The theory of spatiotemporal pattern in nonequilibrium systems
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- 作者：Olemskoi ; Alexander I. ; Klepikov ; Vyacheslav F.
- 关键词：05.70.Fh ; 47.53.+n ; 64.60.Ak ; Order parameter ; Spatiotemporal distribution ; Phase portrait ; Fractal
- 刊名：Physics Reports
- 出版年：2000
- 出版时间：November, 2000
- 年：2000
- 卷：338
- 期：6
- 页码：571-677
- 全文大小：1.41 M
文摘
Stationary states and evolution of the spatially periodic and self-similar structures arising in the course of phase transformations are considered. Stationary inhomogeneous structures are studied in the framework of the thermodynamic approach, investigation of the dynamics of their formation is based on the field representation which makes it possible to describe behavior of the most probable values of stochastic variables. In the framework of unified conceptions, peculiarities of the behavior of thermodynamic, self-organizing and stochastic systems with multiplicative noise are studied. The concepts of the fractional integral and derivative are introduced which allow one to write the equation of evolution of an arbitrary system in a general form. Investigation of stationary inhomogeneous distributions is carried out by using the two- and one-component representations of the order parameter. A qualitative analysis based on the method of phase plane shows that under the transition from the disordered phase into the ordered one the harmonic distribution with a short period transforms into the step-like distribution with a long period. In the framework of the η4-model the stationary distribution of the order parameter reduces to the elliptic cosine and passing to the more complicated η6-model results in the fact that the (approximate) solutions will be the elliptic sine, cosine and delta of amplitude. The first of them describes the order parameter distribution, the second describes the antiphase boundaries and the form of the last one is defined by the two first ones. Creation of incommensurate long-period structures in ordered alloys is connected with the attraction of antiphase boundaries via the optical waves of atomic displacements. On the basis of the synergetic approach it is shown that, with increasing the concentration of the antiphase boundaries the optical-phonon exchange leads to stabilization of the long-period structure. The value of the force of the coherent bond between the boundaries is found. It is shown that creation of the long period is realized by the mechanism of phase transition of the first order. When describing the kinetics of transition into the stationary state, the simplest picture is first studied, in the framework of which the phase transition is represented by a single-order parameter. It is shown that under the parallel regime of phase transition the Debye character of relaxation is transformed into slowly decreasing dependencies of the type of the stretched Kohlrausch exponent, power, logarithmic and double logarithmic ones. The synergetic picture of phase transformations of the first and second orders is investigated, which is represented by the order parameter, conjugate field and control parameter. Analytical and numerical investigations of phase portraits have been carried out in different kinetic regimes. It is shown that owing to the critical increase of the times of relaxation of the order parameter and conjugate field the oscillatory behavior is realized if the initial relaxation time of the control parameter is much larger than its value for other degrees of freedom. In the opposite case, all the phase trajectories quickly converge to the universal zone. For a stochastic system with additive noise a self-consistent evolution of the order parameter and the amplitude of conjugate field fluctuations is considered. Investigation of the corresponding phase portraits shows that, depending on the ratio of the inhomogeneity scales, stable and unstable stationary states are possible. When passing to the multiplicative noise with an amplitude depending on the order parameter, the phase plane is divided into isolated domains of large, intermediate and small values of the order parameter. In the first one, in the course of time the trajectories converge to infinite values and the probability of their realization is vanishing. In the intermediate domain, the configurational point tends to the attractive node which corresponds to the stationary ordered state. Finally, in the region of small values of the order parameter, an absorbing state can arise where the system behaves in a deterministic way. The investigation of quasiperiodic distribution of the type which is observed in quasicrystals is based on the fact that it is generated by the same class of mapping, as incommensurate structures: the long-period structures correspond to points of the monofractal set which is contained in the given multifractal to the maximum extent; the quasicrystal sequence is the most rarely realized. A regular method for constructing this sequence is described and the distribution of wave vectors for which the radiation penetrating the quasicrystal yields the diffraction maxima is found. The mode-locking phenomenon is considered, whose spectra of frequencies and wave vectors represent “the devil's staircase”. In the framework of fractal ideology, the processes are considered which evolve in space–time in a non-local way. It is shown that in the presence of the non-perfect memory the generalized force leads to a flux which is expressed in the form of a fractional integral. Accordingly, increase of the share of dissipative channels leads to the transformation of the wave-type equation into the heat conduction equation. With decrease of the number of channels with the conserved order parameter, a smooth decrease of the order of the spatial derivative occurs. The method developed allows one to obtain not only a linear fractional-order equation of motion, but also to generalize it for the non-linear case. The latter case contains, in particular, such expressions, as the non-linear Schrödinger equation, the Korteweg–de Vries and sine Gordon equations.