$, showing a power-law scaling $b(\varepsilon)\propto\varepsilon^{\gamma}$. (c) Collapse of the curves in panel (a) after rescaling $\tilde{\sigma}=\sigma/\varepsilon^{\gamma}$, indicating scale invariance of the effective-energy fluctuations with respect to $\varepsilon$.
• Figure 3: The largest Lyapunov exponent (a) as a function of the specific energy $\varepsilon$ for different numbers of degrees of freedom $N$ and (b) as a function of the number of degrees of freedom $N$ for different specific energies $\varepsilon$.
• Figure 4: (red curves) Numerical probability density functions (PDFs) for $N = 1024$ and $\varepsilon = 100$ evaluated at different times: (a) $t = 1.0\times10^{5}$, (b) $t = 1.0\times10^{6}$, (c) $t = 1.0\times10^{7}$, and (d) $t = 1.0\times10^{8}$. The blue curves show the best fits to the numerical PDFs, yielding (a) $q = 1.712\pm0.008$, (b) $q = 1.367\pm0.002$, (c) $q = 1.130\pm0.001$, and (d) $q = 1.0224\pm0.0003$. The dashed black curves represent the Gaussian PDFs ($q = 1$) shown for comparison. Note that as $t \to \infty$, $q \to 1$. Table \ref{['tab:qvalues']} reports two quantitative measures of the fit quality for the distributions shown in this figure, namely the mean squared error (MSE) and the mean absolute error (MAE).
• Figure 5: (a)--(c) Entropic index $q$ and (d)--(f) the mean squared error (MSE) as a function of time for different number of degrees of freedom $N$ and different specific energy $\varepsilon$: (a) and (d) $\varepsilon = 10$, (b) and (e) $\varepsilon = 100$, and (c) and (f) $\varepsilon = 1000$. The $q$ values have been obtained via the best fit of the numerical PDFs at each instant of time.