Abstract
We review in this paper the use of the theory of scale relativity and fractal space-time as a tool particularly well adapted to the possible development of a future genuine systems theoretical biology. We emphasize in particular the concept of quantum-type potentials, since, in many situations, the effect of the fractality of space—or of the underlying medium—can be reduced to the addition of such a potential energy to the classical equations of motion. Various equivalent representations—geodesic, quantum-like, fluid mechanical, stochastic—of these equations are given, as well as several forms of generalized quantum potentials. Examples of their possible intervention in high critical temperature superconductivity and in turbulence are also described, since some biological processes may be similar in some aspects to these physical phenomena. These potential extra energy contributions could have emerged in biology from the very fractal nature of the medium, or from an evolutive advantage, since they involve spontaneous properties of self-organization, morphogenesis, structuration and multi-scale integration. Finally, some examples of applications of the theory to actual biological-like processes and functions are also provided.
Highlights
The theory of scale relativity and fractal space-time accounts for a possibly nondifferentiable geometry of the space-time continuum, based on an extension of the principle of relativity to scale transformations of the reference system
The success of this enterprise [7,8] has been completed by obtaining new results: in particular, a generalization of standard quantum mechanics at high energy to new forms of scale laws [9], and the discovery of the possibility of macroscopic quantum-type behavior under certain conditions [10], which may well be achieved in living systems
The laws of motion in scale relativity are obtained by writing the fundamental equation of dynamics in a fractal space
Summary
The theory of scale relativity and fractal space-time accounts for a possibly nondifferentiable geometry of the space-time continuum, based on an extension of the principle of relativity to scale transformations of the reference system. This theory was initially built with the goal of re-founding quantum mechanics on prime principles [4,5,6] The success of this enterprise [7,8] has been completed by obtaining new results: in particular, a generalization of standard quantum mechanics at high energy to new forms of scale laws [9], and the discovery of the possibility of macroscopic quantum-type behavior under certain conditions [10], which may well be achieved in living systems. Scale relativity methods are relevant because they provide new mathematical tools to deal with scale-dependent fractal systems, like equations in scale space and scale-dependent derivatives in physical space This approach is very appropriate for the study of biological systems because its links micro-scale fractal structures with organized form at the level of an organism. For more information the interested reader may consult the two detailed papers [1,2] and references therein
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