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Cantorian Fractal Space-Time Fluctuations in Turbulent Fluid Flows and Kinetic Theory of Gases

A. M. Selvam

Deputy Director (Retired)

Indian Institute of Tropical Meteorology, Pune 411 008, India

Email: selvam@ip.eth.net

Web sites: http://amselvam.tripod.com/index.html

http://www.geocities.com/amselvam

Fluid flows such as gases or liquids exhibit space-time fluctuations on all scales extending down to molecular scales. Such broadband continuum fluctuations characterise all dynamical systems in nature and are identified as selfsimilar fractals in the newly emerging multidisciplinary science of nonlinear dynamics and chaos. A cell dynamical system model has been developed by the author to quantify the fractal space-time fluctuations of atmospheric flows. The earth's atmosphere consists of a mixture of gases and obeys the gas laws as formulated in the kinetic theory of gases developed on probabilistic assumptions in 1859 by the physicist James Clerk Maxwell. An alternative theory using the concept of fractals and chaos is applied in this paper to derive the fundamental equation of the kinetic theory of ideal gases and the Maxwells distribution of molecular speeds.
Keywords: fractals, chaos, kinetic theory of gases, gas laws

1.Introduction

Kinetic theory of ideal gases is a study of systems consisting of a great number of molecules, which are considered as bodies having a small size and mass (Kikoin and Kikoin, 1978). Classical statistical methods of investigation (Dennery, 1972; Yavorsky and Detlaf, 1975; Kikoin and Kikoin, 1978; Rosser, 1985; Guenault, 1988; Gupta, 1990; Ruhla, 1992; Dorlas, 1999; Chandrasekhar, 2000) are employed to estimate average values of quantities characterizing aggregate molecular motion such as mean velocity, mean energy etc., which determine the macro-scale characteristics of gases. The mean properties of ideal gases are calculated with the following assumptions. (1) The intra-molecular forces are completely absent instead of being small. (2) The dimensions of molecules are ignored, and considered as material points. (3) The above assumptions imply the molecules are completely free, move rectilinearly and uniformly as if no forces act on them. (4) The ceaseless chaotic movements of individual molecules obey Newtons laws of motion.
    The observed nonlinear space-time fluctuations of microscopic objects such as atoms and molecules in an ideal gas are now (since 1980s) identified as fractals generic to macro-scale real world dynamical systems in nature such as, fluid flows, stock market price fluctuations, heart beat patterns etc. The apparently chaotic (nonlinear) fractal fluctuations however exhibit self-similarity, i.e., long-range space-time correlations. The identification of the physical laws governing fractal fluctuations is an intensive field of research in the newly (since 1980s) emerging science of Nonlinear Dynamics and Chaos (Gleick, 1987). Mary Selvam (1990) has developed a general systems theory for the simulation and prediction of the observed fractal space-time fluctuations in dynamical systems of all size scales ranging from the microscopic scale of atoms and molecules to macro-scale turbulent fluid flows. The model concepts are applied to derive the following classical relationships for an ideal gas: (1) pressure exerted by an ideal gas (2) the Boltzmann distribution for molecular energies (3) the Maxwell distribution of molecular velocities. The derivation of the above relationships according to classical statistical methods is briefly described followed by a detailed discussion of the fractal concepts applied to derive the same equations.
    The important new contributions of the general systems theory applied to model ideal gases are as follows: (1) fractal fluctuations are signatures of quantum-like chaos on all scales ranging from subatomic dynamics of quantum systems to real world macro-scale fluid flows (2) quantum mechanical laws are applicable to dynamical systems of all size scales.
    The general systems theory concepts used in the derivation of the fundamental equations for the kinetic theory of gases have been applied earlier by the author for the simulation and prediction of both microscopic and macro-scale dynamical systems (Selvam, 1990; Selvam, 1993; Selvam, and Fadnavis, 1998; Selvam and Suvarna Fadnavis, 1999a; Selvam, and Suvarna Fadnavis, 1999b; Selvam, 2001).
    In the following, Sections 2, 3 and 4 deal respectively with application of the model concepts to derive the following three classical relationships for an ideal gas: (1) pressure exerted by an ideal gas (2) the Boltzmann distribution for molecular energies (3) the Maxwell distribution of molecular velocities. The derivation of the above relationships according to classical statistical methods is briefly described followed by a detailed discussion of the fractal concepts applied to derive the same equations. In conclusion Section 5 discusses the universal characteristics of fractal space-time fluctuations, a signature of quantum-like chaos exhibited by dynamical systems of all size scales ranging from sub-atomic dynamics of quantum systems to macro-scale turbulent fluid flows. The model shows that quantum mechanical laws are applicable to macro-scale real world dynamical systems and also provides physically consistent interpretations for wave-particle duality and non-local connection exhibited by microscopic-scale quantum systems which so far do not have a satisfactory explanation on the basis of current concepts in quantum mechanics.

2. Pressure exerted by an ideal gas

2.1 Classical statistical physics

A brief summary of the method for calculating pressure based on classical statistical physics concepts is given in the following The molecular collisions exert a force on the walls of the vessel containing the gas and this force is measured by the parameter pressure, which is equal to the force per unit area perpendicular to the direction of the force.
The pressure   p is calculated as
where n is the number of molecules per unit volume, m is the mass of one molecule and   represents the mean square velocity in any one direction x, y or z. The pressure p may be written as
(1)
In Equation (1) Ek is equal to the mean kinetic energy   of one molecule of a gas. Consequently the pressure of a gas equals two-thirds of the mean kinetic energy of the molecules contained in a unit of its volume. This is one of the most important conclusions of the kinetic theory of an ideal gas. Equation (1) establishes a relationship between molecular quantities, i.e., quantities relating to a separate molecule, and the value of the pressure characterizing a gas as a whole a macroscopic quantity directly measured in experiments. Equation (1) is sometimes called the fundamental equation of the kinetic theory of ideal gases.

2.2 General systems theory

One of the most convincing demonstrations that gases really are made up of fast moving molecules is Brownian motion, the observed constant jiggling around of tiny particles, such as fragments of ash in smoke. This motion was first noticed in 1827 by the British botanist, Robert Brown who initially assumed he was looking at living creatures, but then found the same motion in what he knew to be particles of inorganic material. Einstein showed how to use Brownian motion to estimate the size of atoms (Kikoin and Kikoin, 1978; Fowler, 1997; Lee and Kelvin).
    Chaotic fluctuations such as those executed by molecules in a gas are now identified as fractals generic to space-time fluctuations of dynamical systems in nature (Mandelbrot, 1977; 1983; Gaspard et al., 1998). Identification of the physics of fractal fluctuations and quantification is an intensive field of research in the newly emerging (since 1980s) multidisciplinary science of Nonlinear Dynamics and Chaos (Gleick, 1987). It has been long supposed that the existence of chaotic behaviour in the microscopic motions of atoms and molecules in fluids or solids is responsible for their equilibrium and non-equilibrium properties. Recently this hypothesis of microscopic chaos has been verified experimentally by Gaspard et al. (1998) who found evidence for microscopic chaos in fluid systems, by the observation of Brownian motion of a colloidal particle suspended in water.
    Mary Selvam (1990) has developed a general systems theory (Capra, 1996) for the observed space-time fractal fluctuations in dynamical systems, which enable quantification of large-scale fluctuations in terms of inherent smaller scale fluctuation characteristics. The irregular fractal fluctuations occur on all space-time scales and may be considered to result from the superimposition of a continuum of eddies or waves such as sine waves. An eddy is basically a circular motion characterized by the radius r and r.m.s (root mean square) circulation speed w*. Larger scale fluctuations result from the integration of enclosed smaller scale fluctuations. The relationship between the r.m.s circulation speeds W and w*  of large and small eddy of respective radii R and r is given as (Townsend, 1956; Mary Selvam, 1990)
(2)
The above equation represents the growth of an eddy continuum with formation of a hierarchy of successively larger eddies from enclosed smaller scale eddies. The square of the eddy amplitude, i.e., W2 represents the eddy energy (kinetic). The large eddy energy is the integrated mean of the enclosed smaller scale eddy energies and therefore the eddy energy spectrum will follow statistical normal distribution according to the Central Limit Theorem (Ruhla, 1992). Such a result that the additive amplitudes (W) of eddies, when squared (W2) represent the statistical normal distribution is exhibited by subatomic dynamics of quantum systems such as the electron or photon (Maddox, 1998; 1993).
    By analogy with quantum mechanics the square of the eddy amplitude W2 represents the kinetic energy E given as (from Equation 2)
In the above Equation the parameter n (proportional to 1/R) is the frequency of the large eddy and H is a constant equal to  for the growth of large eddies sustained by constant energy input proportional to w*2 from fixed primary small scale eddy fluctuations. Energy content of eddies is therefore similar to quantum systems which can possess only discrete quanta or packets of energy content hn where h is a universal constant of nature (Planck's constant) and n is the frequency in cycles per second of the electromagnetic radiation.
    The macro-scale eddy continuum represented by Equation (2) obeys quantum-like mechanical laws, a manifestation of quantum-like chaos. The apparent paradox of wave-particle duality exhibited by microscopic scale quantum systems such as an electron or photon is however physically consistent in the context of real world macro-scale dynamical systems as explained in the following. The bi-directional energy flow intrinsic to eddy circulations is associated with bimodal, i.e., formation and dissipation respectively of phenomenological form for manifestation of energy such as the formation of clouds in updrafts and dissipation of clouds in adjacent downdrafts resulting in the observed discrete cellular geometry to cloud structure. The commonplace occurrence of clouds in a row is a manifestation of wave-particle duality in macro-scale atmospheric flows. By analogy, the molecules (atoms) of an ideal gas may be visualised as the manifestation of matter during a half-cycle of an eddy circulation (Mary Selvam, 1990; Selvam and Fadnavis, 1999a). The primary perturbation of r.m.s circulation speed w*  and eddy radius r represents the wave-like structure of a molecule or atom in the ideal gas, the manifestation of matter of molecular mass m occurring during half a cycle of the complete circulation as explained above.
The length scale ratio R/r in the above Equation (2) represents the fractional volume intermittency of occurrence of small-scale (fractal) structures (Mary Selvam, 1993) across unit area of large eddy surface as shown in the following.
Considering two large eddy circulations of respective radii R1 and R2 (R2 being greater than R1) and corresponding r.m.s circulation speeds W1 and W2 which grow from the same small-scale primary perturbation of radius r and r.m.s circulation speed w*, we have from Equation (2)
(3)
Introducing the factor   representing eddy volumes on both sides of the above equation we have
(4)
Therefore
(5)
Substituting for R1/ R2 on the right hand side from Equation (3) we have the following relation for fractional volume intermittency of occurrence of small-scale fluctuations given by the fourth moment about the mean for the relative eddy transports as
(6)
The length scale ratio R1/ R2 is equal to the transport of fractional volume of small-scale fluctuations in the environment of the large eddy (per unit volume of large eddy), basically by eddy mixing process. Considering large and small eddies of respective radii R and r and r.m.s circulation speeds W and w* the corresponding mass transport M of gas across unit area for half cycle of large eddy circulation in terms of molecular mass is equal to  . The molecular mass m corresponds to the small-scale primary eddy perturbation. Multiplying both sides of Equation (2) by nm/2 and rewriting
(7)
    In the above equation the large eddy circulation speed W represents the acceleration since it is computed as an incremental value relative to its earlier stage of eddy growth. The pressure p exerted by the gas is given by the product MW equal to the rate of change of momentum due to molecular impact across unit area of the large eddy surface. Equation (7) may now be written as
(8)
The r.m.s eddy circulation speed w* represents by concept the average molecular speed in any direction and the average kinetic energy of one molecule designated by Ek is equal to  . The above Equation (8) may now be written as
(9)
Equation (9) is almost the same as Equation (1), the fundamental equation of the kinetic theory of ideal gases, namely,  .
The important differences in the physical concepts underlying the derivation of the fundamental equation of the kinetic theory of ideal gases by classical statistical physical methods and the general systems theory for fractal space-time fluctuations are as follows: (1) The general systems theory visualises molecules or atoms as manifestation of matter during half a cycle of eddy circulation. Classical physics visualises molecules and atoms as point objects. (2) The r.m.s velocity w* represents the average velocity for computation of mean molecular kinetic energy in the general systems theory. The mean square velocity of the molecule or atom in any one direction (x, y or z) equal to  is used for computing the molecular kinetic energy in classical physics.

3. Boltzmann distribution for molecular energies in an ideal gas

3.1 Classical physics

For any system large or small in thermal equilibrium at temperature T, the probability P of being in a particular state at energy E is proportional to   where kB is the Boltzmanns constant. This is called the Boltzmann distribution and may be written as
(10)

3.2 General systems theory

The physical concepts of the general systems theory enables to show that precise ordered mathematical patterns underlie the apparently chaotic space-time fluctuations of dynamical systems. The irregular fractal fluctuations of dynamical systems may be visualized to result from the superimposition of an ensemble of eddies, namely an eddy continuum. An eddy continuum by concept consists of a hierarchy of eddies, the larger scale eddies enclosing smaller scale eddies. The larger scale eddies grow by the spatial integration of enclosed smaller scale eddies and the growth trajectory traces an overall logarithmic spiral flow path with the quasiperiodic Penrose tiling pattern for the internal structure (Mary Selvam, 1990; Selvam and Fadnavis, 1998). The ratio of radii (R2/ R1) or r.m.s. circulation speeds (W2/W1) corresponding to the successive growth steps of the large eddy generating the geometry of the quasiperiodic Penrose tiling pattern is equal to the golden mean t (@ 1.618).
(11)
The r.m.s circulation speed W of the large eddy follows a logarithmic relationship with respect to the length scale ratio z equal to R/r as given below
(12)
In Equation (12) the variable k represents for each step of eddy growth, the fractional volume dilution of large eddy by turbulent eddy fluctuations carried on the large eddy envelope and is given as
(13)
Incidentally, Equation (12) represents the observed logarithmic spiral air flow structure in the planetary atmospheric boundary layer and the constant k called the von Karmans constant is determined from observations to be equal to about 0.38 (Mary Selvam, 1990; Selvam and Fadnavis, 1998).
From Equations (11) and (13) it is seen that, for successive large eddy growth steps generating the quasiperiodic Penrose tiling pattern, the value of k is equal to 1/t2 (@0.38) where  t is the golden mean (@1.618). Substituting for k in Equation (12) we have
(14)
Therefore
(15)
The ratio r/R represents the fractional probability P of occurrence of small-scale fluctuations (r) in the large eddy (R) environment. Considering two large eddies of radii R1 and R2 (R2 greater than R1) and corresponding r.m.s circulation speeds W1 and W2 which grow from the same primary small-scale eddy of radius r and r.m.s circulation speed w* we have from Equation (2)
(16)
From Equations (15) and (16)
(17)
The square of r.m.s circulation speed W2 represents eddy kinetic energy. Following classical physical concepts (Kikoin and Kikoin, 1978) the primary (small-scale) eddy energy may be written in terms of the eddy environment temperature T and the Boltzmanns constant kB as
(18)
Representing the larger scale eddy energy as E
(19)
The length scale ratio R1/R2 therefore represents fractional probability P of occurrence of large eddy energy E in the environment of the primary small-scale eddy energy kBT (Equation 18). The expression for P is obtained from Equation (17) as
(20)
The above Equation (20) is the same as the Boltzmanns equation (Equation 10).
The derivation of Boltzmanns equation from general systems theory concepts visualises the eddy energy distribution as follows: (1) The primary small-scale eddy represents the molecules whose eddy kinetic energy is equal to kBT as in classical physics. (2) The energy pumping from the primary small-scale eddy generates growth of progressive larger eddies (Mary Selvam, 1990). The r.m.s circulation speeds W of larger eddies are smaller than that of the primary small-scale eddy (Equation 2). (3) The space-time fractal fluctuations of molecules (atoms) in an ideal gas may be visualised to result from an eddy continuum with the eddy energy E per unit volume relative to primary molecular kinetic energy (kBT) decreasing progressively with increase in eddy size.

4. Maxwell-Boltzmann distribution of molecular speeds

4.1 classical physics

The distribution of molecular speeds was derived by Maxwell based on the probabilistic assumptions, namely (i) uniform distribution in space, (ii) mutual independence of the three velocity components and (iii) isotropy as regards the directions of the velocities (Ruhla, 1992). These assumptions were also used in deriving the fundamental gas law at Equation (1) for a perfect gas. Maxwell's distribution of molecular speeds is given by the following equation.
(21)
where r(v) is the probability density assigned to the speed v, T is the absolute temperature of the perfect gas, m is the mass of a molecule and kB is the Boltzmann's constant. For a given gas at a fixed temperature T, the probability density r(v) may be written as
(22)
A graph of Maxwell's distribution of molecular speeds is shown in Figure 1.

4.2 General systems theory

The steady state upward transport of small-scale fluctuation of speed w* and size scale r in the environment of larger scale fluctuation of speed W and size R is given as (Mary Selvam, 1990; Selvam and Fadnavis, 1998)
(23)
In Equation (23) z is the length scale ratio equal to R/r. Considering three-dimensional fluctuations the fractional contribution (probability density) of smaller length scale r fluctuations in the environment of the larger length scale R fluctuation is given by f 3. The eddy circulation speeds follow the logarithmic law with respect to the length scale ratio z (Equation 12), namely
The eddy circulation speeds are therefore proportional to log z, that is
(24)
A graph of f 3 versus log z will give the probability density distribution for molecular speeds. The cell dynamical system model predicted molecular speed distribution in a perfect gas is shown as crosses in Figure 1. The distributions (Maxwell's and model predicted) are normalized with respect to the maximum speed. There is close agreement between the Maxwell's and model-predicted distributions for molecular speeds in a perfect gas.

Figure 1

5. Conclusions

Dynamical systems of all size scales ranging from microscopic scale quantum systems to macro-scale turbulent fluid flows exhibit self-similar fractal space-time fluctuations. Self-similarity implies long-range space-time correlations or non-local connections such as that observed in quantum systems. A general systems theory developed by the author enables to show quantitatively that the observed fractal space-time fluctuations generic to dynamical systems in nature are signatures of quantum-like chaos. The model concepts for Cantorian fractal space-time fluctuations is applied to derive the fundamental gas law, namely   and also the Maxwells molecular speed distribution for a perfect gas. The model predictions are in agreement with Maxwell's kinetic theory of gases developed in 1859 on probabilistic assumptions.

Acknowledgements

The author is grateful to Dr. A. S. R. Murty for his keen interest and encouragement during the course of the study.

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