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熵概念辨析EntropyCao Zexian中国科学院物理研究所内容提要¾热力学基础知识回顾¾Emergent Phenomenon¾Entropy和熵的字面意思¾熵概念－伤脑筋¾Entropy的数学表达¾Entropy 概念上的伟大成果量子力学的诞生；光子的极化态自旋薛定谔方程的推导；信息论¾Entropy作为过程的判据？¾结束语热力学是怎样的一门学问？我在德国Kaiserslautern大学机械系一间实验室的窗框上读到过这样的一段话，大意是：“热力学是这样的一门课：你学第一遍的时候觉得它挺难，糊里糊涂理不清个头绪，于是，你决定学第二遍；第二遍你觉得好像明白了点什么，这激励你去学第三遍；第三遍你发现好像又糊涂了，于是你只好学第四遍。
等到第四遍，well, 你已经习惯了你弄不懂热力学这个事实了。
”但我们必须理解热力学，因为：¾热力学是真实的。
Nothing in life is certain except death, taxes and the second law of thermodynamics. －Seth Lloyd¾热力学就在身边。
In this house, we obey the laws of thermodynamics! -Dan Castellaneta¾热力学是必备知识。
知冷知热是确立配偶人选的基本判据。
－曹则贤P. W. Anderson: More is different曹则贤，熵非商：the myth of Entropy,《物理》第九期，Entropy的字面意思Tropy的字面意思tropik<tropicus< Gr tropikos,belonging to a turn(of the sun at the solstices)Tropic of Cancer（北回归线）Tropic of Capricorn （南回归线）)Heliotropism: 向日性。
材料科学，主要包含金属、陶瓷、复合材料等，高分子的一般是在化学里面。
1。
acta materialia这个应该算是材料科学中影响力最大的期刊了吧，。
2002 年的if是3.104。
1953年开始的。
原来封面是4个圈的，2004年中间开始多了一个圈（theory）。
paper都很长的，大都在7page以上吧，现在没有letters的。
同一个出版社的还有个scripta materialia。
Acta Materialia's purpose is to publish original papers and occasional critical reviews which advance the understanding of the structural and functional properties of materials: metals and alloys, ceramics, high polymers and glasses. Emphasis is placed on those aspects of the science of materials that are concerned with the relationship between the structure of solids and their properties (mechanical, chemical, electrical, magnetic and optical); with the thermodynamics, kinetics and mechanisms of processes occurring within solids; with experiments and models which help in understanding the macroscopic properties of materials in terms of microscopic mechanisms; and with original work which advances the understanding of structural and functional materials./JournalDetail.html?PubID=221&Precis=DESC/science/journal/135964542。
a r X iv:c o n d -ma t/212223v1[c ond-m at.stat-m ec h]1D ec22Statistical mechanics of two-dimensional vortices and stellar systems Pierre-Henri Chavanis Laboratoire de Physique Quantique,Universit´e Paul Sabatier,118,route de Narbonne 31062Toulouse,France Abstract.The formation of large-scale vortices is an intriguing phenomenon in two-dimensional turbulence.Such organization is observed in large-scale oceanic or atmo-spheric flows,and can be reproduced in laboratory experiments and numerical simula-tions.A general explanation of this organization was first proposed by Onsager (1949)by considering the statistical mechanics for a set of point vortices in two-dimensional hydrodynamics.Similarly,the structure and the organization of stellar systems (glob-ular clusters,elliptical galaxies,...)in astrophysics can be understood by developing a statistical mechanics for a system of particles in gravitational interaction as initiated by Chandrasekhar (1942).These statistical mechanics turn out to be relatively similar and present the same difficulties due to the unshielded long-range nature of the in-teraction.This analogy concerns not only the equilibrium states,i.e.the formation of large-scale structures,but also the relaxation towards equilibrium and the statistics of fluctuations.We will discuss these analogies in detail and also point out the specificities of each system.1Introduction Two-dimensional flows with high Reynolds numbers have the striking property of organizing spontaneously into coherent structures (the vortices)which dominate the dynamics [93](see Fig.1).The robustness of Jupiter’s Great Red Spot,a huge vortex persisting for more than three centuries in a turbulent shear between two zonal jets,is probably related to this general property.Some other coherent structures like dipoles (pairs of cyclone/anticyclone)and sometimes tripoles have been found in atmospheric or oceanic systems and can persist during several days or weeks responsible for atmospheric blocking.Some astrophysicists invoke the existence of organized vortices in the gaseous component of disk galaxies in relation with the emission of spiral density waves [99].It has also been proposed that planetary formation might have begun inside persistent gaseous vortices born out of the protoplanetary nebula [5,121,15,60,33](see Fig.2).As a result,hydrodynamical vortices occur in a wide of geophysical or astrophysical situations and their robustness demands a general understanding.Similarly,it is striking to observe that self-gravitating systems follow a kindof organization despite the diversity of their initial conditions and their en-vironement [9](see Fig.3).This organization is illustrated by morphological classification schemes such as the Hubble sequence for galaxies and by simple2Pierre-Henri Chavanisrules which govern the structure of individual self-gravitating systems.For ex-ample,elliptical galaxies display a quasi-universal luminosity profile described by de Vaucouleur’s R1/4law and most of globular clusters are well-fitted by the Michie-King model.On the other hand,theflat rotation curves of spiral galaxies can be explained by the presence of a dark matter halo with a density profile decreasing as r−2at large distances.The fractal nature of the interstellar medium and the large scale structures of the universe also display some form of organization.Fig.1.Self-organization of two-dimensional turbulentflows into large-scale vortices [93].These vortices are long-lived and dominate the dynamics.The question that naturally emerges is what determines the particular con-figuration to which a self-gravitating system or a large-scale vortex settles.It is possible that their actual configuration crucially depends on the conditions that prevail at their birth and on the details of their evolution.However,in view of their apparent regularity,it is tempting to investigate whether their organi-zation can be favoured by some fundamental physical principles like those of thermodynamics and statistical physics.We ask therefore if the actual states of self-gravitating systems in the universe and coherent vortices in two-dimensional2D vortices and stellar systems3Fig.2.A scenario of planet formation inside large-scale vortices presumably present in the Keplerian gaseous disk surrounding a star at its birth.Starting from a random vorticityfield,a series of anticyclonic vortices appears spontaneously(upper panel). Due to the Coriolis force and to the friction with the gas,these vortices can efficiently trap dust particles passing nearby(lower pannel).The local increase of dust concen-tration inside the vortices can initiate the formation of planetesimals and planets by gravitational instability.This numerical simulation is taken from[15].turbulentflows are not simply more probable than any other possible configura-tion,i.e.if they cannot be considered as maximum entropy states.This statistical mechanics approach has been initiated by Onsager[101]for a system of point vortices and by Chandrasekhar[21]in the case of self-gravitating systems.It turns out that the statistical mechanics of two-dimensional vortices and self-gravitating systems present a deep analogy despite the very different physical nature of these systems.This analogy was pointed out by Chavanis in[29,32,35] and further developed in[54,30,34,36,47,48].In the following,we will essentially discuss the statistical mechanics of2D vortices and refer to the review of Pad-manabhan[103](and his contribution in this book)for more details about the4Pierre-Henri Chavanisrge-scale structures in the universe as observed with the Hubble space tele-scope.The analogy with Fig.1is striking and will be discussed in detail in this paper. statistical mechanics of self-gravitating systems.We will see that the analogy be-tween two-dimensional vortices and(three-dimensional)self-gravitating systems concerns not only the prediction of the equilibrium state,i.e.the formation of large-scale structures,but also the statistics offluctuations and the relaxation towards equilibrium.This paper is organized as follows.In Sec.2,we discuss the statistical me-chanics of point vortices introduced by Onsager[101]and further developed by Joyce&Montgomery[70]and Pointin&Lundgren[107]among others(see a complete list of references in the book of Newton[98]).We discuss the existence of a thermodynamic limit in Sec.2.7and make the connexion withfield theory. Statistical equilibrium states of axisymmetricflows are obtained analytically in Sec.2.8-2.9.The relation with equilibrium states of self-gravitating systems is shown in Sec.2.10.In Sec.3,we discuss the statistics of velocityfluctuations pro-duced by a random distribution of point vortices and use this stochastic approach to obtain an estimate of the diffusion coefficient of point vortices.Application to2D decaying turbulence is considered in Sec.3.4.In Sec.4,we describe the relaxation of a point vortex in a thermal bath and analyze this relaxation in terms of a Fokker-Planck equation involving a diffusion and a drift.In Sec.5,we develop a more general kinetic theory of point vortices.A new kinetic equation is obtained which satisfies all conservation laws of the point vortex system and increases the Boltzmann entropy(H-theorem).We mention the connexion with2D vortices and stellar systems5 the kinetic theory of stars developed by Chandrasekhar[21].In Sec.6,we dis-cuss the violent relaxation of2D vortices and stellar systems.We mention the analogy between the Vlasov and the Euler equations and between the statistical approach developed by Lynden-Bell[90]for collisionless stellar systems and by Kuz’min[83],Miller[95]and Robert&Sommeria[111]for continuous vorticity fields.The concepts of“chaotic mixing”and“incomplete relaxation”are dis-cussed in the light of a relaxation theory in Sec.6.3.Application of statistical mechanics to geophysicalflows and Jupiter’s Great Red Spot are evocated in Sec.6.4.2Statistical mechanics of point vortices intwo-dimensional hydrodynamics2.1Two-dimensional perfectflowsThe equations governing the dynamics of an invisicidflow are the equation of continuity and the Euler equation:∂ρ∂t +(u∇)u=−16Pierre-Henri Chavaniswhere ∆=∂2xx +∂2yy is the Laplacian operator.In an unbounded domain,thisequation can be written in integral form asψ(r ,t )=−12πz × ω(r ′,t )r −r ′∂t +u ∇ω=0.(10)This corresponds to the transport of the vorticity ωby the velocity field u .It is easy to show that the flow conserves the kinetic energyE =u 22 (∇ψ)2d 2r =12 ωψd 2r ,(12)where the second equality is obtained by a part integration with the condition ψ=0on the boundary.Therefore,E can be interpreted either as the kinetic energy of the flow (see Eq.(11))or as a potential energy of interaction between vortices (see Eq.(12)).2.2The point vortex gasWe shall consider the situation in which the velocity is created by a collection of N point vortices.In that case,the vorticity field can be expressed as a sum of δ-functions in the formω(r ,t )=N i =1γi δ(r −r i (t )),(13)where r i (t )denotes the position of point vortex i at time t and γi is its circulation.According to Eqs.(9)(13),the velocity of a point vortex is equal to the sum of the velocities V (j →i )produced by the N −1other vortices,i.e.V i = j =i V (j →i )with V (j →i )=−γj |r j −r i |2.(14)2D vortices and stellar systems7 As emphasized by Kirchhoff[79],the above dynamics can be cast in a Hamil-tonian formγi dx i∂y i,γidy i∂x i,(15) H=−12m).Thisis related to the particular circumstance that a point vortex is not a material particle.Indeed,an isolated vortex remains at rest contrary to a material particle which has a rectilinear motion due to its inertia.Point vortices form therefore a very peculiar Hamiltonian system.Note also that the Hamiltonian of point vortices can be either positive or negative(in the case of vortices of different signs)whereas the kinetic energy of theflow is necessarily positive.This is clearlya drawback of the point vortex model.2.3The microcanonical approach of Onsager(1949)The statistical mechanics of point vortices wasfirst considered by Onsager[101] who showed the existence of negative temperature states at which point vortices of the same sign cluster into“supervortices”.He could therefore explain the formation of large,isolated vortices in nature.This was a remarkable anticipation since observations were very scarce at that time.Let us consider a liquid enclosed by a boundary,so that the vortices are confined to an area A.Since the coordinates(x,y)of the point vortices are canonically conjugate,the phase space coincides with the configuration space and isfinite:dx1dy1...dx N dy N= dxdy N=A N.(17)This striking property contrasts with most classical Hamiltonian systems con-sidered in statistical mechanics which have unbounded phase spaces due to the presence of a kinetic term in the Hamiltonian.As is usual in the microcanonical description of a system of N particles,we introduce the density of statesg(E)= dx1dy1...dx N dy Nδ E−H(x1,y1,...,x N,y N) ,(18)8Pierre-Henri Chavaniswhich gives the phase space volume per unit interaction energy E.The equilib-rium N-body distribution of the system,satisfying the normalization condition µ(r1,...,r N)d2r1...d2r N=1,is given byµ(r1,...,r N)=1T =dS2π i<jγiγj ln|r i−r j|.(22)2D vortices and stellar systems9 Making the change of variable x=r/R,wefind that g(E,V)=V N g(E′,1) with E′=E+1βV 1+ββV 1+γ2(N−1)γ2.(25)We shall see in Sec.2.8that this negative critical inverse temperature is the minimum inverse temperature that the system can achieve.If,on the other hand,we consider a neutral system consisting of N/2vortices of circulationγand N/2vortices of circulation−γ,wefindP=N8π .(26)This result is well-known is plasma physics[113].The critical temperature at which the pressure vanishes is now positiveβc=8π∂t =−Ni=1γ∇δ(r−r i(t))V i.(28)Since V i=u(r i(t),t),we can rewrite the foregoing equation in the form∂ω10Pierre-Henri ChavanisSince the velocity is divergenceless,we obtain∂ω4π i=jγ2 ln|r i−r j|1=−2D vortices and stellar systems 11whereg (r 1,r 2,t )=µ(r 1,...,r N ,t )d 3r 2...d 2r N ,(36)is the two-body distribution function.In the mean-field approximation,whichis exact in a properly defined thermodynamic limit with N →+∞(see Sec.2.7),we haveg (r 1,r 2,t )=P (r 1,t )P (r 2,t ).(37)Accounting that N (N −1)≃N 2for large N ,the average energy takes the formE =−12ω ψd 2r =u 2n i !.(40)The logarithm of this number defines the Boltzmann ing Stirlingformula and considering the continuum limit in which ∆,ν→0,we get the classical formula S =−NP (r )ln P (r )d 2r ,(41)where P (r )is the density probability that a point vortex be found in the sur-face element centered on r .At equilibrium,the system is in the most probablemacroscopic state,i.e.the state that is the most represented at the microscopic level.This optimal state is obtained by maximizing the Boltzmann entropy (41)at fixed energy (39)and vortex number N ,or total circulationΓ=Nγ=ω d 2r .(42)12Pierre-Henri ChavanisWriting the variational principle in the formδS−βδE−αδΓ=0,(43) whereβandαare Lagrange multipliers,it is readily found that the maximum entropy state corresponds to the Boltzmann distributionω =Ae−βγψ,(44) with inverse temperatureβ.We can account for the conservation of angular momentum L= ω r2d2r(in a circular domain)and impulse P= ω yd2r (in a channel)by introducing appropriate Lagrange multipliersΩand U foreach of these constraints.In that case,Eq.(44)remains valid provided that we replace the streamfunctionψby the relative streamfunctionψ′=ψ+Ω2π i<j ln|r i−r j|2D vortices and stellar systems13 where the potential of interaction has been normalized by R.For simplicity, we have ignored the contribution of the images but this shall not affect the final results.We now introduce the change of variables x=r/R and define the functionu(x1,...,x N)=1N2γ2.(49) In terms of these quantities,the density of states can be rewritteng(E)=2πV NNu(x1,...,x N) .(50)The proper thermodynamic limit for a system of point vortices with equal cir-culation in the microcanonical ensemble is such that N→+∞withfixedΛ.Wesee that the box size R does not enter in the normalized energyΛ.Therefore,the thermodynamic limit corresponds to N→+∞withγ∼N−1→0and E∼1. This is a very unusual thermodynamic limit due to the non-extensivity of thesystem.Note that the total circulationΓ=Nγremainsfixed in this process.For sufficiently large N,the density of states can be writteng(E)≃ Dρe NS[ρ]δ Λ−E[ρ]) δ 1− ρ(r)d2r ,(51) withS[ρ]=− ρ(r)lnρ(r)d2r,(52) E[ρ]=−114Pierre-Henri Chavaniswhich is the normalization factor in the Gibbs measure1µ(r1,...,r N)=ln Z.(57)βUsing the notations introduced previously,we can rewrite the integral(55)in the formZ(β)=V N N0... N0N i=1d2x i eηu(x1,...,x N),(58)whereβNγ2η=2D vortices and stellar systems15 and phase transitions occur.This is the case,in particular,for the gravitational problem(see Sec.2.10).Finally,the grand canonical partition function is defined byZ GC=+∞ N=0z N2(∇ξ)2−ρ(r)ξ(r)}d2r=e−12(∇ξ)2d2r+√−βγξ(r),we can easily carry out the summation on N to obtainZ GC= Dφe−12(∇φ)2−µ2eφ(r)},(67)T eff=−βγ2,µ2=−zβγ2.(68) Therefore,the grand partition function of the point vortex gas corresponds to a Liouvillefield theory with an actionS[φ]=12(∇φ)2−µ2eφ(r)}.(69)While the previous description is formally correct if we define Z and Z GC by Eqs.(55)and(64),it must be noted however that the canonical and grand canon-ical ensembles may not have a physical meaning for point vortices.In particular, it is not clear how one can impose a thermal bath at negative temperature. On the other hand,the usual procedure to derive the canonical ensemble from the microcanonical ensemble rests on a condition of additivity which is clearly lacking in the present case.2.8Axisymmetric equilibrium states in a diskLet us consider a collection of N point vortices with circulationγconfined within a disk of radius R.At statistical equilibrium,the streamfunctionψis solution of the Boltzmann-Poisson equation(45).If we work in a circular domain,we must in principle account for the conservation of angular momentum.This can lead to bifurcations between axisymmetric and off-axis solutions[117].We shall,16Pierre-Henri Chavanishowever,ignore this constraint for the moment in order to obtain analytical ex-pressions for the thermodynamical parameters.This is a sufficient approximation to illustrate the structure of the problem,which is our main concern here.If we confine our attention to axisymmetric solutions,the Boltzmann-Poisson equation(45)can be written1dr r dψξddξ =λe−φ,(72)φ(0)=φ′(0)=0,(73)withλ=1ifβ<0andλ=−1ifβ>0.It turns out that this equation can be solved analytically as noticed by a number of authors.With the change of variables t=lnξandφ=2lnξ−z,Eq.(72)can be rewrittend2zdz(λe z).(74)This corresponds to the motion of aficticious particle in a potential V(z)=λe z. This equation is readily integrated and,returning to original variables,wefinally obtaine−φ=18ξ2)2.(75)From the circulation theorem(6)applied to an axisymmetricflow,we have−dψ2πr.(76) whereΓ(r)= r0ω(r′)2πr′dr′is the circulation within r.Taking r=R and introducing the dimensionless variables previously defined,we obtainη≡βγΓπR2(η+4)1η+4r22D vortices and stellar systems 17At positive temperatures (η>0),the vorticity is an increasing function of the distance and the vortices tend to accumulate on the boundary of the domain (Fig.4).On the contrary at negative temperatures (η<0),the vorticity is a decreasing function of the distance and the vortices tend to group themselves in the core of the domain to form a “supervortex”(Fig.5).These results are consistent with Onsager’s prediction [101].We also confirm that statistical equilibrium states only exist for η>ηc =−4,as previously discussed.At this critical temperature,the central vorticity becomes infinite and the solution tends to a Dirac peak:ω (r )→Γδ(r ),for η→ηc =−4.(79)00.20.40.60.81r/R1234567<ω>/ω∗η>0η=0η=20Fig.4.Statistical equilibrium states of point vortices at positive temperatures (ω∗=Γ/πR 2).The vortices are preferentially localized near the wall.The energy defined by Eq.(39)can be written in the dimensionless formΛ≡2πE2η2 α0φ′(ξ)2ξdξ.(80)The integral can be carried out explicitly using Eq.(75).Eliminating αbetweenEqs.(80)and (77),we find that the temperature is related to the energy by the equation of state Λ=1ηln418Pierre-Henri Chavanis00.20.40.60.81r/R246810<ω>/ω∗η<0η=−2η=−3η=−3.5Fig.5.Statistical equilibrium states of point vortices at negative temperatures showing a clustering.For η=−4,the vortices collapse at the center of the domain and the vorticity profile is a Dirac peak.0.250.50.751Λ=2πE/Γ2−10−8−6−4−20246810η=βγΓ/2πΛ0=1/8ηc =−4Fig.6.Equilibrium phase diagram (caloric curve)for point vortices with equal circula-tion confined within a disk.For simplicity,the angular momentum has not been taken into account (Ω=0).2D vortices and stellar systems1900.250.50.751Λ=2πE/Γ2−7−6−5−4−3S /NΛ0=1/8η=0η>0η<0ηc =−4Fig.7.Entropy vs energy plot for a system of point vortices with equal circulation confined within a disk.The entropy (41)can also be calculated easily from the above results.Within an unimportant additive constant,it is given byS ηln 4−1+820Pierre-Henri Chavanisaccount,the density of states is given byg (E,L )=δ E −H (r 1,...,r N ) δ L −N i =1γr 2i N i =1d 2r i ,(83)and the angular velocity of the flow byΩ=2T∂SL ,onefinds the exact resultΩ=2N8π.(85)Therefore,the vorticity field is determined by the Boltzmann distributionω =Ae −βγψ′,(86)where ψ′is the relative streamfunctionψ′≡ψ+Ω4βL(4+η)r 2.(87)For η=0,one hasω =γNΓLr 2.(88)For large r ,the asymptotic behavior of Eq.(86)isω ∼14L(4+η)r 2(r →+∞),(89)where we have used ψ∼−(Γ/2π)ln r at large distances.From Eq.(89),one sees that η≥−4is required for the existence of an integrable solution.Inserting the relation (86)in the Poisson equation (7),we get−∆ψ′=Ae −2πη2πLη(4+η).(90)With the change of variablesξ=γNN 2γ2−2πηdξ2+1dξ=2πηe φ−(4+η),(92)2D vortices and stellar systems21 and the vorticity(86)becomesω =N2γ22πL,if r≤(2L/Γ)1/2,(94)and ω =0otherwise.This vortex patch is the state of minimum energy at fixed circulation and angular momentum.Forη→−4,one has approximatelyω =N2γ2(1−AπηγN4L(4+η)r2.(95)Thefirst factor is an exact solution of Eq.(92)with the second term on the right hand side neglected(see Sec.2.8).The second factor is a correction for large r, in agreement with the asymptotic result expressed by Eq.(89).The parameter A tends to infinity asη→−4and is determined from the condition ω d2r=Γby the formulaπA+ln(πA)=−C−ln 1+η|r j−r i|3,(97)where F(j→i)is the force created by star j on star i.The force can be written as the gradient F=−∇Φof a gravitational potentialΦwhich is related to the stellar densityρ(r,t)=Ni=1mδ(r−r i),(98)by the Poisson equation∆Φ=4πGρ.(99)22Pierre-Henri ChavanisFurthermore,the equations of motion(Newton’s equations)can be put in the Hamiltonian formm d r i∂v i,md v i∂r i,H=1|r i−r j|.(100)In the analogy between stellar systems and two-dimensional vortices,the star densityρplays the role of the vorticityω,the force F the role of the velocity V and the gravitational potentialΦthe role of the streamfunctionψ.The crucial point to realize is that,for the two systems,the interaction is a long-range un-shielded Coulombian interaction(in D=3or D=2dimensions).This makes the connexion between point vortices and stellar systems deeper than between point vortices and electric charges for example.In particular,point vortices can organize into large scale clusters,like stars in galaxies,while the distribution of electric charges in a neutral plasma is uniform.There are,on the other hand, fundamental differences between stars and vortices.In particular,a star creates an acceleration while a vortex creates a velocity.On the other hand,the gravi-tational interaction is attractive and directed along the line joining the particles while the interaction between vortices is rotational and perpendicular to the line joining the vortices.Despite these important differences,the statistical mechanics of2D vortices and stellar systems are relatively similar.Like the point vortex gas,the self-gravitating gas is described at statistical equilibrium by the Boltzmann distri-butionρ =Ae−βΦ,(101) obtained by maximizing the Boltzmann entropy atfixed mass M and energy E. Its structure is therefore determined by solving the Boltzmann-Poisson equation∆Φ=4πGAe−βΦ,(102) where A andβ>0have to be related to M and E.This statistical mechanics approach has been developed principally for globular clusters relaxing towards equilibrium via two-body encounters[9].It is clear that the Boltzmann-Poisson equation(102)is similar to the Boltzmann-Poisson equation(45)for point vor-tices at negative temperatures.The density profile determined by these equations is a decreasing function of the distance,which corresponds to a situation of clus-tering(see Figs.5and8).The similarity of the maximum entropy problem for stars and vortices,and the Boltzmann-Poisson equations(102)(45),is afirst manifestation of the formal analogy existing between these two systems.However,due to the different dimension of space(D=3for stars instead of D=2for vortices),the mathematical problems differ in the details.First of all,the density profile determined by the Boltzmann-Poisson equation(102)in D=3decreases like r−2at large distances leading to the so-called infinite mass2D vortices and stellar systems 23−50510ln(ξ)−20−15−10−50ln (e −ψ)singularsphereξ−2Fig.8.Density profile of the self-gravitating gas at statistical equilibrium.The dashed line corresponds to the singular solution ρ=1/2πGβr 2.problem since M = +∞0ρ4πr 2dr →+∞[19].There is no such problem forpoint vortices in two dimensions:the vorticity decreases like r −4,or even more rapidly if the conservation of angular momentum is accounted for,and the totalcirculation Γ= +∞0ω2πrdr is finite.The infinite mass problem implies thatno statistical equilibrium state exists for open star clusters,even in theory.A system of particles in gravitational interaction tends to evaporate so that the final state is just two stars in Keplerian orbit.This evaporation process has been clearly identified in the case of globular clusters which gradually lose stars to the benefit of a neighboring galaxy.In fact,the evaporation is so slow that we can consider in a first approximation that the system passes by a succession of quasiequilibrium states described by a truncated isothermal distribution function (Michie-King model)[9].This justifies the statistical mechanics approach in that sense.Another way of solving the infinite mass problem is to confine the system within a box of radius R .However,even in that case,the notion of equilibrium poses problem regarding what now happens at the center of the configuration.The equilibrium phase diagram (E,T )for bounded self-gravitating systems is represented in Fig.9.The caloric curve has a striking spiral behavior parametrized by the density contrast R =ρ(0)/ρ(R )going from 1(homogeneous system)to +∞(singular sphere)as we proceed along the spiral.There is no equilibrium state below E c =−0.335GM 2/R or T c =GMm24Pierre-Henri ChavanisΛ=−ER/GM 20.51.52.5η=βG M /R Fig.9.Equilibrium phase diagram for self-gravitating systems confined within a box.For sufficiently low energy or temperature,there is no equilibrium state and the system undergoes gravitational collapse.−1−0.500.51 1.5Λ=−ER/GM 200.511.522.533.5η=βG M /R µ=105µ=104µ=103µ=102µ=10Fig.10.Equilibrium phase diagram for self-gravitating fermions [41].The degeneracy parameter µplays the role of a small-scale cut-offǫ∼1/µ.For ǫ→0,the classical spiral of Fig.9is recovered.2D vortices and stellar systems25 mass goes to zero.Therefore,the singularity contains no mass and this process alone cannot lead to a black hole.Since the T(E)curve has turning points,this implies that the microcanon-ical and canonical ensembles are not equivalent and that phase transitions will occur[103].In the microcanonical ensemble,the series of equilibria becomes un-stable after thefirst turning point of energy(MCE)corresponding to a density contrast of709.At that point,the solutions pass from local entropy maxima to saddle points.In the canonical ensemble,the series of equilibria becomes unsta-ble after thefirst turning point of temperature(CE)corresponding to a density contrast of32.1.At that point,the solutions pass from minima of free energy (F=E−T S)to saddle points.It can be noted that the region of negative specific heats between(CE)and(MCE)is stable in the microcanonical en-semble but unstable in the canonical ensemble,as expected on general physical grounds.The thermodynamical stability of isothermal spheres can be deduced from the topology of theβ−E curve by using the turning point criterion of Katz[75]who has extended Poincar´e’s theory of linear series of equilibria.The stability problem can also be reduced to the study of an eigenvalue equation associated with the second order variations of entropy or free energy as studied by Padmanabhan[102]in the microcanonical ensemble and by Chavanis[37]in the canonical ensemble.This study has been recently extended to other statisti-cal ensembles[44]:grand canonical,grand microcanonical,isobaric....The same stability limits as Katz are obtained but this method provides in addition the form of the density perturbation profiles that trigger the instability at the critical points.It also enables one to show a clear equivalence between thermodynamical stability in the canonical ensemble and dynamical stability with respect to the Navier-Stokes equations(Jeans problem)[37,44].These analytical methods can be extended to general relativity[38].It must be stressed,however,that the statistical equilibrium states of self-gravitating systems are at most metastable: there is no global maximum of entropy or free energy for a classical system of point masses in gravitational interaction[2].Phase transitions in self-gravitating systems can be studied in detail by in-troducing a small-scale cut-offǫin order to regularize the potential.This can be achieved for example by considering a system of self-gravitating fermions (in which case an effective repulsion is played by the Pauli exclusion principle) [62,8,52,41,43]or a hard spheres gas[3,103,120].Other forms of regularization are possible[59,128,45].For these systems,there can still be gravitational collapse but the core will cease to shrink when it feels the influence of the cut-off.The result is the formation of a compact object with a large mass:a“fermion ball”or a hard spheres“condensate”.The equilibrium phase diagram of self-gravitating fermions is represented in Fig.10and has been discussed at length by Chavanis [41]in the light of an analytical model.The introduction of a small-scale cut-offhas the effect of unwinding the classical spiral of Fig.9.For a small cut-offǫ≪1,the trace of the spiral is still visible and the T(E)curve is multivalued (Fig.11).This can lead to a gravitationalfirst order phase transition between a gaseous phase with an almost homogeneous density profile(upper branch)and a。
German mathematicians have made significant contributions to the field of mathematics throughout history.Here are some notable figures and their achievements:1.Carl Friedrich Gauss17771855:Often referred to as the Prince of Mathematicians, Gauss made groundbreaking contributions to number theory,algebra,statistics,analysis, differential geometry,geodesy,geophysics,mechanics,electrostatics,magnetic fields, astronomy,matrix theory,and optics.His work on the Fundamental Theorem of Algebra and the Gaussian distribution are particularly influential.2.Georg Cantor18451918:Cantor is best known for his work on set theory,which has become a fundamental theory in mathematics.He introduced the concept of infinite sets and cardinality,challenging the traditional notion of infinity with his discovery of different sizes of infinity.3.David Hilbert18621943:Hilbert was a prominent mathematician who made significant contributions to algebra,number theory,and geometry.He is famous for his list of23 unsolved problems,presented in1900,which guided much of the mathematical research for the following century.4.Leopold Kronecker18231891:Kroneckers work in algebra,particularly his contributions to number theory and algebraic structure,laid the groundwork for modern algebra.He is known for his belief that God made the integers all else is the work of man.5.Bernhard Riemann18261866:Riemann is renowned for his work in complex analysis and number theory.His most famous contribution is the Riemann Hypothesis,one of the most important unsolved problems in mathematics,which concerns the distribution of prime numbers.6.Felix Klein18491925:Klein made significant contributions to geometry,particularly in the field of nonEuclidean geometry.His Erlangen Program classified geometries by their underlying group of symmetries,which has had a profound impact on the development of modern geometry.7.Sophie Germain17761831:Although not as wellknown as her male counterparts, Germain was a pioneering female mathematician who made important contributions to number theory and elasticity.She is known for her work on Fermats Last Theorem and for her correspondence with Gauss.8.Hermann Minkowski18641909:Minkowski was a key figure in the development of the mathematical theory of relativity.His concept of spacetime,which combines the threedimensions of space with the fourth dimension of time,was crucial for Einsteins theory of special relativity.9.Ernst Eduard Kummer18101893:Kummer made significant contributions to number theory,particularly in the area of prime numbers.His work on the theory of ideals in rings laid the foundation for modern algebraic number theory.10.Max Planck18581947:Although best known for his work in physics and the development of quantum theory,Planck also made contributions to mathematics, particularly in the areas of thermodynamics and statistical mechanics.These mathematicians,among others,have shaped the course of mathematics and continue to influence the field with their theories and discoveries.Their work has not only expanded our understanding of the mathematical universe but has also had practical applications in various scientific and technological advancements.。
高中英语世界著名科学家单选题50题1. Albert Einstein was born in ____.A. the United StatesB. GermanyC. FranceD. England答案：B。
解析：Albert Einstein（阿尔伯特·爱因斯坦）出生于德国。
本题主要考查对著名科学家爱因斯坦国籍相关的词汇知识。
在这几个选项中，the United States是美国，France是法国，England是英国，而爱因斯坦出生于德国，所以选B。
2. Isaac Newton is famous for his discovery of ____.A. electricityB. gravityC. radioactivityD. relativity答案：B。
解析：Isaac Newton 艾萨克·牛顿）以发现万有引力gravity）而闻名。
electricity是电，radioactivity是放射性，relativity 是相对论，这些都不是牛顿的主要发现，所以根据对牛顿主要成就的了解，选择B。
3. Marie Curie was the first woman to win ____ Nobel Prizes.A. oneB. twoC. threeD. four答案：B。
解析：Marie Curie 居里夫人）是第一位获得两项诺贝尔奖的女性。
这题主要考查数字相关的词汇以及对居里夫人成就的了解，她在放射性研究等方面的贡献使她两次获得诺贝尔奖，所以选B。
4. Thomas Edison is well - known for his invention of ____.A. the telephoneB. the light bulbC. the steam engineD. the computer答案：B。
解析：Thomas Edison（ 托马斯·爱迪生）以发明电灯（the light bulb）而闻名。
Chapter6Many-Particle Systemsc 2010by Harvey Gould and Jan Tobochnik8March2010We apply the general formalism of statistical mechanics to systems of many particles and discuss the semiclassical limit of the partition function,the equipartition theorem for classical systems,and the general applicability of the Maxwell velocity distribution.We then consider noninteracting quantum systems and discuss the single particle density of states,the Fermi-Dirac and Bose-Einstein distribution functions,the thermodynamics of ideal Fermi and Bose gases,blackbody radiation,and the specific heat of crystalline solids among other applications.6.1The Ideal Gas in the Semiclassical LimitWefirst apply the canonical ensemble to an ideal gas in the semiclassical limit.Because the thermodynamic properties of a system are independent of the choice of ensemble,we willfind the same thermal and pressure equations of state as we found in Section4.5.Although we will not obtain any new results,this application will give us more experience in working with the canonical ensemble and again show the subtle nature of the semiclassical limit.In Section6.6we will derive the classical equations of state using the grand canonical ensemble without any ad hoc assumptions.In Sections4.4and4.5we derived the thermodynamic properties of the ideal classical gas1 using the microcanonical ensemble.If the gas is in thermal equilibrium with a heat bath at temperature T,it is more natural and convenient to treat the ideal gas in the canonical ensemble. Because the particles are not localized,they cannot be distinguished from each other as were the harmonic oscillators considered in Example4.3and the spins in Chapter5.Hence,we cannot simply focus our attention on one particular particle.The approach we will take here is to treat the particles as distinguishable,and then correct for the error approximately.As before,we will consider a system of noninteracting particles starting from their fundamental description according to quantum mechanics.If the temperature is sufficiently high,we expectCHAPTER6.MANY-PARTICLE SYSTEMS293 that we can treat the particles classically.To do so we cannot simply take the limit →0 wherever it appears because the counting of microstates is different in quantum mechanics and classical mechanics.That is,particles of the same type are indistinguishable according to quantum mechanics.So in the following we will consider the semiclassical limit,and the particles will remain indistinguishable even in the limit of high temperatures.To take the semiclassical limit the mean de Broglie wavelengthλ,thefirst condition will always be satisfied.As shown in Problem6.1,the mean distance between particles in three dimensions isρ−1/3.Hence,the semiclassical limit requires thatλ3≪1(semiclassical limit).(6.1) Problem6.1.Mean distance between particles(a)Consider a system of N particles confined to a line of length L.What is the definition ofthe particle densityρ?The mean distance between particles is L/N.How does this distance depend onρ?(b)Consider a system of N particles confined to a square of linear dimension L.How does themean distance between particles depend onρ?(c)Use similar considerations to determine the density dependence of the mean distance betweenparticles in three dimensions.To estimate the magnitude ofp2/2m= 3kT/2.(We will rederive this result more generally in Section6.2.1.)Henceλ∼h/ p2∼h/√2πmkT 1/2= 2π 22πwill allow us to express the partition function in a convenientform[see(6.11)].The calculation of the partition function of an ideal gas in the semiclassical limit proceeds as follows.First,we assume thatλ≪ρ−1/3so that we can pick out one particle if we make the additional assumption that the particles are distinguishable.(Ifλ∼ρ−1/3,the wavefunctions of the particles overlap.)Because identical particles are intrinsically indistinguishable,we will have to correct for the latter assumption later.With these considerations in mind we now calculate Z1,the partition function for one particle, in the semiclassical limit.As we found in(4.40),the energy eigenvalues of a particle in a cube of side L are given byh2ǫn=CHAPTER 6.MANY-PARTICLE SYSTEMS 294where the subscript n represents the set of quantum numbers n x ,n y ,and n z ,each of which can be any nonzero,positive integer.The corresponding partition function is given byZ 1= n e −βǫn =∞ n x =1∞ n y =1∞ n z =1e −βh 2(n x 2+n y 2+n z 2)/8mL 2.(6.4)Because each sum is independent of the others,we can rewrite(6.4)asZ 1=∞n x =1e −α2n x 2 ∞ n y =1e −αn y 2 ∞ n z =1e −αn z 2 =S 3,(6.5)whereS =∞ n x =1e −α2n x 2.(6.6)andα2=βh 24λ2α ∞0e −u 2du −1=L2πmβh 23/2.(6.10)The result (6.10)is the partition function associated with the translational motion of one particle in a box.Note that Z 1can be conveniently expressed asZ 1=V 2ln β+3h 2.(6.12)CHAPTER6.MANY-PARTICLE SYSTEMS295 The mean pressure due to one particle is given byβ∂ln Z1βV=kTe=−∂ln Z12β=3P=NkTE=3∂T V=3he−βp2/2m.(6.18)The integral over p in(6.18)extends from−∞to+∞.The entropy of an ideal classical gas of N particles.Although it is straightforward to calculate the mean energy and pressure of an ideal classical gas by considering the partition function for one particle,the calculation of the entropy is more subtle.To understand the difficulty,consider the calculation of the partition function of an ideal gas of two particles.Because there are noCHAPTER6.MANY-PARTICLE SYSTEMS296microstate s blue1ǫaǫb2ǫb3ǫcǫaǫa+ǫb5ǫaǫaǫa+ǫc7ǫaǫbǫb+ǫc9ǫbCHAPTER6.MANY-PARTICLE SYSTEMS297 than there are particles(see Problem4.14,page190).(In our simple example,each particle can be in one of only three microstates,and the number of microstates is comparable to the number of particles.)If we assume that the particles are indistinguishable and that microstates with multiple occupancy can be ignored,then Z2is given byZ2=e−β(ǫa+ǫb)+e−β(ǫa+ǫc)+e−β(ǫb+ǫc)(indistinguishable,no multiple occupancy).(6.23)We see that if we ignore multiple occupancy there are three microstates for indistinguishable particles and six microstates for distinguishable particles.Hence,in the semiclassical limit we can write Z2=Z21/2!where the factor of2!corrects for overcounting.For three particles(each of which can be in one of three possible microstates)and no multiple occupancy,there would be one microstate of the system for indistinguishable particles and no multiple occupancy,namely, the microstate a,b,c.However,there would be six such microstates for distinguishable particles. Thus if we count microstates assuming that the particles are distinguishable,we would overcount the number of microstates by N!,the number of permutations of N particles.We conclude that if we begin with the fundamental quantum mechanical description of matter, then identical particles are indistinguishable at all temperatures.If we make the assumption that single particle microstates with multiple occupancy can be ignored,we can express the partition function of N noninteracting identical particles asZ N=Z1NN! 2πmkTN +3h2 +1 .(6.26)In Section6.6we will use the grand canonical ensemble to obtain the entropy of an ideal classical gas without any ad hoc assumptions such as assuming that the particles are distinguish-able and then correcting for overcounting by including the factor of N!.That is,in the grand canonical ensemble we will be able to automatically satisfy the condition that the particles are indistinguishable.Problem6.4.Equations of state of an ideal classical gasUse the result(6.26)tofind the pressure equation of state and the mean energy of an ideal gas.Do the equations of state depend on whether the particles are indistinguishable or distinguishable? Problem6.5.Entropy of an ideal classical gasCHAPTER6.MANY-PARTICLE SYSTEMS298 (a)The entropy can be found from the relations F=E−T S or S=−∂F/∂T.Show thatS(T,V,N)=Nk ln V2ln 2πmkT2 .(6.27)The form of S in(6.27)is known as the Sackur-Tetrode equation(see Problem4.20,page197).Is this form of S applicable for low temperatures?(b)Express kT in terms of E and show that S(E,V,N)can be expressed asS(E,V,N)=Nk ln V2ln 4πmE2 ,(6.28) in agreement with the result(4.63)found using the microcanonical ensemble.Problem6.6.The chemical potential of an ideal classical gas(a)Use the relationµ=∂F/∂N and the result(6.26)to show that the chemical potential of anideal classical gas is given byµ=−kT ln V h2 3/2 .(6.29)(b)We will see in Chapter7that if two systems are placed into contact with different initialchemical potentials,particles will go from the system with higher chemical potential to the system with lower chemical potential.(This behavior is analogous to energy going from high to low temperatures.)Does“high”chemical potential for an ideal classical gas imply“high”or“low”density?(c)Calculate the entropy and chemical potential of one mole of helium gas at standard temperatureand pressure.Take V=2.24×10−2m3,N=6.02×1023,m=6.65×10−27kg,and T= 273K.Problem6.7.Entropy as an extensive quantity(a)Because the entropy is an extensive quantity,we know that if we double the volume and doublethe number of particles(thus keeping the density constant),the entropy must double.This condition can be written formally asS(T,λV,λN)=λS(T,V,N).(6.30) Although this behavior of the entropy is completely general,there is no guarantee that an approximate calculation of S will satisfy this condition.Show that the Sackur-Tetrode form of the entropy of an ideal gas of identical particles,(6.27),satisfies this general condition. (b)Show that if the N!term were absent from(6.25)for Z N,S would be given byS=Nk ln V+3h2 +3CHAPTER6.MANY-PARTICLE SYSTEMS299(a)(b)Figure6.1:(a)A composite system is prepared such that there are N argon atoms in container A and N argon atoms in container B.The two containers are at the same temperature T and have the same volume V.What is the change of the entropy of the composite system if the partition separating the two containers is removed and the two gases are allowed to mix?(b)A composite system is prepared such that there are N argon atoms in container A and N helium atoms in container B.The other conditions are the same as before.The change in the entropy when the partition is removed is equal to2Nk ln2.(c)The fact that(6.31)yields an entropy that is not extensive does not indicate that identicalparticles must be indistinguishable.Instead the problem arises from our identification of S with ln Z as mentioned in Section4.6,page199.Recall that we considered a system withfixed N and made the identification that[see(4.106)]dS/k=d(ln Z+βE).(6.32) It is straightforward to integrate(6.32)and obtainS=k(ln Z+βE)+g(N),(6.33) where g(N)is an arbitrary function only of N.Although we usually set g(N)=0,it is important to remember that g(N)is arbitrary.What must be the form of g(N)in order that the entropy of an ideal classical gas be extensive?Entropy of mixing.Consider two containers A and B each of volume V with two identical gases of N argon atoms each at the same temperature T.What is the change of the entropy of the combined system if we remove the partition separating the two containers and allow the two gases to mix[see Figure6.1)(a)]?Because the argon atoms are identical,nothing has really changed and no information has been lost.Hence,∆S=0.In contrast,suppose that one container is composed of N argon atoms and the other is composed of N helium atoms[see Figure6.1)(b)].What is the change of the entropy of theCHAPTER6.MANY-PARTICLE SYSTEMS300 combined system if we remove the partition separating them and allow the two gases to mix? Because argon atoms are distinguishable from helium atoms,we lose information about the system, and therefore we know that the entropy must increase.Alternatively,we know that the entropy must increase because removing the partition between the two containers is an irreversible process. (Reinserting the partition would not separate the two gases.)We conclude that the entropy of mixing is nonzero:∆S>0(entropy of mixing).(6.34) In the following,we will derive these results for the special case of an ideal classical gas.Consider two ideal gases at the same temperature T with N A and N B particles in containers of volume V A and V B,respectively.The gases are initially separated by a partition.We use(6.27) for the entropy andfindS A=N A k ln V AN B+f(T,m B) ,(6.35b)where the function f(T,m)=3/2ln(2πmkT/h2)+5/2,and m A and m B are the particle masses in system A and system B,respectively.We then allow the particles to mix so that theyfill the entire volume V=V A+V B.If the particles are identical and have mass m,the total entropy after the removal of the partition is given byS=k(N A+N B) ln V A+V BN A+N B−N A ln V AN B (identical gases).(6.37)Problem6.8.Entropy of mixing of identical particles(a)Use(6.37)to show that∆S=0if the two gases have equal densities before separation.WriteN A=ρV A and N B=ρV B.(b)Why is the entropy of mixing nonzero if the two gases initially have different densities eventhough the particles are identical?If the two gases are not identical,the total entropy after mixing isS=k N A ln V A+V B N B+N A f(T,m A)+N B f(T,m B) .(6.38) Then the entropy of mixing becomes∆S=k N A ln V A+V B N B−N A ln V A N B .(6.39) For the special case of N A=N B=N and V A=V B=V,wefind∆S=2Nk ln2.(6.40)CHAPTER6.MANY-PARTICLE SYSTEMS301 Problem6.9.More on the entropy of mixing(a)Explain the result(6.40)for nonidentical particles in simple terms.(b)Consider the special case N A=N B=N and V A=V B=V and show that if we use the result(6.31)instead of(6.27),the entropy of mixing for identical particles is nonzero.This incorrectresult is known as Gibbs paradox.Does it imply that classical physics,which assumes that particles of the same type are distinguishable,is incorrect?6.2Classical Statistical MechanicsFrom our discussions of the ideal gas in the semiclassical limit we found that the approach to the classical limit must be made with care.Planck’s constant appears in the expression for the entropy even for the simple case of an ideal gas,and the indistinguishability of the particles is not a classical concept.If we work entirely within the framework of classical mechanics,we would replace the sum over microstates in the partition function by an integral over phase space,that is,Z N,classical=C N e−βE(r1,...,r N,p1,...,p N)d r1...d r N d p1...d p N.(6.41)The constant C N cannot be determined from classical mechanics.From our counting of microstates for a single particle and the harmonic oscillator in Section4.3and the arguments for including the factor of1/N!on page295we see that we can obtain results consistent with starting from quantum mechanics if we choose the constant C N to be1C N=N! e−βE(r1,...,r N,p1,...,p N)d r1...d r N d p1...d p NCHAPTER6.MANY-PARTICLE SYSTEMS302 For a classical system in equilibrium with a heat bath at temperature T,the meanvalue of each contribution to the total energy that is quadratic in a coordinate equals1f= f(r1,...,r N,p1,...,p N)e−βE(r1,...,r N,p1,...,p N)d r1...d r N d p1...d p NCHAPTER6.MANY-PARTICLE SYSTEMS303whereǫ1=ap21with a equal to a constant.We have separated out the quadratic dependence of the energy of particle one on its momentum.We use(6.45)and express the mean value ofǫ1as∞−∞e−βE(x1,x2,p1,p2)dx1dx2dp1dp2(6.47a)= ∞−∞ǫ1e−β[ǫ1+˜E(x1,x2,p2)]dx1dx2dp1dp2∞−∞e−βǫ1dp1 e−β˜E dx1dx2dp2.(6.47c) The integrals over all the coordinates except p1cancel,and we have∞−∞e−βǫ1dp1.(6.48) As we have done in other contexts[see(4.84),page202]we can writeǫ1=−∂ǫ1=−∂2kT.(6.51)Equation(6.51)is an example of the equipartition theorem of classical statistical mechanics.The equipartition theorem is applicable only when the system can be described classically, and is applicable only to each term in the energy that is proportional to a coordinate squared. This coordinate must take on a continuum of values from−∞to+∞.Applications of the equipartition theorem.A system of particles in three dimensions has 3N quadratic contributions to the kinetic energy,three for each particle.From the equipartition theorem,we know that the mean kinetic energy is3NkT/2,independent of the nature of the interactions,if any,between the particles.Hence,the heat capacity at constant volume of an ideal classical monatomic gas is given by C V=3Nk/2as we have found previously.Another application of the equipartition function is to the one-dimensional harmonic oscillator in the classical limit.In this case there are two quadratic contributions to the total energy andCHAPTER6.MANY-PARTICLE SYSTEMS304 hence the mean energy of a one-dimensional classical harmonic oscillator in equilibrium with a heat bath at temperature T is kT.In the harmonic model of a crystal each atom feels a harmonic or spring-like force due to its neighboring atoms(see Section6.9.1).The N atoms independently perform simple harmonic oscillations about their equilibrium positions.Each atom contributes three quadratic terms to the kinetic energy and three quadratic terms to the potential energy. Hence,in the high temperature limit the energy of a crystal of N atoms is E=6NkT/2,and the heat capacity at constant volume isC V=3Nk(law of Dulong and Petit).(6.52)The result(6.52)is known as the law of Dulong and Petit.This result wasfirst discovered empiri-cally and is valid only at sufficiently high temperatures.At low temperatures a quantum treatment is necessary and the independence of C V on T breaks down.The heat capacity of an insulating solid at low temperatures is discussed in Section6.9.2.We next consider an ideal gas consisting of diatomic molecules(see Figure6.5on page345). Its pressure equation of state is still given by P V=NkT,because the pressure depends only on the translational motion of the center of mass of each molecule.However,its heat capacity differs from that of a ideal monatomic gas because a diatomic molecule has additional energy associated with its vibrational and rotational motion.Hence,we expect that C V for an ideal diatomic gas to be greater than C V for an ideal monatomic gas.The temperature dependence of the heat capacity of an ideal diatomic gas is explored in Problem6.47.6.2.2The Maxwell velocity distributionSo far we have used the tools of statistical mechanics to calculate macroscopic quantities of in-terest in thermodynamics such as the pressure,the temperature,and the heat capacity.We now apply statistical mechanics arguments to gain more detailed information about classical systems of particles by calculating the velocity distribution of the particles.Consider a classical system of particles in equilibrium with a heat bath at temperature T.We know that the total energy can be written as the sum of two parts:the kinetic energy K(p1,...,p N) and the potential energy U(r1,...,r N).The kinetic energy is a quadratic function of the momenta p1,...,p N(or velocities),and the potential energy is a function of the positions r1,...,r N of the particles.The total energy is E=K+U.The probability density of a microstate of N particles defined by r1,...,r N,p1,...,p N is given in the canonical ensemble byp(r1,...,r N;p1,...,p N)=A e−[K(p1,p2,...,p N)+U(r1,r2,...,r N)]/kT(6.53a)=A e−K(p1,p2,...,p N)/kT e−U(r1,r2,...,r N)/kT,(6.53b)where A is a normalization constant.The probability density p is a product of two factors,one that depends only on the particle positions and the other that depends only on the particle momenta. This factorization implies that the probabilities of the momenta and positions are independent. The probability of the positions of the particles can be written asf(r1,...,r N)d r1...d r N=B e−U(r1,...,r N)/kT d r1...d r N,(6.54)and the probability of the momenta is given byf(p1,...,p N)d p1...d p N=C e−K(p1,...,p N)/kT d p1...d p N.(6.55)CHAPTER6.MANY-PARTICLE SYSTEMS305 For notational simplicity,we have denoted the two probability densities by f,even though their meaning is different in(6.54)and(6.55).The constants B and C in(6.54)and(6.55)can be found by requiring that each probability be normalized.We stress that the probability distribution for the momenta does not depend on the nature of the interaction between the particles and is the same for all classical systems at the same temper-ature.This statement might seem surprising because it might seem that the velocity distribution should depend on the density of the system.An external potential also does not affect the velocity distribution.These statements do not hold for quantum systems,because in this case the position and momentum operators do not commute.That is,e−β(ˆK+ˆU)=e−βˆK e−βˆU for quantum systems, where we have used carets to denote operators in quantum mechanics.Because the total kinetic energy is a sum of the kinetic energy of each of the particles,the probability density f(p1,...,p N)is a product of terms that each depend on the momenta of only one particle.This factorization implies that the momentum probabilities of the various particles are independent.These considerations imply that we can write the probability that a particle has momentum p in the range d p asf(p x,p y,p z)dp x dp y dp z=c e−(p2x+p2y+p2z)/2mkT dp x dp y dp z.(6.56) The constant c is given by the normalization conditionc ∞−∞ ∞−∞ ∞−∞e−(p2x+p2y+p2z)/2mkT dp x dp y dp z=c ∞−∞e−p2/2mkT dp 3=1.(6.57)If we use the fact that ∞−∞e−αx2dx=(π/α)1/2(see the Appendix),wefind that c=(2πmkT)−3/2. Hence the momentum probability distribution can be expressed as1f(p x,p y,p z)dp x dp y dp z=2πkT 3/2e−m(v2x+v2y+v2z)/2kT dv x dv y dv z.(6.59) Equation(6.59)is known as the Maxwell velocity distribution.Note that its form is a Gaussian. The probability distribution for the speed is discussed in Section6.2.3.Because f(v x,v y,v z)is a product of three independent factors,the probability of the velocity of a particle in a particular direction is independent of the velocity in any other direction.For example,the probability that a particle has a velocity in the x-direction in the range v x to v x+dv x isf(v x)dv x= mCHAPTER6.MANY-PARTICLE SYSTEMS306Problem6.10.Is there an upper limit to the velocity?The upper limit to the velocity of a particle is the velocity of light.Yet the Maxwell velocity distribution imposes no upper limit to the velocity.Does this contradiction lead to difficulties? Problem6.11.Simulations of the Maxwell velocity distribution(a)Program LJ2DFluidMD simulates a system of particles interacting via the Lennard-Jones poten-tial(1.1)in two dimensions by solving Newton’s equations of motion numerically.The program computes the distribution of velocities in the x-direction among other pare the form of the velocity distribution to the form of the Maxwell velocity distribution in(6.60).How does its width depend on the temperature?(b)Program IdealThermometerIdealGas implements the demon algorithm for an ideal classicalgas in one dimension(see Section4.9).All the particles have the same initial velocity.The program computes the distribution of velocities among other quantities.What is the form of the velocity distribution?Give an argument based on the central limit theorem(see Section3.7) to explain why the distribution has the observed form.Is this form consistent with(6.60)? 6.2.3The Maxwell speed distributionWe have found that the distribution of velocities in a classical system of particles is a Gaussian and is given by(6.59).To determine the distribution of speeds for a three-dimensional system we need to know the number of microstates between v and v+∆v.This number is proportional to the volume of a spherical shell of width∆v or4π(v+∆v)3/3−4πv3/3→4πv2∆v in the limit ∆v→0.Hence,the probability that a particle has a speed between v and v+dv is given byf(v)dv=4πAv2e−mv2/2kT dv,(6.61) where A is a normalization constant,which we calculate in Problem6.12.Problem6.12.Maxwell speed distribution(a)Compare the form of the Maxwell speed distribution(6.61)with the form of the Maxwellvelocity distribution(6.59).(b)Use the normalization condition ∞0f(v)dv=1to calculate A and show thatf(v)dv=4πv2 m v,the most probable speed˜v,and the root-mean-square speed v rmsand discuss their relative magnitudes.(d)Make the change of variables u=v/ π)u2e−u2du,(6.63)where we have again used the same notation for two different,but physically related probability densities.The(dimensionless)speed probability density f(u)is shown in Figure6.2.CHAPTER 6.MANY-PARTICLE SYSTEMS 3070.00.20.40.60.81.00.00.5 1.0 1.5 2.0 2.5 3.0u max uu rmsu f(u)Figure 6.2:The probability density f (u )=4/√u ≈1.13,and theroot-mean-square speed u rms ≈1.22.The dimensionless speed u is defined by u ≡v/(2kT/m )1/2.Problem 6.13.Maxwell speed distribution in one or two dimensionsFind the Maxwell speed distribution for particles restricted to one and two dimensions.6.3Occupation Numbers and Bose and Fermi StatisticsWe now develop the formalism for calculating the thermodynamic properties of ideal gases for which quantum effects are important.We have already noted that the absence of interactions between the particles of an ideal gas enables us to reduce the problem of determining the energy levels of the gas as a whole to determining ǫk ,the energy levels of a single particle.Because the particles are indistinguishable,we cannot specify the microstate of each particle.Instead a microstate of an ideal gas is specified by the occupation number n k ,the number of particles in the single particle state k with energy ǫk .2If we know the value of the occupation number for each single particle microstate,we can write the total energy of the system in microstate s asE s = kn k ǫk .(6.64)The set of n k completely specifies a microstate of the system.The partition function for an ideal gas can be expressed in terms of the occupation numbers asZ (V,T,N )= {n k }e −βP k n k ǫk ,(6.65)CHAPTER6.MANY-PARTICLE SYSTEMS308 where the occupation numbers n k satisfy the conditionN= k n k.(6.66)The condition(6.66)is difficult to satisfy in practice,and we will later use the grand canonical ensemble for which the condition of afixed number of particles is relaxed.As discussed in Section4.3.6,one of the fundamental results of relativistic quantum mechanics is that all particles can be classified into two groups.Particles with zero or integral spin such as4He are bosons and have wavefunctions that are symmetric under the exchange of any pair of particles. Particles with half-integral spin such as electrons,protons,and neutrons are fermions and have wavefunctions that are antisymmetric under particle exchange.The Bose or Fermi character of composite objects can be found by noting that composite objects that have an even number of fermions are bosons and those containing an odd number of fermions are themselves fermions.For example,an atom of3He is composed of an odd number of particles:two electrons,two protons, and one neutron each of spin13In spite of its fundamental importance,it is only a slight exaggeration to say that“everyone knows the spin-statistics theorem,but no one understands it.”See Duck and Sudarshan(1998).CHAPTER6.MANY-PARTICLE SYSTEMS309n1n301011010Table6.2:The possible states of a three-particle fermion system with four single particle energy microstates(see Example6.1).The quantity n1represents the number of particles in the single particle microstate labeled1,etc.Note that we have not specified which particle is in a particular microstate.From Table6.2we see that the partition function is given byZ3=e−β(ǫ2+ǫ3+ǫ4)+e−β(ǫ1+ǫ3+ǫ4)+e−β(ǫ1+ǫ2+ǫ4)+e−β(ǫ1+ǫ2+ǫ3).(6.69)♦Problem6.14.Calculate。
International Journal of Project Management , Volume 28, Issue 3,April 2010, Pages 285-295Paul Bowen, Peter Edwards, Keith Cattell, Ian JayShow preview | Related articles | Related reference work articlesPurchase85Dynamics of R&D networked relationships and mergers and acquisitions in the smart card field Original ResearchArticleResearch Policy , Volume 38, Issue 9, November 2009,Pages 1453-1467 Zouhaïer M’ChirguiClose preview | Related articles | Related reference work articlesAbstract | Figures/Tables | ReferencesAbstractThis paper analyzes how the structure and the evolution of inter-firmagreements have shaped the development of the smart card industry. The aimis to establish a closer connection between the evolution of inter-firmagreements in the smart card industry and the patterns of change of technologyand demand in this new high-tech industry. Based on a proprietary databasecovering both collaborative agreements and mergers and acquisitions (M&As)occurring in this industry over the period 1992–2006, we find that the evolutionof technology and market demand shapes the dynamics of R&D networks andPurchaseM&As are likely to change the industry structure. We also find that a small group of producers – first-movers – still control the industry and technological trajectories. Their position arises not for oligopolistic reasons of marketstructure, but for technological and organizational reasons.Article Outline1. Introduction2. Theoretical background3. The smart card industry: delineating the boundaries and identifying the actors3.1. Defining the smart card3.2. The differentiated market(s)3.3. The actors3.4. The smart card oligopoly: a dual market structure4. Research methods4.1. Methodology4.2. SCIFA database5. Trends in inter-firm agreements and emergence of networks in the smart cardindustry6. The structure of the network6.1. Network evolution6.2. Major players and centrality7. ConclusionAcknowledgementsReferences86The role of industrial maintenance in the maquiladoraindustry: An empirical analysis Original Research ArticleInternational Journal of Production Economics, Volume 114,Issue 1, July 2008, Pages 298-307Shad DowlatshahiClose preview | Related articles | Related reference work articlesPurchaseAbstract | Figures/Tables | ReferencesAbstractThis study explored the role of industrial maintenance in the maquiladora industry. The maquiladora industry is a manufacturing system that utilizes the Mexican workforce and foreign investment and technology on the border region between the United States and Mexico. The issues related to industrial maintenance were studied through a survey instrument and 11 in-depth and extensive field interviews with experts of eight maquiladora industries in El Paso, TX and Juarez, Mexico. Based on an 86% response rate (with 131 usable questionnaires) and four major survey questions, statistical analyses were performed. The survey questions included: collaboration between the maintenance and other functional areas, likely sources of maintenance problems (equipment, personnel, and management), major common losses of maintenance problems, and the role of ISO certification in maintenance. Finally, additional insights and assessment of the results were provided.Article Outline1. Introduction1.1. Review of literature2. Evolution of and various approaches to maintenance3. Historical, operational characteristics and the importance of the maquiladora industry4. Research design4.1. Data collection4.2. The interviews with maquiladora managers5. Analyses of results5.1. Statistical analysis for question 15.2. Statistical analysis for question 25.3. Statistical analysis for question 3 5.4. Statistical Analysis for question 46.Conclusions and assessmentReferences87 A variable P value rolling Grey forecasting model forTaiwan semiconductor industry production OriginalResearch ArticleTechnological Forecasting and Social Change, Volume 72,Issue 5, June 2005, Pages 623-640Shih-Chi Chang, Hsien-Che Lai, Hsiao-Cheng YuClose preview | Related articles |Related reference work articlesAbstract | Figures/Tables | ReferencesAbstractThe semiconductor industry plays an important role in Taiwan's economy. In thispaper, we constructed a rolling Grey forecasting model (RGM) to predictTaiwan's annual semiconductor production. The univariate Grey forecastingmodel (GM) makes forecast of a time series of data without considering possible correlation with any leading indicators. Interestingly, within the RGM there is aconstant, P value, which was customarily set to 0.5. We hypothesized thatmaking the P value a variable of time could generate more accurate forecasts. Itwas expected that the annual semiconductor production in Taiwan should beclosely tied with U.S. demand. Hence, we let the P value be determined by theyearly percent change in real gross domestic product (GDP) by U.S.manufacturing industry. This variable P value RGM generated better forecaststhan the fixed P value RGM. Nevertheless, the yearly percent change in realGDP by U.S. manufacturing industry is reported after a year ends. It cannotserve as a leading indicator for the same year's U.S. demand. We found out thatthe correlation between the yearly survey of anticipated industrial productiongrowth rates in Taiwan and the yearly percent changes in real GDP by U.S. manufacturing industry has a correlation coefficient of 0.96. Therefore, we usedPurchasethe former to determine the P value in the RGM, which generated very accurate forecasts. Article Outline1.Introduction2. The semiconductor industry in Taiwan3. Rolling GM (1,1)4. Forecast Taiwan semiconductor production with RGM (1,1)5. Forecast Taiwan semiconductor production with variable P value RGM (1,1)6. ConclusionsAppendix A. AppendixA.1. 1998 Production forecast for the semiconductor industry under different PvaluesA.2. 1999 Production forecast for the semiconductor industry under different PvaluesA.3. 2000 Production forecast for the semiconductor industry under different PvaluesA.4. 2001 Production forecast for the semiconductor industry under different PvaluesA.5. 2002 Production forecast for the semiconductor industry under different PvaluesReferencesVitae88 Energy demand estimation of South Korea using artificial neural network Original Research ArticleEnergy Policy , Volume 37, Issue 10, October 2009, Pages4049-4054Zong Woo Geem, William E. Roper Close preview | Related articles | Related reference work articlesAbstract | Figures/Tables | ReferencesPurchaseAbstractBecause South Korea's industries depend heavily on imported energy sources (fifth largest importer of oil and second largest importer of liquefied natural gas in the world), the accurate estimating of its energy demand is critical in energy policy-making. This research proposes an artificial neural network model (a structure with feed-forward multilayer perceptron, error back-propagation algorithm, momentum process, and scaled data) to efficiently estimate the energy demand for South Korea. The model has four independent variables, such as gross domestic product (GDP), population, import, and export amounts. The data are obtained from diverse local and international sources. The proposed model better estimated energy demand than a linear regression model (a structure with multiple linear variables and least square method) or an exponential model (a structure with mixed integer variables, branch and bound method, and Broyden–Fletcher–Goldfarb–Shanno (BFGS) method) in terms of root mean squared error (RMSE). The model also forecasted better than the other two models in terms of RMSE without any over-fitting problem. Further testing with four scenarios based upon reliable source data showed unanticipated results. Instead of growing permanently, the energy demands peaked at certain points, and then decreased gradually. This trend is quite different from the results by regression or exponential model.Article Outline1. Introduction2. Artificial neural network model3. Case study of South Korea4. Results of linear regression model5. Results of exponential model6. Results of ANN model7. Validation of the ANN model8.Future estimation with different scenarios9. ConclusionsReferences89 Catching up through developing innovation capability: evidence from China's telecom-equipmentindustry Original Research ArticleTechnovation , Volume 26, Issue 3, March 2006,Pages359-368 Peilei FanShow preview | Related articles | Related reference work articlesPurchase90 Optimization of material and production to develop fluoroelastomer inflatable seals for sodium cooled fastbreeder reactor Original Research ArticleNuclear Engineering and Design , In Press, Corrected Proof, Available online 16 February 2011N.K. Sinha, Baldev RajShow preview | Related articles | Related reference work articlesPurchase Research highlights► Production of thin fluoroelastomer profiles by cold feed extrusion and continuous cure involving microwave and hot air heating. ► Use of peroxide curing in air during production . ► Use offluoroelastomers based on advanced polymer architecture (APA) for the production of profiles. ► Use of the profiles in inflatable seals for critical application of Prototype Fast Breeder Reactor. ► Tailoring of material formulation by synchronized optimization of material and production technologies to ensure that the produced seal ensures significant gains in terms of performance and safety in reactor under synergistic influences of temperature, radiation, air and sodium aerosol.91 The dynamic transfer batch-size decision for thin film transistor –liquid crystal display array manufacturing by artificialneural-network Original Research ArticleComputers & Industrial。
