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Spontaneous entropy decrease and its statistical formula
Xing Xiu-San
(Department of Physics,Beijing Institute of Technology,Beijing 100081,China)
email: xingxs@sohu.com
Abstract
Why can the world resist the law of entropy increase and produce self-organizing structure?
Does the entropy of an isolated system always only increase and never decrease? Can be
thermodymamic degradation and self-organizing evolution united? How to unite? In this paper
starting out from nonequilibrium entropy evolution equation we proved that a new entropy
decrease could spontaneously emerge in nonequilibrium system with internal attractive interaction.
This new entropy decrease coexists with the traditional law of entropy increase, both of them
countervail each other, so that the total entropy of isolated system can be able to decrease. It not
only makes isolated system but also helps open system to produce self-organizing structure. We
first derived a statistical formula for this new entropy decrease rate, and compared it both in
mathematical form and in microscopic physical foundation with the statistical formula for the law
of entropy increase which was derived by us some years ago. Furthermore, we gave the formulas
for the time rate of change of total entropy in isolated system and open system. The former is
equal to the sum of the formula for the law of entropy increase and the formula for the new
entropy decrease rate, the latter is the algebraic sum of the formulas for entropy increase, entropy
decrease and entropy flow. All of them manifest the unity of thermodynamic degradation and
self-organizing evolution. As the application of the new theoretical formulas, we discussed
qualitatively the emergency of inhomogeneous structure in two real isolated systems including
clarifying the inference about the heat death of the universe.
Keywords: entropy evolution equation, internal attractive force, formula for entropy
decrease rate, formula for law of entropy increase, entropy diffusion
1. Introduction
The law of entropy increase[1-4], i.e. the second law of thermodynamics expressed by the
entropy , is a fundamental law in nature. It shows if an isolated system is not in a statistical
equilibrium state, its macroscopic entropy will increase with time, until ultimately the system
reaches a complete equilibrium state where the entropy attains its maximum value. According to
the inference of this law, the universe is as isolated system, it also ought to degrade into a
complete statistical equilibrium state, i.e. the so-called heat death state[3-4]. Then, the entropy and
randomness in the universe are at their maximum, there are only gas molecules with homogeneous
distribution, all macroscopic mechanical energy degrades into heat energy of gas molecules, no
further change occurs. However, the real world is another scene. Everywhere there is order and
structure: stars, galaxies, plants and animals etc. They are always incessantly evolving. When the
law of entropy increase occupies a dominant position, why can an isolated system create order
structure? Why can the life from nothing to some thing with simple atoms and molecules organize
itself into a whole? Why is the universe still able to bring forth stars, galaxies and does not stop at
or dissolve into simple gas? Why can they resist the law of entropy increase and produce
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self-organizing structure? Whether or not because they are also governed by a power of some
unknown entropy decrease inherent in the system? If so, how does this power of entropy decrease
coexist and countervail with the law of entropy increase, and hence both of them form the unity of
opposites between thermodynamic degradation and self-organizing evolution? What is its
dynamical mechanism? And what is its mathematical expression? What is the difference in
mathematical form and microscopic physical foundation between the formulas for this entropy
decrease and the law of entropy increase? Under what condition can an isolated system overcome
the law of entropy increase and produce self-organizing structure? And what is the difference in
entropy condition for growing self-organizing structure between open system and isolated system?
Can all these problems be solved from a nonequilibrium entropy evolution equation in a unified
fashion?
In the late half of the twentieth century, the publications of the theory of dissipative
structures[5], synergetics[6] and the hypercycle[7] marked an important progress of quantitative
theories in self-organization. However, these theories, including the formal entropy theory
decomposing the entropy change into the sum of the entropy flow and the entropy production,
discuss only the problems of open system but not isolated system. From the point of view of
exploring that what system can spontaneously produce a entropy decrease to be a match for the
law of entropy increase, they all have no relation. Of course, they also have no help to clarify the
puzzle of the heat death. In recent years during doing research on the fundamental problems of
nonequilibrium statistical physics, we[8-12] proposed a new equation of time-reveral asymmetry in
place of the Liouville equation of time-reversal symmetry as the fundamental equation of
nonequilibrium statistical physics. That is the anomalous Langevin equation in 6N dimensional
phase space or its equivalent Liouville diffusion equation. Starting from this equation we decided
succinctly the hydrodynamic equations such as diffusion equation, thermal conductivity equation
and Navier-Stokes equation. Furthermore we presented a nonlinear evolution equation of Gibbs
and Boltzmann nonequilibrium entropy density changing with time-space (called for short
nonequilibrium entropy evolution equation), predicted the existence of entropy diffusion, and
obtained a concise statistical formula for the law of entropy increase. In this paper we solved all
above mentioned problems on the basis of these new known works, especially we first proved that
a new entropy decrease can spontaneously emerge in nonequilibrium system with internal
attractive interaction, and derived its statistical formula. Just that the power of this new entropy
decrease countervails the law of entropy increase leads to the total entropy of an isolated system to
be able to decrease, and not only makes isolated system but also helps open system to produce
self-organizing structure.
For the sake of saving space of the paper, the following quantitative expressions are limited
in 6 dimensional phase space.
2. Nonequilibrium entropy evolution equation
According to nonequilibrium statistical physics, nonequilibrium entropy in 6 dimensional
phase space can be defined as[1,13,14]
1
f
10vp0
10
( , )
( )
x
( )( , )ln
t
x
S
ft
S tk f
∫
dSdS
= −+=+
∫
x
xx
(1)
or
11vp
( ) ( , )ln
t
x
( , )
x
S tk fft dS d
= −∫∫
x =x (1a)
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it a brief review. Entropy increase, the third term on the right-hand side, it give a concise statistical
formula for the traditional low of entropy increase, i.e. formula (4). Entropy diffusion, the second
term on the right-hand side, its presence make us to think that the process of tending to
equilibrium is accomplished by entropy diffusing from its high density region to low density
region and finally the distribution of entropy density in all the system reaches to uniformity and
the entropy of the system approaches to maximum. Hence the essence of tending to equilibrium
puzzling physicist for a long time now becomes clear at a glance. Entropy change contributed by
interaction potential energy, the fourth term on the right-hand side, it reveals that a new entropy
decrease can spontaneously emerge in a nonequilibrium system with internal attractive interaction,
and its quantitative expression is formula (6). Obviously, every term among these three terms has
itself important physical meaning. As to entropy drift, i.e. entropy flow of open system, the first
term on the right-hand side, it is well known in existing theory. If we say, formulas (7) and (8)
have united thermodynamic degradation for destroying the order structure and self-organizing
evolution for producing the order structure, then nonequilibrium entropy evolution equation (3)
may be regarded as a basic equation describing the system evolution including the unity of
thermodynamic degradation and self-organizing evolution.
It should be pointed out here that from nonequilibrium entropy evolution equation (3 ) and
the formula for entropy increase rate we have obtained their corresponding dynamic information
evolution equation and the formula for information dissipation rate[17]. Similarly, from the formula
for entropy decrease rate (6) we can obtain its corresponding formula for information production
rate as follows
∫
Where the information density production rate
( ) ( )( )
l f
i
l f
=∫
i
R tdd
x = -x
21
()
1
( ) (
×
)
qq
fq
p
−−
= −∇∇−
1
Nke
q
(9)
( )
il f =- ( )
l f (9a)
Now dynamic information evolution equation corresponding to equation (3) is
2
vp
t
2
q
vp vp1 vp vp
1
()( ln) ( )
l f
i
∂
= −∇+∇−∇ −∇+
∂
I
D
kf
IDI f II
qqq
v
×
(10)
On the basis of this equation it is easy to present the formulas for the time rate of change of total
information in isolated system and open system, which are corresponding to the formulas (7) and
(8). The detailed discussion will publish in another paper, it is omitted here.
5. Brief conclusion
According to the BBGKY diffusion equation of single particle probability density and the
definition of nonequilibrium entropy, many years ago we derived a nonlinear evolution equation
of nonequilibrium entropy density changing with time-space, and from this we gave a statistical
formula for the law of entropy increase. In this paper starting out from this nonequilibrium entropy
evolution equation we proved that the attractive interaction force between microscopic particles
could contribute nonequilibrium system to create a new entropy decrease, and derived a statistical
formula for this entropy decrease rate. When an isolated system with internal attractive interaction
is in nonequilibrium state, the time rate of change of its total entropy is equal to the sum of the
formula for the law of entropy increase and the formula for the new entropy decrease rate . Hence,
its total entropy no more only increases and never decreases, but is able to decease. It shows that
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just this entropy decrease is a dominant power for that an isolated system including the universe
can overcome the law of entropy increase and spontaneously produce self-organizing structure.
Compared with isolated system, the time rate of change of total entropy in open system except
entropy increase and entropy decrease still contains an entropy outflow. Therefore for
spontaneously creating self-organizing structure in open system the new entropy decrease still can
play a promoting role though it is not certainly a dominant power again. Thus, whether isolated
system or open system, its total entropy changes contains both thermodynamic degradation of
entropy increase on the one hand and self-organizing evolution of entropy decrease on the other
hand. Two of them coexist in a theoretical formula and form the unity of opposites.
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