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1.
P. H. Chavanis 《The European Physical Journal B - Condensed Matter and Complex Systems》2006,52(1):61-82
We study the relaxation of a test particle immersed
in a bath of field particles interacting via weak long-range
forces. To order 1/N in the N→+∞ limit, the
velocity distribution of the test particle satisfies a
Fokker-Planck equation whose form is related to the Landau and
Lenard-Balescu equations in plasma physics. We provide explicit
expressions for the diffusion coefficient and friction force in the
case where the velocity distribution of the field particles is
isotropic. We consider (i) various dimensions of space d=3,2 and
1; (ii) a discret spectrum of masses among the particles; (iii)
different distributions of the bath including the Maxwell
distribution of statistical equilibrium (thermal bath) and the step
function (water bag). Specific applications are given for
self-gravitating systems in three dimensions, Coulombian systems in
two dimensions and for the HMF model in one dimension. 相似文献
2.
P. H. Chavanis L. Delfini 《The European Physical Journal B - Condensed Matter and Complex Systems》2009,69(3):389-429
We apply the Nyquist method to the Hamiltonian mean field (HMF) model in order to settle the linear dynamical stability of
a spatially homogeneous distribution function with respect to the
Vlasov equation. We consider the case of Maxwell (isothermal) and Tsallis (polytropic) distributions and show that the system
is stable above a critical kinetic temperature Tc and unstable below it. Then, we consider a symmetric double-humped distribution, made of the superposition of two decentered
Maxwellians, and show the existence of a re-entrant phase in the stability diagram. When we consider an asymmetric double-humped
distribution, the re-entrant phase disappears above a
critical value of the asymmetry factor Δ > 1.09. We also consider the HMF model with a repulsive interaction. In that case,
single-humped distributions are always stable. For asymmetric double-humped distributions, there is a re-entrant phase for
1 ≤ Δ < 25.6, a double re-entrant phase for 25.6 < Δ < 43.9 and no re-entrant phase for Δ > 43.9. Finally, we extend our results
to arbitrary potentials of interaction and mention the connexion between the HMF model, Coulombian plasmas and gravitational
systems. We discuss the relation between linear dynamical stability and formal nonlinear dynamical stability and show their
equivalence for spatially
homogeneous distributions. We also provide a criterion of dynamical stability for spatially inhomogeneous systems. 相似文献
3.
P. H. Chavanis 《The European Physical Journal B - Condensed Matter and Complex Systems》2006,53(4):487-501
Systems with long-range interactions can reach a Quasi
Stationary State (QSS) as a result of a violent collisionless
relaxation. If the system mixes well (ergodicity), the QSS can be
predicted by the statistical theory of Lynden-Bell (1967) based on
the Vlasov equation. When the initial condition takes only two
values, the Lynden-Bell distribution is similar to the Fermi-Dirac
statistics. Such distributions have recently been observed in
direct numerical simulations of the HMF model (Antoniazzi et al. 2006). In this paper, we determine the caloric curve
corresponding to the Lynden-Bell statistics in relation with the
HMF model and analyze the dynamical and thermodynamical stability
of spatially homogeneous solutions by using two general criteria
previously introduced in the literature. We express the critical
energy and the critical temperature as a function of a degeneracy
parameter fixed by the initial condition. Below these critical
values, the homogeneous Lynden-Bell distribution is not a maximum
entropy state but an unstable saddle point. Known stability
criteria corresponding to the Maxwellian distribution and the
water-bag distribution are recovered as particular limits of our
study. In addition, we find a critical point below which the
homogeneous Lynden-Bell distribution is always stable. We apply
these results to the situation considered in Antoniazzi et
al. For a given energy, we find a critical initial
magnetization above which the homogeneous Lynden-Bell distribution
ceases to be a maximum entropy state. For an energy U=0.69, this
transition occurs above an initial magnetization Mx=0.897. In
that case, the system should reach an inhomogeneous Lynden-Bell
distribution (most mixed) or an incompletely mixed state (possibly
fitted by a Tsallis distribution). Thus, our theoretical study
proves that the dynamics is different for small and large initial
magnetizations, in agreement with numerical results of Pluchino et
al. (2004). This new dynamical phase transition may reconcile the
two communities by showing that they study different regimes. 相似文献
4.
R. Salazar A.R. Plastino R. Toral 《The European Physical Journal B - Condensed Matter and Complex Systems》2000,17(4):679-688
We introduce a new nonextensive entropic measure that grows like , where N is the size of the system under consideration. This kind of nonextensivity arises in a natural way in some N-body systems endowed with long-range interactions described by interparticle potentials. The power law (weakly nonextensive) behavior exhibited by is intermediate between (1) the linear (extensive) regime characterizing the standard Boltzmann-Gibbs entropy and (2) the
exponential law (strongly nonextensive) behavior associated with the Tsallis generalized q-entropies. The functional is parametrized by the real number in such a way that the standard logarithmic entropy is recovered when . We study the mathematical properties of the new entropy, showing that the basic requirements for a well behaved entropy
functional are verified, i.e., possesses the usual properties of positivity, equiprobability, concavity and irreversibility and verifies Khinchin axioms
except the one related to additivity since is nonextensive. For , the entropy becomes superadditive in the thermodynamic limit. The present formalism is illustrated by a numerical study of the thermodynamic
scaling laws of a ferromagnetic Ising model with long-range interactions.
Received 24 May 2000 相似文献
5.
P. H. Chavanis 《The European Physical Journal B - Condensed Matter and Complex Systems》2006,52(1):47-59
We present first elements of kinetic theory appropriate
to the inhomogeneous phase of the Hamiltonian Mean Field (HMF)
model. In particular, we investigate the case of strongly
inhomogeneous distributions for T→0 and exhibit
curious behaviour of the force auto-correlation function and
friction coefficient. The temporal correlation function of the
force has an oscillatory behaviour which averages to zero over a
period. By contrast, the effects of friction accumulate with time
and the friction coefficient does not satisfy the Einstein
relation. On the contrary, it presents the peculiarity to increase
linearly with time. Motivated by this result, we provide analytical
solutions of a simplified kinetic equation with a time dependent
friction. Analogies with self-gravitating systems and other systems
with long-range interactions are also mentioned. 相似文献
6.
P. H. Chavanis M. Lemou 《The European Physical Journal B - Condensed Matter and Complex Systems》2007,59(2):217-247
We develop the kinetic theory of point vortices in two-dimensional
hydrodynamics and illustrate the main results of the
theory with numerical simulations. We first consider the evolution of
the system “as a whole” and show that the evolution of the
vorticity profile is due to resonances between different orbits of the
point vortices. The evolution stops when the profile of angular
velocity becomes monotonic even if the system has not reached the
statistical equilibrium state (Boltzmann distribution). In that case,
the system remains blocked in a quasi stationary state with a non
standard distribution. We also study the relaxation of a test vortex
in a steady bath of field vortices. The relaxation of the test vortex
is described by a Fokker-Planck equation involving a diffusion term
and a drift term. The diffusion coefficient, which is proportional to
the density of field vortices and inversely proportional to the shear,
usually decreases rapidly with the distance. The drift is proportional
to the gradient of the density profile of the field vortices and is
connected to the diffusion coefficient by a generalized Einstein
relation. We study the evolution of the tail of the distribution
function of the test vortex and show that it has a front structure. We
also study how the temporal auto-correlation function of the position
of the test vortex decreases with time and find that it usually
exhibits an algebraic behavior with an exponent that we compute
analytically. We mention analogies with other systems with long-range
interactions. 相似文献
7.
L. G. Moyano A. P. Majtey C. Tsallis 《The European Physical Journal B - Condensed Matter and Complex Systems》2006,52(4):493-500
We introduce, and numerically study, a system of N symplectically and globally coupled
standard maps localized in a d=1 lattice array. The global coupling is modulated
through a factor r-α, being
r the distance between maps. Thus, interactions are long-range (nonintegrable) when
0≤α≤1, and short-range (integrable) when α>1.
We verify that the largest Lyapunov exponent λM scales as λM ∝
N-κ(α), where κ(α) is positive when interactions are
long-range, yielding weak chaos in the thermodynamic
limit N↦∞ (hence λM→0). In the short-range case,
κ(α) appears to vanish,
and the behaviour corresponds to strong chaos. We show that, for certain
values of the control parameters of the system, long-lasting metastable states
can be present. Their duration tc scales as tc ∝Nβ(α),
where β(α) appears to be numerically in agreement with the following
behavior: β>0 for 0 ≤α< 1, and zero for α≥1.
These results are consistent with features typically found in nonextensive statistical mechanics.
Moreover, they exhibit strong similarity between the present
discrete-time system, and the α-XY Hamiltonian ferromagnetic model. 相似文献
8.
We have investigated the proof of the H theorem within a
manifestly covariant approach by considering the relativistic
statistical theory developed in [G. Kaniadakis, Phys. Rev. E 66, 056125 (2002); G. Kaniadakis, Phys. Rev. E 72, 036108 (2005)]. As it
happens in the nonrelativistic limit, the molecular chaos hypothesis
is slightly extended within the Kaniadakis formalism. It is shown
that the collisional equilibrium states (null entropy source term)
are described by a κ power law generalization of the
exponential Juttner distribution, e.g.,
,
with θ=α(x)+βμpμ, where α(x) is a
scalar, βμ is a four-vector, and pμ is the
four-momentum. As a simple example, we calculate the relativistic
κ power law for a dilute charged gas under the action of an
electromagnetic field Fμν. All standard results are readly
recovered in the particular limit κ→0. 相似文献
9.
Liyan Liu 《Physica A》2008,387(22):5417-5421
We investigate the general property of the energy fluctuation in the canonical ensemble and the ensemble equivalence in Tsallis statistics. By taking the generalized ideal gas and the generalized harmonic oscillators as examples, we show that, when the particle number N is large enough, the relative fluctuation of the energy is proportional to 1/N in the new statistics, instead of in Boltzmann-Gibbs statistics. Thus the equivalence between microcanonical and canonical ensemble still holds in Tsallis statistics. 相似文献
10.
The present paper develops a Statistical Mechanics approach to the inherent states of glassy systems and granular materials
by following the original ideas proposed by Edwards for granular media. We consider three lattice models (a diluted spin glass,
a system of hard spheres under gravity and a hard-spheres binary mixture under gravity) introduced to describe glassy and
granular systems. They are evolved using a “tap dynamics” analogous to that of experiments on granular media. We show that
the asymptotic states reached in such a dynamics are not dependent on the particular sample history and are characterized
by a few thermodynamical parameters. We assume that under stationarity these systems are distributed in their inherent states
satisfying the principle of maximum entropy. This leads to a generalized Gibbs distribution characterized by new “thermodynamical”
parameters, called “configurational temperatures” (related to Edwards compactivity for granular materials). Finally, we show
by Monte Carlo calculations that the average of macroscopic quantities over the tap dynamics and over such distribution indeed
coincide. In particular, in the diluted spin glass and in the system of hard spheres under gravity, the asymptotic states
reached by the system are found to be described by a single “configurational temperature”. Whereas in the hard-spheres binary
mixture under gravity the asymptotic states reached by the system are found to be described by two thermodynamic parameters,
coinciding with the two configurational temperatures which characterize the distribution among the inherent states when the
principle of maximum entropy is satisfied under the constraint that the energies of the two species are independently fixed.
Received 19 March 2002 and Received in final form 14 June 2002 相似文献
11.
V.?Schw?mmle E. M.F.?Curado F. D.?Nobre 《The European Physical Journal B - Condensed Matter and Complex Systems》2009,70(1):107-116
Consequences of the connection between nonlinear Fokker-Planck equations
and entropic forms are investigated. A particular emphasis is given to
the feature that different nonlinear Fokker-Planck equations can be
arranged into classes associated with the same entropic form and its
corresponding stationary state.
Through numerical integration, the time evolution of the solution
of nonlinear Fokker-Planck equations related to the Boltzmann-Gibbs
and Tsallis entropies are analyzed.
The time behavior in both stages, in a time much smaller than the
one required for reaching the stationary state, as well as
towards the relaxation to the stationary state, are of particular interest.
In the former case, by
using the concept of classes of nonlinear Fokker-Planck equations,
a rich variety of physical behavior may be found, with some curious
situations, like an anomalous diffusion within the
class related to the Boltzmann-Gibbs entropy, as well as a
normal diffusion within the class of equations related
to Tsallis’ entropy. In addition to that, the relaxation
towards the stationary state may present a behavior
different from most of the systems studied in the literature. 相似文献
12.
S. Jain P. Buckley 《The European Physical Journal B - Condensed Matter and Complex Systems》2006,50(1-2):133-136
Persistence is studied in a financial context by mapping the time
evolution of the values of the shares quoted on the London Financial
Times Stock Exchange 100 index (FTSE 100) onto Ising spins.
By following the time dependence
of the spins, we find evidence for power law decay of the proportion
of shares that remain either above or below their 'starting'
values. As a result, we estimate a persistence exponent for the
underlying financial market to be θf∼0.5. 相似文献
13.
E. Bogomolny U. Gerland C. Schmit 《The European Physical Journal B - Condensed Matter and Complex Systems》2001,19(1):121-132
We propose a plasma model for spectral statistics displaying level repulsion without long-range spectral rigidity, i.e. statistics intermediate between random matrix and Poisson statistics similar to the ones found numerically at the critical
point of the Anderson metal-insulator transition in disordered systems and in certain dynamical systems. The model emerges
from Dysons one-dimensional gas corresponding to the eigenvalue distribution of the classical random matrix ensembles by restricting
the logarithmic pair interaction to a finite number k of nearest neighbors. We calculate analytically the spacing distributions and the two-level statistics. In particular we
show that the number variance has the asymptotic form Σ2(L) ∼χL for large L and the nearest-neighbor distribution decreases exponentially when s→∞, P(s) ∼ exp(- Λs) with Λ = 1/χ = kβ + 1, where β is the inverse temperature of the gas (β = 1, 2 and 4 for the orthogonal, unitary and symplectic symmetry class
respectively). In the simplest case of k = β = 1, the model leads to the so-called Semi-Poisson statistics characterized by particular simple correlation functions
e.g.
P(s) = 4s exp(- 2s). Furthermore we investigate the spectral statistics of several pseudointegrable quantum billiards numerically and compare
them to the Semi-Poisson statistics.
Received 13 September 2000 相似文献
14.
T. Dauxois P. Holdsworth S. Ruffo 《The European Physical Journal B - Condensed Matter and Complex Systems》2000,16(4):659-667
It is well known that long-range interactions pose serious problems for the formulation of statistical mechanics. We show
in this paper that ensemble equivalence is violated in a simple mean-field model of N fully coupled classical rotators with repulsive interaction (antiferromagnetic XY model). While in the canonical ensemble the rotators are randomly dispersed over all angles, in the microcanonical ensemble a bi-cluster of rotators separated by angle , forms in the low energy limit. We attribute this behavior to the extreme degeneracy of the ground state. We obtain empirically
an analytical formula for the probability density function for the angle made by the rotator, which compares extremely well
with numerical data and should become exact in the zero energy limit. At low energy, in the presence of the bi-cluster, an
extensive amount of energy is located in the single harmonic mode, with the result that the energy temperature relation is
modified. Although still linear, , it has the slope , instead of the canonical value .
Received 1 February 2000 相似文献
15.
P. H. Chavanis 《The European Physical Journal B - Condensed Matter and Complex Systems》2007,57(4):391-409
We derive the exact expression of the diffusion
coefficient of a self-gravitating Brownian gas in two
dimensions. Our formula generalizes the usual Einstein relation for
a free Brownian motion to the context of two-dimensional gravity. We
show the existence of a critical temperature Tc at which the
diffusion coefficient vanishes. For T < Tc, the diffusion
coefficient is negative and the gas undergoes gravitational
collapse. This leads to the formation of a Dirac peak concentrating
the whole mass in a finite time. We also stress that the critical
temperature Tc is different from the collapse temperature
T* at which the partition function diverges. These quantities
differ by a factor 1-1/N where N is the number of particles in
the system. We provide clear evidence of this difference by
explicitly solving the case N = 2. We also mention the analogy with
the chemotactic aggregation of bacteria in biology, the formation
of “atoms” in a two-dimensional (2D) plasma and the formation of
dipoles or “supervortices” in 2D point vortex dynamics. 相似文献
16.
L. R. Nie D. C. Mei 《The European Physical Journal B - Condensed Matter and Complex Systems》2007,58(4):475-481
The properties of the underdamped Josephson junction subjected to
colored noises were investigated with large and small phase
difference (φ). For the case of the large φ, we found
numerically that: (i) the probability distribution function of
φ exhibits monostability → bistability → monostability
transitions as the autocorrelation rate (λ) of a colored
noise increases; (ii) in the bistability region the multiplicative
noise drives the phase difference to turn over periodically; (iii)
the slope K of the linear response of the junction potential
difference (〈V 〉) can be somewhat reduced by means of tuning an
optimal λ; (iv) the amplitude of φ in response to
external sinusoidal signals changes with λ. For the case of
small φ, after deriving the analytical expressions of the
potential difference amplitude (〈V 〉max) and the K in the
presence of a dichotomous noise, we found nonmonotonic behavior of
〈V 〉max and the slope K as a function of λ. 相似文献
17.
G.-P. Zhang S.-J. Xiong 《The European Physical Journal B - Condensed Matter and Complex Systems》2002,29(3):491-495
We show that the electronic states in a one-dimensional (1D) Anderson model of diagonal disorder with long-range correlation
proposed by de Moura and Lyra exhibit localization-delocalization phase transition in varying the energy of electrons. Using
transfer matrix method, we calculate the average resistivity and investigate how it changes with the size of the system N. For given value of α (> 2) we find critical energies Ec1 and Ec2 such that the resistivity decreases with N as a power law ∝ N
- γ for electron energies within the range of [E
c1, E
c2], and exponentially grows with N outside this range. Such behaviors persist in approaching the transition points and the exponent γ is in the range from 0.92
to 0.96. The origin of the delocalization in this 1D model is discussed.
Received 18 December 2001 / Received in final form 2 May 2002 Published online 14 October 2002
RID="a"
ID="a"e-mail: sjxiong@nju.edu.cn 相似文献
18.
R. M. D'Souza P. L. Krapivsky C. Moore 《The European Physical Journal B - Condensed Matter and Complex Systems》2007,59(4):535-543
The “power of choice” has been shown to radically alter the behavior of a number of
randomized algorithms. Here we explore the effects of choice on models of random tree growth.
In our models each new node has k randomly chosen contacts, where k > 1 is a constant.
It then attaches to whichever one of these contacts is most desirable in some sense, such as its
distance from the root or its degree. Even when the new node has just two choices,
i.e., when k = 2, the resulting tree can be very different from a random graph or tree. For instance,
if the new node attaches to the contact which is closest to the root of the tree, the
distribution of depths changes from Poisson to a traveling wave solution.
If the new node attaches to the contact with the smallest degree, the degree distribution
is closer to uniform than in a random graph, so that with high probability there are no nodes in the
tree with degree greater than O(log log N). Finally, if the new node attaches to the contact
with the largest degree, we find that the degree distribution is a power law with exponent -1
up to degrees roughly equal to k, with an exponential cutoff beyond that; thus, in this case,
we need k ≫ 1 to see a power law over a wide range of degrees. 相似文献
19.
U. Tırnaklı C. Tsallis M. L. Lyra 《The European Physical Journal B - Condensed Matter and Complex Systems》1999,11(2):309-315
Dissipative one-dimensional maps may exhibit special points (e.g., chaos threshold) at which the Lyapunov exponent vanishes. Consistently, the sensitivity to the initial conditions has a
power-law time dependence, instead of the usual exponential one. The associated exponent can be identified with 1/(1-q), where q characterizes the nonextensivity of a generalized entropic form currently used to extend standard, Boltzmann-Gibbs statistical
mechanics in order to cover a variety of anomalous situations. It has been recently proposed (Lyra and Tsallis, Phys. Rev.
Lett. 80, 53 (1998)) for such maps the scaling law , where and are the extreme values appearing in the multifractal function. We generalize herein the usual circular map by considering inflexions of arbitrary power z, and verify that the scaling law holds for a large range of z. Since, for this family of maps, the Hausdorff dimension df equals unity for all z in contrast with q which does depend on z, it becomes clear that df plays no major role in the sensitivity to the initial conditions.
Received 5 February 1999 相似文献
20.
Phase-space Lagrangian dynamics in ideal fluids (i.e., continua) is usually related to the so-called ideal tracer particles. The latter, which can in principle be permitted to have arbitrary initial velocities, are understood as particles of infinitesimal size which do not produce significant perturbations of the fluid and do not interact among themselves. An unsolved theoretical problem is the correct definition of their dynamics in ideal fluids. The issue is relevant in order to exhibit the connection between fluid dynamics and the classical dynamical system, underlying a prescribed fluid system, which uniquely generates its time-evolution.The goal of this paper is to show that the tracer-particle dynamics can be exactly established for an arbitrary incompressible fluid uniquely based on the construction of an inverse kinetic theory (IKT) [M. Tessarotto, M. Ellero, Bull. Am. Phys. Soc. 45 (9) (2000) 40; M. Tessarotto, M. Ellero, AIP Conf. Proc. 762 (2005) 108. RGD24, Italy, July 10-16, 2004; M. Ellero, M. Tessarotto, Physica A 355 (2005) 233; M. Tessarotto, M. Ellero, Physica A 373 (2007) 142, arXiv: physics/0602140; M. Tessarotto, M. Ellero, in: M.S. Ivanov, A.K. Rebrov (Eds.), Proc. 25th RGD, International Symposium on Rarefied gas Dynamics, St. Petersburg, Russia, July 21-28, 2006, Novosibirsk Publ. House of the Siberian Branch of the Russian Academy of Sciences, 2007, p. 1001, arXiv:physics/0611113; M. Tessarotto, C. Cremaschini, Strong solutions of the incompressible Navier-Stokes equations in external domains: Local existence and uniqueness, arXiv:0809.5164v1 [math-ph], 2008]. As an example, the case of an incompressible Newtonian thermofluid is considered here. 相似文献