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Entropy 2016, 18(5), 160; doi:10.3390/e18050160
Abstract: Considering that the motions of the complex system structural units take place on continuous, but non-differentiable curves, in the frame of the extended scale relativity model (in its Schrödinger-type variant), it is proven that the imaginary part of a scalar potential of velocities can be correlated with the fractal information and, implicitly, with a tensor of “tensions”, which is fundamental in the construction of the constitutive laws of material. In this way, a specific differential geometry based on a Poincaré-type metric of the Lobachevsky plane (which is invariant to the homographic group of transformations) and also a specific variational principle (whose field equations represent an harmonic map from the usual space into the Lobachevsky plane) are generated. Moreover, fractal information (which is made explicit at any scale resolution) is produced, so that the field variables define a gravitational field. This latter situation is specific to a variational principle in the sense of Matzner–Misner and to certain Ernst-type field equations, the fractal information being contained in the material structure and, thus, in its own space associated with it.
- The fractal curves (the trajectories of the complex system structural units) are explicitly scale resolution dependent, i.e., their lengths tend to infinity when the scale resolution tends to zero.
- The complex system dynamics is described through fractal variables, i.e., mathematical functions depending on both the space-time coordinates and the scale resolution, since the differential time reflection invariance is broken.
- The differential of the spatial coordinate field is expressed as a sum of two differentials, one of them being scale resolution independent (differential part) and the other one being scale resolution dependent (fractal part).
- The non-differentiable part of the spatial coordinate satisfies a fractal equation.
- The differential time reflection invariance is recovered by means of a complex operator (non-differentiable operator). In particular, by applying this operator to the spatial coordinate field, it results a complex velocity field, the real part being differentiable and scale resolution independent, and the imaginary one is fractal and scale resolution dependent.
- The explanation of the complex operator by means of a generalized statistical fluid-like description (fractal fluid) and its implementation as a covariant derivative (with the status of motion operator in the analysis of complex system dynamics) are given.
- The acceptance of a scale covariance principle implies, by applying the covariant derivative to the complex velocity field, the equations of the structural unit geodesics of the complex system.
- For irrotational motions of the fractal fluid, a case in which the complex velocity field is generated by a complex scalar field, the geodesics equation reduces to a fractal Schrödinger-type equation. Then, the deterministic trajectories are replaced by a “collection of potential routes”, so that both the concept of “definite position” is replaced by that of an ensemble of positions having a definite probability density, and the concept of “particle” (structural units of the complex system) is substituted with the geodesics of the Schrödinger-type themselves.
2. Constitutive Laws of Material by Means of Fractal Information
- The η coefficient from Equation (24) must be a tensor, which accounts for the crystalline lattice symmetry.
- Then, let us observe that the moving velocity of the solid’s particles is not determined as being the temporal derivative of the movement in itself, but as deriving from a potential given by the probability density. This results in the intrinsic movements in the solid being only “sub-products” of the theory, looking like , but raising a question about the deformation as a symmetric gradient of these movements.
- These facts bring forward the time problem. According to the previous observations, the density probability is constant in time and so is the velocity field. Experience contradicts however this deduction, which compels us to change our view of the solid. We conceive thusly the solid as a proper system of particles that are correctly defined by harmonic oscillators, spatially ordered in the crystalline lattice. This means that the harmonic oscillators vibrate around equilibrium positions, i.e., the points of the crystalline lattice. They can be considered identical with respect to vibrational properties, the difference being made only by their position in the crystalline lattice. These oscillators’ ensemble can be described by a probability function ρ, which is constant in time, because the oscillators do not move from their positions. The “time constant” is too an exact term for what we intend to present: due to chaotically-distributed instant spacial states around the equilibrium positions and due to external constraints, of course related to the position they occupy, the oscillators might accidentally interact, low movements and thus repositioning occurring. This repositioning comes as an accommodation of the oscillators to the given conditions or, better put, an accommodation of the crystalline lattice to the internal and external constraints. This information must be present in an equation like Equation (25).
3. Differential Geometry by Means of Fractal Information
4. A Variational Principle by Means of Fractal Information
- Starting from a Schrödinger-type equation as geodesics of a fractal space in the frame of the extended scale relativity theory, it is proven that the velocities’ field is complex. Its real part is the standard velocity, which is differentiable and scale resolution independent, while its imaginary part is the fractal velocity, which is non-differentiable and scale resolution dependent.
- Through a Madelung-type choice of the wavefunction in the Schrödinger geodesic, the following two parts are separated: the real one (which implies the impulse conservation law) and the imaginary one (which implies the states density conservation law). The two conservation laws constitute the fractal hydrodynamic equations system.
- The differential velocity is put in correspondence with the wavefunction phase, while the fractal velocity is put in correspondence with the wavefunction amplitude and, thus, with the states density, through the square amplitude. By such a procedure, the proportionality between the fractal velocity and the fractal information gradient is highlighted.
- The fractal velocity and, thus, the fractal information gradient are responsible (through the fractal potential gradient) for the presence of a force-type term (fractal force) in the impulse conservation law. At equilibrium, this term can be put in correspondence with the divergence of the tensor of “stresses”, and this imposes a constitutive law of material. In other words, the tensor of “stresses” becomes a measure of the fractal information “fluctuations”.
- From the perspective of a two-form of a complex scalar potential of velocities (potential, which contains explicitly the fractal information as its imaginary part), one obtains the correspondence with the metric of the Lobachevsky plane in the Poincaré representation. It thus results in the invariance of the two-form with respect to the homographic transformations group (the Barbilian group), when one defines a parallel transport of vectors in the sense of Levi–Civita (the vector origin moves on geodesics, the angle between the vector and the tangent to the geodesic at the current point being permanently constant). The group is measurable (that is, it possesses a function, which is integrally invariant), so that in the variables space of the group, one can construct Jeans-type probabilistic physical theories based on an elementary probability (for details, see [15,21]). Moreover, the fact that the metric of the Lobachevsky plane can be produced as a Caylean metric of the Euclidean space (for which the absoluteness is a circle with unit radius) proves that one can produce metrics and, thus, fractal information, at any scale resolution, independently of the Einsteinian procedure.
- A variational principle has been constructed. Then, if this variational principle is accepted as a starting point, the main purpose of any field research would be to produce metrics of the Lobachevsky plane, being apart from Einstein’s field equations. The field equations obtained through this variational principle represent a harmonic map from the usual space into the Lobachevsky plane having the metric in Poincaré’s representation. In other words, through this variational principle, one can produce fractal informational that is explicit at any scale resolution. In this sense, if the field variables define a gravitational field, then the variational principle is reducible to a Matzner–Misner-type one, while the field equations are reducible to those of Ernst-type (for details, see [15,21]). From such a perspective, the fractal information is generated together with the material structure, that is together with the induction of the own space associated with the material structure.
Conflicts of Interest
Appendix A. Implications of Non-Differentiability in the Dynamics of Complex Systems
- Any continuous, but non-differentiable curve of the complex system structural units is explicitly scale resolution δt dependent, i.e., its length tends to infinity when δt tends to zero.We mention that, mathematically speaking, a curve is non-differentiable if it satisfies the Lebesgue theorem, i.e., its length becomes infinite when the scale resolution goes to zero . Consequently, in this limit, a curve is as zigzagged as one can imagine. Thus, it exhibits the property of self-similarity in every one of its points, which can be translated into a property of holography (every part reflects the whole) [6,7,10,13].
- The physics of the complex phenomena is related to the behavior of a set of functions during the zoom operation of the scale resolution δt. Then, through the substitution principle, δt will be identified with dt, i.e., , and consequently, it will be considered as an independent variable. We reserve the notation dt for the usual time as in the Hamiltonian complex system dynamics.
- The complex system dynamics is described through fractal variables, i.e., functions depending on both the space-time coordinates and the scale resolution, since the differential time reflection invariance of any dynamical variable is broken. Then, in any point of the fractal curve, two derivatives of the variable field Q(t, dt) can be defined:
- The non-differentiable part of the spatial coordinate field satisfies the fractal equation [6,7]:In our opinion, the processes in complex systems imply a dynamics on geodesics having various fractal dimensions. Precisely, for D = 2, quantum-type processes are generated. For D < 2, correlative-type processes are induced, while for D > 2, non-correlative-type ones can be found (for details, see [6,7,8,9,13]).
- The differential time reflection invariance of any dynamical variable is recovered by combining the derivatives and in the non-differentiable operator:
- In the absence of any external constraint, an infinite number of fractal curves (geodesics) can be found relating any pair of points, and this is true on all scales. Then, in the fractal space, all complex system structural units are substituted with the geodesics themselves (for details, see Appendix B), so that any external constraint is interpreted as a selection of geodesics by the measuring device. The infinity of geodesics in the bundle, their non-differentiability and the two values of the derivative imply a generalized statistical fluid-like description (fractal fluid). Then, the average values of the fractal fluid variables must be considered in the previously-mentioned sense, so the average of is:
- The complex system dynamics can be described through a covariant derivative, the explicit form of which is obtained as follows. Let us consider that the non-differentiable curves are immersed in a three-dimensional space and that X are the spatial coordinate field of a point on the non-differentiable curve. We also consider a variable field and the following Taylor expansion up to the second order:
Appendix B. Geodesics of the Complex System Structural Units
Appendix C. Geodesics of the Complex Systems Dynamics in the Schrödinger-Type Representation
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