arXiv subjects

Weicheng Fu

Publications and source records attributed to Weicheng Fu.

At least 19 recordsLinked to original sources

arXiv2026

Emergence of Gamma-Type Upward-Phase Statistics in the Collatz Map: An Effective Poisson Process Mechanism

The Collatz map is a simple deterministic transformation whose orbit structure remains highly nontrivial. A recent direction-phase decomposition partitions each orbit into upward and downward steps, and numerical observations indicate that the number of upward phases, $N_{\uparrow}$, follows an approximate Gamma distribution. In this work, we provide a mechanistic explanation for this statistical regularity by modeling the occurrence of upward phases in the odd-compressed, or Syracuse, version of the Collatz map as a homogeneous Poisson process. From the mean-field logarithmic balance and the geometric distribution of $2$-adic valuations, we derive closed-form expressions for the Gamma parameters: the scale parameter $\theta = 2/(2-\log_2 3)^2 \approx 11.61$ is constant, whereas the shape parameter $K$ grows logarithmically with the maximal initial value $X_0=2L+1$. We also analyze the closure conditions for periodic orbits, showing that nontrivial cycles are severely constrained, which supports the plausibility of the statistical framework. Numerical validation for $L$ ranging from $10^5$ to $10^{15}$ confirms the theory with relative errors below $3\%$, and a bias-corrected mean estimate reduces the error to $10^{-3}$--$10^{-2}\%$. These results establish a quantitative link between the arithmetic properties of the Collatz map and Gamma-type statistics, and suggest possible extensions to generalized Collatz-type problems.

nlin.CD↗

arXiv2026

Boltzmann Distribution from Invariance of Coarse-Graining-Scale and Energy-Shift

We present a concise derivation of the Boltzmann form for single-particle energy distributions in classical many-body Hamiltonian systems. The derivation relies on two physical facts: coarse-graining-scale invariance of the empirical distribution and invariance under a uniform shift of the energy zero. These conditions uniquely yield the Boltzmann factor, whose parameter is fixed by the mean energy per particle. For separable Hamiltonians, the equilibrium weight factorizes into kinetic and configurational contributions sharing the same parameter, identified from the kinetic part as the inverse kinetic temperature. The principle extends to any physical quantity with a stationary distribution and translational invariance. It is illustrated in a one-dimensional diatomic hard-core gas and a nonlinear lattice chain, where it predicts velocity, energy, spacing, collision-time, and pressure-dependent displacement distributions in agreement with simulations. The lattice model further shows how harmonic elasticity, anharmonic corrections, internal pressure, and thermal expansion emerge from the same exponential equilibrium weights. Finally, the relationships among different ensembles are briefly discussed.

cond-mat.stat-mech↗

arXiv2026

The Fermi-Pasta-Ulam-Tsingou problem after 70 years: toward universal laws of thermalization in lattice systems in the thermodynamic limit

The Fermi--Pasta--Ulam--Tsingou (FPUT) problem provides a paradigmatic framework for understanding thermalization in weakly nonlinear many-body Hamiltonian systems. This focused review summarizes major developments in near-integrable dynamics, wave resonances, and phonon kinetic theory, emphasizing recent results on thermalization-time scaling in nonlinear lattices. A coherent picture emerges in the thermodynamic limit based on the eigenmode properties of an appropriate integrable reference system. If the reference system has extended eigenmodes and the leading resonant or quasi-resonant processes form a sufficiently connected network, the thermalization time follows $T_{\mathrm{eq}}\propto g^{-2}$, where $g$ measures the effective deviation from integrability. This scaling is found broadly in ordered and weakly disordered lattices and is robust to dimensionality, interaction potential, integrability-breaking mechanism, and multimode initial conditions. Identifying the correct integrable reference is essential; in some one-dimensional FPUT-type lattices with cubic interactions, the nearby Toda lattice, rather than the harmonic chain, provides the proper reference. If all reference eigenmodes are localized, spatial-overlap constraints progressively fragment low-order resonance networks as $g$ decreases, and thermalization becomes controlled by higher-order processes. Numerical studies reveal successive regimes $T_{\mathrm{eq}}\propto g^{-\gamma}$ with $\gamma=2,4,6$, together with weak system-size dependence. Whether this hierarchy persists asymptotically and whether a finite thermalization threshold exists remain open questions. We also discuss finite-size effects, strongly nonintegrable dynamics, heat transport, localization, and higher-dimensional lattices.

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arXiv2025

Parallel BiLSTM-Transformer networks for forecasting chaotic dynamics

The nonlinear nature of chaotic systems results in extreme sensitivity to initial conditions and highly intricate dynamical behaviors, posing fundamental challenges for accurately predicting their evolution. To overcome the limitation that conventional approaches fail to capture both local features and global dependencies in chaotic time series simultaneously, this study proposes a parallel predictive framework integrating Transformer and Bidirectional Long Short-Term Memory (BiLSTM) networks. The hybrid model employs a dual-branch architecture, where the Transformer branch mainly captures long-range dependencies while the BiLSTM branch focuses on extracting local temporal features. The complementary representations from the two branches are fused in a dedicated feature-fusion layer to enhance predictive accuracy. As illustrating examples, the model's performance is systematically evaluated on two representative tasks in the Lorenz system. The first is autonomous evolution prediction, in which the model recursively extrapolates system trajectories from the time-delay embeddings of the state vector to evaluate long-term tracking accuracy and stability. The second is inference of unmeasured variable, where the model reconstructs the unobserved states from the time-delay embeddings of partial observations to assess its state-completion capability. The results consistently indicate that the proposed hybrid framework outperforms both single-branch architectures across tasks, demonstrating its robustness and effectiveness in chaotic system prediction.

cs.LG↗

arXiv2025

From Near-Integrable to Far-from-Integrable: A Unified Picture of Thermalization and Heat Transport

Whether and how a system approaches equilibrium is central in nonequilibrium statistical physics, crucial to understanding thermalization and transport. Bogoliubov's three-stage (initial, kinetic, and hydrodynamic) evolution hypothesis offers a qualitative framework, but quantitative progress has focused on near-integrable systems like dilute gases. In this work, we investigate the relaxation dynamics of a one-dimensional diatomic hard-point (DHP) gas, presenting a phase diagram that characterizes relaxation behavior across the full parameter space, from near-integrable to far-from-integrable regimes. We analyze thermalization (local energy relaxation in nonequilibrium states) and identify three universal dynamical regimes: (i) In the near-integrable regime, kinetic processes dominate, local energy relaxation decays exponentially, and the thermalization time $\tau$ scales as $\tau \propto \delta^{-2}$. (ii) In the far-from-integrable regime, hydrodynamic effects dominate, energy relaxation decays power-law, and thermalization time scales linearly with system size $N$. (iii) In the intermediate regime, the Bogoliubov phase emerges, characterized by the transition from kinetic to hydrodynamic relaxation. The phase diagram also shows that hydrodynamic behavior can emerge in small systems when sufficiently far from the integrable regime, challenging the view that such effects occur only in large systems. In the thermodynamic limit, the system's relaxation depends on the order in which the limits ($N \to \infty$ or $\delta \to 0$) are taken. We then analyze heat transport (decay of heat-current fluctuations in equilibrium), demonstrating its consistency with thermalization, leading to a unified theoretical description of thermalization and transport. Our approach provides a pathway for studying relaxation dynamics in many-body systems, including quantum systems.

cond-mat.stat-mech↗

arXiv2025

Impact of on-site potentials on $q$-breathers in nonlinear chains

On-site potentials are ubiquitous in physical systems and strongly influence their heat transport and energy localization. These potentials will inevitably affect the dynamical properties of $q$-breathers (QBs), defined as periodic orbits exponentially localized in normal mode space. By integrating on-site terms into the Fermi-Pasta-Ulam-Tsingou-$\beta$ system, this work utilizes numerical simulations and Floquet analysis to systematically explore the influence of on-site potentials on QB stability. For most QBs, except those at the phonon band edges, the instability is primarily governed by parametric resonance, and effectively described by coupled Mathieu equations. This approach provides a theoretical expression for the instability thresholds, which aligns well with numerical results. We demonstrate that the instability thresholds can be controlled through the strength of on-site potentials, and for a strong enough quadratic on-site potential, the QBs are always stable. Furthermore, the instability threshold is highly sensitive to the seed mode, in stark contrast to systems without on-site potentials. In addition, the instability phase diagrams exhibit joint interplay between different terms in the Hamiltonian, such as the quadratic on-site and quartic inter-site interaction terms, in regulating the QB dynamics. These findings offer valuable insights into QB stability and the manipulation of localized excitations in diverse physical systems with on-site potentials.

cond-mat.stat-mech↗

arXiv2025

Multi-Type Instability Processes of Periodic Orbits in Nonlinear Chains

Nonlinear normal modes are periodic orbits that survive in nonlinear many-body Hamiltonian systems, and their instability is crucial for relaxation dynamics. Here, we study the instability process of the $\pi/3$-mode in the Fermi-Pasta-Ulam-Tsingou-$\alpha$ chain with fixed boundary conditions. We find that three types of bifurcations -- period-doubling, tangent, and Hopf -- coexist in this system, each driving instability at specific reduced wave-number $\tilde{k}$. Our analysis reveals a universal scaling law for the instability time $\mathcal{T} \propto (\lambda - \lambda_{\rm c})^{-1/2}$, independent of bifurcation types and models, where the critical perturbation strength $\lambda_{\rm c}$ scales as $\lambda_{\rm c} \propto (\tilde{k} - \tilde{k}_{\rm c})$, with $\tilde{k}_{\rm c}$ varying across bifurcations. We also observe a double instability phenomenon for certain system sizes, meaning that larger perturbations do not always lead to faster thermalization. These results provide new insights into the relaxation and thermalization dynamics in many-body systems.

cond-mat.stat-mech↗

arXiv2024

The Structure of the Route to the Period-three Orbit in the Collatz Map

This study analyzes the Collatz map through nonlinear dynamics. By embedding integers in Sharkovsky's ordering, we show that odd initial values suffice for full dynamical characterization. We introduce ``direction phases'' to partition iterations into upward and downward phases, and derive a recursive function family parameterized by upward phase counts. Consequently, a logarithmic scaling law between iteration steps and initial values is revealed, demonstrating finite-time convergence to the period-three orbit. Moreover, we establish the equivalence of the Collatz map to a binary shift map, whose ergodicity guarantees universal convergence to attractors, providing additional support for convergence. Furthermore, we identify that basins of attraction follow power-law distributions and find that odd numbers classified by upward phases follow Gamma statistics. These results offer valuable insights into the dynamics of discrete systems and their connections to number theory.

nlin.CD↗

arXiv2024

$q$-Breathers in the diatomic $\beta$-Fermi-Pasta-Ulam- Tsingou chains

$q$-Breathers (QBs) represent a quintessential phenomenon of energy localization, manifesting as stable periodic orbits exponentially localized in normal mode space. Their existence can hinder the thermalization process in nonlinear lattices. In this study, we employ the Newton's method to identify QB solutions in the diatomic Fermi-Pasta-Ulam-Tsingou chains and perform a comprehensive analysis of their linear stability. We derive an analytical expression for the instability thresholds of low-frequency QBs, which converges to the known results of monoatomic chains as the bandgap approaches zero. The expression reveals an inverse square relationship between instability thresholds and system size, as well as a quadratic dependence on the mass difference, both of which have been corroborated through extensive numerical simulations. Our results demonstrate that the presence of a bandgap can markedly enhance QB stability, providing a novel theoretical foundation and practical framework for controlling energy transport between modes in complex lattice systems. These results not only expand the applicability of QBs but also offer significant implications for understanding the thermalization dynamics in complex lattice structures, with wide potential applications in related low-dimensional materials.

cond-mat.stat-mech↗

arXiv2024

A general multi-wave quasi-resonance theory for lattice energy diffusion

In this letter, a multi-wave quasi-resonance framework is established to analyze energy diffusion in classical lattices, uncovering that it is fundamentally determined by the characteristics of eigenmodes. Namely, based on the presence and the absence of extended modes, lattices fall into two universality classes with qualitatively different thermalization behavior. In particular, we find that while the one with extended modes can be thermalized under arbitrarily weak perturbations in the thermodynamic limit, the other class can be thermalized only when perturbations exceed a certain threshold, revealing for the first time the possibility that a lattice cannot be thermalized, violating the hypothesis of statistical mechanics. Our study addresses conclusively the renowned Fermi-Pasta-Ulam-Tsingou problem for large systems under weak perturbations, underscoring the pivotal roles of both extended and localized modes in facilitating energy diffusion and thermalization processes.

cond-mat.stat-mech↗

arXiv2023

Nonintegrability-driven Transition from Kinetics to Hydrodynamics

Nonintegrability plays a crucial role in thermalization and transport processes in many-body Hamiltonian systems, yet its quantitative effects remain unclear. To reveal the connection between the macroscopic relaxation properties and the underlying dynamics, the one-dimensional diatomic hard-point model as an illustrating example was studied analytically and numerically. We demonstrate how the system transitions from kinetic behavior to hydrodynamic behavior as the nonintegrability strength increases. Specifically, for the thermalization dynamics, we find a power-law relationship between the thermalization time and the perturbation strength near integrable regime, whereas in the far from integrable regime, the hydrodynamics dominates and the thermalization time becomes independent of the perturbation strength and exhibits a strong size-dependent behavior. Regarding transport behavior, our results further establish a threshold for the nonintegrable strength of this transition. Consequently, we can predict which behavior dominates the transport properties of the system. Especially, an explicit expression of the thermal conductivity contributed by the kinetics is given. Finally, possible applications were briefly discussed.

cond-mat.stat-mech↗

arXiv2021

The anti-Fermi-Pasta-Ulam-Tsingou problem in one-dimensional diatomic lattices

We study the thermalization dynamics of one-dimensional diatomic lattices (which represents the simplest system possessing multi-branch phonons), exemplified by the famous Fermi-Pasta-Ulam-Tsingou (FPUT)-$\beta$ and the Toda models. Here we focus on how the system relaxes to the equilibrium state when part of highest-frequency optical modes are initially excited, which is called the anti-FPUT problem comparing with the original FPUT problem (low frequency excitations of the monatomic lattice). It is shown numerically that the final thermalization time $T_{\rm eq}$ of the diatomic FPUT-$\beta$ chain depends on whether its acoustic modes are thermalized, whereas the $T_{\rm eq}$ of the diatomic Toda chain depends on the optical ones; in addition, the metastable state of both models have different energy distributions and lifetimes. Despite these differences, in the near-integrable region, the $T_{\rm eq}$ of both models still follows the same scaling law, i.e., $T_{\rm eq}$ is inversely proportional to the square of the perturbation strength. Finally, comparisons of the thermalization behavior between different models under various initial conditions are briefly summarized.

cond-mat.stat-mech↗

arXiv2021

Thermalization in asymmetric harmonic chains

The symmetry of the interparticle interaction potential (IIP) plays a critical role in determining the thermodynamic and transport properties of solids. This study investigates the isolated effect of IIP asymmetry on thermalization. Asymmetry and nonlinearity are typically intertwined. To isolate the effect of asymmetry, we introduce a one-dimensional asymmetric harmonic (AH) model whose IIP possesses asymmetry but no nonlinearity, evidenced by energy-independent vibrational frequencies. Extensive numerical simulations confirm a power-law relationship between thermalization time ($T_{\rm eq}$) and perturbation strength for the AH chain, revealing an exponent larger than the previously observed inverse-square law in the thermodynamic limit. Upon adding symmetric quartic nonlinearity into the AH model, we systematically study thermalization under combined asymmetry and nonlinearity. Matthiessen's rule provides a good estimate of $T_{\rm eq}$ in this case. Our results demonstrate that asymmetry plays a distinct role in enhancing higher-order effects and governing relaxation dynamics.

cond-mat.stat-mech↗

arXiv2021

Effect of pressure on thermalization of one-dimensional nonlinear chains

Pressure plays a vital role in changing the transport properties of matter. To understand this phenomenon at a microscopic level, we here focus on a more fundamental problem, i.e., how pressure affects the thermalization properties of solids. As illustrating examples, we study the thermalization behavior of the monatomic chain and the mass-disordered chain of Fermi-Pasta-Ulam-Tsingou-$\beta$ under different strains in the thermodynamic limit. It is found that the pressure-induced change in nonintegrability results in qualitatively different thermalization processes for the two kinds of chains. However, for both cases, the thermalization time follows the same law -- it is inversely proportional to the square of the nonintegrability strength. This result suggests that pressure can significantly change the integrability of a system, which provides a new perspective for understanding the pressure-dependent thermal transport behavior.

cond-mat.stat-mech↗

arXiv2020

Instability dynamics of nonlinear normal modes in the Fermi-Pasta-Ulam-Tsingou chains

Nonlinear normal modes are periodic orbits that survive in nonlinear chains, whose instability plays a crucial role in the dynamics of many-body Hamiltonian systems toward thermalization. Here we focus on how the stability of nonlinear modes depends on the perturbation strength and the system size to observe whether they have the same behavior in different models. To this end, as illustrating examples, the instability dynamics of the ${N}/{2}$ mode in both the Fermi-Pasta-Ulam-Tsingou (FPUT) -$\alpha$ and -$\beta$ chains under fixed boundary conditions are studied systematically. Applying the Floquet theory, we show that for both models the stability time $T$ as a function of the perturbation strength $\lambda$ follows the same behavior; i.e., $T\propto(\lambda-\lambda_c)^{-\frac{1}{2}}$, where $\lambda_c$ is the instability threshold. The dependence of $\lambda_c$ on $N$ is also obtained. The results of $T$ and $\lambda_c$ agree well with those obtained by the direct molecular dynamics simulations. Finally, the effect of instability dynamics on the thermalization properties of a system is briefly discussed.

nlin.CD↗

arXiv2020

Thermalization of two- and three-dimensional classical lattices

Whether and how a system reaches thermalization is a fundamental issue of statistical physics. While for one-dimensional lattices this issue has been intensively studied in terms of energy equipartition for more than half a century, few work has been performed in the case of two- and three-dimensional lattices, and thus the thermalization dynamics remains unclear for more realistic lattices. In this Letter we investigate analytically and numerically the time-scaling of energy relaxation in these lattices. We show that the equipartition of energy is generally reached following a universal scheme for large enough lattices, regardless of its dimensionality, its specific lattice structure, and whether the system is translation invariant or not. Our results have practical significance in exploring the effect of high-order nonlinearities, i.e., the combining effect of multi-phonon process, in solid materials.

cond-mat.stat-mech↗

arXiv2019

Nonintegrability and thermalization of one-dimensional diatomic lattices

Nonintegrability is a necessary condition for the thermalization of a generic Hamiltonian system. In practice, the integrability can be broken in various ways. As illustrating examples, we numerically studied the thermalization behaviors of two types of one-dimensional (1D) diatomic chains in the thermodynamic limit. One chain was the diatomic Toda chain whose nonintegrability was introduced by unequal masses. The other chain was the diatomic Fermi-Pasta-Ulam-Tsingou-$β$ chain whose nonintegrability was introduced by quartic nonlinear interaction. We found that these two different methods of destroying the integrability led to qualitatively different routes to thermalization, but the thermalization time, $T_{eq}$, followed the same law; $T_{eq}$ was inversely proportional to the square of the perturbation strength. This law also agreed with the existing results of 1D monatomic lattices. All these results imply that there is a universal law of thermalization that is independent of the method of breaking integrability.

cond-mat.stat-mech↗

arXiv2019

Universal law of thermalization for one-dimensional perturbed Toda lattices

The Toda lattice is a nonlinear but integrable system. Here we study the thermalization problem in one-dimensional, perturbed Toda lattices in the thermodynamic limit. We show that the thermalization time, $T_{eq}$, follows a universal law; i.e., $T_{eq}\sim ε^{-2}$, where the perturbation strength, $ε$, characterizes the nonlinear perturbations added to the Toda potential. This universal law applies generally to weak nonlinear lattices due to their equivalence to perturbed Toda systems.

cond-mat.stat-mech↗