Astron. Astrophys. 327, 867-889 (1997)

2. Motivation

2.1. Chaotic systems beyond their horizon of predictability

Consider a strongly chaotic system, i.e., the gap between any couple of trajectories diverges exponentially with time. Let us place ourselves in the reference frame of one trajectory, that we describe as uniform motion on the z axis:

The second trajectory is then described by the equations:

where we have assumed a single Lyapunov exponent for simplicity of the argument. Let us eliminate the time between these equations. We obtain:

As schematized in Fig. 1, this means that the relative motion of one trajectory with respect to another one, when looked at with a very long time resolution (i.e., , right diagram in Fig. 1), becomes non-differentiable at the origin, with different backward and forward slopes. Moreover, the final direction of the trajectory in space is given by the initial "uncertainty vector" . Then chaos achieves a kind of amplification of the initial uncertainty. But the orientation of the uncertainty vector being completely uncontrolable (it can take its origin at the quantum scale itself), the second trajectory can emerge with any orientation with respect to the first. If we now start from a continuum of different values , the breaking point in the slope (Fig. 1) occurs anywhere, and the various trajectories become describable by non-differentiable, fractal paths.

Fig. 1. Schematic representation of the relative evolution in space of two initially nearby chaotic trajectories seen at three different time scales, , 10 and 100 (from Nottale, 1993a).

In the end, beyond the horizon of predictability, the information about the behavior of the trajectory at is completely lost; this strongly suggests that we switch to a statistical description. Indeed, assume that we are looking at the evolution of the system during a very large time scale (of the order of the age of the Universe) with a time-resolution . Each successive event can be considered as totally independent of its preceding one, because of this information loss. Such an independence of the various events leads to describing the trajectories in terms of a Markov process. In other words, even if the basic equations remain deterministic, it is not the case of their solutions. We can then wonder whether the classical equations remain adapted to the physical description on very large time scales, and we are led to suggesting the alternative starting point of inherently statistical systems. Moreover, we shall see that a description in terms of classical probabilities seems to be incomplete, and that, at time resolutions larger than the horizon, we need a quantum-like description in terms of probability amplitudes.

One must keep in mind that such a large time-scale description would be no longer valid at small time-scales, since when going back to (left diagram in Fig. 1), differentiability is recovered. This is in accordance with the scale-relativistic framework, in which physics, including its fundamental equations and their interpretation, is now explicitly scale-dependent. In particular, the physical laws can be subjected to a kind of phase transition around some symmetry breaking scales, as we shall see in what follows.

2.2. Giving up differentiability of space-time

There is a fundamental reason for jumping to a non-deterministic, scale-relativistic physical description at small and large scales. Since more than three centuries, physics relies on the assumption that space-time coordinates are a priori differentiable. However, it was demonstrated by Feynman (see Feynman & Hibbs 1965) that the typical paths of quantum mechanical particles are continuous but non-differentiable. Now, one of the most powerful avenue for reaching a genuine understanding of the laws of nature has been to construct them, not from setting additional hypotheses, but on the contrary by attempting to give up some of them, i.e., by going to increased generality. From that point of view, of which Einstein was a firm supporter, the laws and structures of nature are simply the most general laws and structures that are physically possible. It culminated in the principle of "general" relativity and in Einstein's explanation of the nature of gravitation as the various manifestations of the Riemannian geometry of space-time (i.e., of the giving up of flatness).

However, in the light of the above remark, Einstein's principle of relativity is not yet fully general, since it applies to coordinate transformations that are continuous and at least two times differentiable. The aim of the theory of scale-relativity is to look for the laws and structures that would be the manifestations of still more general transformations, namely, continuous ones (that can be differentiable or not). In such a construction the standard theory will automatically be recovered as a special case, since differentiable spaces are a particular subset of the set of all continuous spaces.

In that quest, the first step consists of realizing that a continuous but non-differentiable space-time is necessarily fractal, i.e., explicitly resolution-dependent (see Appendix A). This leads us to introduce new intrinsic scale variables in the very definition of physical quantities (among which the coordinates themselves), but also to construct the differential equations (in the "scale space") that would describe this new dependence. In other words, the search for the laws of a non-differentiable physics can be brought back to the search of a completion of the laws of motion by new laws of scale and laws of motion/scale coupling.

The remaining of the present section (completed by Appendix B) is aimed at giving to the reader a hint of the general structure of the scale-relativity theory. We shall see that, since the new scale equations are themselves constrained by the principle of relativity, the new concepts fit well established structures. Namely, the so-called symplectic structure of most physical theories (including thermodynamics, see Peterson 1979), i.e., the Poisson bracket / Euler-Lagrange / Hamilton formulation, can be also used to construct scale laws. Under such a viewpoint, scale invariance is recovered as corresponding to the "free" case (the equivalent of what inertia is for motion laws).

2.3. Scale invariance and Galilean scale relativity

Scaling laws have already been discovered and studied at length in several domains of science. A power-law scale dependence is frequently encountered in a lot of natural systems, it is described geometrically in terms of fractals (Mandelbrot 1975, 1983), and algebrically in terms of the renormalization group (Wilson 1975, 1979). As we shall see now, such simple scale-invariant laws can be identified with a "Galilean" version of scale-relativistic laws.

In most present use and applications of fractals, the fractal dimension D is defined from the variation with resolution of the main fractal variable (e.g., the length of a fractal curve which plays here the role of a fractal curvilinear coordinate, the area of a fractal surface, etc...). Namely, if is the topological dimension ( for a curve, 2 for a surface, etc...), the scale dimension is defined, following Mandelbrot, as:

When is constant, we obtain a power-law resolution dependence . The Galilean structure of the group of scale transformation that corresponds to this law can be verified in a straightforward manner from the fact that it transforms in a scale transformation as

This transformation has exactly the structure of the Galileo group, as confirmed by the law of composition of dilations , which writes

with , and .

2.4. Lagrangian approach to scale laws

We are then naturally led, in the scale-relativistic approach, to reverse the definition and the meaning of variables. The scale dimension becomes, in general, an essential, fundamental variable, that remains now constant only in very particular situations (namely, in the case of scale invariance, that corresponds to "scale-freedom"). It plays for scale laws the same role as played by time in motion laws. The resolution can now be defined as a derived quantity in terms of the fractal coordinate and of the scale dimension:

Our identification of standard fractal behavior as Galilean scale laws can now be fully justified. We assume that, as in the case of motion laws, scale laws can be constructed from a Lagrangian approach. A scale Lagrange function is introduced, from which a scale-action is constructed:

The action principle, applied on this action, yields a scale-Euler-Lagrange equation that writes:

The simplest possible form for the Lagrange function is the equivalent for scales of what inertia is for motion, i.e., and (no scale "force", see Appendix B). Note that this form of the Lagrange function becomes fully justified, as in the case of motion laws, once one jumps to special scale-relativity (Nottale 1992) and then goes back to the Galilean limit (see Appendix B). The Lagrange equation writes in this case:

The constancy of means here that it is independent of the scale-time . Then Eq. (8) can be integrated in terms of the usual power law behavior, . This reversed viewpoint has several advantages which allow a full implementation of the principle of scale relativity:
(i) The scale dimension takes its actual status of "scale-time", and the logarithm of resolution its status of "scale-velocity", . This is in accordance with its scale-relativistic definition, in which it characterizes the state of scale of the reference system, in the same way as the velocity characterizes its state of motion.
(ii) This leaves open the possibility of generalizing our formalism to the case of four independent space-time resolutions, .
(iii) Scale laws more general than the simplest self-similar ones can be derived from more general scale-Lagrangians (Appendix B).

It is essentially Galilean scale relativity that we shall consider in the present series of papers. Before developing it further, we recall, however, that the Galilean law is only the simplest case of scale laws that satisfy the principle of scale relativity. We shall, in Appendix B, give some hints about its possible generalizations, since they determine the general framework of which Galilean scale relativity is only a subset.

2.5. Scale-symmetry breaking

An important point concerning the scale symmetry, which is highly relevant to the present study is that, as is well-known from the observed scale-independence of physics at our own scales, and as we shall demonstrate in more detail in Sect. 3, the scale dependence is a spontaneously broken symmetry (Nottale 1989, 1992, 1993a). Let us recall the simple theoretical argument that leads to this result and to its related consequence that space-time is expected to become fractal at small but also at large space-time scales.

In the general framework of a continuous space-time (not necessarily differentiable), we expect a general curvilinear coordinate to be explicitly resolution-dependent (Appendix A), i.e. . We assume that this new scale dependence is itself solution of a differential equation in the scale space. The simplest scale differential equation one can write is a first order equation where the scale variation of depends on only, . The function is a priori unknown but, always taking the simplest case, we may consider a perturbative approach and take its Taylor expansion. We obtain the equation:

Disregarding for the moment the quadratic term, this equation is solved in terms of a standard power law of power , broken at some scale , as illustrated in Fig. 2 ( appears as a constant of integration):

Fig. 2. Typical behavior of the solutions to the simplest linear scale differential equation (see Sect. 3.1.1). One obtains an asymptotic fractal (power-law resolution-dependent) behavior at either large or small scales, and a transition to scale-independence toward the classical domain (intermediate scales). The transitions are given by the Compton-de Broglie scale in the microscopic case and by the typical static radius of objects (galaxy radii, cluster cores) in the macroscopic case. Note that the microscopic and macroscopic plots actually correspond to two different kinds of experiments: in the microscopic case, the "window" is kept constant while the "resolution" is changed, leading to an increase toward small scales, ; in the macroscopic case, the fractal behavior shows itself by increasing the window for a fixed resolution , this leading to an increase toward large scales, .

Depending on the sign of , this solution represents either a small-scale fractal behavior (in which the scale variable is a resolution), broken at larger scales, or a large-scale fractal behavior (in which the scale variable would now represent a changing window for a fixed resolution ), broken at smaller scales.

The symmetry between the microscopic and the macroscopic domains can be even more directly seen from the properties of Eq. (12). Let us indeed transform the two variables and by inversion, i.e. and , we find that Eq. (12) becomes:

This is exactly the same equation up to the exchange of the constants a and c. In other words, Eq. (12) is covariant (i.e. form invariant) under the inversion transformation, which transforms the small scales into the large ones and reciprocally, but also the upper symmetry breaking scale into a lower one. Hence the inversion symmetry, which is clearly not achieved in nature at the level of the observed structures, may nevertheless be an exact symmetry at the level of the fundamental laws. This is confirmed by directly looking at the solutions of Eq. (12) keeping now the quadratic term, since they may include two transitions separating the scale space into 3 domains.

The symmetry breaking is also an experimental fact. The scale symmetry is indeed broken at small scales by the mass of elementary particles, i.e., by the emergence of their de Broglie length:

and at large length-scales by the emergence of static structures (galaxies, groups, cluster cores) of typical sizes

beyond which the general scale dependence shows itself in particular by the expansion of the Universe but also by the fractal-like observed distribution of structures in the Universe. The effect of these two symmetry breakings is to separate the scale space into three domains (see Figs. 2 and 5), a microphysical quantum domain (scale-dependent), a classical domain (scale-independent), and a "cosmological" domain (scale-dependent again). Remark that the existence of the classical, scale-independent domain does not disprove the universality of the principle of scale relativity, since this intermediate domain actually plays for scale laws the same role as statics plays for motion laws: namely, it corresponds to a degeneration of the scale laws. It is easy to include the symmetry breaking in our description, by accounting for the fact that the origin of a fractal coordinate is arbitrary. As we shall see in more detail in what follows, the spontaneous symmetry breaking is the result of translation invariance, i.e., of the coexistence of scale laws and of motion/displacement laws (see Sect. 3.1.1 and Fig. 3).

Fig. 3. The quantum-microscopic to classical transition is understood in the scale-relativistic approach as a spontaneous symmetry breaking: the "classical" term becomes dominant beyond some upper scale while the "fractal" term (here of critical fractal dimension 2) is dominant toward the small scale in absolute value, though it vanishes in the mean. Note that in the cosmological-macroscopic case which is the subject of the present paper, there is an additional transition to classical laws toward the small scales, while the upper classical domain is sent to infinity (since , with ).

Simply replacing the fractal coordinate by in the pure scale-invariant law , we recover the broken law

which becomes scale-independent for when , and for when . We then expect the three domains and the two transitions of Figs. 2 and 5(see Sect. 3 for more detail). Note that in these figures, the scale dimension is an effective scale dimension which includes the transition in its definition, i.e. .

© European Southern Observatory (ESO) 1997

Online publication: April 6, 1998
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