Fractional dynamics of systems with long-range space interaction and temporal memory

Field equations with time and coordinate derivatives of noninteger order are derived from a stationary action principle for the cases of power-law memory function and long-range interaction in systems. The method is applied to obtain a fractional generalization of the Ginzburg-Landau and nonlinear Schrödinger equations. As another example, dynamical equations for particle chains with power-law interaction and memory are considered in the continuous limit. The obtained fractional equations can be applied to complex media with/without random parameters or processes.     

Nonequilibrium statistical mechanics of systems with long-range interactions

Systems with long-range (LR) forces, for which the interaction potential decays with the interparticle distance with an exponent smaller than the dimensionality of the embedding space, remain an outstanding challenge to statistical physics. The internal energy of such systems lacks extensivity and additivity. Although the extensivity can be restored by scaling the interaction potential with the number of particles, the non-additivity still remains. Lack of additivity leads to inequivalence of statistical ensembles. Before relaxing to thermodynamic equilibrium, isolated systems with LR forces become trapped in out-of-equilibrium quasi-stationary states (qSSs), the lifetime of which diverges with the number of particles. Therefore, in the thermodynamic limit LR systems will not relax to equilibrium. The qSSs are attained through the process of collisionless relaxation. Density oscillations lead to particle–wave interactions and excitation of parametric resonances. The resonant particles escape from the main cluster to form a tenuous halo. Simultaneously, this cools down the core of the distribution and dampens out the oscillations. When all the oscillations die out the ergodicity is broken and a qSS is born. In this report, we will review a theory which allows us to quantitatively predict the particle distribution in the qSS. The theory is applied to various LR interacting systems, ranging from plasmas to self-gravitating clusters and kinetic spin models.     

Fokker-Planck equation with fractional coordinate derivatives

Using the generalized Kolmogorov-Feller equation with long-range interaction, we obtain kinetic equations with fractional derivatives with respect to coordinates. The method of successive approximations, with averaging with respect to a fast variable, is used. The main assumption is that the correlation function of probability densities of particles to make a step has a power-law dependence. As a result, we obtain a Fokker-Planck equation with fractional coordinate derivative of order 1<α<2.     

Statistical mechanics and dynamics of solvable models with long-range interactions

For systems with long-range interactions, the two-body potential decays at large distances as V(r)1/r^α, with α≤d, where d is the space dimension. Examples are: gravitational systems, two-dimensional hydrodynamics, two-dimensional elasticity, charged and dipolar systems. Although such systems can be made extensive, they are intrinsically non additive: the sum of the energies of macroscopic subsystems is not equal to the energy of the whole system. Moreover, the space of accessible macroscopic thermodynamic parameters might be non convex. The violation of these two basic properties of the thermodynamics of short-range systems is at the origin of ensemble inequivalence. In turn, this inequivalence implies that specific heat can be negative in the microcanonical ensemble, and temperature jumps can appear at microcanonical first order phase transitions. The lack of convexity allows us to easily spot regions of parameter space where ergodicity may be broken. Historically, negative specific heat had been found for gravitational systems and was thought to be a specific property of a system for which the existence of standard equilibrium statistical mechanics itself was doubted. Realizing that such properties may be present for a wider class of systems has renewed the interest in long-range interactions. Here, we present a comprehensive review of the recent advances on the statistical mechanics and out-of-equilibrium dynamics of solvable systems with long-range interactions. The core of the review consists in the detailed presentation of the concept of ensemble inequivalence, as exemplified by the exact solution, in the microcanonical and canonical ensembles, of mean-field type models. Remarkably, the entropy of all these models can be obtained using the method of large deviations. Long-range interacting systems display an extremely slow relaxation towards thermodynamic equilibrium and, what is more striking, the convergence towards quasi-stationary states. The understanding of such unusual relaxation process is obtained by the introduction of an appropriate kinetic theory based on the Vlasov equation. A statistical approach, founded on a variational principle introduced by Lynden-Bell, is shown to explain qualitatively and quantitatively some features of quasi-stationary states. Generalizations to models with both short and long-range interactions, and to models with weakly decaying interactions, show the robustness of the effects obtained for mean-field models.     

A physically based connection between fractional calculus and fractal geometry

We show a relation between fractional calculus and fractals, based only on physical and geometrical considerations. The link has been found in the physical origins of the power-laws, ruling the evolution of many natural phenomena, whose long memory and hereditary properties are mathematically modelled by differential operators of non integer order. Dealing with the relevant example of a viscous fluid seeping through a fractal shaped porous medium, we show that, once a physical phenomenon or process takes place on an underlying fractal geometry, then a power-law naturally comes up in ruling its evolution, whose order is related to the anomalous dimension of such geometry, as well as to the model used to describe the physics involved. By linearizing the non linear dependence of the response of the system at hand to a proper forcing action then, exploiting the Boltzmann superposition principle, a fractional differential equation is found, describing the dynamics of the system itself. The order of such equation is again related to the anomalous dimension of the underlying geometry.     

Transition to chaos in discrete nonlinear Schrödinger equation with long-range interaction

Discrete nonlinear Schrödinger (DNLS) equation describes a chain of oscillators with nearest-neighbor interactions and a specific nonlinear term. We consider its modification with long-range interaction through a potential proportional to 1/l^1+α$1/l^{1+\alpha }$ with fractional α<2$\alpha <2$ and l   as a distance between oscillators. This model is called α$\alpha$DNLS. It exhibits competition between the nonlinearity and a level of correlation between interacting far-distanced oscillators, that is defined by the value of α$\alpha$. We consider transition to chaos in this system as a function of α$\alpha$ and nonlinearity. It is shown that decreasing of α$\alpha$ with respect to nonlinearity stabilize the system. Connection of the model to the fractional generalization of the NLS (called FNLS) in the long-wave approximation is also discussed and some of the results obtained for α$\alpha$DNLS can be correspondingly extended to the FNLS.     

The fractional variational iteration method using He's polynomials

In this Letter, by introducing He's polynomials in the correct functional, we propose a new fractional variational iteration method to solve nonlinear time-fractional partial differential equations involving Jumarie's modified Riemann-Liouville derivative. Several examples have been solved to illustrate the proposed method is quite effective and convenient for solving kinds of nonlinear fractional order problems.     

Discrete chaos in fractional sine and standard maps

Fractional standard and sine maps are proposed by using the discrete fractional calculus. The chaos behaviors are then numerically discussed when the difference order is a fractional one. The bifurcation diagrams and the phase portraits are presented, respectively.     

On the solution of the fractional nonlinear Schrödinger equation

We present the nonlinear Schrödinger (NLS) equation of fractional order. The fractional derivatives are described in the Caputo sense. The Adomian decomposition method (ADM) in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of fractional differential equations. In these schemes, the solution constructed in power series with easily computable components.     

Transition on the relationship between fractal dimension and Hurst exponent in the long-range connective sandpile models

The relationships between the Hurst exponent H and the power-law scaling exponent B in a new modification of sandpile models, i.e. the long-range connective sandpile (LRCS) models, exhibit a strong dependence upon the system size L. As L decreases, the LRCS model can demonstrate a transition from the negative to positive correlations between H- and B-values. While the negative and null correlations are associated with the fractional Gaussian noise and generalized Cauchy processes, respectively, the regime with the positive correlation between the Hurst and power-law scaling exponents may suggest an unknown, interesting class of the stochastic processes.     

Internal degrees of freedom, long-range interactions and nonlocal effects in perturbed Klein-Gordon equations

We investigate different mechanisms for the excitation of soliton internal degrees of freedom and for the existence of long-range interactions between solitons. We will study a nonlocal Klein-Gordon equation that is used as a model for Josephson junctions in thin films. We will show the connections between nonlocality, nonlinearity, internal degrees of freedom, long-range interactions and power-law behaviors.     

Thermodynamics and dynamics of systems with long-range interactions

We review simple aspects of the thermodynamic and dynamical properties of systems with long-range pairwise interactions (LRI), which decay as 1/r^d+σ at large distances r in d dimensions. Two broad classes of such systems are discussed. (i) Systems with a slow decay of the interactions, termed “strong” LRI, where the energy is super-extensive. These systems are characterized by unusual properties such as inequivalence of ensembles, negative specific heat, slow decay of correlations, anomalous diffusion and ergodicity breaking. (ii) Systems with faster decay of the interaction potential, where the energy is additive, thus resulting in less dramatic effects. These interactions affect the thermodynamic behavior of systems near phase transitions, where long-range correlations are naturally present. Long-range correlations are often present in systems driven out of equilibrium when the dynamics involves conserved quantities. Steady state properties of driven systems with local dynamics are considered within the framework outlined above.     

On bandlimitedness and lag-limitedness of fractional Gaussian noise

The contributions of this short paper are two-fold. We shall show two interesting properties of fractional Gaussian noise (fGn), namely, its bandlimitedness and lag-limitedness. The computation formulas for the maximum frequency of bandlimited fGn and the maximum lag of lag-limited fGn are proposed. In addition, we will give a new explanation of the statistical dependences of fGn based on the present bandlimitedness and lag-limitedness of fGn.     

Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus

The fractional complex transform is suggested to convert a fractional differential equation with Jumarie?s modification of Riemann-Liouville derivative into its classical differential partner. Understanding the fractional complex transform and the chain rule for fractional calculus are elucidated geometrically.     

Hamiltonian and Brownian systems with long-range interactions: IV. General kinetic equations from the quasilinear theory

We develop the kinetic theory of Hamiltonian systems with weak long-range interactions. Starting from the Klimontovich equation and using a quasilinear theory, we obtain a general kinetic equation that can be applied to spatially inhomogeneous systems and that takes into account memory effects. This equation is valid at order 1/N in a proper thermodynamic limit and it coincides with the kinetic equation obtained from the BBGKY hierarchy. For N→+∞, it reduces to the Vlasov equation governing collisionless systems. We describe the process of phase mixing and violent relaxation leading to the formation of a quasistationary state (QSS) on the coarse-grained scale. We interpret the physical nature of the QSS in relation to Lynden-Bell’s statistical theory and discuss the problem of incomplete relaxation. In the second part of the paper, we consider the relaxation of a test particle in a thermal bath. We derive a Fokker-Planck equation by directly calculating the diffusion tensor and the friction force from the Klimontovich equation. We give general expressions of these quantities that are valid for possibly spatially inhomogeneous systems with long correlation time. We show that the diffusion and friction terms have a very similar structure given by a sort of generalized Kubo formula. We also obtain non-Markovian kinetic equations that can be relevant when the auto-correlation function of the force decreases slowly with time. An interesting factor in our approach is the development of a formalism that remains in physical space (instead of Fourier space) and that can deal with spatially inhomogeneous systems.     

The improved fractional sub-equation method and its applications to the space-time fractional differential equations in fluid mechanics

By introducing a new general ansätz, the improved fractional sub-equation method is proposed to construct analytical solutions of nonlinear evolution equations involving Jumarie?s modified Riemann-Liouville derivative. By means of this method, the space-time fractional Whitham-Broer-Kaup and generalized Hirota-Satsuma coupled KdV equations are successfully solved. The obtained results show that the proposed method is quite effective, promising and convenient for solving nonlinear fractional differential equations.     

Hamiltonian and Brownian systems with long-range interactions: III. The BBGKY hierarchy for spatially inhomogeneous systems

We study the growth of correlations in systems with weak long-range interactions. Starting from the BBGKY hierarchy, we determine the evolution of the two-body correlation function by using an expansion of the solutions of the hierarchy in powers of 1/N in a proper thermodynamic limit N→+∞, where N is the number of particles. These correlations are responsible for the “collisional” evolution of the system beyond the Vlasov regime due to finite N effects. We obtain a general kinetic equation that can be applied to spatially inhomogeneous systems and that takes into account memory effects. These peculiarities are specific to systems with unshielded long-range interactions. For spatially homogeneous systems with short memory time like plasmas, we recover the classical Landau (or Lenard-Balescu) equations. An interest of our approach is to develop a formalism that remains in physical space (instead of Fourier space) and that can deal with spatially inhomogeneous systems. This enlightens the basic physics and provides novel kinetic equations with a clear physical interpretation. However, unless we restrict ourselves to spatially homogeneous systems, closed kinetic equations can be obtained only if we ignore some collective effects between particles. General exact coupled equations taking into account collective effects are also given. We use this kinetic theory to discuss the processes of violent collisionless relaxation and slow collisional relaxation in systems with weak long-range interactions. In particular, we investigate the dependence of the relaxation time with the system size N and try to provide a coherent discussion of all the numerical results obtained for these systems.     

Hamiltonian and Brownian systems with long-range interactions: V. Stochastic kinetic equations and theory of fluctuations

We developed a theory of fluctuations for Brownian systems with weak long-range interactions. For these systems, there exists a critical point separating a homogeneous phase from an inhomogeneous phase. Starting from the stochastic Smoluchowski equation governing the evolution of the fluctuating density field of Brownian particles, we determine the expression of the correlation function of the density fluctuations around a spatially homogeneous equilibrium distribution. In the stable regime, we find that the temporal correlation function of the Fourier components of density fluctuations decays exponentially rapidly, with the same rate as the one characterizing the damping of a perturbation governed by the deterministic mean field Smoluchowski equation (without noise). On the other hand, the amplitude of the spatial correlation function in Fourier space diverges at the critical point T=T_c (or at the instability threshold k=k_m) implying that the mean field approximation breaks down close to the critical point, and that the phase transition from the homogeneous phase to the inhomogeneous phase occurs sooner. By contrast, the correlations of the velocity fluctuations remain finite at the critical point (or at the instability threshold). We give explicit examples for the Brownian Mean Field (BMF) model and for Brownian particles interacting via the gravitational potential and via the attractive Yukawa potential. We also introduce a stochastic model of chemotaxis for bacterial populations generalizing the deterministic mean field Keller-Segel model by taking into account fluctuations and memory effects.     

聚对苯乙炔的能带结构和关联能计算

在考虑电子-电子相互作用情况下,对基态非简并聚合物聚对苯乙炔（PPV）的长程关联能进行了理论计算,并用长程关联能对带隙进行修正,最后讨论了长程关联能对带隙的影响。在对PPV链的计算中发现,长程关联能先会随着链长N的增加而减小,但当链长大于10后长程关联能会达到饱和。在链长为20个单元的PPV计算中,长程关联能的修正会降低由Hartree-Fock(HF)方法所得的能隙宽度。修正后的能隙宽度能很好的接近实验结果。