Tendency towards maximum complexity in a nonequilibrium isolated system
Xavier Calbet
Instituto de Astrofı ´sica de Canarias Vı ´a La ´ctea, s/n, E-38200 La Laguna, Tenerife, Spain
Ricardo Lo
´
pez-Ruiz*
Departamento de Fı ´sica Teo ´rica Facultad de Ciencias, Edificio A, Universidad de Zaragoza, E-50009 Zaragoza, Spain
Received 30 January 2001; published 22 May 2001
The time evolution equations of a simplified isolated ideal gas, the ‘‘tetrahedral’’ gas, are derived. The
dynamical behavior of the Lo ´pez-Ruiz–Mancini–Calbet complexity R. Lo ´pez-Ruiz, H. L. Mancini, and X.
Calbet, Phys. Lett. A 209, 321 1995 is studied in this system. In general, it is shown that the complexity
remains within the bounds of minimum and maximum complexity. We find that there are certain restrictions
when the isolated ‘‘tetrahedral’’ gas evolves towards equilibrium. In addition to the well-known increase in
entropy, the quantity called disequilibrium decreases monotonically with time. Furthermore, the trajectories of
the system in phase space approach the maximum complexity path as it evolves toward equilibrium.
DOI: 10.1103/PhysRevE.63.066116 PACS numbers: 89.75.Fb, 05.45.-a, 02.50.-r, 05.70.-a
I. INTRODUCTION
Several definitions of complexity, in the general sense of
the term, have been presented in the literature. These can be
classified according to their calculation procedure into two
broad and loosely defined groups.
One of these groups is based on computational science
and consists of all definitions based on algorithms or au-
tomata to derive the complexity. Examples are the logical
depth 1, the -machine complexity 2, and algorithmic
complexity 3. These definitions have been shown to be
very useful in describing symbolic dynamics of chaotic
maps, but they have the disadvantage of being very difficult
to calculate.
Another broad group consists of those complexities based
on the measure of entropy or entropy rate. Among these, we
may cite the metric or K -S entropy rate 4,5, the thermody-
namic depth 6, the effective measure complexity 7, and
the simple measure for complexity 8. These definitions
have also been very useful in describing symbolic dynamical
maps, the latter having been applied to a nonequilibrium
Fermi gas 9. They suffer the disadvantage of either being
very difficult to calculate or having a simple relation to the
regular entropy.
New definition types of complexity have recently been
introduced. These are based on quantities that can be calcu-
lated directly from the distribution function describing the
system. One of these is based on ‘‘metastatistics’’ 10 and
the other on the notion of ‘‘disequilibrium’’ 11. This latter
definition will be referred to hereafter as the Lo
´
pez-Ruiz–
Mancini–Calbet LMC complexity. These definitions, to-
gether with the simple measure for complexity 8 described
above, have the great advantage of allowing easy calcula-
tions within the context of kinetic theory and of permitting
their evaluation in a natural way in terms of statistical me-
chanics.
The disequilibrium-based complexity is easy to calculate
and shows some interesting properties 11, but suffers from
the main drawback of not being very well behaved as the
system size increases, or equivalently, as the distribution
function becomes continuous 12. Feldman and Crutchfield
tried to solve this problem by defining another equivalent
term for disequilibrium, but ended up with a complexity that
was a trivial function of the entropy.
Whether these definitions of complexity are useful in non-
equilibrium thermodynamics will depend on how they be-
have as a function of time. There is a general belief that,
although the second law of thermodynamics requires average
entropy or disorder to increase, this does not in any way
forbid local order from arising 13. The clearest example is
seen with life, which can continue to exist and grow in an
isolated system for as long as internal resources last. In other
words, in an isolated system the entropy must increase, but it
should be possible, under certain circumstances, for the com-
plexity to increase.
In this paper we will examine how LMC complexity
evolves with time in an isolated system and we will show
that it indeed has some interesting properties. The
disequilibrium-based complexity defined in Ref. 11 actu-
ally tends to be maximal as the entropy increases in a Bolt-
zmann integrodifferential equation for a simplified gas.
In Sec. II LMC complexity definition is reviewed. We
proceed to calculate the distributions which maximize and
minimize the complexity and its asymptotic behavior, and
also introduce the basic concepts underlying the time evolu-
tion of LMC complexity in Sec. III. Later, in Sec. IV, by
means of numerical computations following a restricted ver-
sion of the Boltzmann equation, we apply this to a special
system, which we shall term ‘‘tetrahedral gas.’’ Finally, in
Sec. V, the results and possible future lines of investigation
are discussed, together with their possible applications. Ana-
lytical and numerical demonstrations of the results of the
numerical calculations for the tetrahedral gas are shown in
the appendixes.
*Present address: Area de Ciencias de la Computacion, Facultad
de Ciencias, Edificio B, Universidad de Zaragoza, E-50009 Zara-
goza, Spain.
PHYSICAL REVIEW E, VOLUME 63, 066116
1063-651X/2001/636/0661169/$20.00 ©2001 The American Physical Society 63 066116-1