Scale-covariant derivative

We can describe the elementary displacement dX as the sum of a mean, <dx_+/-> = v_+/- dt, and a fluctuation about this mean, dx_+/-, which is then by definition of zero average: <dx_+/-> = 0, i.e.:

  (1)

Consider first the average displacements. The fundamental irreversibility of the description is now apparent in the fact that the average backward and forward velocities are in general different. So mean forward and backward derivatives, d₊/dt and d_-/dt, are defined. Once applied to the position vector x, they yield the forward and backward mean velocities, and .

Concerning the fluctuations, the generalization of the fractal behavior to three dimensions writes

  (2)

standing for a fundamental parameter that characterizes the new scale law at this simple level of description. The dx(t)'s are of mean zero and mutually independent. If one assumes them to be also Gaussian, our process becomes a standard Wiener process. But such an assumption is not necessary in our theory, since only the property (2) will be used in the calculations.

Our main tool now consists of recovering local time reversibility in terms of a new complex process (Nottale 1993a): we combine the forward and backward derivatives in terms of a complex derivative operator

  (3)

which, when applied to the position vector, yields a complex velocity

 (4)

The interest of such a choice is that, at the classical limit (), The real part V of the complex velocity identify itself with the classical velocity, while its imaginary part, U, vanishes. The latter is a new quantity arising from non-differentiability. The splitting of the velocity field, v --> (V, U), is the new essential physical behaviour, of which complex numbers and complex product are only a representation. Another kind of product could have been chosen, but one can show that the complex product achieves the simplest (in other words, covariant) representation.

Equation (2) now allows us to get a general expression for the complex time derivative . Consider a function f(x(t),t). Contrarily to what happens in the differentiable case, its total derivative with respect to time contains finite terms up to higher order (Einstein, 1905). In the special case of fractal dimension 2, only the second order intervenes. Indeed its total differential writes

 (5)

Classically the term dX_i dX_j  / dt  is infinitesimal, but here we have:

dX_i dX_j = dx_i dx_j + dx_i dx_j + dx_i dx_j + dx_i  dx_j

Let us now take the average of (5). its average reduces to <dx_i  dx_j>/dt, so that the last term of Eq. (5) will amount to a Laplacian thanks to Eq. (2). Then

   (6)

By inserting these expressions in (3), we finally obtain the expression for the complex time derivative operator (Nottale 1993a):

   (7)

The passage from classical (differentiable) mechanics to the new nondifferentiable mechanics can now be implemented by a covariant prescription: Replace the standard time derivative d/dt by the new complex operator . In other words, this means that  plays the role of a scale-covariant derivative (in analogy with Einstein's general relativity where the basic tool consists of replacing by the covariant derivative ). In this replacement, one must obviously be aware of the fact that combines first and second order derivatives, in particular concerning its Leibniz rule (see Pissondes 1998 about developments of this point).