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The three-body Lennard-Jones system on the plane has a transition state, which is the straight conformation located at a saddle point of the potential energy landscape. We show that the transition state can be dynamically stabilized by excited vibration of particle distances. The stabilization mechanism is explained theoretically, and is verified by performing molecular dynamics simulations. We also examine whether the dynamical stabilization gives an impact on the reaction rate between the two isomers of equilateral triangle conformations by comparing with the transition state theory., Comment: 12 pages, 9 figures

We propose a novel method to reconstruct phase dynamics equations from responses in macroscopic variables to weak inputs. Developing linear and nonlinear response theories in coupled phase-oscillators, we derive formulae which connect the responses with the system parameters including the time delay in interactions. We examine our method by applying it to two phase models, one of which describes a mean-field network of the Hodgkin--Huxley type neurons with a nonzero time delay. The method does not require much invasiveness nor microscopic observations, and these advantages highlight its broad applicability in various fields., Comment: 12 pages, 5 figures

In a Vlasov equation, the destabilization of a homogeneous stationary state is typically described by a continuous bifurcation characterized by strong resonances between the unstable mode and the continuous spectrum. However, when the reference stationary state has a flat top, it is known that resonances drastically weaken, and the bifurcation becomes discontinuous. In this article, we use a combination of analytical tools and precise numerical simulations to demonstrate that this behavior is related to a codimension-two bifurcation, which we study in details., Comment: 14 pages, 10 figures

We consider conformation of a chain consisting of beads connected by stiff springs, where the conformation is determined by the bending angles between the consecutive two springs. A conformation is stabilized or destabilized not only by a given bending potential but also the fast spring motion, and stabilization by the spring motion depends on their excited normal modes. This stabilization mechanism has been named the dynamically induced conformation in a previous work on a three-body system. We extend analyses of the dynamically induced conformation in many-body chain-like bead-spring systems. The normal modes of the springs depend on the conformation, and the simple rule of the dynamical stabilization is that the lowest eigenfrequency mode contributes to the stabilization of the conformation. The high the eigenfrequency is, the more the destabilization emerges. We verify theoretical predictions by performing numerical simulations., Comment: 19 pages, 11 figrues

We present dynamical effects on conformation in a simple bead-spring model consisting of three beads connected by two stiff springs. The conformation defined by the bending angle between the two springs is determined not only by a given potential energy function depending on the bending angle, but also fast motion of the springs which constructs the effective potential. A conformation corresponding with a local minimum of the effective potential is hence called the dynamically induced conformation. We develop a theory to derive the effective potential by using multiple-scale analysis and the averaging method. A remarkable consequence is that the effective potential depends on the excited normal modes of the springs and amount of the spring energy. Efficiency of the obtained effective potential is numerically verified., Comment: 15 pages, 8 figures

A coupled phase-oscillator model consists of phase-oscillators, each of which has the natural frequency obeying a probability distribution and couples with other oscillators through a given periodic coupling function. This type of model is widely studied since it describes the synchronization transition, which emerges between the non-synchronized state and partially synchronized states. The synchronization transition is characterized by several critical exponents, and we focus on the critical exponent defined by coupling strength dependence of the order parameter for revealing universality classes. In a typical interaction represented by the perfect graph, an infinite number of universality classes is yielded by dependency on the natural frequency distribution and the coupling function. Since the synchronization transition is also observed in a model on a small-world network, whose number of links is proportional to the number of oscillators, a natural question is whether the infinite number of universality classes remains in small-world networks irrespective of the order of links. Our numerical results suggest that the number of universality class is reduced to one and the critical exponent is shared in the considered models having coupling functions up to the second harmonics with unimodal and symmetric natural frequency distributions., Comment: 9 pages, 8 figures

We propose a classification of bifurcations of Vlasov equations, based on the strength of the resonance between the unstable mode and the continuous spectrum on the imaginary axis. We then identify and characterize a new type of generic bifurcation where this resonance is weak, but the unstable mode couples with the Casimirs, which are constants of motion, to form a size 3 Jordan block. We derive a three-dimensional reduced noncanonical Hamiltonian system describing this bifurcation: coupling with the Casimirs controls the phase space portrait. Comparison of the reduced dynamics with direct numerical simulations on a test case gives excellent agreement. We finally discuss the relevance of this bifurcation to specific physical situations., Comment: 6 pages, 2 figures. Supplemental material available at the URL https://www.idpoisson.fr/barre/publications-et-preprints/

We develop a linear response theory by computing the asymptotic value of the order parameter from the linearized equation of continuity around the nonsynchronized reference state using the Laplace transform in time. The proposed theory is applicable to a wide class of coupled phase oscillator systems and allows for any coupling functions, any natural frequency distributions, any phase-lag parameters, and any values for the time-delay parameter. This generality is in contrast to the limitation of the previous methods of the Ott--Antonsen ansatz and the self-consistent equation for an order parameter, which are restricted to a model family whose coupling function consists of only a single sinusoidal function. The theory is verified by numerical simulations.

For a mean-field classical spin system exhibiting a second-order phase transition in the stationary state, we obtain within the corresponding phase space evolution according to the Vlasov equation the values of the critical exponents describing power-law behavior of response to a small external field. The exponent values so obtained significantly differ from the ones obtained on the basis of an analysis of the static phase-space distribution, with no reference to dynamics. This work serves as an illustration that cautions against relying on a static approach, with no reference to the dynamical evolution, to extract critical exponent values for mean-field systems., Comment: 12 pages, 8 figures; v2: added text and figure, Accepted for publication in Physical Review E (2019); v3: published version

Synchronization among rhythmic elements is modeled by coupled phase-oscillators each of which has the so-called natural frequency. A symmetric natural frequency distribution induces a continuous or discontinuous synchronization transition from the nonsynchronized state, for instance. It has been numerically reported that asymmetry in the natural frequency distribution brings new types of bifurcation diagram having, in the order parameter, oscillation or a discontinuous jump which emerges from a partially synchronized state. We propose a theoretical classification method of five types of bifurcation diagrams including the new ones, paying attention to the generality of the theory. The oscillation and the jump from partially synchronized states are discussed respectively by the linear analysis around the nonsynchronized state and by extending the amplitude equation up to the third leading term. The theoretical classification is examined by comparing with numerically obtained one., Comment: 18 pages, 14 figures

In this study, we propose a universal scenario explaining the $1/f$ fluctuation, including pink noises, in Hamiltonian dynamical systems with many degrees of freedom under long-range interaction. In the thermodynamic limit, the dynamics of such systems can be described by the Vlasov equation, which has an infinite number of Casimir invariants. In a finite system, they become pseudoinvariants, which yield quasistationary states. The dynamics then exhibit slow motion over them, up to the timescale where the pseudo-Casimir-invariants are effective. Such long-time correlation leads to $1/f$ fluctuations of collective variables, as is confirmed by direct numerical simulations. The universality of this collective $1/f$ fluctuation is demonstrated by taking a variety of Hamiltonians and changing the range of interaction and number of particles., Comment: 13 pages, 12 figures

The linear response is studied in globally coupled oscillator systems including the Kuramoto model. We develop a linear response theory which can be applied to systems whose coupling functions are generic. Based on the theory, we examine the role of asymmetry introduced to the natural frequency distribution, the coupling function, or the coupling constants. A remarkable difference appears in coexistence of the divergence of susceptibility at the critical point and a nonzero phase gap between the order parameter and the applied external force. The coexistence is not allowed by the asymmetry in the natural frequency distribution but can be realized by the other two types of asymmetry. This theoretical prediction and the coupling-constant dependence of the susceptibility are numerically verified by performing simulations in $N$-body systems and in reduced systems obtained with the aid of the Ott-Antonsen ansatz.

We propose a new mechanism of long-range coupling to excite low-frequency discrete breathers without the on-site potential. This mechanism is universal in long-range systems irrespective of the spatial boundary conditions, of topology of the inner degree of freedom, and of precise forms of the coupling functions. The limit of large population is theoretically discussed for the periodic boundary condition. Existence of discrete breathers is numerically demonstrated with stability analysis., Comment: 5 pages, 4 figures

The linear response is investigated in a long-range Hamiltonian system from the view point of dynamics, which is described by the Vlasov equation in the limit of large population. Due to existence of the Casimir invariants of the Vlasov dynamics, an external field does not drive the system to the forced thermal equilibrium in general, and the linear response is suppressed. With the aid of a linear response theory based on the Vlasov dynamics, we compute the suppressed linear response in a system having two order parameters, which introduce the conjugate two external fields and the susceptibility matrix of size two accordingly. Moreover, the two order parameters bring three phases and the three types of second-order phase transitions between two of them. For each type of the phase transitions, all the critical exponents for elements of the susceptibility matrix are computed. The critical exponents reveal that some elements of the matrices do not diverge even at critical points, while the mean-field theory predicts divergences. The linear response theory also suggests appearance of negative off-diagonal elements, in other words, an applied external field decreases the value of an order parameter. These theoretical predictions are confirmed by direct numerical simulations of the Vlasov equation., Comment: 14 pages, 6 figures

We study transitions in the Kuramoto model by shedding light on asymmetry in the natural frequency distribution, which has been assumed to be symmetric in many previous studies. The asymmetry brings two nonstandard bifurcation diagrams, with the aid of bimodality. The first diagram consists of stationary states, and has the standard continuous synchronization transition and a subsequent discontinuous transition as the coupling strength increases. Such a bifurcation diagram has been also reported in a variant model, which breaks the odd symmetry of the coupling function by introducing the phase lag. The second diagram includes the oscillatory state emerging from the partially synchronized state and followed by a discontinuous transition. This diagram is firstly revealed in this study. The two bifurcation diagrams are obtained by employing the Ott-Antonsen ansatz, and are verified by direct $N$-body simulations. We conclude that the asymmetry in distribution, with the bimodality, plays a similar role to the phase lag, and diversifies the transitions., Comment: The axes in figures have been fixed

We numerically exhibit strange scaling and temporal evolution of finite-size fluctuation in thermal equilibrium of a simple long-range interacting system. These phenomena are explained from the view point of existence of the Casimirs and their nonexactness in finite-size systems, where the Casimirs are invariants in the Vlasov dynamics describing the long-range systems in the limit of large population. This explanation expects appearance of the reported phenomena in a wide class of isolated long-range systems. The scaling theory is also discussed as an application of the strange scaling., Comment: 5 pages, 5 figures

We investigate response to an external magnetic field in the Hamiltonian mean-field model, which is a paradigmatic toy model of a ferromagnetic body and consists of plane rotators like the XY spins. Due to long-range interactions, the external field drives the system to a long-lasting quasistationary state before reaching thermal equilibrium, and the susceptibility tensor obtained in the quasista- tionary state is predicted by a linear response theory based on the Vlasov equation. For spatially homogeneous stable states, whose momentum distributions are asymmetric with zero-means, the theory reveals that the susceptibility tensor for an asymptotically constant external field is neither symmetric nor diagonalizable, and the predicted states are not stationary accordingly. Moreover, the tensor has no divergence even at the stability threshold. These theoretical findings are confirmed by direct numerical simulations of the Vlasov equation for the skew-normal distribution functions., Comment: 10 pages, 8 figures

An external force dynamically drives an isolated mean-field Hamiltonian system to a long-lasting quasistationary state, whose lifetime increases with population of the system. For second order phase transitions in quasistationary states, two non-classical critical exponents have been reported individually by using a linear and a nonlinear response theories in a toy model. We provide a simple way to compute the critical exponents all at once, which is an analog of the Landau theory. The present theory extends universality class of the non-classical exponents to spatially periodic one-dimensional systems, and shows that the exponents satisfy a classical scaling relation inevitably by using a key scaling of momentum., Comment: 7 pages

Predicting the long-lasting quasistationary state for a given initial state is one of central issues in Hamiltonian systems having long-range interaction. A recently proposed method is based on the Vlasov description and uniformly redistributes the initial distribution along contours of the asymptotic effective Hamiltonian, which is defined by the obtained quasistationary state and is determined self-consistently. The method, to which we refer as the rearrangement formula, was suggested to give precise prediction under limited situations. Restricting initial states consisting of spatially homogeneous part and small perturbation, we numerically reveal two conditions that the rearrangement formula prefers: One is no Landau damping condition for unperturbed homogeneous part, and the other comes from the Casimir invariants. Mechanisms of these conditions are discussed. Clarifying these conditions, we inform validity to use the rearrangement formula as the response theory for an external field, and we shed light on improving the theory as a nonequilibrium statistical mechanics., Comment: 11pages, 4 figures

A nonlinear response theory is provided by use of the transient linearization method in the spatially one-dimensional Vlasov systems. The theory inclusively gives responses to external fields and to perturbations for initial stationary states, and is applicable even to the critical point of a second order phase transition. We apply the theory to the Hamiltonian mean-field model, a toy model of a ferromagnetic body, and investigate the critical exponent associated with the response to the external field at the critical point in particular. The obtained critical exponent is nonclassical value 3/2, while the classical value is 3. However, interestingly, one scaling relation holds with another nonclassical critical exponent of susceptibility in the isolated Vlasov systems. Validity of the theory is numerically confirmed by directly simulating temporal evolutions of the Vlasov equation., Comment: 15 pages, 8 figures, accepted for publication in Phys. Rev. E, Lemma 2 is corrected

A phenomenological theory is proposed to analyze the asymptotic dynamics of perturbed inviscid Kolmogorov shear flows in two dimensions. The phase diagram provided by the theory is in qualitative agreement with numerical observations, which include three phases depending on the aspect ratio of the domain and the size of the perturbation: a steady shear flow, a stationary dipole, and four traveling vortices. The theory is based on a precise study of the inviscid damping of the linearized equation and on an analysis of nonlinear effects. In particular, we show that the dominant Landau pole controlling the inviscid damping undergoes a bifurcation, which has important consequences on the asymptotic fate of the perturbation., Comment: 9 pages, 7 figures

We consider the one-dimensional Vlasov equation with an attractive cosine potential, and its non homogeneous stationary states that are decreasing functions of the energy. We show that in the Sobolev space $W^{1,p}$ ($p>2$) neighborhood of such a state, all stationary states that are decreasing functions of the energy are stable. This is in sharp contrast with the situation for homogeneous stationary states of a Vlasov equation, where a control over strictly more than one derivative is needed to ensure the absence of unstable stationary states in a neighborhood of a reference stationary state [Z.Lin and C.Zeng, Comm.Math.Phys. {\bf 306}, 291-331 (2011)]., Comment: 17 pages, 4 figures

The mean-field theory tells that the classical critical exponent of susceptibility is the twice of that of magnetization. However, the linear response theory based on the Vlasov equation, which is naturally introduced by the mean-field nature, makes the former exponent half of the latter for families of quasistationary states having second order phase transitions in the Hamiltonian mean-field model and its variances. We clarify that this strange exponent is due to existence of Casimir invariants which trap the system in a quasistationary state for a time scale diverging with the system size. The theoretical prediction is numerically confirmed by $N$-body simulations for the equilibrium states and a family of quasistationary states., Comment: 6 pages, 3 figures

We investigate the dynamics of a small long-range interacting system, in contact with a large long-range thermal bath. Our analysis reveals the existence of striking anomalies in the energy flux between the bath and the system. In particular, we find that the evolution of the system is not influenced by the kinetic temperature of the bath, as opposed to what happens for short-range collisional systems. As a consequence, the system may get hotter also when its initial temperature is larger than the bath temperature. This observation is explained quantitatively in the framework of the collisionless Vlasov description of toy models with long-range interactions and shown to be valid whenever the Vlasov picture applies, from cosmology to plasma physics., Comment: 7 pages, 5 figures, revised version

We investigate the asymptotic damping of a perturbation around inhomogeneous stable stationary states of the Vlasov equation in spatially multi-dimensional systems. We show that branch singularities of the Fourier-Laplace transform of the perturbation yield algebraic dampings. In two spatial dimensions, we classify the singularities and compute the associated damping rate and frequency. This 2D setting also applies to spherically symmetric self-gravitating systems. We validate the theory using a toy model and an advection equation associated with the isochrone model, a model of spherical self-gravitating systems., Comment: 37 pages, 10 figures

We investigate the asymptotic behavior of a perturbation around a spatially non homogeneous stable stationary state of a one-dimensional Vlasov equation. Under general hypotheses, after transient exponential Landau damping, a perturbation evolving according to the linearized Vlasov equation decays algebraically with the exponent -2 and a well defined frequency. The theoretical results are successfully tested against numerical $N$-body simulations, corresponding to the full Vlasov dynamics in the large $N$ limit, in the case of the Hamiltonian mean-field model. For this purpose, we use a weighted particles code, which allows us to reduce finite size fluctuations and to observe the asymptotic decay in the $N$-body simulations., Comment: 26 pages, 8 figures; text slightly modified, references added, typos corrected

Systems with long-range interactions display a short-time relaxation towards Quasi Stationary States (QSSs) whose lifetime increases with system size. With reference to the Hamiltonian Mean Field (HMF) model, we here show that a maximum entropy principle, based on Lynden-Bell's pioneering idea of "violent relaxation", predicts the presence of out-of-equilibrium phase transitions separating the relaxation towards homogeneous (zero magnetization) or inhomogeneous (non zero magnetization) QSSs. When varying the initial condition within a family of "water-bags" with different initial magnetization and energy, first and second order phase transition lines are found that merge at an out--of--equilibrium tricritical point. Metastability is theoretically predicted and numerically checked around the first-order phase transition line., Comment: Physical Review Letters (2007)

We study finite size effects in a family of systems in which a parameter controls interaction-range. In the long-range regime where the infinite-size free energy is universal, we show that the finite size effects are not universal but depend on the interaction-range. The finite size effects are observed through discrepancies between time-averages of macroscopic variables in Hamiltonian dynamics and canonical averages of ones with infinite degrees of freedom. For a high energy regime, the relation to a pair of the discrepancies is theoretically predicted and numerically confirmed. We also numerically show that the finite-size effects of macroscopic variables in the canonical ensemble are close to ones in the dynamical systems., Comment: Proceedings of the workshop Comlexty and Nonextensivity - New Trends in Statistical Mechancs - (CN-Kyoto)

We show that the quasi-stationary states observed in the $N$-particle dynamics of the Hamiltonian Mean-Field (HMF) model are nothing but Vlasov stable homogeneous (zero magnetization) states. There is an infinity of Vlasov stable homogeneous states corresponding to different initial momentum distributions. Tsallis $q$-exponentials in momentum, homogeneous in angle, distribution functions are possible, however, they are not special in any respect, among an infinity of others. All Vlasov stable homogeneous states lose their stability because of finite $N$ effects and, after a relaxation time diverging with a power-law of the number of particles, the system converges to the Boltzmann-Gibbs equilibrium.

For perturbed ordinally differential equations, a procedure of renormalization group method is proposed. To uniquely obtain renormalized solutions for given initial conditions, the procedure assumes that the extra integral constants yielded by perturbative expansion are determined as they cancel out all the prescribed secular terms. A sufficient condition for the renormalized solutions coinciding with exact solutions is also stated., Comment: 4 pages

Using the renormalization group method, we improved the first order solution of the long-wavelength expansion of the Einstein equation. By assuming that the renormalization group transformation has the property of Lie group, we can regularize the secular divergence caused by the spatial gradient terms and absorb it to the background seed metric. The solution of the renormalization group equation shows that the renormalized metric describes the behavior of gravitational collapse in the expanding universe qualitatively well., Comment: 12 pages, 1 figure, typeset by REVTeX

Renormalization group method is one of the most powerful tool to obtain approximate solutions to differential equations. We apply the renormalization group method to Hamiltonian systems whose integrable parts linearly depend on action variables. We show that the renormalization group method gives the same approximate solutions as canonical perturbation theory up to the second order of a small parameter with action-angle coordinates., Comment: LaTeX2e, 6 pages

We investigate perturbed Hamiltonian systems with two degrees of freedom by renormalization group method, which derives a reduced equation called renormalization group equation (RGE) by handling secular terms. We found that RGE is not always a Hamiltonian system. The necessary and sufficient condition that RGE becomes a Hamiltonian system up to the second leading order of a small parameter is that the original system is separable by Cartesian coordinates. Moreover, RGE is integrable when it is a Hamiltonian system. These statements are partial generalizations of our previous paper., Comment: 21 pages, LaTeX (58 kb)

We investigate Hamiltonian systems with two degrees of freedom by using renormalization group method. We show that the original Hamiltonian systems and the renormalization group equations are integrable if the renormalization group equations are Hamiltonian systems up to the second leading order of a small parameter., Comment: 7 pages, No figures, LaTeX (19 kb)

We introduce a model of Poincar\'e mappings which represents hierarchical structure of phase spaces for systems with many degrees of freedom. The model yields residence time distribution of power type, hence temporal correlation remains long. The power law behavior is enhanced as the system size increases., Comment: 6 pages, 3 Encapsulated Postscript figures, LaTeX (58 kb)

A new universality of Lyapunov spectra {\lambda_i} is shown for Hamiltonian systems. The universality appears in middle energy regime and is different from another universality which can be reproduced by random matrices in the following two points. One is that the new universality appears in a limited range of large i/N rather than the whole range, where N is degrees of freedom. The other is Lyapunov spectra do not behave linearly while random matrices give linear behavior even on 3D lattice. Quadratic terms with smaller nonlinear terms of potential functions play an intrinsic role in the new universality., Comment: 19 pages, 16 Encapsulated Postscript figures, LaTeX (100 kb)

Temporal evolutions toward thermal equilibria are numerically investigated in a Hamiltonian system with many degrees of freedom which has second order phase transition. Relaxation processes are studied through local order parameter, and slow relaxations of power type are observed at the critical energy of phase transition for some initial conditions. Numerical results are compared with results of a phenomenological theory of statistical mechanics. At the critical energy, the maximum Lyapunov exponent takes the largest value. Temporal evolutions and probability distributions of local Lyapunov exponents show that the system is highly chaotic rather than weakly chaotic at the critical energy. Consequently theories for perturbed systems may not be applied to the system at the critical energy in order to explain the slow relaxation of power type., Comment: 16 pages, LaTeX, 13 Postscript figures