arXiv subjects

Z. C. Tu

Publications and source records attributed to Z. C. Tu.

At least 19 recordsLinked to original sources

arXiv2026

Universal Construction of Generalized Lyapunov Functions for Nonlinear Dynamical Systems Using Physics-Informed Neural Networks

A scalar potential landscape provides an intuitive description of the stability, transitions, and global organization of dynamical systems. For non-gradient dynamics, however, constructing global Lyapunov-type functions for nonlinear flows with recurrent structures remains a formidable challenge. Here, we introduce the generalized Lyapunov function (GLF), a scalar potential that is monotonically non-increasing along deterministic trajectories, as a unifying framework for nonequilibrium potentials. Conventional Lyapunov functions, Freidlin--Wentzell quasi-potentials, and potentials arising from Ao-type decompositions naturally emerge as special cases within this framework. Furthermore, we develop a data-free physics-informed neural network (PINN) approach in which the Lyapunov inequality and a weak divergence-scale compatibility condition are directly embedded into the loss function. The framework is demonstrated on diverse systems, including linear dynamics, the Hopf normal form, the van der Pol oscillator, a three-dimensional Hopf-link flow, and symmetric and asymmetric competitive Lotka--Volterra systems. The learned landscapes agree closely with analytical results where available and, more generally, successfully reveal invariant sets as low- or constant-potential structures. In particular, the van der Pol oscillator, Hopf-link flow, and asymmetric Lotka--Volterra system demonstrate that coherent GLFs can be constructed without relying on known closed-form expressions, establishing a versatile route to potential-landscape construction for complex nonlinear non-gradient systems.

nlin.CD↗

arXiv2026

Geometric Bounds on the Finite-Time Performance of Active Machines

Optimizing energy conversion in active matter remains a central challenge in nonequilibrium physics. Here, we develop a unified thermodynamic framework that characterizes the finite-time performance of interacting active machines. We show that cyclic work admits a geometric decomposition into an antisymmetric thermodynamic curvature, governing work extraction, and a symmetric metric, controlling dissipation. Minimal-dissipation protocols follow geodesics in parameter space, while optimal work extraction deviates from them due to a curvature-induced, Lorentz-like effect. This geometric structure directly determines the finite-time scaling of work and dissipation, enabling a mapping onto Onsager-type quasi-linear current--force relations. We show that both the maximal efficiency and the efficiency at maximum power are governed by an asymmetry parameter and a figure of merit, establishing a formal correspondence between active machines and thermoelectric devices with broken time-reversal symmetry. Our results reveal a fundamental geometric origin of energy-conversion performance and provide a general framework for optimizing active machines.

cond-mat.stat-mech↗

arXiv2026

Axisymmetric membrane shapes in the Willmore and Helfrich models: first integrals and a hyperbolic formulation

The equilibrium shapes of fluid lipid membranes are governed by nonlinear differential equations derived from the Helfrich bending energy. In the tensionless, pressure-free, and zero-spontaneous- curvature limit, the governing equation reduces to the Willmore equation. We show that the combination of the Zheng--Liu and Langer--Singer first integrals reduces the third-order axisymmetric Willmore equation to a first-order ordinary differential equation. This formulation recovers the sphere, axisymmetric minimal surfaces, and the Clifford torus as special cases, while organizing the local solution space according to two integration constants. For nonzero spontaneous curvature, the axisymmetric Helfrich functional is reformulated as the energy of an inhomogeneous hyperbolic elastica with a preferred curvature proportional to the distance from the axis. This formulation identifies a constant-mean-curvature branch and a logarithmic branch associated with a biconcave disk profile. The resulting exact relations provide analytical constraints and benchmark solutions for configurations of axisymmetric vesicles and biomembranes.

math-ph↗

arXiv2026

Shortcuts to state transitions for active matter

Shortcut schemes can accelerate quasi-static processes in passive systems by adding auxiliary controls to realize swift transitions between equilibrium states. In active systems, however, inherently directed motion driven by free energy consumption continually drives the system away from equilibrium. In this work, we develop a shortcut framework to realize swift state transitions for active systems operating in the weak activity regime. An auxiliary potential is introduced to guide the system along a predefined distribution path, allowing it to reach the target state within a finite time. Considering unavoidable energy cost in such a finite-time process, we derive a thermodynamic metric from the dissipative work to induce a Riemann manifold on the space spanned by the control parameters. The optimal protocol with minimum dissipative work is then identical to the geodesic path in the geometric space. We demonstrate this framework by considering active systems confined in an external harmonic trap and interacting via two distinct internal potentials, respectively: an attractive harmonic coupling and a repulsive pairwise Gaussian-core coupling. The strengths of both the external trap and the internal interactions are controllable. For the latter case, since the auxiliary potential can not be derived precisely, we adopt a variational method to obtain an approximate auxiliary control. Compared to linear protocols, the geodesic protocols can effectively reduce dissipation.

cond-mat.stat-mech↗

arXiv2026

Macroscopic Thermodynamic Framework for the Mpemba Effect

The counterintuitive Mpemba effect, wherein a hotter system cools faster, critically lacks a general macroscopic theory. Here, starting from linear irreversible thermodynamics, we formulate a generalized Newton's cooling law, $\mathrm{d}T/\mathrm{d}t = -[\gamma_0 + \mathcal{M}Q(t)](T-T_r)$, for a system at temperature $T$ relaxing in a thermal reservoir at $T_r$, where the bare relaxation rate $\gamma_0$ is modified by an initial-state memory term, $Q(t) \propto T(0)-T_r$. Arising from the interplay between heat flux and structural evolution, the coefficient $\mathcal{M}$ governs anomalous relaxation behaviors, where $\mathcal{M} > 0$ ($\mathcal{M} < 0$) induces the (inverse) Mpemba effect. This universal thermodynamic framework maps out phase diagram to provide general criteria for the Mpemba effect in complex systems, offering a macroscopic picture that bridges disparate microscopic approaches.

cond-mat.stat-mech↗

arXiv2025

A minimal model with stochastically broken reciprocity

We introduce a minimal model consisting of a two-body system with stochastically broken reciprocity (i.e., random violation of Newton's third law) and then investigate its statistical behaviors, including fluctuations of velocity and position, time evolution of probability distribution functions, energy gain, and entropy production. The effective temperature of this two-body system immersed in a thermal bath is also derived. Furthermore, we heuristically present an extremely minimal model where the relative motion adheres to the same rules as in classical mechanics, while the effect of stochastically broken reciprocity only manifests in the fluctuating motion of the center of mass.

cond-mat.stat-mech↗

arXiv2024

Weighted average temperature as the effective temperature of a system in contact with two thermal baths

We investigate the effective temperature of a harmonic chain whose two ends are coupled to two baths at different temperatures. We propose to take the weighted average temperature as the effective temperature of the system. The weight factors are related to the couplings between the system and two baths as well as the asymmetry of interactions between oscillators. We revisit the thermodynamics of nonequilibrium steady states based on the weighted average temperature. It is found that the fundamental thermodynamic relations in nonequilibrium steady states possess similar concise forms as those in equilibrium thermodynamics, provided that we replace the temperature in equilibrium with the weighted average temperature in steady states. We also illustrate the procedure to explicitly calculate the effective temperatures via three examples.

cond-mat.stat-mech↗

arXiv2024

Thermodynamic Geometric Control of Active Matter

Active matter represents a class of non-equilibrium systems that constantly dissipate energy to produce directed motion. The thermodynamic control of active matter holds great potential for advancements in synthetic molecular motors, targeted drug delivery, and adaptive smart materials. However, the inherently non-equilibrium nature of active matter poses a significant challenge in achieving optimal control with minimal energy cost. In this work, we extend the concept of thermodynamic geometry, traditionally applied to passive systems, to active matter, proposing a systematic geometric framework for minimizing energy cost in non-equilibrium driving processes. We derive a cost metric that defines a Riemannian manifold for control parameters, enabling the use of powerful geometric tools to determine optimal control protocols. The geometric perspective reveals that, unlike in passive systems, minimizing energy cost in active systems involves a trade-off between intrinsic and external dissipation, leading to an optimal transportation speed that coincides with the self-propulsion speed of active matter. This insight enriches the broader concept of thermodynamic geometry. We demonstrate the application of this approach by optimizing the performance of an active monothermal engine within this geometric framework.

cond-mat.stat-mech↗

arXiv2023

Engineering Ratchet-Based Particle Separation via Shortcuts to Isothermality

Microscopic particle separation plays vital role in various scientific and industrial domains. In this Letter, we propose a universal non-equilibrium thermodynamic approach, employing the concept of Shortcuts to Isothermality, to realize controllable separation of overdamped Brownian particles. By utilizing a designed ratchet potential with temporal period $\tau$, we find in the slow-driving regime that the average particle velocity $\Bar{v}_s\propto\left(1-D/D^*\right)\tau^{-1}$, indicating that particles with different diffusion coefficients $D$ can be guided to move in distinct directions with a preset $D^*$. Furthermore, we reveal that there exists an extra energetic cost with a lower bound $W_{\rm{ex}}^{(\rm{min})}\propto\mathcal{L}^{2}\Bar{v}_s$, alongside a quasi-static work consumption. Here, $\mathcal{L}$ is the thermodynamic length of the driving loop in the parametric space. We numerically validate our theoretical findings and illustrate the optimal separation protocol (associated with $W_{\rm{ex}}^{(\rm{min})}$) with a sawtooth potential. This study establishes a bridge between thermodynamic process engineering and particle separation, paving the way for further explorations of thermodynamic constrains and optimal control in ratchet-based particle separation.

cond-mat.stat-mech↗

arXiv2023

Brownian motion of a particle with higher-derivative dynamics

The Brownian motion of a particle with higher-derivative dynamics (HDD) coupling with a bath consisting of harmonic oscillators is investigated. The Langevin equation and corresponding Fokker-Planck equation for the Brownian motion of the HDD particle are derived. As a case study, we particularly consider a stochastic Pais-Uhlenbeck oscillator. It is found that the Boltzmann distribution is pathological while this distribution is the steady solution to the Fokker-Planck equation.

cond-mat.stat-mech↗

arXiv2023

Polarons in Binary Bose-Einstein Condensates

Bose polarons are quasiparticles formed through the interaction between impurities and Bose-Einstein condensates. In this paper, we derive an effective Fröhlich Hamiltonian using the generalized Bogoliubov transformation. The effective Fröhlich Hamiltonian encompasses two types of effective interactions: impurity-density (ID) coupling and impurity-spin (IS) coupling. Furthermore, we employ the Lee-Low-Pines variational approach to investigate the relevant properties of Bose polarons induced by the ID and IS coupling. These properties include the ground state energy, effective mass, and average number of virtual phonons. Our findings reveal that the contribution resulting from IS couplings to the ground energy decreases to zero near the miscible-immiscible boundary. Additionally, the increase of the IS coupling induces a greater number of virtual phonons, impeding the movement of impurities and leading to a significant increase in the effective mass of Bose polarons.

cond-mat.quant-gas↗

arXiv2022

Dynamic path dependence of phase behaviors in dense active system

There are rich emergent phase behaviors in non-equilibrium active systems. Flocking and clustering are two representative dynamic phases. The relationship between these two phases is still unclear. In the paper, we numerically investigate the evolution of flocking and clustering in a system consisting of self-propelled particles with active reorientation. We consider the interplay between flocking and clustering phases under different initial states, and observe an unstable domain in order parameters phase diagrams due to initial states even in the absence of an explicit attraction. This point is different from the previous finding that active angular fluctuations lead to an earlier breakdown of collective motion and the emergence of a new bi-stable regime in the aligned active particles [R.Grossmann et al, New J. Phys.073033,14 (2012)]. In particular, we find that the existence of bi-stable states is due to the diversity of dynamic paths arising from different initial states. By increasing (decreasing) the initial degree of ordering, the bi-stable state can be shifted to a more ordered flocking (disordered clustering) state. These results enlighten us pave the way to manipulate emergent behaviors and collective motions of active system.

cond-mat.soft↗

arXiv2022

Exploring fundamental laws of classical mechanics via predicting the orbits of planets based on neural networks

Neural networks have provided powerful approaches to solve various scientific problems. Many of them are even difficult for human experts who are good at accessing the physical laws from experimental data. We investigate whether neural networks can assist us in exploring the fundamental laws of classical mechanics from data of planetary motion. Firstly, we predict the orbits of planets in the geocentric system using the gate recurrent unit, one of the common neural networks. We find that the precision of the prediction is obviously improved when the information of the Sun is included in the training set. This result implies that the Sun is particularly important in the geocentric system without any prior knowledge, which inspires us to gain Copernicus' heliocentric theory. Secondly, we turn to the heliocentric system and make successfully mutual predictions between the position and velocity of planets. We hold that the successful prediction is due to the existence of enough conserved quantities (such as conservations of mechanical energy and angular momentum) in the system. Our research provides a new way to explore the existence of conserved quantities in mechanics system based on neural networks.

astro-ph.EP↗

arXiv2022

Escape rate of an active Brownian particle in a rough potential

We discuss escape problem with the consideration of both the activity of particles and the roughness of potentials. we derive analytic expressions for the escape rate of a Brownian particle (ABP) in two types of rough potentials by employing the effective equilibrium approach and the Zwanzig method. We find that activity enhances the escape rate, but both the oscillating perturbation and the random amplitude hinder escaping.

cond-mat.stat-mech↗

arXiv2022

Microscopic low-dissipation heat engine via shortcuts to adiabaticity and shortcuts to isothermality

We construct a microscopic model of low-dissipation engines by driving a Brownian particle in a time-dependent harmonic potential. Shortcuts to adiabaticity and shortcuts to isothermality are introduced to realize the adiabatic and isothermal branches in a thermodynamic cycle, respectively. We derive an analytical expression of the efficiency at maximum power for this kind of engines. This expression satisfies the universal law of efficiency at maximum power up to the second order of the Carnot efficiency. We also analyze the issue of power at any given efficiency for general low-dissipation engines, and then obtain the supremum of the power in three limiting cases respectively.

cond-mat.stat-mech↗

arXiv2021

Deep learning Local Reduced Density Matrices for Many-body Hamiltonian Estimation

Human experts cannot efficiently access the physical information of quantum many-body states by simply "reading" the coefficients, but have to reply on the previous knowledge such as order parameters and quantum measurements. In this work, we demonstrate that convolutional neural network (CNN) can learn from the coefficients of local reduced density matrices to estimate the physical parameters of the many-body Hamiltonians, such as coupling strengths and magnetic fields, provided the states as the ground states. We propose QubismNet that consists of two main parts: the Qubism map that visualizes the ground states (or the purified reduced density matrices) as images, and a CNN that maps the images to the target physical parameters. By assuming certain constraints on the training set for the sake of balance, QubismNet exhibits impressive powers of learning and generalization on several quantum spin models. While the training samples are restricted to the states from certain ranges of the parameters, QubismNet can accurately estimate the parameters of the states beyond such training regions. For instance, our results show that QubismNet can estimate the magnetic fields near the critical point by learning from the states away from the critical vicinity. Our work illuminates a data-driven way to infer the Hamiltonians that give the designed ground states, and therefore would benefit the existing and future generalizations of quantum technologies such as Hamiltonian-based quantum simulations and state tomography.

quant-ph↗

arXiv2021

Equilibrium free energy differences from a linear nonequilibrium equality

Extracting equilibrium information from nonequilibrium measurements is a challenge task of great importance in understanding the thermodynamic properties of physical, chemical, and biological systems. The discovery of the Jarzynski equality illumines the way to estimate the equilibrium free energy difference from the work performed in nonequilibrium driving processes. However, the nonlinear (exponential) relation causes the poor convergence of the Jarzynski equality. Here, we propose a concise method to estimate the free energy difference through a linear nonequilibrium equality which inherently converges faster than nonlinear nonequilibrium equalities. This linear nonequilibrium equality relies on an accelerated isothermal process which is realized by using a unified variational approach, named variational shortcuts to isothermality. We apply our method to an underdamped Brownian particle moving in a double-well potential. The simulations confirm that the method can be used to accurately estimate the free energy difference with high efficiency. Especially during fast driving processes with high dissipation, the method can improve the accuracy by more than an order of magnitude compared with the estimator based on the nonlinear nonequilibrium equality.

cond-mat.stat-mech↗

arXiv2020

Abstract models for heat engines

We retrospect three abstract models for heat engines which include a classic abstract model in textbook of thermal physics, a primary abstract model for finite-time heat engines, and a refined abstract model for finite-time heat engines. The detailed models of heat engines in literature of finite-time thermodynamics may be mapped into the refined abstract model. The future developments based on the refined abstract model are also surveyed.

cond-mat.stat-mech↗