E-infinity as a fiber bundle and its thermodynamics
Introduction
Nowadays our understanding of fractals is undergoing essential transformation. The structure of matter, high energy physics and all related subjects are calling for fractals [1], [2], [3], [4], [5]. Fractal sets with a large number of characteristic scales present the most attractive model for representation of multi-scaled phenomena.
Multi-scaled dynamic processes present a particular case of fractal sets. They represent the sets of V4 generated by dynamic laws of motion. Unstable chaotic motion destroys initial smooth structure of V4. It converts continuum V4 into discretuum E(∞). Thinking in this way it is not difficult to conclude that phase spaces of different multi-scaled processes can be treated from a unified dynamic point of view. They present particular realizations of fractal space–time in the frame of E-infinity theory. This idea was developed recently by El Naschie in terms of E-infinity technique for dealing with fractality.
In connection with the above there arises a question. Does fractal space–time represents a particular physical system or is it a frame only? We are inclined to adopt the positive answer for the first part of above-mentioned question. Our argument consists of a hypothetical statement that E-infinity space–time represents a pure form of transfinite structure. It can be investigated independently. Other fractals should be produced starting from the previous one. Proceeding in this way we treat different states of fractal space–time as deformations of the pure form of fractal transfinite discretuum. From this point of view the pure form corresponds to equilibrium state of fractal space–time. We hope to develop this idea as a novel approach to the investigation of fractals and this paper is the first step in this direction.
It seems that the structure of fractals is generated by specific laws of energy transformations. In order to reconstruct all of them we need characteristics that would be able to represent the details of energy exchange. At the same time such characteristics must produce a description of the stable states of fractal organization that correspond to stationary processes. In the first paragraph of the paper proceeding in the spirit of thermodynamics we define fractal generating function that introduces variables for determination of stable fractal states. The transitions between particular fractal states are treated as processes of deformations of fractal organization. For that all the considerations are embedded into the framework of E-infinity theory.
We follow the geometrical way for studying thermodynamics of fractals. New problems arise from the determination of proper geometrical technique. In fact, Euler description implies all the variables to be functions of large-scaled coordinates of space–time V4 accepted in general relativity theory. This would not be proper in the case of fractals. Therefore we change the character of description. An appropriate technique is presented using the theory of fiber bundle spaces. The base of those spaces is identified with V4. It can be interpreted as large-scale reduction of fractal space–time. The layer of the bundle spaces is introduced in connection with fractal specific character. It can be interpreted as meso-, micro- and submicro-scaled reductions of fractal space–time.
Applying the bundle space version of the theory of fractal media we are able to construct particular types of fractal configurations. The way to produce such sets in connection with turbulence was shown in [6]. Following this method we introduce geometrical objects that determine fractal dynamics. These objects can be interpreted in terms of thermodynamical concepts. For example, lifting of geodesics from the base V4 into the bundle space gives a representation for vortex sheets that accompany the thermodynamic processes. Other modifications take place as well.
The consecutive development demands proper definition and we start with that.
Section snippets
Thermodynamic variables of fractals
Let us assume a phase space of some multi-scaled hierarchically constructed processes. We suppose it to be a Hilbert space H(∞) in which we choose some basis {ψn(x), n = 1, … , ∞}. The particular process ψ(t, x) can be considered as decompositionin which the amplitudes cn present time dependence of the process
Let us assume the amplitudes to be scaled variables. So, the process (1.2) represents fractal set in phase space H(∞). We imply H(∞) to be a particular case of E-infinity
The fiber bundle space structure of fractal space–time
In this section we consider a novel representation for fractal space–time E(∞). Thereupon let us note that earlier in the previous section we provided E(∞) with the structure of a Hilbert space H(∞). This was a particular case of fractal space–time that corresponded to microscopic level of resolution of dynamical events. Now we consider another model for E(∞). It corresponds to the microscopic level of description as well. Nevertheless it allows us to introduce some mathematical technique for
Thermodynamic processes in fractal space–time
Consider a particular integral curve of Eq. (2.13). Let it be a lifting of a geodesic fixed determined in connection with the large-scaled structure of space–time V4. Then, we have equationsIt is easy to conclude that (3.1), (3.2) is not a closed system of equations. In order to improve the matter we need equations that would determine the behavior of parameters of deformation, but these imply thermodynamic treatments. Indeed, parameters an
Conclusion
The paper deals with the thermodynamics of fractals and proceeds from there to putting E-infinity theory on a firm conventional mathematical basis by viewing E-infinity as a fiber bundle manifold.
References (15)
Elementary prerequisites for E-infinity
Chaos, Solitons & Fractals
(2006)Intermediate prerequisites for E-infinity theory
Chaos, Solitons & Fractals
(2006)Advanced prerequisites for E-infinity theory
Chaos, Solitons & Fractals
(2006)Towards a gauge theory of turbulence
Chaos, Solitons & Fractals
(2006)The two-slit experiment as the foundation of E-infinity of high energy physics
Chaos, Solitons & Fractals
(2005)- He Ji-Huan, El-Naschie MS. Transfinite Physics. Shanghai, PR China, 2005. ISBN...
The fractal geometry of nature
(1982)
Cited by (4)
Meso-structures of dynamical chaos and E-infinity theory
2009, Chaos, Solitons and FractalsCitation Excerpt :It gives rise to mixing processes breaking up the partition between micro and macro appearances. It seems that the second scenario leads to indistinguishibility conditions, golden ratio, scaling and etc that determine standard structure of E-infinity proposed by El Naschie [5,9,14–16]. It seems interesting to study some intermediate cases of the previous scenario.
Conjectures regarding kissing spheres hierarchy and quantum gravity unification
2008, Chaos, Solitons and FractalsThe issue of primitive indefinite-integral in the theory of fractal space-time
2009, Romanian Reports in PhysicsStructures of induced chaos
2008, Russian Physics Journal