A.M.Selvam

Indian Institute of Tropical Meteorology, Pune 411 008, India

(Retired email: selvam@ip.eth.net)

Web site: http://www.geocities.com/CapeCanaveral/Lab/5833

**Abstract**

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 these fundamental
gas laws.

**1. Introduction**

The kinetic theory of gases is based on the statistical method of investigation (Yavorsky and Detlaf, 1975; Ruhla and Barton, 1992).

The basic gas law for a perfect gas is

(1)

** V** = volume of the gas

and

(2)

where (3)

represents the average kinetic energy of a molecule
of mass ** m** in any direction, i.e. the average for the three
Cartesian co-ordinates

The distribution of molecular speeds was derived
by *Maxwell* based on three 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 and Barton, 1992). These assumptions were also used in deriving
the fundamental gas law at Eq.(1) for a perfect gas. *Maxwell*'s distribution
of molecular speeds is given by the following equation.

(4)

where For a given gas at a fixed temperature ** T**
, the probability density r

*r (v) µ
exp(-v ^{2} ) v^{2}*

(5)

A graph of

**2**. **Cell dynamical system model for kinetic
theory of gases**

The above equations for the kinetic theory of gases
can be derived directly from the cell dynamical system model (Selvam *et
al.*, 1984a,b ; 1992; 1996; Sikka *et al*., 1984; Selvam and Murty,
1985; Selvam, 1988; 1989; 1990;1993; 1997; 1998; 1999; 2000; Selvam and
Joshi, 1995; Fadnavis and Selvam, 1997; Selvam and Fadnavis, 1998; 1999a,b,cReferences)
as follows. The random thermal agitation (fluctuation) of molecular speeds
is analogous to a continuum of eddy circulations, that is a hierarchy of
eddy fluctuations where, the larger scale fluctuations enclose smaller
scale fluctuations.

*2.1 The fundamental gas law for a
perfect gas*

The root mean square (r.m.s.) circulation speed ** W**
over length scale

(6)

Therefore

(7)

(8)

(9)

*2.2 Distribution of molecular speeds*

The steady state upward transport of small-scale
fluctuation of speed *w***_{*
}**and
size scale

(10)

where ** z** is the size scale ratio equal
to

(11)

where ** k** is a constant equal to

*W »
log z*

(12)

A graph of **3. Conclusion**

The concept of Cantorian fractal spacetime fluctuations
is applied to derive the fundamental gas law, namely ** PV=RT**
and also the molecular speed distribution for a perfect gas. The model
predictions are in agreement with

Selvam, A. M., A.S.Ramachandra Murty, G.K.Manohar,
S.S.Kandalgaonkar, Bh.V.Ramana Murty 1984a: A new mechanism for the maintenance
of fair weather electric field and cloud electrification, *Proc. VII
International Conference on Atmospheric Electricity*, June 3-8, Albany,N.Y.(American
Meteorological Society),154-159. http://xxx.lanl.gov/abs/physics/9910006
.

Sikka, P., A.Mary Selvam, A.S.RamachandraMurty, Bh.V.RamanaMurty,
1984: Possible solar influence on atmospheric electric field,* Preprint
Volume VII International Conference on Atmospheric Electricity*, June
3-8, Albany, N.Y., American Meteorological Society, Boston, Mass., USA.
http://xxx.lanl.gov/abs/chao-dyn/9806014
.

Selvam, A. M., A.S.Ramachandra Murty and Bh.V. Ramanamurty,
1984b: A New hypothesis for the vertical distribution of atmospheric aerosols,
*Proc.of
the XI Int. Conf. on Atmospheric Aerosols*, *Condensation and Ice
Nuclei*, 2-7 Sept., Budapest, Hungary, 77-81. http://xxx.lanl.gov/html/physics/9912014
.

Selvam, A. M. and A.S.R Murty, 1985: Numerical simulation
of warm rain process *Proc. of the 4th WMO Scientific Conf. on Weather
Modification* 12-14 August, Honolulu, Hawaii, 503-506. http://xxx.lanl.gov/abs/physics/9911021
.

Selvam A. M., 1988: The dynamics of non-linear effects
in optical processes, *Journal of Luminescence* **40 & 41**,
535-536. http://xxx.lanl.gov/abs/chao-dyn/9807005.

Selvam A.M., 1989: A cell dynamical system model
for thundercloud electrification, *Proc., Int'l . Conf. Lightning and
Static Electricity*, 26-28 September, University of Bath U.K. http://xxx.lanl.gov/abs/chao-dyn/9809002
.

Selvam, A. M. 1990: Deterministic chaos, fractals
and quantum-like mechanics in atmospheric flows,*Canadian J.Physics***68**,
831-841.

Selvam, A. M., J.S.Pethkar and M.K.Kulkarni, 1992:
Signatures of a universal spectrum for atmospheric interannual variability
in rainfall time series over the Indian region, *Int'l J.Climatol*.**12**,
137-152.

Selvam, A. M. 1993: Universal quantification for
deterministic chaos in dynamical systems, *Applied Mathematical Modelling*
, **17**, 642-649.

Selvam, A. M. and R.R.Joshi, 1995: Universal spectrum
for interannual variability in COADS global air and sea surface temperatures,
*Int'l.
J.Climatol*. **15**, 613 - 623.

Selvam, A. M., J.S. Pethkar, M.K. Kulkarni and R.
Vijayakumar, 1996: Signatures of a universal spectrum for atmospheric interannual
variability in COADS surface pressure time series, *Int'l. J. Climatol*.
**16**,
393 - 404.

Selvam, A.M. 1997: Universal Quantification for Self-Organized
Criticality in Atmospheric Flows *Proc. Conf. Patterns, Nonlinear Dynamics
and Stochastic Behaviour in Spatially Extended Complex Systems*, October
24-28, Budapest, Hungary. . http://xxx.lanl.gov/abs/chao-dyn/9710004
.

Fadnavis, S. and A.M.Selvam, 1997: Universal spectrum
for interannual variability of rainfall over India and Scotland, *Proceedings
, National Space Science Symposium*, Physical Research Laboratory, Ahmedabad,
India, November 25-28. http://xxx.lanl.gov/abs/chao-dyn/9806028
.

Selvam, A.M., 1998: Quasicrystalline pattern formation
in fluid substrates and phyllotaxis In *Symmetry in Plants*, D. Barabe
and R.V. Jean (Editors), World Scientific Series in Mathematical Biology
and Medicine, Vol.4., Singapore, pp.795-809. http://xxx.lanl.gov/abs/chao-dyn/9806001
.

Selvam, A. M. and S. Fadnavis, 1998: Signatures of
a universal spectrum for atmospheric interannual variability in some disparate
climatic regimes, *Meteorology and Atmospheric Physics*,** 66**,
87-112, (Springer-Verlag, Austria) http://xxx.lanl.gov/abs/chao-dyn/9805028
.

Selvam, A. M. and Suvarna Fadnavis, 1999a: The dynamics
of fullerene structure formation : order out of chaos phenomenon, Accepted
for publication in *FRACTALIA*, Romania . http://xxx.lanl.gov/abs/physics/9909052
.

Selvam, A. M., 1999: Cantorian fractal spacetime
and quantum-like chaos in neural networks of the human brain, *Chaos,
Solitons and Fractals ***10(1)** , 25 - 29 . http://xxx.lanl.gov/abs/chao-dyn/9809003
.

Selvam, A. M., and Suvarna Fadnavis, 1999b: Cantorian
fractal spacetime, quantum-like chaos and scale relativity in atmospheric
flows, *Chaos, Solitons and Fractals ***10(9)**, 1577 - 1582.http://xxx.lanl.gov/abs/chao-dyn/9808015
.

Selvam, A.M. and Suvarna Fadnavis, 1999c: A superstring
theory for fractal spacetime, chaos and quantumlike mechanics in atmospheric
flows, *Chaos, Solitons and Fractals* **10(8)**, 1321-1334. http://xxx.lanl.gov/abs/chao-dyn/9806002
.

Selvam, A. M., D. Sen and S. M. S. Mody, 2000: Critical
fluctuations in daily incidence of acute myocardial infarction, *Chaos,
Solitons and Fractals* (Accepted for publication in *Chaos, Solitons
and Fractals *2000). http://xxx.lanl.gov/abs/chao-dyn/9810017
.

Yavorsky, B. and A.Detlaf, 1975: *Handbook of Physics*
, Mir Publishers, Moscow, pp.965.