## 2. Motivation## 2.1. Chaotic systems beyond their horizon of predictabilityConsider 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 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
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description in terms of One must keep in mind that such a large time-scale description would be no longer valid at small time-scales, since when going back to (left diagram in Fig. 1), differentiability is recovered.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-timeThere is a fundamental reason for jumping to a non-deterministic,
scale-relativistic physical description at small 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 relativityScaling 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 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 lawsWe 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 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: 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 breakingAn 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 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):
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 This is exactly the same equation up to the exchange of the
constants 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
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 © European Southern Observatory (ESO) 1997 Online publication: April 6, 1998 |