Maximum entropy production principle in physics, chemistry and biology

Physics Reports 426(2006)1–45

https://www.doczj.com/doc/6e16433832.html,/locate/physrep

Maximum entropy production principle in physics,chemistry

and biology

L.M.Martyushev a ,b ,?,V .D.Seleznev b

a Institute of Industrial Ecology,Russian Academy of Sciences,S.Kovalevskoi 20A,Ekaterinburg,620219,Russia

b Ural State Technical University,Ekaterinburg,620002,Russia

Accepted 5December 2005

Available online 23January 2006

editor:H.Orland

Abstract

The tendency of the entropy to a maximum as an isolated system is relaxed to the equilibrium (the second law of thermodynamics)has been known since the mid-19th century.However,independent theoretical and applied studies,which suggested the maximization of the entropy production during nonequilibrium processes (the so-called maximum entropy production principle,MEPP),appeared in the 20th century.Publications on this topic were fragmented and different research teams,which were concerned with this principle,were unaware of studies performed by other scientists.As a result,the recognition and the use of MEPP by a wider circle of researchers were considerably delayed.The objectives of the present review consist in summation and analysis of studies dealing with MEPP.The ?rst part of the review is concerned with the thermodynamic and statistical basis of the principle (including the relationship of MEPP with the second law of thermodynamics and Prigogine’s principle).Various existing applications of the principle to analysis of nonequilibrium systems will be discussed in the second part.

?2005Elsevier B.V .All rights reserved.

PACS:05.70.Ln;05.20.?y;05.65.+b

Keywords:Maximum entropy production principle (MEPP);Ziegler’s and Prigogine’s principles;MEPP applications

Contents

0.Introduction ...........................................................................................................

21.MEPP in nonequilibrium thermodynamics .................................................................................

41.1.Fundamentals of linear nonequilibrium thermodynamics .................................................................

41.2.Critical review and development of Ziegler’s approach ..................................................................

61.2.1.Formulation of Ziegler’s principle .............................................................................

61.2.2.Some arguments for substantiation of Ziegler’s principle ..........................................................

91.2.3.Deduction of Onsager’s principle from Ziegler’s principle .........................................................

101.2.4.MEPP and the second law of thermodynamics ...................................................................

111.2.5.On the possible “paradox”of the use of the variational approach ....................................................

111.2.6.The relation of Ziegler’s maximum entropy production principle and Prigogine’s minimum entropy production principle ....

121.3.Brief conclusion (13)

?Corresponding author.Tel.:+73433493516;fax:+73433743771.E-mail address:mlm@ecko.uran.ru (L.M.Martyushev).

0370-1573/$-see front matter ?2005Elsevier B.V .All rights reserved.

doi:10.1016/j.physrep.2005.12.001

2L.M.Martyushev,V.D.Seleznev/Physics Reports426(2006)1–45

2.The MEPP in nonequilibrium statistical physics (13)

2.1.MEPP and kinetic theory of gas (13)

2.2.MEPP and general statistical theory of nonequilibrium processes(linear response theory,etc.) (16)

2.3.The most probable trajectory of the evolution and the information theory (20)

2.3.1.The method of the most probable path of the evolution (20)

2.3.2.Introduction to Jaynes’method (22)

2.3.3.The variational principle for the most probable state (23)

2.3.4.The second derivation of MEPP using Jaynes’method (24)

2.4.Brief conclusion (25)

3.The use of MEPP in different sciences (26)

3.1.MEPP in hydrodynamics.Transfer in the atmosphere and oceans (26)

3.1.1.Convective transfer.Paltridge’s method (26)

3.1.2.Statistical model of turbulence (30)

3.2.MEPP in a crystal growth and other solid state transformations (31)

3.3.MEPP and transfer of electrical charge,radiation,etc (36)

3.4.MEPP in chemistry and biology (37)

3.4.1.Berthelot and Shahparonov’s principles (37)

https://www.doczj.com/doc/6e16433832.html,ws of biological evolution (39)

4.Conclusion (41)

References (42)

He who gains time,gains everything.

G.B.Molière

0.Introduction

The notions of entropy and its production in equilibrium and nonequilibrium processes not only form the basis of modern thermodynamics and statistical physics,but have always been at the core of various ideological discussions concerned with the evolution of the world,the course of time,etc.These issues were raised by many outstanding scientists,including Clausius,Boltzmann,Gibbs and Onsager.As a result,today we have thousands of books,reviews and papers dedicated to properties of the entropy in different systems.The present review deals with the entropy production behavior in nonequilibrium processes.This topic is not new.What was the impetus to this study? Attempts to?nd some universal function,whose extremum would determine the development of a system,have been made at all times.A certain success was achieved in optics(Fermat’s principle),mechanics(the principle of least action), and some other disciplines.Entropy,which was assigned a semimystic signi?cance in the“world management”nearly at the moment it appeared,has been doomed to be a quantity describing the progress of nonequilibrium dissipative process.Great contribution has been done in this respect by two scientists,namely Clausius,who in1854–1862 introduced the notion of entropy in physics and advanced the concept of the thermal collapse of the Universe,and Prigogine.In1947the latter researcher proved the so-called minimum entropy production principle and then spent a number of years developing and popularizing the apparatus of nonequilibrium thermodynamics and his principle as applied to description of various nonequilibrium processes in physics,chemistry and biology.However,applications of this principle are relatively narrow as it was acknowledged by Prigogine himself and his opponents.Nevertheless, two essentially extreme opinions have been formed in the literature.Some scientists glorify the principle and think it is capable of describing various nonequilibrium processes to a certain extent.Other researchers,who observed weak points of the principle and unceasing efforts by Prigogine and his progeny to generalize it,are very skeptic about the possibility to formulate universal entropy principles,which would govern so diverse and dissimilar nonequilibrium processes.

The so-called maximum entropy production principle(MEPP)is known much less(even among specialists in physics of nonequilibrium processes).This antipode,as its name seemingly means,of Prigogine’s principle has been overshadowed by its more famous twin.MEPP was independently proposed and used by several scientists throughout the 20th century when they dealt with general theoretical issues of thermodynamics and statistical physics or solved speci?c

L.M.Martyushev,V.D.Seleznev/Physics Reports426(2006)1–453 problems.By this principle,a nonequilibrium system develops so as to maximize its entropy production under present constraints.The rigorous formulation,the origin and corollaries of this principle will be analyzed comprehensively in what follows.Here we shall only mention two principal features concerning the relationship between MEPP and the other two known statements about entropy.

1.The second law of thermodynamics,as it was worded by Clausius,states that for an arbitrary adiabatic process entropy(S)of the?nal state is larger than or equal to entropy of the initial state.In terms of the entropy production ( ),this means that 0.In this case,MEPP obviously represents new additional statement meaning that the entropy production is not just positive,but tends to a maximum.Thus,apart from the direction of the evolution,which follows from the Clausius’s formulation,we have information about the movement rate of a system.However,if we consider the statistical interpretation of entropy,which follows from studies by Boltzmann and Gibbs,entropy not only tends to increase,but will increase to a maximum value,which is allowed by imposed constraints.Here,the?nal(equilibrium) state is most probable and is described by a maximum number of microscopic states.On the strength of this statistical interpretation of the second law,MEPP may be viewed as the natural generalization of the Clausius–Boltzmann–Gibbs formulation of the second law and,in some cases,even as a corollary.Indeed,let us take an isolated system in some nonequilibrium state.The system will reach equilibrium with time(of the order of the relaxation time)and,among a great number of possible states,it will be in a state,for which entropy is a maximum.Therefore,the change of entropy within a given time interval will also be a maximum among possible values and,since the system is isolated, the entropy production will be the largest.This relationship between the second law of thermodynamics and MEPP was noted earlier(see,for example[1–3]).

2.The relationship between the minimum entropy production principle and MEPP is not simple.It has been the subject of long-standing discussions and will be considered in p.1.2.6.Here we shall note the following points.These variation principles are absolutely different.Although the extremum of one and the same function—the entropy production—is sought,these principles include different constraints and different variable parameters.These principles should not be mutually opposed since they are applicable to different stages of the evolution of a nonequilibrium system.It should be noted also that Prigogine[4](see,[5]too)stated more than once and adduced examples when a nonequilibrium system behaved oppositely to his principle of the minimum(Benard’s effect and structure instability during the biochemical evolution).He thought however that this situation is possible only in systems,which are far from the equilibrium.

It will be demonstrated in this review that it is MEPP,rather than Prigogine’s principle,can pretend to be a universal principle governing the evolution of nonequilibrium dissipative systems.The present review is the?rst generalizing treatise on this topic.One more factor,which adds signi?cance to this study,is that publications on this topic are fragmented and different research teams,which are concerned with this principle,are unaware of studies performed by other scientists.1Therefore,the recognition and a wider use of MEPP in different?elds of science have been considerably delayed.

In the?rst and second chapters we shall describe in suf?cient detail the available thermodynamic and statistical basis of this principle.The third chapter will deal with diverse applications of MEPP.The most interesting and signi?cant thing is that many researchers,who used MEPP,were unaware of general theoretical approaches,which are presented in the?rst two chapters.They were guided exclusively by the common sense and utility of the principle for settling of some speci?c problems existing in physics and interdisciplinary sciences.A feature in common of those studies was that MEPP proved to be the missing link,which allowed understanding the evolution of a nonequilibrium system.On account of the diversity of the reading matter and the unwillingness to change traditional well-established designations of quantities,we recommend to read each section attentively so as not to misunderstand a particular symbol.In closing, we shall present conclusions and formulate problems,which,in our opinion,require attention in later studies.

Since one of the objectives of this review was to attract attention of a wide circle of people working in various domains of science,we had to proceed,?rst and foremost,from a maximum accessibility and a relative simplicity of the presentation of the material.In some cases,therefore,the rigor is impaired and some details were excluded from the review.2In our opinion,such presentation of the material,which refers to different areas of knowledge,is most reasonable today.

Notice also that in this review we deliberately concentrated on evolution criteria for nonequilibrium systems,which are directly connected with entropy and MEPP.This is because we wanted to show that the idea about the relationship 1Important results were published not only in English,but also in other languages(Russian,German and French).

2These drawbacks can be eliminated if the reader refers to original papers,which are cited in the review.

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between entropy and the evolution of nonequilibrium systems,which has been known for over150years,still deserves attention and is very useful.We think also that MEPP is relatively simple,elegant and evident as compared to other plentiful hypotheses-criteria,most of which represent just some mathematical structures that are not obvious in physical terms and are not illustrative.

1.MEPP in nonequilibrium thermodynamics

The?rst part of this section presents necessary notions and some existing variational principles of nonequilibrium thermodynamics,including Prigogine’s principle.In the second part we shall consider Ziegler’s approach,which is based on the deductive formulation of nonequilibrium thermodynamics using MEPP.

1.1.Fundamentals of linear nonequilibrium thermodynamics

The?rst law of thermodynamics or the law of conservation of energy states that the heat Q,which is put into a system,is consumed for the work W done by the system over external bodies and the change of the internal energy of the system,d U[5,6]:

Q= W+d U.(1.1) The second law of thermodynamics,which has many wordings[5,6],states that any equilibrium thermodynamic system has a unique state function or entropy S:

T d S= Q,(1.2)

where T is the temperature.

As applied to nonequilibrium processes in an isolated system,the second law states that entropy cannot decrease. The entropy is one of main notions in thermodynamics and analysis of the behavior of this function or its derivatives is very important as it will be shown below in the text.

Most processes around us are nonequilibrium processes and a local equilibrium is assumed for the use of a relationship like(1.2).This assumption is valid if it is possible to distinguish two characteristic times,that is,the time required to reach the equilibrium in the entire system and the time required to reach the equilibrium in some volume,which is small as compared to the size of a system under study.The?rst time should be much longer than the second time.

A relationship like(1.2)can be applied to each of these small elements.The assumption on the local equilibrium is, of course,very strong,but it holds for many systems[5,6].

The variation of entropy with time t in a local unit volume can be written in the form[5–8]

j s

= ?div(j s),(1.3) j t

=

X i J i,(1.4) i

where s is the entropy density; is the entropy production density3(on the strength of second law of thermodynamics, the entropy production is always larger than or equal to zero);j s is the entropy?ux density,which additively depends on thermodynamic?ux densities J i;X i is thermodynamic forces(depending on particular problem conditions,the subscript i refers to different?uxes(forces)and vector components).

In1931–1932Onsager,who theoretically generalized empirical laws established by Fourier,Ohm,Fick and Navier, proposed a linear relationship between thermodynamic?uxes and forces[5–9]in the form

L ik X k,(1.5) J i=

k

L ik=L ki,(1.6) where L ik is the matrix of kinetic coef?cients,which are independent of J i and X k.

3Further in the text referred to as“the entropy production”for brevity.

L.M.Martyushev,V .D.Seleznev /Physics Reports 426(2006)1–455

The relationships (1.5)and (1.6)form the basis of the apparatus of linear nonequilibrium thermodynamics.They close the system of equations for the transfer of energy,momentum and mass,making it possible to solve the system.It should be emphasized that these expressions are valid only when thermodynamic forces are relatively small (in this case,a nearly linear relationship may be assumed between thermodynamic ?uxes and forces).However,the spectrum of problems,which can be solved in terms of this simple formalism,is suf?ciently wide.At the same time,examples are known when calculations by the linear relationships (1.5)and (1.6)may prove to be incorrect (speci?cally when chemical reactions are concerned).

In 1931Onsager proposed a variational principle,which can yield relationships (1.5)and (1.6)of linear nonequilib-rium thermodynamics [6,9].Thus,he gave the ?rst deductive formulation of linear nonequilibrium thermodynamics.This principle is worded as follows [9]:if values of irreversible forces X i are assigned,true ?uxes J i maximize the expression [ (X i ,J k )? (J i ,J k )],i.e.,

J [ (X i ,J k )? (J i ,J k )]X =0,

(1.7) (J i ,J k )=12 i,k R ik J i J k .(1.8)

Here and henceforth J [...]X denotes the variation of the expression in square brackets with respect to J at constant X , is the dissipative function (it is postulated that >0),and R ik is the coef?cient matrix (it will be shown later in the text that this matrix is inverse to the L ik matrix).

Let us deduce Eqs.(1.5)and (1.6)from Eqs.(1.7)and (1.8).

We shall show ?rst that the tensor R ik may be viewed as symmetric.Since an arbitrary tensor R ik can be presented at all times as the sum of symmetric S ik and antisymmetric A ik tensors,expression (1.8)takes the form

=12 i,k

(S ik J i J k +A ik J i J k ).(1.9)If we take the second term,rearrange its members,and consider that A ii =0and A ik =?A ki for the antisymmetric tensor,it is easy to see that the antisymmetric part of the tensor R ik does not contribute to the dissipative function.Therefore,the tensor R ik is symmetric (R ik =R ki ).

Substituting Eqs.(1.4)and (1.8)into Eq.(1.7)and replacing the variance by the derivative with respect to the corresponding ?uxes,we have j j J j ?? i X i J i ?12 i,k R ik J i J k ?

?X

=0.(1.10)The differentiation of Eq.(1.10)transforms it to

X j =12 k (R jk +R kj )J k = k R jk J k .(1.11)

Since is a positively de?ned quadratic form,the solution to the (1.11)for unknown J k is existent and unique:J j =

k R ?1jk X k ≡ k

L jk X k .(1.12)The designation R ?1jk ≡L jk has been introduced.Since R ik is a symmetric matrix,the matrix R ?1jk ,which is inverse

to it,is symmetric too.

Thus,it was shown that relationships (1.5),(1.6)can be obtained from the variational formulation (1.7),(1.8).It was demonstrated in the foregoing that the expression (X i ,J k )? (J i ,J k )has the only extremum point Eq.(1.11)or Eq.(1.12).Since is a homogeneous quadratic positively de?ned function of ?uxes,the obtained point is the maximum point.

Onsager’s variational principle was formulated in the space of thermodynamic ?uxes.An alternative formulation of this principle in the force space was proposed by Gyarmati [9,10].His formulation was worded as follows:if values of thermodynamic ?uxes J i are assigned,actual irreversible forces X i maximize the expression (X i ,J i )?Y (X i ,X k ),

6L.M.Martyushev,V.D.Seleznev/Physics Reports426(2006)1–45

i.e.

X[ (X i,J i)?Y(X i,X k)]J=0,(1.13)

Y(X i,X k)=1

2

i,k

L ik X i X k,(1.14)

where Y(X i,X k)>0is the dissipative function in the force representation.

Since entropy production is a symmetric bilinear form,then,as Gyarmati demonstrated,the formulations(1.7),(1.8) and(1.13),(1.14)are equivalent.

Let us turn to Prigogine’s principle(theorem)(or the minimum entropy production principle),which is formulated as follows[8].

Let Eqs.(1.5)and(1.6)be ful?lled in a system,irreversible forces X i(i=1,...,j;j

This principle can be proved rather easily[8].It is necessary to substitute the expressions(1.5)and(1.6)into Eq.(1.4)and differentiate it with respect to un?xed forces.The obtained expressions are equal to within constants to ?uxes having the numbers j+1,...,n and,therefore,are zero,because entropy production is a minimum under the theorem condition.

The corollary of this statement is that Prigogine’s system is stationary.

A slightly different formulation of Prigogine’s principle is frequently considered and proved[6,10].It states that in the stationary nonequilibrium state,which is consistent with external restrictions(constant irreversible forces X i, where i=1,...,j;j

It is seen from the above reasoning that Prigogine’s theorem is a simple corollary of Onsager–Gyarmati’s principle, because this principle is used to deduce the relationships(1.11)and(1.12).Prigogine’s result is obtained from these relationships using additional restrictions.

It should be noted that other variational formulations of linear nonequilibrium thermodynamics are available(see, for example,[6,10,11]),which are equivalent to some extent to the aforementioned formulations.

The apparatus of linear nonequilibrium thermodynamics is used extensively for the following reasons:

(1)The relationship(1.5)allow solving a system of equations for the transfer of mass,momentum and energy, because in this case the number of equations is equal to the number of unknowns.

(2)It is possible to describe cross-coupling?uxes(by means of off-diagonal coef?cients L ik)of chemical,electrical and other kinetic processes,which are characteristic of a wide range of physical,chemical and biological systems. (3)Additional information about values of kinetic coef?cients can be obtained using,for example,the circumstance that entropy production is a positively de?ned quadratic form,or reciprocal relation(1.6).

(4)Quantities(for example,entropy production),which have extremum values in the nonequilibrium state,also provide additional information about a system.

Thus,linear nonequilibrium thermodynamics is very useful within its applications.It should be noted that linear nonequilibrium thermodynamics described the majority of engineering problems encountered in the?rst half of the 20th century and,therefore,the progress and the popularization of nonlinear equilibrium thermodynamics were very slow and were viewed by many researchers as a theoretical whim.As it is known,for a system to be described in terms of linear nonequilibrium thermodynamics,thermodynamic forces should be small.However,sometimes this condition proves to be too rough,speci?cally,with respect to chemical reactions.As a result,linear nonequilibrium thermodynamics cannot explain and describe many,and often principal,problems encountered today.They include self-organization,oscillation processes,etc.

The possible generalization of Onsager’s thermodynamics(linear thermodynamics)to a nonlinear case using the maximum entropy production principle will be discussed in the section below.

1.2.Critical review and development of Ziegler’s approach

1.2.1.Formulation of Ziegler’s principle

Let us work in the?ux space{J k}and assume that we know the form of expression(1.4)for entropy production in this space.Our main task is to?nd X k as a function of{J k}.A local equilibrium is assumed for the system.

L.M.Martyushev,V .D.Seleznev /Physics Reports 426(2006)1–457

s

s Fig.1.1.Variation of entropy s with time t for two possible trajectories of the development (two thermodynamic ?uxes at a preset force are possible).In this case,expression (1.4)transforms to

(J i )=

k X k (J i )J k .(1.15)

To ?nd X k (J k )in the explicit form,Ziegler [1,12–15]proposed the maximum entropy production principle:If irreversible force X i is prescribed ,the actual ?ux J i ,which satis?es the condition (J i )= i X i J i ,maximizes the entropy production (1.15).

A similar principle originally appeared and was widely used in the theory of plasticity where it is called the principle of the maximum dissipation rate of mechanical energy or Mises’s principle [12,16,17].The wording of this principle is that the dissipation rate of mechanical energy in a unit volume during plastic deformation is a maximum in a truly stressed state among all stressed states allowed by the given condition of plasticity (the deformation rate is assumed to be constant)[17].Ziegler essentially extended this principle of the theory of plasticity to all nonequilibrium thermodynamics [1,13].

In isolated systems d s/d t = and the maximum entropy production principle re?ects the fact that an isolated system tends to the state with maximum entropy along the shortest possible path (by the fastest means).Fig.1.1gives an example illustrating the possible behavior of a system.

The maximum entropy production principle can be presented mathematically as

J (J k )?

(J k )?

i X i J i X =0,(1.16)

where is a Lagrangean multiplier.It was already mentioned in the foregoing that the subscripts may denote either different thermodynamic forces (?uxes)or their spatial components.

This principle can be also formulated in the force space when entropy production depends only on X k and ?uxes are prescribed,i.e.analogs of expressions (1.15),(1.16)are written simply substituting J for X .

Expression (1.16)is interpreted geometrically as described below.Entropy production in the ?ux space represents some surface (J k ).This surface is intersected by the plane i X i J i and ?uxes,which maximize (J k ),are chosen on their intersection line.The simplest form of entropy production is shown in Fig.1.2for two thermodynamic ?https://www.doczj.com/doc/6e16433832.html,ing the formulated principle,we ?nd the explicit expression for the thermodynamic force.For this purpose,transform Eq.(1.16):

?????????j j J i (J k )? (J k )? i X i J i X, =0,j j (J k )? (J k )? i X i J i X,J

=0.(1.17)

8L.M.Martyushev,V.D.Seleznev/Physics Reports426(2006)1–45

J1

Fig.1.2.Simplest dissipative surface during a linear irreversible process.Here J1and J2are two thermodynamic?uxes and0denotes the point corresponding to the equilibrium state.

Since forces are assumed to be assigned,Eq.(1.17)may be written in the form

???j (J k)

j J i

?

j (J k)

j J i

?X i

=0,

(J i)=

i

X i J i.

(1.18)

Let us denote =( ?1)/ and rearrange the?rst expression in Eq.(1.18)as

X i= j /j J i,(1.19) where the proportionality coef?cient is obtained using the second expression in Eq.(1.18):

=

i

j

j J i J i

?1

.(1.20)

Expressions(1.19)and(1.20)are known as orthogonality conditions,because their geometrical meaning is that the thermodynamic force X i,which corresponds to the?ux J i,is orthogonal to the surface (J k)=const(the line A in Fig.1.2).The analog of the orthogonality condition in the theory of the plasticity is called the associated law[17]. The relationship between?uxes and forces,which is given by Eqs.(1.19)and(1.20),is not unique.In a particular case,when Eqs.(1.19)and(1.20)have a unique solution(only one vector J corresponds to a preset vector X),the orthogonality condition determines only one extremum point.One can prove that this point corresponds to the maximum of entropy production[1,13].However,this is clear from geometrical considerations4too( >0and,if one takes into account the imposed restrictions,entropy production cannot be in?nitely large;it is reasonable to expect therefore that the only extremum point corresponds just to the maximum).In this particular case,the maximum entropy production principle and the orthogonality condition are equivalent.In the general case,the orthogonality condition obviously determines all extremum points in the corresponding area of the surface (J k),whereas the variational principle selects the point,at which the entropy production has the largest value.

4See,for example,Fig.1.2.

L.M.Martyushev,V .D.Seleznev /Physics Reports 426(2006)1–459

Although the maximum entropy production principle has been postulated,we shall adduce some arguments for its substantiation.

1.2.2.Some arguments for substantiation of Ziegler’s principle

Let entropy production be in the mathematical sense a suf?ciently good function of ?uxes Eq.(1.15).We shall expand it into a series and restrict ourselves to linear expansion terms:5

= i

j i J i .On the other hand,if thermodynamic forces are assumed to be unchanged during variation,then,according to Eq.(1.4),the change of entropy production can be written as =

i

X i J i .In accordance with the last two expressions,forces can be de?ned as

X i = j /j J i ,

where the constant can be determined,as before,using Eq.(1.4).

The orthogonality condition can be substantiated more rigorously for a narrower class of functions determining entropy production.Let entropy production (J i )be a homogeneous function of degree r .Then,according to Euler’s theorem (see,for example,[18]),the expression

r = i

j j J i J i or

=1r i j j J i J i (1.21)

holds.From the comparison of the last expression and (J i )= i X i J i it is reasonable and the simplest to take (de?ne)thermodynamic forces in the form 6

X i =1r j j J i .(1.22)

It is seen that Eqs.(1.21)and (1.22)coincide with Eqs.(1.19)and (1.20),but it should be noted that in this case is equal to constant 1/r .

Let us demonstrate the transition from the orthogonality condition (1.22)to the variational principle (1.16).For this purpose,we shall rearrange the last expression to the form j j J i r i X i J i ? (J k ) X,r

=0.(1.23)Let r = /( ?1),where is some number.Then Eqs.(1.23)and (1.4)can be written as ???????j j J i (J k )? (J k )?

i X i J i X, =0,j j (J k )? (J k )? i X i J i X,J

=0.(1.24)

5This is possible if the described nonequilibrium system is not too far from equilibrium.6In nonequilibrium thermodynamics,the choice of thermodynamic forces and ?uxes is not unique (for details see Section 1.2.5).

10L.M.Martyushev,V .D.Seleznev /Physics Reports 426(2006)1–45

Expression (1.24)obviously coincides with Eq.(1.16),i.e.the variational principle can be deduced from the orthog-onality condition.

Let us consider the simplest type of entropy production as a homogeneous function of degree 2: = i,k

R ik J i J k .(1.25)

It was shown in the foregoing (see Eqs.(1.8),(1.9))that the tensor in is symmetric,R ik =R ki .Using the orthogonality relationship (1.22),we shall obtain the expression for forces,

X k =12 i (R ik +R ki )J i = i

R ki J i .(1.26)Thus,Eq.(1.26)coincides with Eq.(1.11),i.e.entropy production of the form Eq.(1.25)corresponds,in accordance with Ziegler’s principle,to the case of Onsager’s linear thermodynamics Eqs.(1.5),(1.6).

So,main relationships of linear nonequilibrium thermodynamics can be deduced from Ziegler’s principle if is selected as a homogeneous function of degree 2.

Let us note a remarkable feature.In the case at hand,entropy production is a homogeneous function and,as it was demonstrated in the foregoing,the orthogonality condition can be proved rigorously enough using properties of homogeneous functions (rather than deduced from Ziegler’s postulate).Since Eq.(1.26)de?nes a unique set of thermodynamic ?uxes,the orthogonality condition and the maximum entropy production principle are equivalent.We have demonstrated therefore that if forces are prescribed,the entropy production is always maximized in a system with nonequilibrium processes described by linear nonequilibrium thermodynamics.

It should be specially emphasized in conclusion that the rigorous substantiation of Ziegler’s principle using relatively simple intuitive assumptions and just in terms of thermodynamic representations seems to us very dif?cult or impossible.

1.2.3.Deduction of Onsager’s principle from Ziegler’s principle

Let us demonstrate that Onsager’s principle can be obtained from Zirgler’s principle in the linear case.It was shown in the foregoing that in this case entropy production is a quadratic function of ?uxes (Eq.(1.25)).Substituting Eq.(1.25)into Eq.(1.17),rearrange Ziegler’s principle to the form ???????j j J i i,k R ik J i J k ? i,k R ik J i J k ? i X i J i X, =0, i,k R ik J i J k = i

X i J i .

(1.27)The ?rst relationship (1.27)gives

X i =2( ?1) k R ik J k .(1.28)

Substituting Eq.(1.28)into the second relationship (1.27),we have 2( ?1)/ =1,i.e. =2.Therefore,Eq.(1.27)can be written as J ?? i,k R ik J i J k ?2?? i,k R ik J i J k ? i X i J i ????X

=0.(1.29)

Collecting terms and considering Eq.(1.4),recast Eq.(1.29)as J ?? (X i ,J k )?12 i,k R ik J i J k ?

?X =0.(1.30)

Here the second term coincides with the dissipative function in the ?ux representation (Eq.(1.8)):

J [ (X i ,J k )? (J i ,J k )]X =0.(1.31)

L.M.Martyushev,V.D.Seleznev/Physics Reports426(2006)1–4511 Thus,Onsager’s variational principle can be obtained from Ziegler’s principle.

1.2.4.MEPP and the second law of thermodynamics

The relationship between MEPP and the second law of thermodynamics,which is supplemented with statistical arguments due to Boltzmann and Gibbs,has been already mentioned in the Introduction.We shall only consider if the second law can be deduced from MEPP.In doing so,we shall use exclusively the thermodynamic interpretation of the entropy and its production.

The second law of thermodynamics was used to prove that entropy production,which was determined from orthog-onality conditions,was not a minimum,but a maximum(Section1.2.1).However,if the maximum entropy production principle is postulated ab initio,the second law of thermodynamics can be obtained as a corollary.

We shall assume in line with[13]that entropy production is convex as a function of?uxes and the maximum principle must provide the biunique correspondence between?uxes and forces.The geometrical interpretation of the principle (see Section1.2.1and Fig.1.2)will be used.Since entropy production is a convex function and tends to zero as?uxes tend to zero,all the surface (J i)will be located above or below the plane (J i)=0,i.e.the sign of (J i)is?xed.Let us assume that (J i) 0.If forces have arbitrary values,the line of intersection between the surface (J i)and the plane

i X i J i will be located in the negative region and the entropy production maximum corresponding to this line will be

zero.However,the unique correspondence between?uxes and forces is disturbed at this point(a zero?ux corresponds to any force).Thus,the assumption on the negative value of entropy production leads to contradiction.It is proved therefore that (J i)>0.

It is remarkable that even if convexity and uniqueness,which were used by Ziegler,are not assumed,the second law of thermodynamics can be obtained as a corollary of MEPP.Indeed,let the entropy production in some imaginary system take the negative values at some preset force.Then,on the strength of the postulated principle,the physically realizable?ux will be such that entropy production is the largest,i.e.entropy production in the system will be equal to a maximum positive value out of those possible.If it is assumed that the system cannot?nd any?ux satisfying the preset force such that entropy production is greater than zero,then in any case the system has the variant to take the?ux equal to zero,since this value is a maximum in this extravagant example.Therefore,from the maximum entropy production principle it follows that physically realizable states with negative entropy production are impossible whatsoever.

1.2.5.On the possible“paradox”of the use of the variational approach

Some type of entropy production is postulated in the variational construction of nonequilibrium thermodynamics. However,?uxes and forces are chosen arbitrarily to a certain extent.We shall illustrate this arbitrariness for the case when two forces X1and X2,which are known functions of?uxes J1and J2,are present in a system.

By the de?nition(Eq.(1.15)),entropy production in this system is written as

(J1,J2)=X1(J1,J2)J1+X2(J1,J2)J2.(1.32) Let us determine the forces using the orthogonality condition(1.19),(1.20)and designate them as X?1and X?2: X?k= j /j J k;k=1,2.(1.33)

=

2

k=1

j

j J k J k

?1

.(1.34)

Rearranging Eqs.(1.33),(1.34)and taking into account Eq.(1.32),we have correspondingly X?1=X1+ /J1,(1.35) X?2=X2? /J2,(1.36)

=? J1J2

X1J1

j X1

j J2

?X2J2

j X2

j J1

+X1J2

j X2

j J2

?X2J1

j X1

j J1

.(1.37)

The forces X1and X2in expression(1.32)and the forces X?1and X?2,which are determined from the orthogonality condition,differ by (the discrepancy)divided by the corresponding?ux.It is seen that both sets of forces satisfy the entropy production(1.32).

12L.M.Martyushev,V.D.Seleznev/Physics Reports426(2006)1–45

If the discrepancy is equated to zero,it is possible to determine the condition when the set of forces is uniquely determined by known entropy production:

X2(J1,J2)j (J1,J2)

j J1

=X1(J1,J2)

j (J1,J2)

j J2.(1.38)

Eq.(1.38)de?nes the class of entropy production functions when forces,which are determined using Eq.(1.32)and orthogonality conditions,coincide.The search for and analysis of these functions represent a separate very interesting problem.We shall only note that for the quadratic function of entropy production(1.25)condition(1.38)is ful?lled if Onsager’s reciprocity relations hold.

This feature was also noted in[19].The conclusion,which is drawn herein,is not a drawback or even a refutation of Ziegler’s approach.It is rather a speci?c feature of all nonequilibrium thermodynamics,which ab initio is based on the equation of balance of entropy,energy,momentum and matter,and also on the?rst two laws of thermodynamics.

1.2.6.The relation of Ziegler’s maximum entropy production principle and Prigogine’s minimum entropy production principle

If one casts a glance at the heading,he may think that the two principles are absolutely contradictory.This is not the case.It follows from the above discussion that both linear and nonlinear thermodynamics can be constructed deductively using Ziegler’s principle.This principle yields,as a particular case(Section1.2.3),Onsager’s variational principle,which holds only for linear nonequilibrium thermodynamics.Prigogine’s minimum entropy production principle(see Section1.1)follows already from Onsager–Gyarmati’s principle as a particular statement,which is valid for stationary processes in the presence of free forces.Thus,applicability of Prigogine’s principle is much narrower than applicability of Ziegler’s principle.7

These differences can be also explained in terms of a less formalized language.Let us consider a system with known entropy production as a function of?uxes(forces).If thermodynamic forces(?uxes)are assigned,the system will adjust,in accordance with Ziegler’s principle,its thermodynamic?uxes(forces)such that entropy production is a maximum.If entropy production is a quadratic function,the relation between?uxes and forces takes the form(1.5)as a result of this adjustment(Section1.2.2).Further,if the system acquires a stationary weakly nonequilibrium state,but some thermodynamic forces remain free,?uxes,which are formed under Ziegler’s principle(relationships(1.5)),will decrease thermodynamic forces and these forces will decrease,in turn,?uxes to a minimum of entropy production. Thus,some hierarchy of the processes is observed.If the time is short,the system maximizes entropy production at preset?xed forces at a given moment.As a result,linear relationships(1.5)are valid.If the time is long,the system changes free thermodynamic forces so as to decrease entropy production.

Let us dwell on one more important issue relevant to this problem.What is the rate at which a nonequilibrium system tends to its?nal state?As it was discussed in Section1.2.1(Fig.1.1),the maximum entropy production principle suggests that a system tends to its state with maximum entropy at the largest rate.At each moment of time the system chooses its?ux at?xed forces such that the change of entropy is the largest and,hence,the tendency to the?nal state is the fastest.This adjustment of parameters may be continuous or stepwise(at bifurcation points)depending on speci?c features of the system.In the last case,one force may correspond simultaneously to several?uxes and the?ux satisfying the principle is chosen out of these?uxes.This circumstance does not contradict the principle in mathematical terms, because the relation between?uxes and forces is ambiguous in the general case(see Eqs.(1.19),(1.20)).How the choice is made from different states satisfying Eqs.(1.19),(1.20)requires more studies.8Notice that the consequence of this ambiguity of?uxes and forces was not considered by Ziegler(who only discussed a biunique case).

7Considering what has been said above,MEPP is applicable to both linear and nonlinear systems.The popular opinion(see,for example,[20]) that Prigogine’s principle is valid for linear systems and MEPP holds for nonlinear systems is wrong.Of course,the application of MEPP to nonlinear systems presents the greatest interest.

8Studies by Sawada(1981–1984)[3,21–23]present interest in this respect.He independently advanced a hypothesis that the most stable state against perturbation among the possible(metastable)states should correspond to the state of the maximum entropy production.To support his postulate,Sawada reported results of computation of the dissipative structure in charged-?uid layer systems(electroconvection),a nonlinear chemical reaction system(Brusselator),and a growing random pattern.Similar conclusions were drawn by Shimokawa et al.[24],who made calculations using the oceanic general-circulation model.The use of MEPP for selection of the most stable state among several possible states in nonequilibrium processes will be discussed in Section3.2.

L.M.Martyushev,V.D.Seleznev/Physics Reports426(2006)1–4513 1.3.Brief conclusion

When Ziegler formulated his principle,he aimed mainly at the deductive foundation of nonequilibrium thermody-namics,which would hold in both linear and nonlinear cases.The main idea of his approach was that the nature disposes of its freedom at preset thermodynamic forces and the dissipation law so as to maximize the entropy production.The principal result is that he obtained,as a corollary of the principle,all results,which were known for the linear case,and the second law of thermodynamics.

It should be acknowledged that Ziegler presented his material in a formalized theoretical manner and applications of his principle were exempli?ed by some problems of the theory of plasticity and chemical kinetics[1,13–15],which could be solved by alternative methods.This factor,imperfect understanding of nonlinear phenomena,and other variational formulations in thermodynamics,which are valid for nonlinear cases(for example,Biot’s principle[10,11]),had the result that Ziegler’s principle was overlooked and did not receive wide acceptance.However,as compared to other variational formulations of thermodynamics,which are applicable to the nonlinear region,Ziegler’s formulation is most evident and simplest in our opinion.This is con?rmed by the fact that many researchers intuitively advanced similar statements both before and after Ziegler’s publications.However,their formulations were not so complete and general as in studies by Ziegler and represented just statements,which could help solve a speci?c problem.The principles themselves were formulated proceeding from the common sense and practicability for solution of particular problems, rather than the general logic of formation of nonequilibrium thermodynamics(for more details see Section3).

The thermodynamic discussion of MEPP raises the question as to how this principle shows itself at the microscopic level.The second section of the present study deals with this question.

2.The MEPP in nonequilibrium statistical physics

In this section we shall consider MEPP from the viewpoint of different directions in modern nonequilibrium statistical physics,such as the kinetic theory of gases,the theory of random processes,the linear response theory,etc.

2.1.MEPP and kinetic theory of gas

The kinetic theory of gases deals with determination of the one-particle velocity distribution function and calculation of all necessary characteristics of a system(kinetic coef?cients,?uxes,etc.)using this function.We shall brie?y recall this theory foundation.

Let us consider a suf?ciently rare?ed monatomic gas in a nonequilibrium state,whose properties can be expressed as the one-particle distribution function f(r,c,t),where r is the particle radius-vector,c is the particle velocity vector, and t is the time.The full change of f(r,c,t)with time can be written as[25–28]

d f d t =

j f

j t

+

c

j f

j r

+1

m

F

j f

j c

=I(f),(2.1)

where(c j f/j r)+1/m(F j f/j c)is the change of the distribution function,which is caused by collisionless motion of particles in the?eld of external forces(m being the molecule mass and F the external force acting on molecules);I(f) is the collision integral characterizing the change of the distribution function resulting from collisions of molecules. If the mean free path of particles is much larger than the effective radius of intermolecular forces,then,on the one hand,only binary collisions may be considered and,on the other hand,the hypothesis of molecular chaos can be used.If it is assumed further that collisions represent instantaneous events occurring at a single point in the space,the collision integral can be written in the form[25–28]

I(f)=

f f ??ff?

gb d b d d c?,(2.2)

where f=f(r,c,t)and f?=f(r,c?,t)are distribution functions of particles having velocities c and c?before the collision;f =f(r,c ,t)and f ?=f(r,c ?,t)are distribution functions of particles having velocities c and c ?after the collision(these velocities can be expressed as velocities of particles before the collision in terms of the collision

14L.M.Martyushev,V .D.Seleznev /Physics Reports 426(2006)1–45

law);g =|c ?c ?|is the relative velocity of colliding molecules.The integration in Eq.(2.2)is taken over the impact parameter b ,the azimuth angle ,and all possible velocities of one of the colliding molecules.

Eq.(2.1)and the collision integral (2.2)are called the Boltzmann equation.

If the gas approaches the locally equilibrium state and the ratio between the mean free path and the characteristic size of the system is small,Eq.(2.2)is often simpli?ed [25–27].It is assumed that in this case the sought-for distribution function is nearly equal to the Maxwell local equilibrium function f 0:

f =f 0(1+ (c )),

| |>1,(2.3)where f 0=n m 2 kT 3/2exp ?mC 2

2kT

is a locally equilibrium distribution function of particles having the mass m ,the temperature T (r ,t),the peculiar velocity C(r ,t),and the number density n(r ,t).The function (c )should meet the normalization conditions,which follow from the assumption on the local equilibrium.As a result,this function should not contribute to the density,the mean velocity and the temperature:

f 0 d c = f 0 C d c = f 0 mC 2

2

d c =0,(2.4)wher

e C =c ?u ,u is the local hydrodynamic velocity.

In this case,using Eqs.(2.3)and (2.4),the collision integral (2.2)can be written in the form [25–27]I (f )= f 0f 0?( + ?? ? ?)gb d b d d c ?=?? ,(2.5)

where designations of the functions are similar to those of f above.The integral operator ? is introduced for compactness of subsequent https://www.doczj.com/doc/6e16433832.html,ing Eq.(2.5),it is possible to obtain the following properties of the operator ? [25–27,29]:(1)the operator ?

is linear,i.e.? ( F + G)= ? F + ? G ,(2.6)where and are some numbers,while F and G are some functions of the same variables as is.(2)the operator ? is self-adjoint,i.e. F ? G d c = G ? F d c .Introducing the so-called integral brackets as F ?

G d c ≡[F,? G ],the reduced property can be recast to the form [F,?

G ]=[G,? F ].(2.7)(3)[G,?

G ] 0.(2.8)

Eq.(2.1)with the collision integral (2.5)can be written as

Z =?? ,(2.9)where the symbol Z denotes the left-hand side of the linearized Eq.(2.1).

Multiply the left-and right-hand sides of Eq.(2.9)by and integrate it over c .Then,using the designations introduced above,we can write

[ ,Z ]=?[ ,? ].(2.10)

Let us prove the following variational principle (see,for example,[25–27,29–33]).Amongst all the functions satis-fying the condition (2.10),the function ,which maximizes [ ,?

],is the solution to Eq.(2.9).Proof.Let there be another function Y ,which is not a solution to Eq.(2.9),but satis?es condition (2.10):

?[Y,? Y ]=[Y,Z ].(2.11)

L.M.Martyushev,V .D.Seleznev /Physics Reports 426(2006)1–4515

Then,considering Eq.(2.8),one can write for the function ?Y :

[ ?Y,?

( ?Y )] 0.(2.12)

Since the operator ?

is linear and self-adjoint (Eqs.(2.6),(2.7)),then Eq.(2.12)can be transformed to [ ,?

]+[Y,? Y ]?[ ,? Y ]?[Y,? ]=[ ,? ]+[Y,? Y ]?2[Y,? ] 0(2.13)Using Eqs.(2.9)and (2.11),inequality (2.13)takes the form

[ ,?

]+[Y,? Y ]+2[Y,Z ]=[ ,? ]+[Y,? Y ]?2[Y,? Y ]=[ ,? ]?[Y,? Y ] 0(2.14)Thus,it is proved that the inequality

[ ,? ] [Y,? Y ](2.15)holds for any trial function Y ,which satis?es condition (2.10).

The proved variational principle is widely used to solve the linearized Boltzmann equation [25–27].The calculation algorithm is as follows:

1.Introduce a trial function Y constructed of known functions i ,which include some number of arbitrary variation parameters a i :

Y =

i

a i i .(2.16)Hermite polynomials,etc.are chosen as i .However,associated Legendre polynomials (or Sonine polynomi-als)are most convenient since they provide the best ef?ciency from the viewpoint of the algebraic manipulation simplicity [27].2.Substituting Eq.(2.16)into the functionals [Y,? Y ]and [Y,Z ]and ?nding the extremum of [Y,? Y ]subject to condition (2.10),we obtain a i values.

3.Considering the proved variational principle,the function Y ,which is determined by this means,approaches best the true solution of the Boltzmann equation.

The solution of the linearized Boltzmann equation,which was obtained by this method,can be used for calculation of kinetic coef?cients by standard methods [25–27].

The variational principle in the above-discussed form represents just some mathematical tool for the approximate solution of the linearized Boltzmann equation.Since this equation can be solved by other methods as well,one may get an impression of the secondary importance and physical insigni?cance of this principle.But this is not the case

[29–31].Let us multiply Eq.(2.1),which includes the linearized collision integral (2.5),by ?k ·ln f and integrate over the whole space of velocities.Then we have

j s j t

+div j s =k [ ,? ],(2.17)where s =?k f ln f d c is the local density of the entropy,j s =?k c f ln f d c is the entropy ?ow density,and k is

the Boltzmann constant.

If we use the nomenclature adopted in nonequilibrium thermodynamics,the right side of Eq.(2.17)may be called the density of the local entropy production ,i.e.

=k [ ,? ].(2.18)

Thus,the above principle can be worded as follows [26,29–32]:the velocity distribution function for nonequilibrium gas systems is such that the entropy production density is a maximum at preset gradients of the temperature ,the concentration and the mean velocity .

This principle can be formulated in terms of the thermodynamic formalism when the entropy production (J i )equals k X k J k [29–32].Since the temperature,the concentration,etc.are preset at each point of the system,the

16L.M.Martyushev,V.D.Seleznev/Physics Reports426(2006)1–45

corresponding thermodynamic forces X k(gradients of the temperature,the concentration,etc.)are known.Therefore, the entropy production(2.18)is maximized at constant forces.Then the thermodynamic formulation of the principle is

,

( )+

( )?

k

X k J k( )

X

=0,(2.19)

where the Lagrange multiplier is introduced to take into account the additional restriction on the sought-for function in accordance with Eq.(2.10).

It is seen that the mathematical notation of this principle formally is analogous to Ziegler’s principle(see the previous chapter),but the variation in Eq.(2.19)actually is not over?uxes J k,as it is in Ziegler’s principle,but over the distribution function.

Let us note one corollary of Eq.(2.19).It is known that if forces are?xed,the entropy production in the linear approximation proves to be only a function of the kinetic coef?cients L ik: =L ik X i X k.Therefore,by?nd-ing the velocity distribution function,which maximizes the entropy production,we actually maximize the kinetic coef?cients L ik.

Enskog[32]and Hellund and Uehling[33]were the?rst to propose and use the variational principle for the solution of the Boltzmann equation.In1948Kohler[31]formulated another principle,which is similar to the principle discussed above,but contains slightly different additional conditions,which,in his opinion,are more illustrative from the physical viewpoint.According to his calculations,both principles yielded the same results for the kinetic coef?cients.Kohler probably was among those who?rst used the variational approach for description of the transport of electrons in metals (a study by Sondheimer[34]should be mentioned too).Ziman[30]was the?rst to give a thermodynamic interpretation of the variational principle and assign it the status of a physical law,rather than just a mathematical method for the solution of the Boltzmann equation.He voiced an idea that Boltzmann’s H-theorem represented a kind of a proof(or demonstration)of the second law of thermodynamics.By analogy,the variational theorem under consideration points to some additional rather general statement about the behavior of the entropy production in nonequilibrium systems (the maximum entropy production principle).He also pointed out that similar,but more particular,propositions were made by Rayleigh[35]for mechanical systems and by Jeans[36]for electrical systems.

Today this variational principle is viewed as one of the main ef?cient methods for solving the Boltzmann equa-tion not only in classical gas systems[25–27],but also for the study of the electron and phonon transport in solids [29–31,37–39].9Different generalizations of this principle(beyond the region of linear response,etc.)are available, but they are not used widely since they represent some mathematical structures and are less intuitively understood in physical terms.10The discussion of those principles is beyond the scope of the present review.

We conclude this section by emphasizing the following point.The kinetic approach and,speci?cally,results of analysis of the Boltzmann equation represent the basis for the microscopic argumentation of the second law of thermodynamics. However,as it was shown above,this equation has one more feature,which received little attention.The solution to the Boltzmann equation obeys MEPP—a thermodynamic principle,which was introduced independently in the most complete form by Ziegler.One may expect therefore that MEPP is not just a“freak of mind”of theoreticians,who need it for a generalized formulation of thermodynamics or the variational solution to the Boltzmann equation,but is

a regularity inherent in the nature.

2.2.MEPP and general statistical theory of nonequilibrium processes(linear response theory,etc.)

The classical kinetic theory,which is brie?y described in the foregoing,proves to be inapplicable to relatively dense systems with a strong interparticle interaction.Therefore,it is necessary to develop a nonequilibrium microscopic theory capable of describing such systems.One of the main tasks of this theory consists in deduction of equations for the transfer of energy,momentum,mass,etc.and calculation of kinetic coef?cients for various systems(gases,

9The authors hold to the opinion that the further expansion of applications of this approach(for example,solution of the Boltzmann equation taking into account the gas–surface interaction)is interesting and fruitful[40].

10By way of example,the reader is referred to studies by Kikuchi[41]and Blount[42].The last author also holds to the opinion that the relationship between the entropy production and the variational principle in the solution to the kinetic equation in the general case is not suf?ciently close to justify Ziman’s interpretation.

L.M.Martyushev,V.D.Seleznev/Physics Reports426(2006)1–4517 liquids,solids)directly from equations of classical and quantum mechanics.Such statistical theory has been developing vigorously starting from the middle of the20th century(see reviews[43–47]).

Onsager may be reputed as one of the initiators of this general approach.He made the following statement:the time evolution of the?uctuation of a given physical value in an equilibrium system obeys the same laws on the average as the change of the corresponding macroscopic variable in a nonequilibrium system[47–51].The essence of this assumption is that,being in a nonequilibrium state,a system“does not know”how it got to this state:thanks to a ?uctuation or an external effect.Therefore,its subsequent response should be the same.As a consequence,relaxation of the nonequilibrium system near the equilibrium and dissipation of?uctuations obey the same laws.

If we assume that?uctuations of a i dissipate near the equilibrium state by linear laws(in proportion to thermodynamic forces)and?uctuations arising in the system are ergodic,it is possible not only to obtain reciprocity relations,but also express kinetic coef?cients L ij as time correlation functions for time variation of the corresponding values?a i[48–50]:

L ij～ ∞

?

a i(t)?a j(0)

d t,(2.20)

where ··· means averaging over an equilibrium ensemble of?uctuations having the distribution function P(a):

P(a)～exp

?

S(a)

k

,(2.21)

where k is the Boltzmann constant, S(a)=S eq?S is the change of entropy during a?uctuation,S eq is entropy of the system in the equilibrium,and a(a1,...,a i,...,a j...)is a set of values characterizing the system.

Formula(2.20)has the following physical meaning:the longer a?uctuation exists(the slower the correlation function decays),the larger the kinetic coef?cient is[46].

Let us use Onsager’s hypothesis and show how we can obtain the statement about the maximum entropy production when a nonequilibrium isolated system is relaxed to the equilibrium[52].11Let a system be in a nonequilibrium state with entropy S0at a moment of time t0.Assume that during the next time interval t(which is much longer than the time between two collisions,but is shorter than the relaxation time)the system can change to one of states with entropies S1,...,S N(S1<···

The modern theory of nonequilibrium processes is characterized by a variety of approaches,but basic ideas un-derlying these approaches are similar(see reviews[43–47]).Let us dwell on one of those approaches(the method of nonequilibrium statistical operator),which is widely used currently.In doing so,we shall focus on issues directly related to the topic of the present review.The Liouville equation[43–46],which was deduced for classical systems,14 is chosen as the basis:

j

j t

+i L =0,(2.22)

where (q,p,t)is the distribution function of many particles,q and p denote the coordinate and the momentum of the system in the6N-dimensional phase space,t is the time,i is an imaginary unit,L is the Liuoville linear operator,which

11Woo[53]proposed a little different procedure for deduction of MEPP basing on Eq.(2.21).He supposed that the probability of transition from one state to another in an arbitrary(nonlinear)case has a slightly more complicated form than Eq.(2.21).The goal was to substantiate that Onsager’s variational principle(see Eqs.(1.7)and(1.8))can be used in cases when instability and pattern formation can occur.According to Woo,the so-called generalized entropy production is maximized and it is reduced to the ordinary(thermodynamic)entropy production at a linear constitutive relation.

12The number of these states and their entropy values are naturally determined by both the initial state and the time interval t.

13Rigorously speaking,the validity of this statement requires additional veri?cation.

14Only classical systems will be considered below,but the given equations can be easily generalized to quantum systems[43–46].

18L.M.Martyushev,V .D.Seleznev /Physics Reports 426(2006)1–45

is de?ned in terms of the Poisson brackets as i L ={ ,H }= k j j q k j H j p k ?j j p k j H j q k ,(2.23)

where is some function and H =H (q,p,t)is the Hamiltonian of the system.

A considerable problem encountered in nonequilibrium statistical mechanics constructed on the basis of Eq.(2.22)is the deduction of time-irreversible transfer equations from the reversible Liouville equation.A detailed discussion of this principal issue is beyond the scope of this review.We shall note only that this is achieved by sacri?cing completeness of the description inherent in the distribution function and converting to a more concise description of nonequilibrium states [43–46].We shall assume therefore that the nonequilibrium macroscopic state is only described by a set of observed values P m t representing averages of the corresponding basis dynamic variables P m (for example,energy,number of particles and momentum).Further we shall seek for such solutions of Eq.(2.22),which only depend on these observed values.Obviously,values P m t do not uniquely de?ne .Amongst the whole multitude,we shall choose the distribution function,which corresponds to the principle of maximum information entropy (naturally at preset P m t ).15As a result,it is possible to ?nd the so-called quasi-equilibrium distribution function q in the form

q (t)=exp ? (t)?

m

F m (t)P m ,(2.24)where (t )is the Massieu–Planck function,which is determined from normalization conditions: (t)=ln exp ? m F m (t)P m d ,

(2.25)

while the Lagrange multipliers F m (t)are chosen considering the so-called self-consistency condition,according to which true averages of the set of values P m should be equal to their quasi-equilibrium averages: P m t = P m t q ≡

q P m d .(2.26)The integration is taken over the whole phase volume and d =d q d p/(N !h 3N )(N being the number of particles and h the Planck constant).

Thus,considering Eq.(2.26),averages over the quasi-equilibrium ensemble (Eq.(2.24))coincide with true values of the observed macroscopic values.

We shall assume further that the equality

(t )= q (t )

(2.27)is ful?lled at some initial moment of time t .Then the formal solution of Eq.(2.22)becomes [44,45]

(t)=e ?i (t ?t )L q (t ).(2.28)Since classical phase trajectories are considerably unstable,the behavior of a macroscopic system over time intervals,which are not too short,should be independent of microscopic details of its initial state.It is necessary therefore to exclude the considerable dependence of Eq.(2.28)on initial conditions.In line with [44,45]we shall assume that the evolution can begin with an equal probability from any state q (t )over the interval from t 0to t and the true nonequilibrium distribution (t)is equal to the average taken over initial moments of time t with respect to the distribution (2.28):

(t)=1t ?t 0 t t 0

e ?i (t ?t )L q (t )d t .(2.29)15This principle will be discussed in detail in Section 2.3.2.Here we shall only note in advance that the maximum entropy production principle can be directly obtained from the principle o

f maximum information entropy (see Sections 2.3.3–2.3.4).

L.M.Martyushev,V .D.Seleznev /Physics Reports 426(2006)1–4519

It should be noted that (t)enters both sides of Eq.(2.29),because parameters of the quasi-equilibrium distribution on the right are determined from the self-consistency condition (2.26).Expression (2.29)can be rearranged [44,45]to the form (t)=lim →+0

t ?∞

e ? (t ?t )e ?i (t ?t )L q (t )d t .(2.30)It is remarkable [44,45]that the pre-limit statistical distribution t ?∞e ? (t ?t )e ?i (t ?t )L q (t )d t satis?es the

Liouville equation with an in?nitely small source on the right,which disturbs the symmetry of Eq.(2.22)relative to the time reversal:

j j t

+i L =? { (t)? q (t)}.(2.31)Although the source tends to zero ( →0),it selects “delayed solutions”of the Liouville equation,which describe the irreversible evolution of a system.

Relations (2.30),(2.31)form the basis of the so-called nonequilibrium statistical operator method,which can help,depending on selection of basic variables,derive kinetic,hydrodynamic or relaxation equations for description of the evolution of a nonequilibrium system on different time scales [43–46].It should be also emphasized that ideas forming the basis of this approach and the majority of derived equations are much like ideas and results of other existing approaches to the construction of the general theory of nonequilibrium processes using ?rst principles [43–46].

When the equilibrium-disrupting perturbation is suf?ciently weak,general equations derived from Eqs.(2.30),(2.31)[43–46]can be simpli?ed if we only ?nd linear corrections to equilibrium values (the so-called linear response theory).16Assume that a perturbation (an additional term in the Hamiltonian of an unperturbed system)can be written as:17

? j h j B j ,(2.32)where h j denotes some stationary (to be selected exclusively for more simplicity)external ?elds and B j stands for their conjugate dynamic variables.For example,if a system in a magnetic ?eld is considered,components of the magnetizing force may be taken as h j and projections of the magnetic moment as B j .In this case,using Eqs.(2.24)–(2.26)and (2.30),(2.31),it is possible to obtain [45]stationary equations for response parameters of the system F n to the disturbance (2.32)in the form m = n

D mn F n ,(2.33)

where D mn is the generalized probability of the transition (an analog of the collision integral)and m is the so-called drift term.Here D mn = P m ;˙P n i

, m = j

P m ;˙B j i h j .(2.34)The last expression includes the following designations: A ;B i = ∞0

e ? t (A(t),B)d t, →+0;(A,B)= A B eq ;

A(t)=e i tL A ;

A =A ? A eq ;

B =B ? B eq ; eq = eq d ,where eq is the equilibrium distribution function.

16It was shown [43–45]that the Kubo method,which is widely used in the linear response theory,and the method of nonequilibrium statistical operator yield similar expressions for the corresponding nonequilibrium values,but the second method is more convenient for solution of speci?c problems.17We only deal with so-called mechanical disturbances.However,the approach can be generalized to so-called thermal disturbances [43–45].

20L.M.Martyushev,V .D.Seleznev /Physics Reports 426(2006)1–45

The derived Eq.(2.33)can be solved using the variational principle,which is very much like the principle described in Section 2.1for solving the linearized Boltzmann equation.Indeed,let {F m }be a trial set of response parameters satisfying the condition (see Eq.(2.33))

m F m m = mn F m D mn F n .(2.35)

Then it is possible to formulate and prove the following principle [45]:response parameters ,which represent solutions of Eq.(2.33),maximize ,amongst all functions {F m }obeying the condition (2.35),the entropy production of a system .19

This principle considerably generalizes the results under Section 2.1since the selected ?uxes (response parameters)maximize the entropy production at preset forces,but no assumptions are made with respect to the rarefaction of the system.This generalization was ?rst made by Nakano in 1959–1960,who noted not only the maximum entropy production,but also the accompanying maximization of transfer coef?cients,which are calculated in terms of the linear response theory [54–56].A similar generalization of the variational method for solving the Boltzmann equation was performed by Christoph and R?pke [57]too.

Thus,methods of solving some statistical model equations by means of MEPP are described in Sections 2.1,2.2.Although these results adduce microscopic arguments in favor of the given principle,the reasons for this regularity are still not clear.For example,microscopic reasons for the second law of thermodynamics are instability of the motion in a system with many particles.As a result,the system realizes all possible (a maximum number)microstates with time.A question arises if MEPP is based on a similar logic?In the next section we shall present attempts to answer this question.

2.3.The most probable trajectory of the evolution and the information theory

Since nonequilibrium evolution of a system is too complex and,as a rule,unstable,its microscopic characteristics may be viewed as random variables.The maximum entropy principle is used in statistical physics to ?nd the distribution of these variables in equilibrium.If nonequilibrium processes are described,the knowledge of distributions of random variables at a given moment of time is insuf?cient,because it is important to know additionally the rate (the probability)at which a system passes from one state to another.One more postulate is needed to determine this quantity.Is it possible to use some analog of the second law of thermodynamics by introducing,similarly to the Boltzmann entropy,some quantity (for example,entropy of the evolution (trajectory probabilities)or entropy production)and determine probabilities of the transition from the maximum of this quantity?We shall dwell on this issue in more detail below.

2.3.1.The method of the most probable path of the evolution

Filyukov and Karpov [58–60]proposed the method of the most probable path of the evolution for description of nonequilibrium stationary systems.This approach is very interesting and is intimately connected with the topic of the present review.The studies by Filyukov and Karpov did not attract attention at their time,but the method,which was proposed and developed by these researchers,has much in common with approaches,which were advanced much later,and evoked great interest (for details see Section 2.3.4).

In what follows we shall describe the essence of their approach.Let us have a nonequilibrium system,in which a stationary ?ux is established.It is assumed for simplicity that the evolution of the system can be described by the Markovian chain with a discrete time and a ?nite number of the system states.The symbol p ij ( )means the conditional probability of the transition during the time step from the state i to the state j and the symbol p i denotes the stationary probability of the state i .In the case of stationary Markovian chains,these probabilities should satisfy the following

18It holds not only in the classical stationary case under consideration,but also in quantum nonstationary case.19Here it equals j ˙B j t h j to within the positive constant multiplier.

北师大教育学原理作业标准答案

北师大《教育学原理》在线作业及答案 一、单选题 1、从课程设计的活动阶段来看，包括两个基本阶段：确定课程目标和：（B） A、确定教学策略 B、选择和组织课程内容 C、选择教学媒体 D、确定评价方法 2、强调内部动机和有意义学习的学习理论是：（C) A、行为主义 B、建构主义 C、认知主义 D、人本主义 3、两种比较有影响的教育定位理论，分别把学生定位在教育活动的边缘和中心位置的是：B A、“教师中心论”和“学生中心论” B、“教师中心论”和“儿童中心论” C、“家长中心论”和“儿童中心论” D、“学校中心论”和“学生中心论” 4、蔡元培最有价值和最有影响的教育思想是关于：(D) A、小学教育思想 B、初中教育思想 C、高中教育思想 D、大学教育思想 5、学校的简单要素不包括：（A ) A、组织结构 B、组织目标 C、参与者 D、技术 6、一种典型的以学科知识为中心的课程理论是：（D)

A、人本主义 B、概念重建主义 C、后现代主义 D、结构主义 7、欧洲大陆上最早开展中等教育综合化的国家是：（A） A、法国 B、芬兰 C、英国 D、德国 二、多选题 1、文艺复兴与宗教改革时期有哪些类型的学校：（B.C.D) A、修道院学校 B、人文主义学校 C、新教学校 D、天主教学校 2、从课程管理和开发主体的角度可以把课程划分为:（A.C.D.) A、国家课程 B、综合课程 C、地方课程 D、校本课程 3、德国综合中学的形式包括:（A.D） A、合作式综合中学 B、探究式综合中学 C、协作式综合中学 D、一体化综合中学 4、课程评价按照评价进行的时间可分为：（B.D） A、目标本位评价 B、形成性评价 C、目标游离评价

灰度阈值分割算法

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公开 取消收藏 窗体底端 分享资讯 传PPT/文档 提问题 写博客 传资源 创建项目 创建代码片 设置昵称编辑自我介绍，让更多人了解你帐号设置退出 社区 博客 论坛 下载 Share 极客头条 服务 CODE 英雄会 活动 CSTO 俱乐部 CTO俱乐部 高校俱乐部 奋斗斌斌的专栏

解决项目中的琐碎细节问题b.zhao_npu@https://www.doczj.com/doc/6e16433832.html, 目录视图 摘要视图 订阅 有奖征资源，博文分享有内涵人气博主的资源共享：老罗的Android之旅微软Azure?英雄会编程大赛题关注CSDN社区微信，福利多多社区问答：叶劲峰游戏引擎架构 灰度图像阈值化分割常见方法总结及VC实现 分类：图像处理OpenCV 2011-11-11 23:20 7427人阅读评论(14) 收藏举报 算法图形byte图像处理扩展 目录(?)[+] Otsu法最大类间方差法 一维交叉熵值法 二维OTSU法 参考文献 在图像处理领域，二值图像运算量小，并且能够体现图像的关键特征，因此被广泛使用。将灰度图像变为二值图像的常用方法是选定阈值，然后将待处理图像的每个像素点进行单点处理，即将其灰度值与所设置的门限进行比对，从而得到二值化的黑白图。这样一种方式因为其直观性以及易于实现，已经在图像分割领域处于中心地位。本文主要对最近一段时间作者所学习的阈值化图像分割算法进行总结，全文描述了作者对每种算法的理解，并基于OpenCV和VC6.0对这些算法进行了实现。最终将源代码公开，希望大家一起进步。（本文的代码暂时没有考虑执行效率问题） 首先给出待分割的图像如下： 1、Otsu法（最大类间方差法） 该算法是日本人Otsu提出的一种动态阈值分割算法。它的主要思想是按照灰度特性将图像划分为背景和目标2部分，划分依据为选取门限值，使得背景和目标之间的方差最大。（背景和目标之间的类间方差越大，说明这两部分的差别越大，当部分目标被错划分为背景或部分背景错划分为目标都会导致这两部分差别变小。因此，使用类间方差最大的分割意味着错分概率最小。）这是该方法的主要思路。其主要的实现原理为如下： 1）建立图像灰度直方图（共有L个灰度级，每个出现概率为p） 2）计算背景和目标的出现概率，计算方法如下： 上式中假设t为所选定的阈值，A代表背景（灰度级为0~N）,根据直方图中的元素可知，Pa为背景出现的概率，同理B为目标，Pb为目标出现的概率。 3）计算A和B两个区域的类间方差如下：

宇宙的熵增

宇宙会消亡吗 主流猜想——宇宙”热寂” 宇宙热寂的由来 热力学第二定律：能量可以转化，但是无法100%利用。在转化过程中，总是有一部分能量会被浪费掉。 这部分浪费掉能量命名为熵。熵不断在增加。 热力学第二定律也叫熵增原理。就是孤立热力学系统的熵不减少，总是增大或者不变。用来给出一个孤立系统的演化方向。说明一个孤立系统不可能朝低熵的状态发展即不会变得有序。 考虑到宇宙的能量总和是一个常量，而每一次能量转化，必然有一部分”有效能量”变成”无效能量”（即”熵”），因此不难推论，有效能量越来越少，无效能量越来越多。直到有一天，所有的有效能量都变成无效能量，那时将不再有任何能量转化，这就叫宇宙的”热寂”。 结论宇宙就会进入一个死寂的状态。 我的观点，宇宙不会”热寂”。 宇宙是无限的，动态循环的。这个宇宙是总称，不是有人提的这里一个宇宙，那里一个，多少年前或者远处还有一个宇宙。这种分法应该叫天体，为什么还要叫宇宙呢？ 宇宙存在自发的熵减的过程。 物质合久必分，分久必合。基本微观粒子质子、电子、中子、光子等聚合原子、分子等物质，这种聚合靠物质自身的引力、电磁力等自发完成。分子组成有序多样的宏观世界。这种聚合可以在某处相对孤立系统中完成，不需要外部干涉。这个过程是熵减的。 星系是宇宙的常见的单元。星系的核心是恒星，恒星的质量足够大，引力也足够大。最初恒星的物质含有的内能（主要是核能）充足，反应活跃与引力平衡，释放出光芒。当能量消耗到一定程度，内能不足以抵抗引力，不能再向外释放能

量。恒星不断收缩、坍塌，并且不断吸收外界物质，此时“恒星”只进不出，形成“黑洞”，黑洞缓慢吸收飞来的各种物质和能量，质量超大，大到吞噬星系的大部分行星和尘埃，甚至是相邻的星系的一部分。黑洞也有生命周期，这个也许比恒星的生命更长。黑洞碾碎了大部分物质（分子原子）分解成基本粒子，喷射出去。这一过程是熵增的。 粒子聚合成原子分子直至形成新的星系或者被其他星系吸收，形成循环。 宇宙各处随机重复着这一过程。每一个循环周期都是极漫长的。 物质的多样性是基本粒子聚合的随机性造成的。如同生物的多样性，生物演化出动、植物、细菌等等。 生物的生成发展也主要是熵减的过程。但是这种熵减的体量相对于星系的熵增是微不足道的。生物不改变星系毁灭的历程。 星系中存在生物也许是偶然的。智慧生命逃出一个星系的衰变毁灭，应该是概率很小的。 路过万丈红尘

流体最小熵产生原理与最小能耗率原理_

2003年6月 水 利 学 报 SHUILI XUEBAO 第6期 收稿日期:2002 09 22 基金项目:国家自然科学基金资助项目(59979020) 作者简介:徐国宾(1956-),男,河北石家庄人,高级工程师,博士,主要从事水力学及河流动力学研究工作。 文章编号:0559 9350(2003)06 0043 05 流体最小熵产生原理与最小能耗率原理( ) 徐国宾 1,2 ,练继建 1 (1 天津大学建筑工程学院,天津 300072;2 水利部天津水利水电勘测设计研究院,天津 300222) 摘要:本文是 流体最小熵产生原理与最小能耗率原理 的第 篇。在这篇中,一是阐明了最小熵产生原理等价于最小能耗率原理;二是基于最小熵产生原理,利用流体力学的3个基本方程,即连续方程、运动方程和能量方程以及热力学的吉布斯公式,推导出了流体最小能耗率原理数学表达式。该式适用于:(1)具有稳定边界的任何开放的流体系统,如河流;(2)恒定非均匀流或均匀流;(3)层流或紊流。关键词:流体;河流;开放系统;最小能耗率原理;最小熵产生原理;非平衡态热力学中图分类号:TV131 文献标识码:A 作者在 流体最小熵产生原理与最小能耗率原理( ) 一文中介绍了非平衡态热力学中的最小熵产生原理。本文基于该原理推导出了流体最小能耗率原理数学表达式。 1 最小熵产生原理与最小能耗率原理等价 以普利高津为首的布鲁塞尔学派在推导最小熵产生原理时,使用了系统的局域熵产生 [1] ,但是 利用能量耗散函数 也能得出同样结论。局域熵产生 和能量耗散函数 有下列关系[2,3] : =T (1) 式中:T 为绝对温度; 称为能量耗散函数,具有单位时间单位体积能量的量纲,表示系统在单位时 间内单位体积耗散掉的能量,它是由不可逆过程引起的。 类似于局域熵产生可以写成广义力和广义流乘积的总和,能量耗散函数也可写成[3] : = m j =1 J j X j (2) 式(2)中广义力和广义流的选取原则与文献[4]式(26)中的广义力和广义流选取原则相同,只不过是广义力和广义流的乘积必须具有能量耗散函数 的量纲。 根据式(1),可得到能耗率 与熵产生P 之间的关系: = V d V =T V d V = TP (3) 那么,现在就可以用能耗率 替代熵产生P 来表示最小熵产生原理,可得: d d t 0(4)所以,线性区的最小熵产生原理亦可称为最小能耗率原理,二者是等价关系。即在非平衡线性区,一个开放系统内的不可逆过程总是向熵产生或能耗率减小的方向进行,当熵产生或能耗率减小至最小值时,系统的状态不再随时间变化。此时,系统处于与外界约束条件相适应的非平衡定态。 43

《教育学原理》作业及答案

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南京工程学院信息论参考试卷D

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熵增原理

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熵产生原理与不可逆过程热力学简介

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教育管理原理 名词解释

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信息论习题

前三章习题 选择题 1、离散有记忆信源],[21x x X =，12()()0.5P x P x ==，其极限熵H ∞ C 。 A 、1bit > B 、1bit < C 、1bit = D 、不能确定 2、任意离散随机变量X 、Y 、Z ， C 必定成立 A 、)|()|(XZ Y H YZ X H = B 、)()()()(Z H Y H X H XYZ H ++= C 、)|()|(Y X H YZ X H ≤ D 、0)|;(=Z Y X I 3、|Y X P 给定时，(;)I X Y 是X P 的 A 函数。 A 、上凸 B 、下凸 C 、上升 D 、下降 4、使(;)I X Y 达到最大的 D 称为最佳分布。 A 、联合分布 B 、后验分布 C 、输出分布 D 、输入分布 5、离散平稳无记忆信源],[21x x X =，且bit X H 1)(=，则=)(1x P D 。 A 、41 B 、2 C 、1 D 、2 1 6、=);(Y X I C 。 A 、)|()(X Y H X H - B 、)|()(Y X H Y H + C 、)|()(X Y H Y H - D 、)()(X H XY H - 7、通常所说的“连续信源”是指 B 信源。 A 、时间连续且取值连续的 B 、取值连续 C 、时间离散且取值连续的 D 、时间连续 8、已知信道，意味着已知 B 。 A 、 先验分布 B 、转移概率分布 C 、 输入输出联合概率分布 D 、输出概率分布 9、已知X Y P |，可求出 B A 、)(XY H B 、 )|(X Y H C 、);(Y X I D 、)|(i j x y I 10、连续信源的输出可用 D 来描述 A 、常量 B 、变量 C 、离散随机变量 D 、连续随机变量 11、101 )(=i x P ，则=)(i x I D 。 A 、bit 10ln B 、dit 10ln C 、dit 1 D 、dit 10log 12、信道容量表征信道的 A 。 A 、最大通过能力 B 、最大尺寸 C 、最小通过能力 D 、最小尺寸 13、DMS 的信息含量效率等于信源的实际熵 C 信源的最大熵。 A 、乘以 B 、减去 C 、除以 D 、加上 14、下面信道矩阵为准对称信道的是 。

教育学原理及答案

一、填空（每空1分，共20分） １、教育学是以研究＿＿＿＿＿＿＿＿为对象的一门＿＿＿科学。 ２、原始形态教育的特点是＿＿＿＿＿＿＿＿＿、＿＿＿＿＿＿＿＿＿＿＿、＿＿＿＿＿＿＿＿＿＿。 3、１９２２年通过的“＿＿＿”学制、基本参照＿＿国的学制，通常又称“＿＿＿＿”学制，这是旧中国使用时间最长的学制。 4、教育学的产生和发展大体上经历了＿＿＿＿＿阶段、＿＿＿＿＿＿＿阶段和＿＿＿＿＿阶段。 5、教育有两大基本规律，一是＿＿＿＿＿＿、一是＿＿＿＿＿。 6、影响人的身心发展的基本因素是＿＿＿、＿＿＿和＿＿＿＿。 7、《＿＿＿＿＿》是教育史上最早的教育专著，约写于中国的战国末年，比欧州昆体良所著的《＿＿＿＿＿＿＿》约早＿＿＿年。 8．我国社会主义教育目的的理论基础是____。 9．捷克民主主义教育家____所著的《____》标志着教育学成为一门独立的学科。 10．____是一切教育工作的出发点和归宿。教育目的 11.广义的教育包括、学校教育和。 12.三级课程管理是、和学校三级共同管理基础教育课程体系。 13.教师是学校教育工作的主要实施者，根本任务是。 14.我国新型的师生关系包括尊师爱生、教学相长、和。 15.课堂教学的主要形式是。 16．古希腊提出问答法的哲学家和思想家是____苏格拉底_____。 18．马卡连柯说，“要尽量多地要求一个人，也要尽可能多尊重一个人”。这句话说明的德育原则是尊重学生与严格要求学生相结合＿。 19．布卢姆认为最低水平的认知学习结果是__知识__，最高水平的认知学习结果是_评价_。20．皮亚杰的研究表明，在10岁以前，儿童的道德主要处于___他律道德发展阶段。二、单选题。（每小题1分，共20分。） 1．世界上最早的一部教育专著是（） A．《学记》 B．《论语》 C．《大教学论》 D．《普通教育学》答案：A 2．教育的本体功能之一是（） A．减少人口数量、提高人口素质 B．促进生产发展，服务经济建设 C．对政治经济有巨大的影响作用 D．加速年轻一代身心发展与社会化进程答案：D 3．遗传素质是人身心发展的（） A．主导因素 B．决定因素 C．物质前提 D．内部动力答案：C

基于阈值的图像分割方法

基于阈值的图像分割方法

课程结业论文 课题名称基于阈值的图像分割方法姓名湛宇峥 学号1412202-24 学院信息与电子工程学院专业电子信息工程 指导教师崔治副教授 2017年6月12日

湖南城市学院课程结业论文诚信声明 本人郑重声明：所呈交的课程结业论文，是本人在指导老师的指导下，独立进行研究工作所取得的成果，成果不存在知识产权争议，除文中已经注明引用的内容外，本论文不含任何其他个人或集体已经发表或撰写过的作品成果。对本文的研究做出重要贡献的个人和集体均已在文中以明确方式标明。本人完全意识到本声明的法律结果由本人承担

目录 摘要 (1) 关键词 (1) ABSTRACT (2) KEY WORDS (2) 引言 (3) 1基于点的全局阈值选取方法 (4) 1.1最大类间交叉熵法 (5) 1.2迭代法 (6) 2基于区域的全局阈值选取方法 (7) 2.1简单统计法 (8) 2.3 直方图变化法 (9) 3局部阈值法和多阈值法 (10) 3.1水线阈值算法 (11) 3.2变化阈值法 (12) 4仿真实验 结论 (12) 参考文献 (13) 附录

基于阈值的图像分割方法 摘要：图像分割多年来一直受到人们的高度重视，至今这项技术也是趋于成熟，图像分割方法类别也是不胜枚举，近年来每年都有上百篇有关研究报道发表。图像分割是由图像处理进到图像分析的关键环节，是指把图像分成各具特性的区域并提取出有用的目标的技术和过程。在日常生活中，人们对图片的要求也是有所提高，在对图像的应用中，人们经常仅对图像中的某些部分感兴趣，这些部分就对应图像中的特定的区域，为了辨识和分析目标部分，就需要将这些有关部分分离提取出来，因此就要应用到图像分割技术。 关键词：图像分割；阈值；matlab

信息论与编码试卷A

南京工程学院 试题评分标准及参考答案 一 填空题（本题15空,每空1分,共15分 ） 1 一个消息来自于四符号集{a ，b ，c ，d}，四符号等概出现。由10个符号构成的消息“abbbccddbb ”所含的信息量为（ 20 ）bit ，平均每个符号所包含的信息量为（ 2 ）bit 。 2 两个二元信道的信道转移概率矩阵分别为 ????? ?????=??????=2/16/13/13/12/16/16/13/12/1,10000121P P ，则此信道的信道容量C1=（1bit/符 号 ），信道2的信道容量C2=（ 0.126bit/符号 ）。两信道串联后，得到的 信道转移概率矩阵为（??? ? ??=2/16/13/16/13/12/1P ），此时的信道容量C= （0.126bit/符号 ）。 3 条件熵H(Y/X)的物理含义为（唯一地确定信道噪声所需要的平均信息量），所以它又称为（噪声熵或散布度 ）。 4 一袋中有5个黑球、10个白球，以摸一个球为一次实验，摸出的球重新放进袋中。第一次实验包含的信息量为（0.915bit/符号）；第二次实验包含的信息量为（0.915bit/符号）。 5 线性分组码的伴随式定义为（S=EH T ），其中错误图案E 指的是（E=R-C (mod M)）。二进制码中，差错个数可等效为（收码和发码的汉明距离 ）。 6 设有一个二元等概信源：u={0，1}，P 0=P 1=1/2，通过一个二进制对称信道BSC ，

其失真函数d ij 与信道转移概率P ji =p(v j /u i )分别定义为 ?? ?=≠=j i j i d ij 01， ???=-≠=j i j i P ji ε ε1，则失真矩阵[d ij ]=（ ??????0110 ），平均失真D=（ ε ）。 二 判断题（本题10小题,每小题1分,共10分） 1．√2．√3．×4．×5．×6．√7．×8．√9．×10．√ （1） I(p i ) = -logp i 被定义为单个信源消息的非平均自信息量，它给出某个具体 消息信源的信息度量。 （ ） （2） 异前置码一定是唯一可译码。 （ ） （3） 无记忆离散消息序列信道，其容量C ≥各个单个消息信道容量之和。（ ） （4） 冗余度是表征信源信息率多余程度的物理量，它描述的是信源的剩（ ） （5） 当信道固定时，平均互信息),(Y X I 是信源分布的∪型凸函数。 （ ） （6） BCH 码是一类线性循环码，其纠错能力强、构造方便。 （ ） （7） R(D)被定义为在限定失真为D 的条件下，信源的最大信息速率。（ ） （8） 设P 为某马尔可夫信源的转移概率矩阵，若存在正整数N 使得N P 中的 元素全都为0，则该马尔可夫信源存在稳态分布。 （ ） （9） 信道容量随信源概率分布的变化而变化。 （ ） （10）如果两个错误图样e1、e2的和是一个有效的码字，则它们具有相同的 伴随式。（ ） 三名词解释（本题4小题,每小题5分,共20分） 1 极限熵 序列长度趋于无限大时，序列的平均符号熵称为极限熵，又称极限信息量。 2 最佳变长码 变长编码中，所有编出的唯一可译码中平均码长最短的码即为紧致码。 3 限失真信源编码 离散无记忆信源X 的信息率失真函数为R(D)，当信息率大于R(D)时，只要信源序列的长度足够长，一定存在一种编码方法，其译码失真小于或等于D+ε；反之，则无论采用什么方法，其译码失真必大于D 。 4 信道容量 平均互信息I （X ；Y ）在转移概率p(y/x)一定时，关于X 的概率分布是上凸函数，因此有极大值存在，这个极大值定义为信道容量) ;(max ) (Y X I C xi p =

阈值分割

在图像处理领域，二值图像运算量小，并且能够体现图像的关键特征，因此被广泛使用。将灰度图像变为二值图像的常用方法是选定阈值，然后将待处理图像的每个像素点进行单点处理，即将其灰度值与所设置的门限进行比对，从而得到二值化的黑白图。这样一种方式因为其直观性以及易于实现，已经在图像分割领域处于中心地位。本文主要对最近一段时间作者所学习的阈值化图像分割算法进行总结，全文描述了作者对每种算法的理解，并基于OpenCV和VC6.0对这些算法进行了实现。最终将源代码公开，希望大家一起进步。（本文的代码暂时没有考虑执行效率问题） 首先给出待分割的图像如下：

1、Otsu法（最大类间方差法） 该算法是日本人Otsu提出的一种动态阈值分割算法。它的主要思想是按照灰度特性将图像划分为背景和目标2部分，划分依据为选取门限值，使得背景和目标之间的方差最大。（背景和目标之间的类间方差越大，说明这两部分的差别越大，当部分目标被错划分为背景或部分背景错划分为目标都会导致这两部分差别变小。因此，使用类间方差最大的分割意味着错分概率最小。）这是该方法的主要思路。其主要的实现原理为如下： 1）建立图像灰度直方图（共有L个灰度级，每个出现概率为p） 2）计算背景和目标的出现概率，计算方法如下： 上式中假设t为所选定的阈值，A代表背景（灰度级为0~N）,根据直方图中的元素可知，Pa为背景出现的概率，同理B为目标，Pb为目标出现的概率。 3）计算A和B两个区域的类间方差如下：

第一个表达式分别计算A和B区域的平均灰度值； 第二个表达式计算灰度图像全局的灰度平均值； 第三个表达式计算A、B两个区域的类间方差。 4）以上几个步骤计算出了单个灰度值上的类间方差，因此最佳分割门限值应该是图像中能够使得A与B的类间灰度方差最大的灰度值。在程序中需要对每个出现的灰度值据此进行寻优。 本人的VC实现代码如下。 [cpp]view plaincopyprint? 1. /***************************************************************************** 2. * 3. * \函数名称： 4. * OneDimentionOtsu() 5. * 6. * \输入参数: 7. * pGrayMat: 二值图像数据 8. * width: 图形尺寸宽度 9. * height: 图形尺寸高度 10. * nTlreshold: 经过算法处理得到的二值化分割阈值 11. * \返回值: 12. * 无