Coherences in the Dynamics of Physical Systems from a Multifractal Perspective of Motion

Abstract

Using an analogy between the multi-fractal Schrödinger equation and the dumped oscillator equation through a special ansatz, Stoler-type coherences in the dynamics of physical systems are highlighted. Such a result implies a Ricatti-type gauge, a process that can be considered a calibration of the difference between the kinetic and potential energy of a Lagrangian, specified as a perfect square in generic coordinates.

1. Introduction

Unlike classic theories (fluid mechanics, General Relativity Theory, etc.), in which the dynamics of the physical systems are described through continuous and differentiable curves, the Scale Relativity Theory [1,2,3] operates in the description of the dynamics of physical systems with continuous and non-differentiable curves (fractal/multifractal curves). Because any fractal/multifractal curve exhibits the property of self-similarity in any one of its points (every part reflects the whole and vice versa), the holographic-type implementations are thus fundamental in the description of the dynamics of physical systems [4,5,6]. In such a context, instead of operating with a single variable (a strictly non-differentiable mathematical function), it is possible to operate only with approximations of the mathematical function, with said approximations resulting from averaging it at different-scale resolutions. Thus, any physical variable used to describe the dynamics of complex systems will operate as the limit of a family of mathematical functions, with the function being non-differentiable for zero-scale resolution and differentiable for non-zero scale resolution.

In such an approach, many dynamical problems have been analyzed, both at a small scale (Kepler-type problem in the fractal hydrodynamic model, harmonic oscillator-type problem in the fractal hydrodynamic model, elastic- and plastic-type behaviors in the fractal theory of motion, etc. [2,3,7,8,9,10,11]) and at a large scale (solar system dynamics, Hubble effect, etc. [4]).

In the present paper, using an analogy between the multifractal Schrödinger equation and the dumped oscillator equation through a special ansatz, Stoler-type coherences in the dynamics of physical systems are highlighted.

2. Correspondences between Multifractal Schrödinger Equation and Dumped Oscillator through a Special Ansatz

The multifractal Schrödinger equation is considered for a constant scalar potential $U$ [2,3]:

$${\alpha }^2{\partial }_l{\partial }^l\mathsf{\Psi }+i\alpha {\partial }_t\mathsf{\Psi }+\frac{U}{2}\mathsf{\Psi }=0$$

(1)

where

$$\alpha =\lambda {\left(dt\right)}^{\left[\frac{2}{f\left(\alpha \right)}\right]-1}$$

(2)

and

$${\partial }_l=\frac{\partial }{\partial x^l}, {\partial }_l{\partial }^l=\frac{\partial }{\partial x^l}\frac{\partial }{\partial x^l}, {\partial }_t=\frac{\partial }{{\partial }_t}, i,l=1,2,3$$

(3)

The quantities from Equations (1)–(3) have the following meaning: $x^l$ is a non-differentiable spatial coordinate, t is a differentiable temporal coordinate with the role of an affine parameter of the non-differentiable motion curves (fractal/multifractal curves), $\mathsf{\Psi }$ is the state function, U is the external scalar potential, dt is the resolution scale, $\lambda$ is a constant coefficient associated with the differentiable–non-differentiable scale transition, and $f\left(\alpha \right)$ is the singularity spectrum of order $\alpha$, where $\alpha =\alpha \left(D_F\right)$ and $D_F$ is the fractal dimension of the motion curves. For details see [2,3].

By admitting the functionality of the ansatz [6,12]:

$$\xi =a_{1}x+a_{2}y+a_{3}z-i{a}_{4}t$$

(4)

Equation (1) with substitutions:

$$M={\alpha }^2\left({a_1}^2+{a_2}^2+{a_3}^2\right), 2R=\alpha a_4, K=\frac{U}{2}, q\equiv \mathsf{\Psi }$$

(5)

is formally reduced to a damped oscillator, i.e.,

$$M\ddot{q}+2R\dot{q}+Kq\equiv 0$$

(6)

We must mention that the meaning of ansatz (4) will be given at a later time.

In what follows, the physical significances will first be used for the classic case (i.e., for the damping oscillator) and then for the SRT case.

Regarding the classic case, $q$ is the relevant coordinate of the motion. We prefer this equation to other formalisms from several points of view, but mainly due to the fact that the physical parameters that characterize the harmonic oscillator—the coefficients of the equation—have always had in history, often even explicitly, an explanation through interaction. $R$ is the motion damping coefficient, and it is a characteristic of the force—the proportionality with the instantaneous speed—that opposes instantaneous motion. K-elasticity or elastic stiffness, although usually considered as an intrinsic property of the oscillator, is actually a property of the structure for which the harmonic oscillator is a physical component, i.e., of the local universe, and as such is also an interaction property. The same can be said about mass M, only that the fact is not so obvious: the history of physics still has considerations regarding different types of masses, determined by different types of interactions that a particle can support (electromagnetic mass, gravitational mass, inertial mass, etc.). In Equation (6) it is assumed that we have to work with inertial mass. As such, we retain here the idea that mass is an expression of global circumstances of the existence of matter, as elastic stiffness is an expression of local circumstances.

The circumstances, always left unspecified in such a problem, are therefore summarized in the explanation of these physical parameters by interactions. In the usual problems we are only interested in their fixed values, and these values intuitively explain the movement described by Equation (6). However, there are situations that require an explanation of these parameters by interaction and, when physics comes back for such an explanation, it is without exception either completely unsatisfactory or, at best, partially. We want to put things in order, at least as far as the place of this physical explanation is concerned. We are sure that only after this, things will naturally settle in their natural places. The dynamics analysis that follows takes into consideration the mathematical methodology from [13].

Equation (6) can be written as a system:

$$\dot{p}=-2\frac{R}{M}p-\frac{K}{M}q;\dot{q}=p$$

(7)

The second of these equations is obviously only a definition of momentum. System (7) is not yet a Hamiltonian system, as would be expected when discussing about momentum and coordinate, because its matrix is not an involution (it has no zero trace). This fact is much more obvious if we put the system in matrix form:

$$\left(\begin{array}{c}\dot{p}\\ \dot{q}\end{array}\right)=\left(\begin{array}{cc}-2\frac{R}{M}& -\frac{K}{M}\\ 1& 0\end{array}\right)\left(\begin{array}{l}p\\ q\end{array}\right)$$

(8)

As long as the physical parameters contained in the $2\times 2$ matrix of this system are constant, we can put the system in an equivalent form that highlights the position of the energy, and so of the Hamiltonian system (of course, in the particular instances where it can be identified with energy). Indeed, from (8) we can immediately obtain the equation

$$\frac{1}{2}M(p\dot{q}-q\dot{p})=\frac{1}{2}\left(M{P}^2+2Rpq+K{q}^2\right)$$

(9)

which proves that the energy—the quadratic form on the right—is the rate of variation of the physical action represented by the elementary area in the phase plane $(q,p)$. This was, incidentally, always the case in physics, so Equation (9) does not represent anything new. However, we want to emphasize here the fact that energy is not required to be conserved in order to be taken as the rate of a physical action. All that is required is that the action be adequately defined as an area in the phase plane. Equation (9) is actually a Riccati-type equation for a certain frequency, because it can be put into the shape

$$M\dot{\omega }+M{\omega }^2+2R\omega +K=0;\omega =\frac{p}{q}$$

(10)

It is not necessary to ask now whether this frequency has a physical meaning or not. We note only that the solution of the above equation is given by the ratio of the solutions of the Hamiltonian system corresponding to (8), i.e.,

$$\left(\begin{array}{c}\dot{p}\\ \dot{q}\end{array}\right)=\left(\begin{array}{cc}-\frac{R}{M}& -\frac{K}{M}\\ 1& \frac{R}{M}\end{array}\right)\left(\begin{array}{l}p\\ q\end{array}\right)$$

(11)

This is, moreover, a general characteristic of the relationship between Riccati’s equation and Hamiltonian dynamics [14]. We can return to Equation (9) above by simply constructing from (11) the 1-differential form that characterizes the elementary area in the phase plane. As far as Equation (10), it can be easily integrated to show us that energy is no longer conserved, but we have the much more complicated conservation law discovered by H. H. Denman [15].

$$\begin{array}{rr}& \frac{1}{2}\left(M{p}^2+2Rpq+K{q}^2\right)\cdot \exp \left\{\frac{2R}{\sqrt{MK-R^2}}\right.\\ & \left.\times {\tan }^{-1}\left(\frac{Mp+Rq}{q\sqrt{MK-R^2}}\right)\right\}=\mathrm{const}.\end{array}$$

(12)

It can be seen from here that energy is conserved in the classical sense only if the coefficient of damping is zero or the movement in the phase plane is made on a straight line passing through the origin, with the slope determined by the ratio between the damping coefficient and the mass. Cancellation of the damping coefficient is usually associated with the absence of dissipative forces. This makes physical sense, but our chief concern here is the identity of these dissipative forces. As such, we will focus on the significance of the Riccati equation, Equation (10), and of the associated Hamiltonian system (11).

What would be the possibility of “agreement” required for the measurement? Note that the classical equation of motion (6) is the expression of a variational principle related to the Lagrangian

$$L(q,\dot{q},t)=\frac{1}{2}M\left({\dot{q}}^2-K{q}^2\right)\exp \left(\raisebox{1ex}{$2RT$}\!\left/ \!\raisebox{-1ex}{$M$}\right.\right)$$

(13)

which represents a harmonic oscillator with explicitly time-varying parameters. Integral on one finite interval $\left[t_0,t_1\right]$ of this function is the physical action of the oscillator on that interval of time, i.e., the difference between the average kinetic energy and the average potential energy. For obtaining Equation (6) it is necessary to take the variation in this action under the mandatory condition that the variation in the coordinate at the ends of the time interval should be zero:

$${\left.\delta q\right|}_{t_0}={\left.\delta q\right|}_{t_1}=0$$

(14)

However, to obtain a closed trajectory, we need to ask even more, as the coordinate values at the ends of the time interval should be the same:

$$q\left(t_0\right)=q\left(t_1\right)$$

(15)

If this trajectory must be closed in the phase plane, it is obvious that the values of the speed at the ends of the time interval must be the same. These circumstances allow us to define the measurement process with the help of the harmonic oscillator in a slightly more precise way. Indeed, let us direct our thought along the following line: from the point of view of the variational principle and the resulting equation of motion, the Lagrangian (13) is defined up to an additive function to be time derivative of another function. The process is extensively used in various branches of theoretical physics to define the so-called gauge transformations. Let us then proceed as usual and define a gauge in whose Lagrangian is a perfect square. This fact is well known and widely exploited in control theory [14], so that we are not in unknown territories. The procedure is to add to the Lagrangian (13) the term

$$\frac{1}{2}\frac{d}{dt}\left[w\exp \left(\frac{2R}{M}t\right)Q^2\right]$$

(16)

where $w$ is a continuous function of time, requiring that the final Lagrangian be a perfect square. The variation of the function under the derivative operator is zero due to the conditions in Equation (14), so that the equation of motion does not change. The new Lagrangian, expressed in generic coordinates, turns out to be

$$L(Q,\dot{Q},t)=\frac{1}{2}M\exp \left(\raisebox{1ex}{$2Rt$}\!\left/ \!\raisebox{-1ex}{$M$}\right.\right){\left(\dot{Q}+\frac{w}{M}Q\right)}^2$$

(17)

provided that $w$ satisfies the following Riccati-type differential equation:

$$\dot{w}=\frac{1}{M}{w}^2-2\frac{R}{M}w+K$$

(18)

The Lagrangian (17) will be taken here to represent the energy of the measurement results. As before, there is a relationship between the Riccati equation, Equation (18), and the Hamiltonian dynamics. We find that the analog of Equation (11) is

$$\left(\begin{array}{c}\dot{\eta }\\ \dot{\xi }\end{array}\right)=\left(\begin{array}{cc}-\frac{R}{M}& \frac{K}{M}\\ -1& \frac{R}{M}\end{array}\right)\left(\begin{array}{c}\eta \\ \xi \end{array}\right);w\equiv \frac{\eta }{\xi }$$

(19)

which obviously represents a Hamiltonian system. Therefore, we can, strictly speaking, conveniently identify the factors of $w$ with the coordinates in the phase plane. It can be said that “the Lagrangian of measurement” (17) represents a set of oscillators along a certain Hamiltonian evolution in the phase plane, an evolution given by Equation (19). The problem now is the physical meaning of this assembly. However, before giving an answer to this problem, a little digression is in order.

One can ask the natural question: what is so special about the quadratic Lagrangian that represents the measurement results? We have no other answer than a philosophical one, deduced from our theoretical experience. First let us recall that free particle mechanics can be described by considering the kinetic energy as a Lagrangian—a quadratic Lagrangian—which leads directly to the equations of motion of the free particle [16]. Therefore, we say, why not reverse the process, and say that a quadratic Lagrangian actually represents an elementary physical structure, like the harmonic oscillator, as free as its circumstances permit. In order to be able to say, “this oscillator made this measurement”, we must be able first to discern that oscillator as a stand-alone structure within the physics structure of which it is a part. When we refer to the classical free particle, we identify it by the fact that it only has kinetic energy—and so is a quadratic Lagrangian—so we will indicate a harmonic oscillator by a quadratic Lagrangian leading to its equations of motion. Therefore, such a Lagrangian will be taken as representing the measurement with the harmonic oscillator. This purely philosophical conclusion will finally be demonstrated rationally with the help of Stoka’s theorem.

Equation (18) shows us that $w$ is a dissipation coefficient, more precisely a variation rate of the mass (in the case that mass is not constant), and this mass variation refers to the oscillator harmonic that performs the measurement. For obvious physical reasons it is therefore important to find the most general solution of that equation. José Carineña and Arturo Ramos offer us a pass in a short but modern and pertinent review of the integrability of Riccati’s equation [16,17]. For our current needs it is enough to note that the complex numbers

$$w_0\equiv R+iM\mathsf{\Omega },w_0^{*}\equiv R-iM\mathsf{\Omega };{\mathsf{\Omega }}^2=\frac{K}{M}-{\left(\frac{R}{M}\right)}^2$$

(20)

are the roots of the quadratic polynomial on the right side of Equation (18). Therefore, first we perform the homographic transformation:

$$z=\frac{w-w_0}{w-w_0^{*}}$$

(21)

and now it can easily be seen by direct calculation that $z$ is a solution of the linear and homogeneous equation of the first order

$$\dot{z}=2i\mathsf{\Omega }z\therefore z\left(t\right)=z\left(0\right)e^{2i\mathsf{\Omega }t}.$$

(22)

Therefore, if we conveniently express the initial condition $z\left(0\right)$, we can give the general solution of the Equation (18) by simply inverting the transformation (21), with the result

$$w=\frac{w_0+r{e}^{2i\mathsf{\Omega }\left(t-t_r\right)}{w}_0^{*}}{1+r{e}^{2i\mathsf{\Omega }\left(t-t_r\right)}}$$

(23)

where $r$ and $t_r$ are two real constants that characterize the solution. Using Equation (20) we can put this solution in real terms, i.e.,

$$\begin{array}{rr}z=& R+M\mathsf{\Omega }\left(\frac{2r\sin \left[2\mathsf{\Omega }\left(t-t_r\right)\right]}{1+r^2+2r\cos \left[2\mathsf{\Omega }\left(t-t_r\right)\right]}\right.\\ & \left.+i\frac{1-r^2}{1+r^2+2r\cos \left[2\mathsf{\Omega }\left(t-t_r\right)\right]}\right)\end{array}$$

(24)

which highlights a frequency modulation through what we would call a Stoler transformation (coherences of Stoler type) [18,19] and leads us to a complex form of this parameter. More than that, if we make the notation

$$r\equiv \coth \tau$$

(25)

Equation (24) becomes

$$z=R+M\mathsf{\Omega }h$$

(26)

where $h$ is given by

$$h=-i\frac{\cosh \tau -e^{-2i\mathsf{\Omega }\left(t-t_m\right)}\sinh \tau }{\cosh \tau +e^{-2i\mathsf{\Omega }\left(t-t_m\right)}\sinh \tau }$$

(27)

Now, through Equation (27), correspondences between the ansatz (4) and the concept of skyrmions can be found (see Appendix A). We note that h from (27) appears as a solution of the equations

$$\begin{gathered} \left(h-h^{*}\right)\nabla \left(\nabla h\right)=2{\left(\nabla h\right)}^2 \\ \left(h-h^{*}\right)\nabla \left(\nabla h^{*}\right)=2{\left(\nabla h^{*}\right)}^2 \end{gathered}$$

equations which result through harmonic mappings of the Poincaré metric

$$d{s}^2=\frac{dhd{h}^{*}}{{\left(h-h^{*}\right)}^2}=\frac{(du{)}^2+(dv{)}^2}{v^2}$$

(28)

For details, see ref. [6] from Appendix A.

Moreover, we must mention that a Hamiltonian theory can be built for $h=u+iv$. For details, see Appendix B.

Taking into account that h is the solution for the above-presented equations, in Figure 1a–d we show this solution, which can be associated to various Stoler-type coherences, in the form of period-doubling (Figure 1a), damped oscillations (Figure 1b), quasi-periodicity (Figure 1c), and intermittences (Figure 1d). These figures were obtained by means of |h| for different values of the pulsation-type characteristic ω (1, 1.42, 10, and 15) and for three different r values (0.1, 0.5, and 0.9). The period-doubling and quasi-periodicity behaviors can be verified by means of the Fast Fourier Transform.

Let us note that the measurement process appears here as a frequency modulation process. More precisely, this process is a calibration of the difference between kinetic and potential energy—the Lagrangian classic definition—which brings this quantity to a perfect square. The physical meaning of the perfect square Lagrangian is that it describes a fundamental physical unit in the interior of a complicated system, as kinetic energy describes the free particle in space. As expected, the quadratic Lagrangian actually corresponds to an ensemble of fundamental physical units: a set of oscillators of the same frequency.

3. Conclusions

The main conclusions of the present paper are the following:

i.

An analogy between the multifractal Schrödinger equation and the dumped oscillator equation through a special ansatz is established;

ii.

Using a Ricatti-type gauge, Stoler-type coherences in the dynamics of physical systems are highlighted;

iii.

This Ricatti-type gauge was assimilated to a calibration process of the difference between the kinetic and the potential energy of a Lagrangian, specified as a perfect square in generic coordinates.