The dynamics of quasistationary states of long-range interacting systems with N particles can be described by kinetic equations such as the Balescu-Lenard and Landau equations. In the case of one-dimensional homogeneous systems, two-body contributions vanish as two-body collisions in one dimension only exchange momentum and thus cannot change the one-particle distribution. Using a Kac factor in the interparticle potential implies a scaling of the dynamics proportional to N(δ) with δ=1 except for one-dimensional homogeneous systems. For the latter different values for δ were reported for a few models. Recently it was shown by Rocha Filho and collaborators [Phys. Rev. E 90, 032133 (2014)] for the Hamiltonian mean-field model that δ=2 provided that N is sufficiently large, while small N effects lead to δ≈1.7. More recently, Gupta and Mukamel [J. Stat. Mech. (2011) P03015] introduced a classical spin model with an anisotropic interaction with a scaling in the dynamics proportional to N(1.7) for a homogeneous state. We show here that this model reduces to a one-dimensional Hamiltonian system and that the scaling of the dynamics approaches N(2) with increasing N. We also explain from theoretical consideration why usual kinetic theory fails for small N values, which ultimately is the origin of noninteger exponents in the scaling.
Systems with long-range interactions, while relaxing towards equilibrium, sometimes get trapped in long-lived non-Boltzmann quasistationary states (QSS) which have lifetimes that grow algebraically with the system size. Such states have been observed in models of globally coupled particles that move under Hamiltonian dynamics either on a unit circle or on a unit spherical surface. Here, we address the ubiquity of QSS in long-range systems by considering a different dynamical setting. Thus, we consider an anisotropic Heisenberg model consisting of classical Heisenberg spins with mean-field interactions and evolving under classical spin dynamics. Our analysis of the corresponding Vlasov equation for time evolution of the phase space distribution shows that in a certain energy interval, relaxation of a class of initial states occurs over a timescale which grows algebraically with the system size. We support these findings by extensive numerical simulations. This work further supports the generality of occurrence of QSS in long-range systems evolving under Hamiltonian dynamics.
We consider a long-range interacting system of N particles moving on a spherical surface under an attractive Heisenberg-like interaction of infinite range and evolving under deterministic Hamilton dynamics. The system may also be viewed as one of globally coupled Heisenberg spins. In equilibrium, the system has a continuous phase transition from a low-energy magnetized phase, in which the particles are clustered on the spherical surface, to a high-energy homogeneous phase. The dynamical behavior of the model is studied analytically by analyzing the Vlasov equation for the evolution of the single-particle distribution and numerically by direct simulations. The model is found to exhibit long-lived nonmagnetized quasistationary states (QSSs) which in the thermodynamic limit are dynamically stable within an energy range where the equilibrium state is magnetized. For finite N, these states relax to equilibrium over a time that increases algebraically with N. In the dynamically unstable regime, nonmagnetized states relax fast to equilibrium over a time that scales as lnN. These features are retained in presence of a global anisotropy in the magnetization.
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.
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 nonclassical 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 the universality class of the nonclassical 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.
The duration of the quasistationary states (QSSs) emerging in the
d-dimensional classical inertial $\ensuremath{\alpha}$-Heisenberg model, i.e.,
N three-dimensional rotators whose interactions decay with distance
rij as $1/{r}_{ij}^{\ensuremath{\alpha}}$ ($\ensuremath{\alpha}\ensuremath{\ge}0$), is studied through first-principle molecular dynamics. These QSSs appear for the very-long-range interaction regime ($0\ensuremath{\le}\ensuremath{\alpha}/d\ensuremath{\le}1$), for an average energy per rotator
U<Uc (
Uc=5/6), and they do not exist for
U>Uc. They are characterized by a kinetic temperature
TQSS, before a crossover to a second plateau occurring at the Boltzmann-Gibbs temperature
TBG>TQSS. We investigate here the behavior of their duration
tQSS when
U approaches
Uc from below, for large values of
N. The QSS gradually disappears as $U\ensuremath{\rightarrow}{U}_{c}$, while its duration undergoes a critical phenomenon, namely ${t}_{\text{QSS}}\ensuremath{\propto}{({U}_{c}\ensuremath{-}U)}^{\ensuremath{-}\ensuremath{\xi}}$. Universality is found for the critical exponent $\ensuremath{\xi}=1\text{.}67\ifmmode\pm\else\textpm\fi{}0\text{.}02$ throughout the very-long-range interaction regime.
The present paper is devoted to the study of the few–body quantum mechanical problem with singular finite rank cluster interactions. The corresponding Hamiltonians play an important role in mathematical physics, since few body Hamiltonians with more regular interactions present considerable difficulties which make impossible a detailed analytic study [ 13 ]. Also a numerical study of such Hamiltonians using Faddeev equations present very hard problems, since the interaction between the particles does not vanish at large distances. On the other hand few-body problems with singular cluster interaction are useful and intensively studied in statistical physics, since the corresponding Hamiltonians can be analyzed in detail even if the number of particles is very large. Models describing one dimensional particles are of particular interest, since the eigenfunctions of the many body Hamiltonians can often be calculated using Bethe Ansatz [ 14 ]. In fact the well-known Yang-Baxter equation was first written in connection with the study of system of several one-dimensional particles with pairwise delta interactions. Similar methods were used in atomic physics to study collisions of several particles [ 12 ] (in three dimensions). Few-body systems in applications are usually investigated by combining the classical description of the dynamics of heavy atoms with the quantum description of the dynamics of electrons and other light particles. The first attempt to construct a three body Hamiltonian describing three quantum particles in R 3 interacting through pairwise delta functional potentials is due to G.V. Skorniakov and K.A.Ter-Martirosian [ 29 ]. R.A.Minlos and L.D.Faddeev proved that the corresponding Hamiltonian is not bounded from below and therefore cannot be used in 278 the originally intended physical applications [ 24 , 25 ]. The operator was defined using the method of self–adjoint perturbations used by F.A.Berezin and L.D.Faddeev for investigating Schödinger operators with delta potential in R 3 [ 10 ]. Almost three decades later semibounded three-body operators in dimension three with generalized two-body interactions were constructed by extending the standard Hilbert space of square integrable functions in R 9 [ 17 , 18 , 19 , 27 , 30 ]. The most interesting model considered used the theory of generalized extensions suggested by B.Pavlov [ 7 , 26 ]. Different aspects of these models were analyzed recently [ 8 , 23 , 20 ]. A general approach to these operators is described in [ 7 , 22 ]. Let us mention that a realization of a many-body lower bounded Hamiltonian with point interactions for particles in R 3 has been obtained in the original Hilbert space by using the theory of Dirichlet forms [ 2 ]. The few-body systems of two dimensional particles with two-body interactions was studied in [ 11 ], where it was proven that the corresponding Hamiltonian is semibounded.