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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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where

()

x q, p

=

is the state vector in 6 dimensional phase space, q and p are the position

and momentum of the single particle,

11

()(

ftf

x,q, p,

=

) t is the single particle probability

density at time t ,

( )

x is its equilibrium probability density.

10

f

0 S is equilibrium entropy in 6

dimensional phase space ,k is the Boltzmanm constant,

vp

S is entropy density in 6 dimensional

phase space. The reason why we used the expression (1) but not(1a) as the definition of

nonequilibrium entropy will explain in the following section 2.

Just the same as that the BBGKY equation hierarchy[14,15] can be derived from the Liouville

equation, the BBGKY diffusion equation[8-11] of the single-particle probability density can also be

derived from the Liouville diffusion equation as follows

2

q

121

( , )

x

()( ,

x x

, )

t d

( , )

x

ftNfDft

tm

f

∂

∂

+ ∇ +

q

×

∇=∇∇+∇

∫

pqp11

p

Fx

××

(2)

Where

21

f ( t)

x,x , is two particle joint probability density at time t ,F is external force,

N is particle number in the system, D is self-diffusion coefficient of the particle,

1

ff (||)

qq

=-

is the two-particle interaction potential. The first term on the right-hand side of

equation (2) is commonly referred to as the collision integral, now we call it interaction term. We

shall see below that the new entropy decrease of an isolated system with internal attractive

interaction just is emerged from this term. The second term on the right-hand side of equation (2)

comes from the diffusion term of the Liouville diffusion equation, when there is no this term,

equation (2) is just the BBGKY equation of single-particle probability density.

Differentiating both side of formula (1) with respect to time t and using the BBCKY

diffusion equation (2)(now assume F = 0), we obtained nonequilibrium entropy evolution

equation in 6 dimensional phase space as follows [9-12]

2

vp

t

2

q

vp vp1 vp vp

1

S

∂

( S )

v

×

S(ln)SS( )

D

kf

Df

l f

∂

= −∇+∇+∇−∇+

qqq

(3)

where

m

=

v p /

is the particle velocity. Equation (3) shows that the time rate of change of

nonequilibrium entropy density (on the left-hand side)originates together from its drift in space

( the first term on the right-hand side), diffusion (the second term on the right-hand

side),increase(the third term on the right-hand side) and the entropy density change contributed by

interaction potential energy(the fourth term on the right-hand side)(the discussion on physical

meaning of this equation is remained in the end of this letter). Both entropy diffusion and entropy

increase come from the diffusion term, the second term on the right–hand side of the BBGKY

diffusion equation (2), whose microscopic mechanism is stochastic motion of the particle. The

entropy density change rate

( )

l f contributed by the interaction potential between microscopic

particles originates from the first term on the right-hand side of the BBGKY diffusion equation (2),

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which is the core of this paper.

In order to investigate ( )

l f and its countervailing with the law of entropy increase, at first let

us consider the formula for the law of entropy increase. Integrating the third term on the

right-hand side of entropy evolution equation (3) with respect to 6 dimensional phase space, we

obtained the formula for entropy increase rate of nonequilibrium system as follows[8-12]

2

2

1

f

11

10

( , )

( )

x

( ) ( , )

x

ln()0

ft

P tkD ftdkD

q

q

=∇=∇≥

∫

q

x

x

（4）

Where

10

10

10

1

10

1

1

ln ln

w

w

w

w

q

∆

−≈−==

f

f

can be defined as the percentage departure from

equilibrium of the number density of micro-states of the nonequilibrium system in 6 dimensional

phase space (

1

w and

10

w

for nonequilibrium and equilibrium states), or called the percentage

departure from equilibrium for brevity. It can play a role of an independent physical parameter to

describe quantitatively how far a nonequilibrium system is from equilibrium as if that the strain or

elongation percentage

0

lnl

l

∈=

describes the deformation of solid materials in solid mechanics.

As explained at the beginning of this paper, although people know long ago that the law of

entropy increase is a fundamental law in nature, however, what is its microscopic physical basis?

Which physical parameter and how does it change with? Can it be described by a quantitative

concise formula? This is all along an important problem to be solved in statistical thermodynamics

especially in nonequilibrium statistical physics. Formula (4) is the concise statistical formula for

entropy increase rate in 6 dimensional phase space derived by us. This is also the quantitative

concise statistical formula for the law of entropy increase. It shows that the entropy increase

rate ( )

P t equals to the product of diffusion coefficientD , the average value of the square of

space gradient of the percentage departure from equilibrium

2

()

q

q

Ñ

and Boltzmann constant k .

It can be seen that the macroscopic entropy increase in nonequilibrium system is caused by

spatially stochastic and inhomogeneous departure from the equilibrium of the number density of

micro-states. Obviously for nonequilibrium

) 0 (

1≠

q

, spatially inhomogeneous system

1

0

()

q ≠

q

Ñ

with stochastic diffusion

0

D ≠

, the entropy always increases( 0)

P >

. Conversely

for equilibrium system

) 0(

1=

q

or that nonequilibrium but spatially homogeneous system

1

0

()

q =

q

Ñ

or that system only with deterministic but no stochastic motion(0)

D ≠

, there all

are no entropy increase(0)

P =

.

3. Formula for entropy decrease rate

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Now let us discuss the core topic: a new entropy decrease can spontaneously emerge in

nonequilibrium system with internal attractive interaction. Before proving this proposition, let us

consider the relation among a famous kinetic equation — Boltzmann equation, Landau equation

and BBGKY diffusion equation (2). In fact the former two is a variety of the latter (when there is

no diffusion term) and is applied to describe the motion of a dilute and weakly coupled gas

system[14,15] with internal short-range repulsive interaction or collision. More concretely saying,

that is to change the first term with

21

( ,

x x

, )

tf

, the two particle interaction term, on the right

hand side of equation (2) into collision term only with

1( , )

f

x

t

of dilate and weakly coupled gas

particle. All other terms do not change. As a result, entropy of this nonequilibrium gas system

increases due to the existence of short-range repulsive interaction or collision. In other words,

entropy increase in the system described by Boltzmann equation and Landau equation originates

from short-range repulsive interaction or collision[14,15] between internal particles. This enlights

us：when the interaction force among internal particles is attractive but not repulsive, a new

entropy decrease should spontaneously emerge in nonequilibrium system. In view of that all

closed kinetic equations should be changed from the unclosed BBGKY equation (2), and up to

now we did not know what closed kinetic equation should be applied to describe a nonequilibrium

statistical system with internal attractive interaction, so the kinetic equation to describe galactic

dynamics is yet BBGKY equation [16]. This is the reason why our discussion is based on BBGKY

diffusion equation (2). Thus, our proof reduces to: when the internal two-particle interaction

potential

0

f <

, the time change rate of entropy

( )

R t contributed by it and described by the

fourth term on the right-hand side of equation(3) should be negative. That is ( )( )

R td0

l f

=∫

x <

.

After some operation, we obtained its mathematical expression

( )( )

R td

l f

=∫

x

1

f

1

f

201211

1010

( , )

( )

x

( , )

( )

x

() ( ,

x x

)( ,

x x

, ) 1 ln

t

ftft

Nkff d d

x x

f

=∇∇−∇+

∫

qpp

xx

×

=

20

( ,

x x

1

, )

t

1

f

1

f

211

1021 10

( ,

x x

) ( , )

( )

x

( , )

( )

x

( ,

x x

, )(

t

) ln

fftft

Nk fd d

x x

f

f

∇∇−∇

∫

qpp

xx

×

(5)

As mentioned above, since this expression is too complicated to be understood clearly its physical

meaning, previously we paid no attention to and wrote it as

vp

qJ

- Ñ ×

all along in

nonequilibrium entropy evolution equation(3). Now let us simplify expression (5) and investigate

its physical meaning. Similar to introducing

1 q into formula (4), here we also define

20

2

20

2

2

lnln

w

w

q

−==

f

f

as the percentage departure from equilibrium of nonequilibrium system

in 12 dimensional phase space. Substituting

1 q and

2

q into last line of expression (5) and through

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