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Fractional Calculus View of Complexity : Tomorrow's Science
The view point that kept asserting itself is a resurgence of the Natural Philosophy of the seventeenth
and eighteenth centuries. A perspective in which the pursuit of scientific knowledge and understanding has wisdom as its ultimate goal, and not merely the ordered accumulation of empirical facts necessary for a rational model of the world.
于此相对，某些人的想法可能是：着什么急啊，慢慢研究，否则研究完了之后做什么啊。
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Many believe, that in the last half of the twentieth century, we passed through one of the plateaus predicted by de Solla Price, and are now in the process of developing a new way of thinking about science. We have passed through what can be mastered by the formation of separate and distinct scientific disciplines and now enter a phase where synthesis and integration in the manner of da Vinci are necessary for further progress. I hope to convince the reader that this new way of thinking is facilitated by an old, but not widely known, form of quantitative reasoning, that being the fractional calculus.
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我们日常把复杂问题简化为若干线性问题的叠加，这个思想源于18世纪Bernoulli父子提出的Principle of Superposition。后来在19世纪被推广为the general form of the superposition principle。这个思想不仅影响了物理学，还影响了社会科学和生命科学。20世纪这种思想又被应用于微观----量子力学。量子力学可以被视为无穷维度空间里的线性问题。
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泛函分析就是研究无穷维度空间的理论；所以泛函分析的诞生根源于Bernoulli的Principle of Superposition。
为什么是无穷空间，类比一下：对于超越函数，我们没有它的解析表达式，所以不得已用无穷级数去近似表达之(逼近之)。
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数学史本身就很有趣了，如今跟随作者以批判的态度来看数学史，更赞。
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Principle of Superposition解决了那么多问题，为什么后来有些问题解决不了了呢？
看Principle of Superposition是如何得出来的：
给定n维向量X(t) = (X1 , ..., Xn)，动力学系统由方程dX(t)/dt= AX(t)来描述。如果X能通过相似变换S变换为Y：Y(t) = SX(t)，那么原来的动力学系统变为：dY(t)/dt = ΛY(t)，这意味着解λn和yn(t)可以描述系统,这也就是Principle of Superposition。注意这里有一个假设：如果X能通过相似变换S变换为Y 。
我们在使用Principle of Superposition时，我们的解决方法里包含了”X是可以通过相似变换S变换为Y的”这个条件。在后来的很多问题上，求得的解和实际不符，所以对于这些问题，有理由认为”X无法通过相似变换S变换为Y”。
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linear superposition 扩展到 continuous spatial phenomena,就是Sturm-Liouville theory
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the physical world not being simply linear ---> 比如friction (dissipation) 。Dissipation is the
irreversible transfer of useful energy into waste heat
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Fermi-Pasta-Ulam (FPU) problem：
how macroscopic irreversibility was a consequence of the deviation of the microscopic interactions from linearity.
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the equi-partition theorem：
for a given temperature of the environment the system ought to equilibrate in such a way that each degree of freedom (normal mode) has the same amount of energy, that being, ½ kB T
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FPU：
the failure of a nonlinear dynamical system to relax to a state in which each of the dynamic modes has the same amount of energy, therefore apparently violating the energy equi-partition theorem
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finite dimensional nonlinear equations of motion may be replaced by infinite-order sets of linear
rate equations
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The dynamics of complex nonlinear phenomena demands that we extend our horizons beyond analytic functions and classical analysis.
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Complex phenomena require new ways of thinking and the fractional calculus provides one framework for that thinking
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since the nineteenth century was the period when the quantitative measurement of phenomena became the same as knowledge. Consequently, if precision is knowledge, then variability must be ignorance or error; and this is what was believed.
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This bias is the belief that there is a special kind of validity associated with being able to characterize a phenomenon with a number or set of numbers. This particular bias is one that is shared by most physical scientists, such as physicists or chemists. I point this out to alert the reader that numbers
are representations of facts and are not facts in themselves, what da Vinci called experience. It is the underlying facts that are of importance and not necessarily the numbers associated with them.
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Historically it was left to statistical physics to restore to the mechanical description of complex systems the uncertainty observed in actual measurements and to construct the associated PDF as a measure of that uncertainty. Probability theory provided the first universally accepted systematic treatment of physical complexity and was the mathematical foundation of kinetic theory.
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Boltzmann,believed that such microscopic dynamics should be described by continuous but non-differentiable functions such as the one developed by Weierstrass and which is discussed subsequently herein.
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Technically the dimension of an object is determined by how it is measured,
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When a process is continuous, but no where differentiable, its evolution must be described by the fractional calculus.
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It is worth re-emphasizing that there is no single fractional calculus, just as there is no single geometry.Different definitions of fractional operators, differentials and integrals, have been constructed to satisfy various needs, desires and constraints.
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The effect of memory is
modeled by means of a fractional difference equation that in the continuous limit yields a fractional time derivative resulting in the fractional Langevin equation.
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One of the most significant results presented in this chapter is the realization that fractal functions, whose integer derivatives diverge, do not have ordinary equations of motion. However, fractional derivatives of fractal functions can converge as in the case of the GWF, suggesting that the equations of motion for fractal phenomena are provided by the fractional calculus.
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Our interest is providing fractional calculus descriptions of complex phenomena is two fold. On the one hand, we want to understand the behavior of the phenomena and this is the science. On the other hand we also want to control the behavior of the phenomena and this is the engineering.
fractional control processes
Figure 5.1 Eq.(5.12) 很好地解释了population overshoot的现象
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fractional harmonic oscillator变成了阻尼振动
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5.6 Fractional Leibniz Rule
fractional derivatives有很多种定义，包括Riemann-Liouville, Caputo, Marchaud, Weyl, Reisz and others [22, 39]，但它们都不满足 Leibniz rule
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On the other hand, Weberszpil [47] presented a counter argument to show that if a function is non-differentiable then in a coarse-grained medium its fractional derivative is compatible with the Leibniz rule. He agreed that Eqs.(5.144) and (5.145) are appropriate for the fractional derivatives identified, but maintained that the MRL fractional derivative introduced by Jumarie [18, 19] satisfies the fractional Leibniz rule:
Following the logic of Tarasov, Weberszpil also assumed that a given function satisfied the Leibniz rule and was defined in a Holder space. He pointed out that a Holder space and nowhere differentiable functions are related, giving a Weierstrass function as an example.
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The generalization of the Taylor expansion to non-differentiable functions：
if the Leibniz rule is not violated there can be no fractional derivative. so this violation of the rule is one of the characteristic properties of fractional derivatives
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5.7 After Thoughts
three sources of complexity: statistical fluctuations, nonlinearity ，fractional dynamics
因为the nonlinearity may be incorporated into an infinite-order linear description. 所以，我们期待 infinite-order linear也可以描述fractional differential equations，但实际上是不行的(只能解决特殊的情况)。
This result suggests that the fractional calculus might be used in the future to replace the equations of motion for highly nonlinear dynamic networks, with an equivalent set of fractional linear dynamic stochastic equations.
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Historically the two ways of describing the changing behavior of complex phenomena
- the Langevin equation for the dynamic variables,
- the phase space equation for the PDF.
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The complexity of many physical and virtually all social and life science phenomena suggest that the mechanical model of the world ought to be abandoned, or at least significantly modified.
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He determined
that a classical Lagrangian could be generalized to include friction by adding a fractional derivative of order one-half to the equations of motion.(摩擦力可用1/2阶导数表示)
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he [65] motivated his development of the formalism with the observation that space and time are not the continuous featureless processes first assumed by Newton.
The fact that such general principles as Noether’s theorem for the problems of the calculus of variation with fractional derivatives have been obtained [1] strongly suggests the existence of symmetry and/or
conservation laws for general systems from which fractional equations of motion can be generated.
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Copot et al. suggest that the fractional-orders models avoid the high degree of inter-patient variability and nonlinearity resulting in the use of linear, rather than nonlinear, control techniques.
Hamilton’s equations, could be generalized to the fractional calculus. This extension allows us to set aside the clockwork universe and accept a less preordained view of dynamics; one that could be adopted in the social and biological sciences without overly constraining them.
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complexity can only be addressed through the introduction of randomness into the dynamics.
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The extension of the Hamiltonian formalism rested on introducing a new interpretation of time, one local to the system of interest and the other available to the greater world of experiment. In this way the extended theory was shown to be compatible with the fractional calculus and to lead to fractional equations of motion.
statistical fluctuations were necessary to capture the multi-modal nature of complexity, particularly in FLEs。
the FLE has as its natural description in phase space fractal equations for the PDF.
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fractional calculus will dominate the science of the future
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Ockham‘s principle (assumption)：
physical processes are adequately modeled as being linear, or approximately so
this assumption was based on the notion of continuous isotropic time , and homogeneous space of infinite extent,
本书，对complexity的阐述，实际上否定了这个假设。
The modern direction of science is to understand and even to embrace complexity。
This was no where more evident than with the introduction of uncertainty into science, without leaving
behind quantifiability.Making a statistical description compatible with the mechanical model of the universe was truly an amazing accomplishment, and enabled the random scatter of measurements to be explained, while preserving the notion of predictability.
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The most surprising result of this chapter has to do with how the dynamics of a complex network influences the behavior of an individual within a
large network.
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The collapse of the ten thousand dimensional system of equations onto a linear fractional equation is the result of two general features of the dynamics:
(1) the network dynamics subordinate the dynamics of the individual by transforming the dynamics in operational time into dynamics in chronological time and
(2) the dynamics of the network are critical, which is to say, the nearest neighbor interactions become long-range for a critical value of the control parameter.
This suggests that the fractional equations of motion are a natural consequence of criticality of the underlying dynamics.
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The L ́evy statistics of velocity fluctuations in the wind field are described by fractional time derivatives, whereas the spatial energy cascades are described by fractional derivatives in space.
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8.2 Conformation Bias and Scientific Truth
If it is not quantitative, it is not scientific.
<---------> Qualitative descriptions of phenomena can be as important, if not more important, than quantitative descriptions in science.
Galileo condensed into the three tenants:
-Description is the pursuit of science, not causation.
-Science should follow mathematical, that is, deductive reasoning.
-First principle comes from experiment, not the intellect
Physical observables are represented by analytic functions
<---------> Many phenomena are singular (fractal) in character and cannot be represented by analytic functions.
The final state of a physical systems is predictable by means of its equation of motion and the given initial state.
<---------> Nonlinear deterministic equations of motion, whether discrete or continuous, do not necessarily have predictable final states due to the sensitivity of the solutions on initial conditions.
Physical systems can be characterized by fundamental scales, such as those of length and time.
<--------->Natural phenomena do not necessarily have fundamental scales and may be described by scaling relations.
-----REF-----------------------------------
[36] Magin, R.L. 2006. Fractional Calculus in Bioengineering, begell house
inc., New York.
West, B.J., M. Bologna and P. Grigolini. 2003. Physics of Fractal
Operators, Springer, Berlin.
[42]Miller, K.S. and B. Ross. 1993. An Introduction to the Fractional
Calculus and Fractional Differential Equations, John Wiley & Sons, New
York.
[56] Samko, S.G., A.A. Kilbas and O.I. Marichev. 1993. Fractional Integrals
and Derivatives, Gordon and Breach Science Pub., USA.
[66] West, B.J., M. Bologna and P. Grigolini. 2003. Physics of Fractal
Operators, Springer, Berlin.
52] Mackay, C. 1852. Memoirs of Extraordinary Popular Delusions and the
Madness of Crowds, 2 nd Ed., London, Office of the National Illustrated
Library.
[115] Whyte, W.H. 1956. The Organization Man, Simon & Schuster, NY
[65] Stanislavsky, A.A. 2006. Eur. Phys. J. B 49, 93
