a r X i v :c o n d -m a t /9610059v 3 [c o n d -m a t .s t a t -m e c h ] 9 O c t 1997Reaction-Di?usion Processes from Equivalent Integrable Quantum Chains Malte Henkel a ,Enzo Orlandini b,c and Jaime Santos
b a Laboratoire de Physique des Mat′e riaux 1,Universit′e Henri Poincar′e Nancy I,BP 239,F -54506Vand?uvre-l`e s-Nancy Cedex,France ,b Theoretical Physics,Department of Physics,University of Oxford,1Keble Road,Oxford OX13NP,UK
c Service de Physique Th′e orique,CEN Saclay,F -91191Gif-sur-Yvette Cedex,France 2Abstract One-dimensional reaction-di?usion systems are mappe
d through a similarity transformation onto integrabl
e (and a priori non-stochastic)quantum chains.Time-dependent properties o
f these chemical models can then be found exactly.The reaction-di?usion processes related to free fermion systems with site-independent interactions are classi?ed.The time-dependence of the mean particle density is calculated.Furthermore new integrable stochastic processes related to the Heisenber
g XXZ chain are identi?ed and the relaxation times for the particle density and density correlation for these systems are found.PACS:02.50.Ga,05.40.+j,82.20.Mj
1Introduction
The physics of interacting particles out of thermodynamic equilibrium is an intriguing and challenging problem.For important reaction-di?usion processes which display a wide variety of interesting and unexpected phenomena,see[1,2,3,4,5]and references therein.They are not only of experimental importance but also pose a theoretical problem of considerable di?culty as there is no general framework for their treatment,such as detailed balance for equilibrium systems.While the classic approach, involving phenomenological rate equations,is su?cient to the describe the time-dependent behavior of non-equilibrium systems in su?ciently high spatial dimensions,in low dimensions the di?usive mixing is ine?ective and new phenomena appear.
As an example,consider a system consisting of a single species of particles A undergoing di?usion and subject to the two chemical reactions A+A→A with rate r c and A+A→inert with rate r a.For this system one expects,starting from an initial particle densityρ0,at late times an algebraic decay of the mean particle concentration[6]
ρA(t)～A t?y;t→∞(1.1) In one-dimensional systems,one has y=1
group(see e.g.[13]),improved mean-?eld approaches or exact calculations,see[2]for a collection of reviews.It is the aim of this paper to explore some of the possibilities of obtaining exact results.
A convenient starting point for an exact description of a stochastic process is the formulation of the problem in terms of a master equation.The underlying assumption of this approach,viz.the absence of memory e?ects,is at least approximately true for many real systems.A master equation(see section 2)is a?rst order linear di?erential equation in time for the probability distribution of the stochastic variables describing the https://www.doczj.com/doc/4314319090.html,ing standard procedures,see[14,16,15,17]it may be conveniently written as a matrix equation and thus its solution is turned into an eigenvalue problem3.The crucial point is that it has been realized in recent years that for many models of interest the matrix appearing in this eigenvalue problem is the quantum Hamiltonian(or the transfer matrix respectively,in the case of discrete time dynamics)of some integrable quantum chain[18,19,20,21,22,23,24,25,26,27,28, 29,30,31,32,33,34].This insight has made available the tool box of integrable models(e.g.Bethe ansatz[35,36,37])for these interacting particle systems far from equilibrium and has led to many new exact results for their dynamical and stationary properties.Clearly,the models tractable by such means are usually rather simple,but the observed universality of many quantities of interest make these models both theoretically and experimentally interesting.
Most integrable stochastic models studied so far are directly given in terms of an integrable quantum chain or transfer matrix.In these cases,“just by looking”at the problem one recognizes some known integrable system.However,it has been observed recently that one may obtain from a given stochastic quantum Hamiltonian by a similarity transformation the Hamiltonian of some other stochastic process, see[38,39,10,40].This relates a system which is not obviously integrable to another system which is and in this way exact results may be obtained.At the same time it was also realized that indeed one may obtain by a similarity transformation a stochastic Hamiltonian from some integrable Hamiltonian which does not describe a stochastic process[25,10].This is implicit in many earlier treatments of reaction-di?usion processes,but has not yet been exploited systematically.It shows that many more stochastic systems than previously thought are actually integrable.
In this paper,our strategy is as follows.We start from some(a priori non-stochastic)integrable quantum chain H.The examples we shall consider can be analyzed using Bethe ansatz and/or free fermion techniques.Next we perform a similarity transformation and ask under which conditions on the parameters of the original integrable quantum chain H and on the transformation matrix one obtains
a stochastic Hamiltonian.In this way we identify several classes of new integrable stochastic models. Finally,we use the knowledge provided by the integrability to obtain some exact results for these processes,in particular,the average density at time t,and the(longest)relaxation time in the system.
Pursuing this program in full generality appears to be a formidable task.Therefore we shall re-strict ourselves to systems which are(a)translationally invariant,(b)involve particles of only one kind with hard core repulsion preventing double occupancy of a single site and(c)which interact only via e?ective nearest neighbor interaction.We shall make at various points comments on generalizations, e.g.to other boundary conditions,but we shall not work out any details in these cases.Finally,we shall consider only similarity transformations which are translationally invariant and which preserve the locality of all observables.This excludes e.g.sublattice transformations[41]or duality transfor-mations[42,43]which also give rise to transformations between di?erent stochastic Hamiltonians or between non-stochastic and stochastic Hamiltonians.We would like to stress that we shall consider only continous-time dynamics.However,all our results apply also to models with suitably chosen discrete time dynamics.
The paper is organized as follows:in section2we review the Hamiltonian formulation for non-equilibrium stochastic systems and its connection with integrable quantum Hamiltonians of magnetic systems.We also specify the similarity transformations which will be used to link stochastic systems with integrable ones and comment brie?y on the relationship with continuum?eld theory.In section 3,we give the full classi?cation of the stochastic systems which can be reduced to a free fermion Hamiltonian with site-independent interactions.In section4,we?nd stochastic systems which have the same spectrum as a XXZ chain with real matrix elements.In section5,we use the link with the free fermion case to calculate explicitly the time-dependent particle density.Section6gives our conclusions. More technical material is relegated to the appendices.In appendix A,we extended the standard technique of Lieb-Schultz-Mattis for the diagonalization of a hermitian free fermion Hamiltonian to the non-hermitian case.Appendix B gives further details for the classi?cation of those stochastic systems similar to a free fermion Hamiltonian.In appendix C we recall the relationship of one of the stochastic systems related to free fermions to the biased voter model.Appendix D gives some further details for the construction of stochastic processes from the XXZ chain.Appendix E describes the details of the caculation of the long-time behaviour of particle densities and two-point correlators from both the Bethe ansatz and the truncated equations of motion.Appendix F recalls the relationship between the Hamiltonian spectra for periodic and free boundary conditions.
2Quantum chain formulation of stochastic processes
In this section,we review the reformulation of a non-equilibrium stochastic system de?ned by some master equation in terms of the spectral properties of an associated quantum Hamiltonian H.To be speci?c,we only consider systems de?ned on a chain with L sites and two allowed states per site.We represent the states of the system in terms of spin con?gurations{σ}={σ1,σ2,...,σL}whereσ=+1
corresponds to an empty site andσ=?1corresponds to a site occupied by a single particle.We are interested in the probability distribution function P(σ;t)of the con?gurations{σ}.Our starting point is the master equation for the P(σ;t)
?t P(σ;t)= τ=σ w(τ→σ)P(τ;t)?w(σ→τ)P(σ;t) (2.1) where w(τ→σ)are the transition rates between the con?gurations{τ}and{σ}and are assumed to be given from the phenomenology.In order to rewrite this as a matrix problem,we introduce a state vector
|P = σP(σ;t)|σ (2.2) and eq.(2.1)becomes
?t|P =?H|P (2.3) where the matrix elements of H are given by
σ|H|τ =?w(τ→σ)ifτ=σ, σ|H|σ = τ=σw(σ→τ)(2.4) The operator H describes a stochastic process since all the elements of the columns add up to zero. This conservation of probability,viz. σP(σ;t)=1,is equivalent to the relation
s|H=0(2.5) where s|= σ σ|is a left eigenvector of H with eigenvalue0.Correspondingly,H has at least one right eigenvector with eigenvalue0.Such a vector does not evolve in time and therefore corresponds to a steady state distribution of the system.Since in general H is not symmetric,this steady state vector may be highly non-trivial.Note that all this is completely general and applies to any stochastic process de?ned by a master equation.With a view on the processes that we shall study we call H a quantum Hamiltonian and this formulation of the master equation the quantum Hamiltonian formalism.The reason for this choice of language is the fact that for the processes studied below(and,in fact,many
other processes as well)H is the Hamiltonian of some quantum system such as the Heisenberg XXZ Hamiltonian.The steady state of a stochastic system corresponds in this mapping to the ground state of the quantum system.
Now,it is straightforward to translate the well-known theorems[44]about the solution of the master equation(2.1)to the Hamiltonian formulation at hand.In particular,the real parts of the eigenvalues E i of H are non-negative,?E i≥E0=0,and there are no purely imaginary eigenvalues,which excludes the
possibility of undamped oscillations.Furthermore,if the initial distribution P0({σ})=P(σ;0)= σ|P0 satis?es0≤P0({σ})≤1and s|P0 =1,it follows that since
|P =exp(?Ht)|P0 (2.6) the relations0≤ σ|P ≤1and s|P =1hold for all times t.This guarantees the probabilistic interpretation.Then time-dependent averages of an observable F are given by the matrix element
s| H=0, σ| H|τ ≤0for allσ=τ(2.8) it follows that H is the generator of a stochastic process[44].These conditions will play a major role later on.We shall refer to the?rst as probability conservation condition and to the second as positivity conditions.
In what follows,we shall be mainly interested in averages of particle numbers n j at site j and their correlators.These can be expressed in the quantum spin formulation in terms of the projector
1
?n j=
4We stress that the structure of these matrix elements is quite distinct from expectation values in ordinary quantum mechanics where one always considers matrix elements between right and left states that are hermitean conjugates.
hand,if there is in the L→∞limit a continuous spectrum down to E0=0,one expects an algebraic approach of the correlators to the steady state.
The recent interest in this setup comes from the integrability of quantum Hamiltonians in one dimension[19,21,23,24,25,26,27]5.As the paradigmatic example,consider the asymmetric simple exclusion process where particles are di?using to the right,A+?→?+A with rate1/q and di?using to the left?+A→A+?with rate q.The Hamiltonian generating the time evolution of the system can,for free boundary conditions,be written in the form[24]
H=?1
2 q+q?1 σz jσz j+1?1 ?1
2
and2depending on the
particle density[21,47,27].The basic mechanism behind this is explained in appendix F.
In fact,integrable quantum Hamiltonians can be found for much more general systems.We restrict our attention here to models with nearest-neighbor interactions and only consider a single species of particles.Unfortunately,there is no standard notation for the various rates.In table1we give the possible reaction-di?usion processes together with their rates,providing at the same time a dictionary between the notations used by di?erent authors.In this paper,we shall use the notation employed in [10].
For the time being and for purposes of illustration let us consider besides di?usion only those reactions which irreversibly reduce the number of particles(that is,βL,R=σL,R=ν=0).For temporary convenience we rescaleα→Dαand de?ne also
D= γLγRδLδR D LγLδL 5Or equally,from the integrability of transfer matrices for discrete time dynamics[20,22]
di?usion to the left D L w1,1(1,0)
A+?→?+A a23Γ1001 pair annihilation2αw1,1(0,0)
A+A→?+A a24Γ1101 coagulation to the leftγL w0,1(1,0)
A+?→?+?a13Γ1000 death at the rightδR w0,1(0,0)
?+A→A+A a42Γ0111 decoagulation to the rightβR w0,1(1,1)
?+?→A+A a41Γ0011 creation at the rightσR w0,1(0,1)
?+?→A+?a31Γ0010
[10][24]
2 q+q?1 (1+δ?γ)?α,h=1
2L?1
j=1 σx jσx j+1+σy jσy j+1+? σz jσz j+1?1
+h σz j+σz j+1?2 ?1
2
(σx±iσy)are the one-particle annihilation/creation operators.
Remarkably,the spectrum of H is independent of the terms contained in (2.16),that is spec(H )=spec(DH XXZ (
h,?
,δ
))[24].To see this,recall that the XXZ Hamiltonian conserves the number of particles while the reaction terms irreversibly decrease the total particle number.Thus,H can be written in a block diagonal form
H = N 0X δX αN 1X γ,δ
X αN 2X γ,δ...
...... (2.17)
where N n refers to the n -particle states and X are the reaction matrix elements.Because of the identity
det A X 0B =det A det B (2.18)
it is clear that the elements of (2.16)do not enter into the characteristic polynomial 6of H .The phase diagram for H XXZ (h,?,δ)is known [49,50].For our purposes,we need the following.From (2.13),only the portion of the phase diagram where h +?≥1is important for us.The spectrum always has a ?nite gap when h +?>1,which is realized whenever δ=0or q =1.Then the ground state is a trivial ferromagnetic frozen state.The spectrum is gapless for ?+h =1,where the system undergoes a Pokrovsky-Talapov transition.This situation occurs for δ=0and q =1.We have thus identi?ed the cases where the model approaches the steady state exponentially (non-vanishing gap)or algebraically (gapless).A special point is ?=0where H XXZ becomes a free fermion system and thus in certain cases not only the spectrum may be obtained,but also correlation functions can be calculated explicitly (see below).
While large classes of reaction-di?usion problems are now recognised as being integrable,see [23,25,26]for lists including systems with more than one kind of particles 7,it is still far from obvious how to exploit the algebraic structure hidden beneath it in order to get explicit expressions for the desired particle number correlators for ?=0.A very interesting alternative was proposed in [48],where subsets of observables were identi?ed for which closed equation of motion can be obtained,thus leading to partially integrable systems.In any case,it is important to realize that the knowledge about the spectrum (and,in certain cases such as the free fermion case ?=0,about correlators)does not
t .
come from the assumption that the Hamiltonian is stochastic,but from its integrability.It is therefore natural to ask whether one can?nd new integrable non-equilibrium systems from transformations of known integrable quantum chains.
Speci?cally,we shall investigate the transformation[38,39,10]
H=B H B?1,B=L j=1B j(2.19) where H is a quantum Hamiltonian with known properties, H a stochastic Hamiltonian and B j is the transformation matrix B acting on the site j.From now on,we shall focus on translationally invariant systems,i.e.we shall consider periodic boundary conditions.To study the e?ect of the transformation B,it is su?cient to consider the e?ect on a two-particle Hamiltonian H j,j+1,that is we write
H=
L
j=111?···?1j?1?H j,j+1?1j+2···?1L(2.20)
where1?is the unit matrix acting on the site?.For the model(2.14)with left-right symmetry,that is q=1,we have for example
H j,j+1=D
0?δ?δ?2α
01+δ?1?γ
0?11+δ?γ
0002(α+γ)
(2.21)
Transformations of the type(2.19)have been mainly studied when both H and H represent stochastic systems.Besides,an equivalent way of relating stochastic systems was developed[52]in the context of the Lagrangian formulation of the model(see below),which is more suitable for?eld theory techniques. In particular,these transformations allow to treat the general model(2.21),since the observables in the two systems are simply related[10,38,39,40,52].The results for the long-time behaviour of the one-and two-point functions for translationally invariant initial conditions are collected in table2.For the derivation,see appendix E.
From the spectrum of H,we would naively expect exponential factors e?2kδt to be present in the k-point correlator C k,but we also see that for example algebraic prefactors are not readily predicted from the spectrum alone.This expectation,however,is only valid provided there are no bound states in the system.This is so as long the model stays su?ciently close to the caseδ=α+γ?1,where the exact spectrum of H can be found from free fermion techniques.In general,however,one can show from the Bethe ansatz[53,35,36,37]that there does exist a bound state in the two-particle sector,which has the energy4δ+4?2??2/?.8While the results given in table2are valid for homogeneous initial
δ
exp(?2δt)t?1/2exp(?4δt)
>α+γ
9The above analysis assumes that the associated amplitudes are non-vanishing.This is not always so,as can be seen in the calculation of the correlator
10The radioactive decay of particles is described in this setting by the conditionsα=0andγ=δ.
11We limit ourselves to these two types of integrable systems for pragmatic reasons.In fact,for the second type of integrable system we consider here we merely use the integrability of the spectrum and do not even ask whether or not the entire system might be integrable.A su?cient criterion to establish integrability of the full Hamiltonian H is provided by showing that the H j,j+1satisfy a Hecke algebra.It can be shown that forδ=0in the model(2.14)the entire non-symmetric Hamiltonian matrix and not just the spectrum-determining part is integrable[24].See[23]for a list of integrable stochastic Hamiltonians through the Hecke algebra technique.
fermions.We shall give in the next section the complete list of stochastic systems with spatially constant reaction rates which can be transformed into this form.
2.The most general Hamiltonian with the spectrum of the XXZ chain and with real matrix elements.
Here we shall obtain a new3-parameter manifold(after normalization)of integrable stochastic processes.
Before starting this,however,it might be useful to recall brie?y how results obtained from the quan-tum Hamiltonian formulation of reaction-di?usion systems presented here compare with the Lagrangian techniques,as developed in[15,17,56].See[13]for a recent review.As for the quantum Hamiltonian formulation,the starting point is the master equation.The creation and destruction of particles is conveniently described in a Fock space formalism,where the state vector becomes
|P(t) = σp(n1,n2,...;t) a?1 n1 a?2 n2···|0 (2.22) where a?i creates a particle at site i and p(n1,n2,...;t)is the probability of having n1particles at site 1,n2particles at site2and so on at time t.Consider for example the case when only di?usion and annihilation occur.Then the Hamiltonian is
H=D (i,j)(a?i?a?j)(a i?a j)?2α i a2i?a?2i a2i → D(?a?)(?a)?2α a2?a?2a2 d d x(2.23) where a formal continuum limit is taken12.One then goes over to a path integral representation involving ?elds a(x,t)and a?(x,t)and an action S[a,a?].In order to reduce the calculation of avarages to the usual QFT expectation values,it is customary to perform a shift a?=1+ˉa.Then the action becomes [15,17,56]
S[a,ˉa]= ˉa?t a+D?ˉa?a+4αˉa a2+2αˉa2a2 dtd d x(2.24) Remarkably,since the particle number can only decrease,the diagrammatics of the associated?eld theory is greatly simpli?ed.In particular,there are no loop corrections to the propagator and thus no wave function renormalization[15,17,56]and only the reaction rateαneeds to be renormalized.This can be done by summing up the corresponding diagrams to all orders,with the result that the one-loop beta function is exact.From the renormalization group equations it then follows that the upper critical dimension is at d?=2and that the exponent y describing the decay of the mean particle number(1.1)
is given by y=min(1,d/2),in agreement with previous heuristic arguments[6,59].In particular,it transpires that the whole probability distribution for?uctuations becomes universal in the late time regime[13].
What is the analogue of the similarity transformation(2.19)in the Lagrangian formulation?To see this,consider the process with only di?usion,annihilation and coagulation present[52].The action takes the form,where V describes the interactions of the particles undergoing a reaction
Sλ= dtdxˉa ˙a?D?2a + dtdxdx′V(|x?x′|)[ˉa(x)ˉa(x′)+λˉa(x)]a(x)a(x′)(2.25) such that forλ=1only coagulation and forλ=2only annihilation reactions occur.Following [52],consider the potentials Vα,γfor pure annihilation and coagulation,respectively,to be of the form Vα(x)=αV(x)and Vγ(x)=γV(x).Then
2α+γ
λ=
which is diagonalizable through a Jordan-Wigner transformation followed by a Bogoliubov transfor-mation or by a canonical transformation as described in appendix A.Here,D 1,2,η1,2,h 1,2are free parameters.In the following,we write h =h 1+h 2,g =h 1?h 2and η2=η1η2.We shall consider in this section η=0.The case η=0is a special case of a Hamiltonian with XXZ spectrum and will be con-
sidered in the next https://www.doczj.com/doc/4314319090.html,ing (2.20),we can write down the matrix describing the nearest-neighbor interactions
H j,j +1=? h ?C 00η10g ?C D 100D 2?g ?C 0η200?h ?C (3.2)and we look for a matrix H
through the transformation H =B H B ?1,B =L j =1B j (3.3)
and describing a stochastic system.Following the logic described in the preceding section we want to calculate correlation functions of the stochastic system in terms of correlators of the non-stochastic model which we can solve.We therefore rewrite a correlator
s |?n x 1(t 1)...?n x k (t k )|P 0 = s |?n x 1exp ? H (t 1?t 2) ?n x 2...?n x k exp
? Ht k |P 0 (3.4)
in terms of operators transformed using B ,thus reducing this correlator to a matrix element computable through free fermion techniques.Inverting this strategy,one may ?rst calculate matrix elements in the free fermion system,and then obtain the corresponding quantity for all stochastic processes related to this system by choosing an appropriate B .This is the strength of this approach,illustrated in section 5for the density ?ρ(t )as a function of time for an uncorrelated initial state with initial density ?ρ0.A single calculation gives ?ρ(t )for all stochastic processes related to the free fermion system (3.1).
In order to ?nd a convenient parametrization for B we ?rst ?x the determinant of B to be 1which
is no loss of generality since B must be non-singular and H is independent of det B .We can then write B in a parametrization analogous to the Euler angles (alternatives are discussed in appendix B)
B = √a ?1
cosh φsinh φsinh φcosh φ b 00b ?1 (3.5)with transformation parameters a,b,φ.Of these,we can always arrange for b =1,since otherwise,its e?ect could be absorbed by rede?ning η1→b 4η1and η2→b ?4η2.So we are left with just two transformation parameters a and φ.Below,we shall work with Y de?ned by
Y =e 2φ
(3.6)
In order to ensure that H is stochastic we shall have to satisfy the conditions(2.8).Let C m= 4k=1( H j,j+1)k,m denote the sum of the elements of the m-th column of H.Then,up to boundary terms inσz1,L(which are absent for periodic boundary conditions)and up to an additive constant in H (which will be?xed later by specifying the constant C),probability conservation(2.8)is implemented by demanding that
C1= C4=1
2(?1+a+Y+aY)2,η2=
(D1+D2?2h)(?1+a+Y+aY)2
2(?L+?R),?′=
1
In this paper,we study exclusively periodic boundary conditions.Thus it is su?cient to impose the positivity conditons(2.8)on the parity symmetric combinations of the matrix elements of H and on two other combinations of transformed matrix elements in which the terms containing x,y cancel.This reduces the number of inequalities one has to satisfy by two.It is important to keep this distinction between the transformed matrix elements(which occur in H j,j+1)and the actual rates of the stochastic system(which occur in H phys j,j+1)which have all to be positive.
Next,we have to write down the elements of the transformed matrix H j,j+1in terms of the matrix elements of H j,j+1and the transformation parameters.To simplify expressions,we make use of the freedom to take the normalization of time scale
D1+D2=2(3.12) and we shall consider the other possible case,when D1+D2=0,in appendix https://www.doczj.com/doc/4314319090.html,ing the convention of(3.9),we get for the parity symmetric matrix elements
?2 α=a2(?1+a?Y2?aY2)·F1·N
? γ=1
(a2?1)(Y2?1)·F2·N
2a
? δ=a(Y2?1)(1?a+Y2+aY2)·F3·N
? β=?(Y2?1)(?1+a+Y2+aY2)·F3·N
? D=1
It is remarkable that the matrix elements can be factorised in terms of only very few building blocks. This observation is going to be essential in the classi?cation of the possible stochastic systems.Whether this factorization property has a deeper algebraic meaning is still open.Note that if we use the scaling 1
a(Y2?1)
8Y
σ′+x=(D1?D2+2g)1
a(Y2?1)
8Y
? β′?x=(D1?D2?2g)1
(Y2+1)(3.15)
4Y
In order to get a stochastic system from the matrix elements(3.13,3.15),we must implement the positivity conditions(2.8)on the parity symmetric rates α, γ, ν, σ, δ, β, D.These conditions completely determine the transformation matrix B.The physical systems thus found will have left-right symmetric reaction-di?usion rates.Then,in a second step and using the extra freedom coming from adding the divergence terms to H phys j,j+1as done in(3.11),we shall identify free fermion systems with left-right biased di?usion and reaction rates.
We now turn to the analysis of the positivity conditions(2.8).
3.2Reality of the transformation parameters a,Y
Before embarking on checking the positivity of the symmetric rates(3.13),it is useful to break the problem into smaller pieces.It is a necessary condition to positivity that all matrix elements of H are real.This reality condition allows us to distinguish a few subcases.First,for Y2=1,the transformation from H to H merely amounts to the diagonal transformation encountered before[61].So from now on we always assume Y=1,?1.Second,since the ratios γ′/ σ′and δ′/ β′must be real,it follows from (3.15)that
a2real(3.16) and we distinguish the following.
1.a real.Then,since2 α/ γis real,it follows that
a(?1+a?Y2?aY2)
and it follows that Y2must be real.In the?rst case,Y itself is real,and reality of H is guaranteed if h,g and D1?D2are real.In the second case,Y is purely imaginary and reality of the elements of H is assured if D1?D2,h and g are all imaginary.
2.a imaginary.We write a=iA with A real.Since2 ν/ σmust be real,it follows from(
3.13)that
?1+a+Y2+aY2
Y2?1=0(3.19) or,letting Y2=P+iQ,we?nd that P2+Q2=1and thus have
Y2 2=1if a is imaginary(3.20) In summary,we have to consider the following cases separately
A)a real and Y real,B)a real and Y imaginary,C)a imaginary and|Y|=1(3.21) and we now ask when the conditions α, β, γ, δ, ν, σ, D≥0can be simultaneously met.
3.3A no-go theorem
In this section,we prove the lemma stated below.The reader not interested in the details of the proof can go on to the next section,where the implications will be discussed.
Lemma:For the parity-symmetric reaction rates α, γ, νand σfrom eq.(3.13)with the scaling1
a=a(Y)such that the rates ?vanish.These curves are readily identi?ed from the factorized form of the matrix elements in eq.(3.13).
Case A:a,Y both real.
For h?xed,the changes of sign of αare along the curves
1+Y2
a=
1?h2
Y=0(3.23) γchanges sign along the curves a
±= h 1?Y2 ?1 h(1+Y2)+2Y 1±√
1+Y2
a±= h?4Y+6hY2?4Y3+hY4 ?1 1?Y2 h 1+Y2 ?2Y 1±√
1?h2 a=±1,a=0
Y=±1,Y=0(3.26) Next,we have to distinguish between the cases h2>1and h2≤1.In the?rst case,the a±given above are complex and will not play a role in dividing the(a,Y)plane into regions where the ?are positive or negative,respectively.In fact,in this case the boundaries are completely independent of h.In?gure1, we show the excluded(shaded)regions for the four rates α, γ, ν, σ.Comparing them,it follows that no region in the(a,Y)plane remains where the four inequalities(3.22)are all satis?ed.
In the second case,the a±are all real and must be taken into account.In order to illustrate the mutual exclusion of the allowed regions,we show in?gure2the excluded(shaded)regions for the four
rates for the special value h=1 3.In comparing the excluded regions for the di?erent rates,note that because of the factor structure of the matrix elements,the boundary curves in the di?erent?gures coincide.Again,it is apparent that(3.22)cannot be satis?ed.
Case B:a real and Y imaginary.
We write Y=iy and h=ik where y,k are both real.We?x k and look for the curves in the(a,y) plane where the ?have simple zeroes or poles.For α,these occur at
1?y2
a=
1+k2)
y=0(3.27) γchanges sign at a= k(1+y2) ?1 k(1?y2)+2y(1±√
1+y2
a= k?4y?6ky2+4y3+ky4 ?1 (1+y2) k(1?y2)?2y(1±√
1+k2) a=±1,a=0
y=0(3.30) In analogy what was done in case A,one can show that each region in the(a,y)plane is excluded by at least one of the four rates and thus(3.22)cannot be satis?ed.
Case C:a imaginary and|Y|=1.
We write a=iA and Y=cos q+i sin q with A,q real.We?x h and look for the simple zeroes or poles
52门户网站实训指导（一） 实训目标： 1 掌握门户网站的设计技巧。 2 掌握库项目和模板的制作及使用方法。 实训步骤： 一创建站点。 在Dreamweaver CS3中定义一个本地站点，命名为“52menhu”，存储位置为：D：\sample\52menhu\，并将本项目素材文件夹下的图像文件夹image复制到网站的根文件夹下。然后，选择“窗口”→“资源”命令或者按F11键，打开“资源”面板。如图1。 图1 资源面板图2 新建库项目 二创建库文件。 单击“资源”面板右下角的按钮，新建一个库，然后在列表框中输入库的名称banner并加以确认。使用同样的方法创建名称为foot
的库文件。 二、制作主页的Banner 基本步骤: 1.在“资源”面板中双击打开，找到库文件banner.lbi。 2.插入一个3行1列，宽为760像素的表格。 3.设置表格属性(具体步骤参照书讲解)。 4.插入图片，输入文本。 最终效果如图3所示: 图3 banner效果图 制作页脚 基本步骤： （1）在资源面板中打开库文件foot.lbi； （2）插入一个3行1列,宽为760像素的表格； （3）设置表格属性，参数如图4； 图4 表格属性 （4）输入文本，并保存文件。效果如图5所示。
图5 页脚效果图 ●四、库项目的应用 ●1.插入库文件 新建页面myindex.html，依据前面做的banner.lib和foot.lib 文件充实首页面myindex.html。打开“资源”面板，点击左下角的“插入”按钮，分别将两个库项目分别插入到页面对应位置。 2.编辑库项目 在资源面板中单击“编辑”按钮，可打开库项目进行编辑。编辑的方法同编辑普通页面一样。 任务2 制作女性频道子页 1、使用模板建立子页 操作步骤： （1）将素材中的首页复制到站点中。选择“文件”→“另存为模板”命令，将已有页面另存为模板，命名为moban1。 2、定义可编辑区域 操作步骤： （1）选择“插入”→“常用”命令，单击“可编辑区域”按钮或者选择“插入”→“模板对象”→“可编辑区域”命令，打开“新建可编辑区域”对话框。
创建PowerPoint 2007 模板 全部隐藏 要应用新的或已经存在的其他PowerPoint 2007 模板，请参阅应用PowerPoint 2007 模板。 本文内容 何为PowerPoint 2007 模板？ 模板不同于设计模板 创建模板的最佳做法 创建新模板 何为PowerPoint 2007 模板？ Microsoft Office PowerPoint 2007 模板是您另存为.potx 文件的一张幻灯片或一组幻灯片的图案或蓝图。模板可以包含版式（版式：幻灯片上标题和副标题文本、列表、图片、表格、图表、形状和视频等元素的排列方式。）、主题颜色（主题颜色：文件中使用的颜色的集合。主题颜色、主题字体和主题效果三者构成一个主题。）、主题字体（主题字体：应用于文件中的主要字体和次要字体的集合。主题字体、主题颜色和主题效果三者构成一个主题。）、主题效果（主题效果：应用于文件中元素的视觉属性的集合。主题效果、主题颜色和主题字体三者构成一个主题。）、背景样式，甚至可以包含内容。 您可以创建自己的自定义模板，然后存储、重用并与他人共享它们。您还可以在Office Online和其他合作伙伴网站上找到数百种不同类型的免费模板。 下面是Office Online 上包括（但不限于）的一些模板示例： 议程奖状小册子 预算名片日历 内容幻灯片合同数据库 设计幻灯片图表信封
费用报表传真表传单 表单礼品证书贺卡 存货单邀请函发票 标签信函清单 备忘录议定书新闻稿 计划规划表明信片 采购订单收据报表 简历日程表日程表 声明信纸时间表 模板可以包括以下组件： 主题特定的内容，例如结业证书、足球和足球图像 背景格式设置，例如图片、纹理、渐变或纯色填充色和透明度。此示例演示浅蓝色的纯色填充背景 颜色、字体、效果（三维、线条、填充、阴影，等等）和主题设计元素（例如词“足球”中的颜色和渐变效果） 占位符（占位符：一种带有虚线或阴影线边缘的框，绝大部分幻灯片版式中都有这种框。在这些框内可以放置标题及正文，或者是图表、表格和图片等对象。）中提示人们输入特定信息的文本，例如运动员姓名、教练姓名、演示日期和任何变量，如年份(2007) 返回页首
第八讲使用模板和库项目 在Dreamweaver中，利用模板和库项目能够创建具有统一风格的网页，利用这一功能还有助于简化对网站的维护，你可以在短时间内重新设计自己的网站并对数以百计的网页做出修改。 一、模板的应用 1．模板概述 模板实际上是一种特殊网页，可以通过模板页面产生许多形式相似的页面，和CSS样式表相区别的是，CSS样式表主要是对文字、超级链接等网页元素进行规范统一设置，而模板是对整个页面来说的，包括图片、表格、文字等等。由模板产生的页面和模板有着附属关系，模板的改变会导致由模板产生的页面发生变化。 模板的使用是有一定的条件的，只有在制作过程中包含有大量的网页、而且页面布局比较相近的情况下，才考虑使用模板。因为如果一页一页地制作，不仅浪费时间，还容易出错；在多人分工的情况下，如果不使用模板，每人制作的页面会有很大差别，而调用同样的模板，能够制作出一致效果的页面。 所谓模板，就是指一个文档，可以将该文档用作其它要创建的文档的基础。在创建模板的时候，可以指定哪些元素保持不变(也就是不可编辑)，哪些元素可以进行修改。举例说，如果准备发布在线杂志，报头应该是不会再发生什么变化了，但是每一期的标题和特写故事的内容肯定是要有变化的。为了指定特写故事的风格和位置，可以使用占位符文本，并将其定义为可编辑区域。当要添加新的特写时，只要选取占位符文本，并输入文
章取而代之即可。 即使模板已经被用来创建文档了，也是可以修改的。这样，在更新应用模板的文档时，文档中不可编辑的区域就会进行更新，继续同模板保持一致。 2．创建模板 可以利用现有的HTML文档进行必要的修改制作出符合需要的模板，当然也可以从空白文档开始创建模板。 创建的模板会自动存放在位于本地机器上的站点的根目录中的模板文件夹Templates中，扩展名为.dwt。如果该文件夹还不存在，Dreamweaver 会在保存新模板的时候自动创建。不要任意更改移动该文件夹，也不要更改模板文件的扩展名。 ⑴将现有的文档保存为模板 当网站中存在布局和样式的设计都比较满意的网页时，可以将这样的网页保存为模板，使其他页面的设计和修改都与此页保持同样的风格。 ①选择【文件】→【打开】，打开选定的文档。 ②选择【文件】→【另存为模板】。 ③在随后出现的对话框中，选取站点，并在另存为对话框中输入模板的名字。 ④点击保存。 ⑵创建新模板 ①选择【窗口】→【资源】→【模板】。 ②在模板面板上，执行操作：
实验名称：层叠样式、模板和库实验学时： 3学时 班级： 学号： 姓名： 指导教师： 实验时间：
一、实验目的和要求 1.掌握附加和编辑外部样式表的方法 2.掌握CSS样式的各种特殊效果的方法 3.掌握创建和应用CSS样式及CSS样式表的方法 4.掌握模板的创建、编辑和应用的方法 5.掌握对象的创建、编辑和应用的方法 二、实验内容及步骤 1.CSS样式表的创建、编辑和应用 （1）在网页编辑窗口中，打开本地站点My site\html 文件夹中的art.htm 文件。修改文档的标题为“生活和艺术”，文档每段文符改成首行缩进两个字样。 在本地根文件夹My site中，创建名为docformat.css的层叠样式表文件，并在其中创建背景色为#99FFCC的层叠样式docformat（打开该样式表，在文档窗口输入代码“docfornat{color:#99FFCC;}”）。 （2）创建名为.char1的层叠样式，并将这个样式定义在docformat.css的层叠样式表文件中。将其【类型】的参数设置：【字体】为方正舒体、【大小】为36像素、【样式】为斜体、【颜色】#FF0000。在art.htm文档中，将样式应用于标题文字“生活和艺术”。方法：打开“CSS”浮动面板，选择“CSS样式”中的“全部”，可以看到已经创建好的docformat.css层叠样式表，在该处单击右键，选择“新建”命令，在弹出的对话框内创建层叠样式，并为之设置参数（也可以通过编写代码的方式创建），创建完成后，执行“文件”—>“保存”，即可完成。 （3）将名为.char1的样式复制成名为.char2的层叠样式，将其【类型】的参数修改为：【字体】为楷体、【大小】为18像素、【样式】为正常、【颜色】为#000000。在art.htm文档中，将样式应用于第１段文字。 （4）新建名为 .char3的层叠样式，将其【类型】的参数设置为：【字体】为仿宋体、【大小】为18像素、【样式】为正常、【颜色】为＃333333。在art.htm 文档中，将样式应用于第2段文字，浏览网页观察效果。 （5）将名为.char3的样式修改为：【字体】为方正舒体、【大小】为16像素、【样式】为偏斜体、【颜色】为＃000099。修改方法：1.改变代码中相应
适用对象： 1， SW版本高于（含）2013者，可以完全适配本设计库及模板，低于2013亦可使用材质库，焊接库等。 2，零部件属性值设置乱，工程图属性链接失败的SW零部件。 3，做机械的基本功是仔细，没有仔细的功夫，不知道细心阅读的设计，请跳过。4，本站有很多台湾朋友使用繁体系统，遇到问题的时候，请首先考虑路径问题。内容说明： （为匹配贵公司的其他图纸，黄色部分请自行修改） 1. 零件，装配体，工程图（A1，A2，A3，A4）模板 零件模板： ?a) 自定义属性：设变版本，名称，设计者，设计日期 ?b) 配置特定属性：图号，型号，材料，质量，工艺，加工费，成本价，素材（如钣金下料尺寸1mm，如法兰需20mm钢板），包装，喷涂，图纸记录，备注（）；购入价，来源（此两项需要购入判断） ?c) 文档绘图标准基于ISO标准修改。（由于SW对GB绘图标准设置较为粗糙，有很多显示错误，所以我们推荐使用ISO） 装配体模板： ?d) 自定义属性：设变版本，名称，设计者，设计日期 ?e) 配置特定属性：图号，型号，材料，质量，成本价，包装，图纸记录，备注；购入价，来源（此两项需要购入判断） 工程图模板：（我们做了2套工程图模板，V1包含修订表，但我们还是推荐V1.5版本） ?g) 属性：制图，绘图日期，修订版本 ?h) 字体选用汉仪长仿宋体（为了让工程图转化为其他、如dwg的时候，最大程度的兼容，推荐选用宋体，或仿宋；这个长仿宋体，原本就不是什么国标要求）?i) 工程图表格属性值，除制图，和绘图日期链接到工程图属性外，其他属性均通过公式连接到零件及装配体。 ?j) 添加设计变更修订表。 ?k) 采用第三视角视图 2. 焊件轮廓： 3. DWG映射模板： ?用于直接导出到DWG文件，并区分图层，对应上色。 4. 材质库： 5. 标注样式库（快速增加公差）常用公差类型可以在制图时添加文件夹位置。