Introduction

In the previous chapters we have studied waves which can be represented either as exact solutions of the field equations (as in the case of simple waves) or as propagating surfaces of discontinuities, for which exact transport laws can be obtained. These classes of waves, although very important for testing the mathematical structure of the theory, have a limited range of applicability. In particular, simple waves are restricted to one dimensional propagation into a constant state whereas propagating surfaces can model only impulsive waves. This leaves out the vast class of harmonic waves propagating into an arbitrary nonuniform state, which comprises most of the interesting applications. In order to treat the latter case it is necessary, in general, to resort to perturbation methods. Most of the perturbation methods used for this purpose are based on the so-called geometrical optics approximation, or variants thereof such as the high-frequency expansion or the method of multiple scales.

The idea underlying these methods is that there are at least two widely different length (or time) scales in the problem, the length L characteristic of the variation of the background state into which the wave is propagating, and the mean wavelength λ of the wavetrain, with the ordering λ « L. One then introduces a parameter ɛ = λ/L into the problem (usually through the appearance of one or several rapidly varying phase functions and also through other stretched variables, depending on ɛ) and seeks asymptotic solutions of the field equations as appropriate power series in ɛ.

Introduction

In the previous chapter we introduced the concept of propagating singular surfaces (or singular hypersurfaces in space-time) and derived the compatibility relations among the jumps of the field variables. When only derivatives of the field variables can be discontinuous across a propagating singular surface, we are dealing with weak discontinuities. When the fields themselves can be discontinuous we have strong discontinuities (among which shock waves are of paramount importance).

Whereas shock waves can be produced from an initially smooth solution as a consequence of nonlinear steepening and breaking, weak discontinuities can only be produced as a result of discontinuities in initial or boundary conditions. For instance, in gas dynamics, a jump in the derivatives of the velocity can appear in a flow along a solid obstacle with angles. Also, a jump in the derivatives of the pressure can appear (among other discontinuities like shocks and contact discontinuities) when the initial condition consists of two adjoining masses of gas compressed to different pressures (Landau and Lifshitz, 1959a). Although conceptually these two examples remain valid also for relativistic fluid motion, for relativistic flow in an astrophysical context only the latter is meaningful. For instance, the case of a cloud moving relativistically in a jet has been considered by Blandford and Königl (1979) in the context of models for the rapid variations in extragalactic radio sources' emissions.

An extremely important application of the concept of weak discontinuity waves is to the study of impulsive gravitational waves.

Relativistic fluid models are of considerable interest in several areas of astrophysics, plasma physics, and nuclear physics. Here we will mention briefly some of these areas and emphasize the problems which form the physical motivations for the theories expounded in this book.

Theories of gravitational collapse and models of supernova explosions (Van Riper, 1979; Chevalier, 1981; Shapiro and Teukolsky, 1983) are based on a relativistic fluid model for the star. In most models a key feature is the occurrence of an outward propagating relativistic shock. The precise conditions under which the shock forms at some point with exactly the necessary strength to expel the bulk of the star but still leave behind a remnant remain to be studied in detail and are the subject of current investigation. The effects of deviations from spherical symmetry due to an initial angular momentum and magnetic field must also be assessed. This requires the use of relativistic magneto-fluid dynamical models (Yodzsis, 1971; Maeda and Oohara 1982; Sloan and Smarr, 1986). The problem of the shock stability when traversing regions where the equation of state softens could be of interest for supernova models.

In the theories of galaxy formation, relativistic fluid models have been used in order to describe the evolution of perturbations of the baryon and radiation components of the cosmic medium (Peebles, 1980). Other components consisting of collisionless particles (such as massive neutrinos or photinos) are usually treated within a kinetic framework (Peebles, 1980).

Introduction

The stability of relativistic shock waves is a fundamental problem of relativistic fluid dynamics and its resolution might have interesting consequences for astrophysics, plasma physics, and nuclear physics. In particular, in models of gravitational collapse and supernova explosions, the bounce shock traverses regions of varying state equations. The stability of this shock is essential for the overall validity of the model. Present numerical codes, being restricted to spherical symmetry, are not capable of detecting these kinds of instabilities. In high-energy heavy ion collisions, the formation of a quark-gluon plasma might occur through a relativistic shock and its instability might signal the transition (e.g., by observing shock splitting; Barz et al., 1985).

In nonrelativistic fluid dynamics the stability of plane shock fronts has been investigated by D'Yakov (1956) and Erpenbeck (1962), using linear stability methods.

The problem has also been studied by Gardner (1963) in the framework of shock splitting methods and more recently by Fowels (1981, and references therein) who considered the stability of the front against an impinging acoustic wave from behind. By requiring that the perturbations decay sufficiently fast at infinity these authors obtained restrictions on the equation of state of a fluid in order for it to sustain plane fronted shock waves.

The linear stability analysis has been extended by Gardner and Kruskal (1964) to the fast magnetoacoustic shock and by Lessen and Deshpande (1967) to the slow magnetoacoustic shock.

Introduction

Simple waves are exact solutions of quasi-linear equations representing traveling waves. They are the most natural nonlinear analog of the plane traveling waves of the linear theory. Although they correspond to special initial conditions, they are still sufficiently general to be of physical interest (as witnessed by Friedrichs' theorem, which, loosely speaking, states that any one dimensional smooth solution neighboring a constant state must be a simple wave). Simple waves are exact analytical solutions which show clearly some of the main features of nonlinear wave propagation in general, such as steepening, breaking, and shock formation. They are also sufficiently complex to be useful as benchmarks against which to test numerical codes.

In classical fluid dynamics simple waves are of paramount importance for several reasons. Simple waves are the basic ingredients for constructing analytical solutions to the Riemann problem (or shock-tube problem: the evolution of an initial state corresponding to two adjacent fluids at different pressures). These solutions are among the standard tests for numerical hydrodynamical codes (Sod, 1978) because they comprise some of the key features of general hydrodynamical behavior (nonlinear steepening and the occurrence of discontinuities). Also, analytical solutions to the Riemann problem can be used in order to construct sophisticated numerical codes able to deal very accurately with shock front tracking (Plohr, Glimm, and McBryan, 1983). Simple waves are also used in order to construct approximate analytical solutions to the problem of weak shock decay (Landau and Lifshitz, 1959a; Whitham, 1974; Courant and Friedrichs, 1976) and in Whitham's theory of geometric shock dynamics (Whitham, 1974).

Most of the matter in the universe can, in some form or other, be treated as a fluid and in several instances (supernova explosions, jets in extragalactic radio sources, accretion onto neutron stars and black holes, high-energy particle beams, high-energy nuclear collisions, etc.) undergoes relativistic motion. This consideration alone should be sufficient to motivate research in relativistic fluids. In the past most of the results on relativistic fluids have been obtained in a piecemeal way, in relation to a particular problem under consideration and using ad hoc techniques. Although this approach is perfectly legitimate in the process of research in the various areas of applications (astrophysics, plasma physics, nuclear physics), in the long run it is unsatisfactory because it tends to obscure the underlying unity of the subject and of the relevant techniques. In fact, a problem tailored approach (instead of a systematic and general one) necessarily precludes utilizing, in a particular area, results obtained in another area, and therefore hinders the cross fertilization of various techniques, a method which has been fruitful in several areas of science.

In 1967 the French mathematician André Lichnerowicz published a masterful monograph on relativistic fluid dynamics and magneto-fluid dynamics, which covered mainly existence and uniqueness results. Since then there has been no other attempt at a systematic development of the subject, although there have been several important developments in the field (particularly in shock wave theory).

Introduction

All macroscopic systems are subject to irregular perturbations that cannot be modelled in a deterministic description. Following common usage, we shall call these perturbations ‘noise’. These may be ‘internal fluctuations’, due to the complex interaction of parts of a composite system, or they may be ‘external noise’, due to an irregular influence imposed on the system from the environment to which it is coupled. In this chapter we shall be interested particularly in the latter type of fluctuations and their effects on the macroscopic behavior of open nonlinear systems. It is well known now that nonlinear systems, such as the laser (Haken, 1978), the Couette system (Di Prima and Swinney, 1981), and the Belousov–Zhabotinskii reaction (Field and Burger, 1985), can display a rich variety of transition phenomena leading to temporal (and spatial) structuring: bistability, limit cycle oscillations, period doubling behavior, aperiodic behavior, etc. These transitions are due to the nonequilibrium constraints imposed by the environment on the system. It is under nonequilibrium conditions that the intrinsic nonlinearities of the system come fully into play; at and near equilibrium the behavior of these macroscopic systems can be well described by linear laws.

In light of this fact, it is particularly important to take into account that environments of open systems, and thus the external constraints imposed on the system, fluctuate in general more or less strongly. Rapid random fluctuations are always present in natural systems and their amplitude is not necessarily small, contrary to laboratory systems.

Introduction

During the long development time of the theory of stochastic processes in nonlinear dynamical systems many approximate methods of calculation of various stochastic characteristics have been worked out for the Markov and non-Markov cases. There are two different equivalent approaches to the problem of the Markov case. Stochastic differential equations are used in the first approach, and the Fokker–Planck equation for time-dependent probability density is applied in the second. One must know how to pass from the stochastic differential equation to the Fokker–Planck equation and vice versa, so choosing the most appropriate method of solving the problem. Therefore at the beginning of this chapter we consider the rules governing the choice of method mentioned above and the different interpretations of stochastic differential equations are also considered.

It is a well-known fact that there are two basic interpretations, each of them having its own advantages. The first is the Ito interpretation. Using this interpretation one can easily write the drifts if the stochastic differential equation is known (generally speaking, the rules of averaging are simpler in this interpretation). This is the advantage of the Ito interpretation. The other interpretation is the S-interpretation. In it the stochastic differential equation is more simply transformed when the nonlinear change of variables is made, and the form of the stochastic S-equation coincides with the form of the pre-limit stochastic differential equation. This is the advantage of the S-interpretation.

Introduction

The electrohydrodynamic (EHD) instability in nematic liquid crystals (NLCs) has received considerable attention in recent years for two main reasons. The first is that it enables one to study the formation process of a dissipative structure including the onset of chaos (Hirakawa and Kai, 1977; Kai, Araoka, Yamazaki and Hirakawa, 1979a; Kai and Hirakawa, 1977,1978; Kai, Wakabayashi and Imasaki, 1986). The second is that it allows experimental access to a multiplicative stochastic process (Schenzle and Brand, 1979) because it is easy to obtain a fluctuating control parameter, the so-called multiplicative noise (Kai, Kai, Takata and Hirakawa, 1979b; Kawakubo, Yanagita and Kabashima, 1981). These two aspects of the EHD instability involve a wide range of interesting new phenomena that need to be accounted for. The onset of dissipative structures (pattern formations) can be observed corresponding to instabilities relaxing from unstable to stable states of thermodynamically excited systems. The dynamical phenomena near such instability points have been studied in various fields, such as hydrodynamic instabilities (Croquette and Pocheau, 1984; Greenside and Coughran, 1984; Greenside and Cross, 1985; Pocheau and Croquette, 1984; Pomeau and Zaleski, 1983; Tesauro and Cross, 1986); phase separations (Furukawa, 1985; Gunton, San Miguel and Sohni, 1983; Komura, Osamura, Fujii and Takeda, 1984); oscillatory instabilities in electrical circuits (Kabashima, Itsumi, Kawakubo and Nagashima, 1975); and in laser radiation (Arecchi and Degiorgio, 1971).

All macroscopic physical systems are subject to fluctuations or noise. One of the most useful and interesting developments in modern statistical mechanics has been the realization that even complex nonequilibrium systems can often be reduced to equivalent ones of only a few degrees of freedom by the elimination of dynamically nonrelevant variables. Theoretical descriptions of such contracted systems necessarily begin with a set of either continuous or discrete dynamical equations which can then be used to describe noise driven systems with the inclusion of random terms. Studies of these stochastic dynamical equations have expanded rapidly in the past two decades, so that today an exuberant theoretical activity, a few experiments, and a remarkably large number of applications, some with challenging technological implications, are evident.

The purpose of these volumes is twofold. First we hope that their publication will help to stimulate new experimental activity by contrasting the smallness of the number of existing experiments with the many research opportunities raised by the chapters on applications. Secondly, it has been our aim to collect together in one place a complete set of authoritative reviews with contributions representative of all the major practitioners in the field. We recognize that as an inevitable consequence of the intended comprehensiveness, there will be few readers who will wish to digest these volumes in their entirety. We trust, instead, that readers will be stimulated to choose from the many possibilities for new research represented herein, and that they will find all the specialized tools, be they experimental or theoretical, that they are likely to require.

Introduction

The study of non-linear stochastic processes is usually characterized by considerable difficulties, thereby rendering it necessary to have recourse to computers for the theoretical predictions to be tested. This in turn involves interminable times of calculation with a great waste of money.

Analogue simulation by means of electronic devices allows us to bypass these difficulties. Although the precision of the analogue technique is modest compared to that of computer calculations, it must be remarked that in many cases it is more significant to have available, immediate results rather than very accurate numerical precision.

It is important to underline that the analogue experiment does not completely replace computer simulation; rather, a comparison of the two techniques, wherever possible, has proven to be a powerful method for studying non-linear stochastic phenomena.

In order to give a short description of electronic simulation, a few basic elements of the techniques will be given, then we shall discuss some significant applications. Finally, we shall describe how it is possible to obtain important conclusions via a joint use of computer and analogue simulation. This has been the object of a recent investigation on the role of coloured noise in some dynamical systems.

The basis of the analogue simulation via electronic devices

The integration operation

This operation is usually obtained using an RC-circuit; nevertheless, a Miller integrator, assembled in the configuration shown in Figure 8.1(a) is a better choice because of the following advantages:

1. (a) The high input impedance allows us to add many input signals.

2. […]