Digital Object Identi?er (DOI)10.1007/s00220-005-1462-y Commun.Math.Phys.(2005)
Communications in Mathematical Physics
Nonequilibrium Energy Pro?les for a Class of 1-D Models J.-P.Eckmann 1,2, ,L.-S.Young 3,
1
D′e partement de Physique Th′e orique,Universit′e de Gen`e ve,Gen`e ve,Switzerland 2
Section de Math′e matiques,Universit′e de Gen`e ve,Gen`e ve,Switzerland 3Courant Institute for Mathematical Sciences,New York University,251Mercer Street,New York,NY 10021,USA
Received:11March 2005/Accepted:21May 2005Published online:2005–?Springer-Verlag 2005
Abstract:As a paradigm for heat conduction in 1dimension,we propose a class of models represented by chains of identical cells,each one of which contains an energy storage device called a “tank”.Energy exchange among tanks is mediated by tracer particles,which are injected at characteristic temperatures and rates from heat baths at the two ends of the chain.For stochastic and Hamiltonian models of this type,we develop a theory that allows one to derive rigorously –under physically natural assump-tions –macroscopic equations for quantities related to heat transport,including mean energy pro?les and tracer densities.Concrete examples are treated for illustration,and the validity of the Fourier Law in the present context is discussed.
Contents
1.Introduction ................................
..2.Main Ideas .................................
..2.1General setup ............................
..2.2Proposed program:from local rules to global pro?les .......
..3.Stochastic Models .............................
..3.1The “random-halves”model ....................
..3.2Single-cell analysis .........................
..3.2.1Single cell in equilibrium with 2identical heat baths....
..3.2.2Chain of N cells in equilibrium with 2identical heat baths.
..3.3Derivation of equations of macroscopic pro?les ..........
..3.4Simulations .............................
..3.5Interpretation of results .......................
..3.6A second example ..........................
..
JPE is partially supported by the Fonds National Suisse. LSY is partially supported by NSF Grant #0100538.
2J.-P.Eckmann,L.-S.Young
4.Hamiltonian Models .................
............4.1Rotating disks models ..............
............4.1.1Dynamics in a closed cell........
............4.1.2Coupling to neighbors and heat baths..
............4.2Single-cell analysis ...............
............4.3Derivation of equations of macroscopic pro?les
............4.3.1Comparisons of models.........
............4.4Ergodicity issues ................
............4.5Results of simulations ..............
............4.6Related models .................
............
1.Introduction Heat conduction in solids has been a subject of intensive study ever since Fourier’s pio-neering work.An interesting issue is the derivation of macroscopic conduction laws from the microscopic dynamics describing the solid.A genuinely realistic model of the solid would involve considerations of quantum mechanics,radiation and other phenomena.In this paper,we address a simpler set of questions,viewing solids that are effectively 1-dimensional as modeled by chains of classical Hamiltonian systems in which heat transport is mediated by tracer particles.Coupling the two ends of the chain to unequal tracer-heat reservoirs and allowing the system to settle down to a nonequilibrium steady state,we study the distribution of energy,heat ?ux,and tracer ?ux in this context.
We introduce in this paper a class of models that can be seen as an abstraction of certain types of mechanical models.These models are simple enough to be amenable to analysis,and complex enough to have fairly rich dynamics.They have in common the following basic set of characteristics:Each model is made up of an array of iden-tical cells that are linearly ordered.Energy is carried by two types of agents:storage receptacles (called “tanks”)that are ?xed in place,and tracer particles that move about.Direct energy exchange is permitted only between tracers and tanks.The two ends of the chain are coupled to in?nite reservoirs that emit tracer particles at characteristic rates and characteristic temperatures;they also absorb those tracers that reach them.To allow for a broad range of examples,we do not specify the rules of interaction between tracers and tanks.All the rules considered in this paper have a Hamiltonian character,involving the kinetic energy of tracers.Formally they may be stochastic or purely dynamical,resulting in what we will refer to as stochastic and Hamiltonian models .
Via the models in this class,we seek to clarify the relation among several aspects of conduction,including the role of conservation laws,their relation to the dynamics within individual cells,and the notion of “local temperature”.We propose a simple recipe for deducing various macroscopic pro?les from local rules (see Sect.2.2).Our recipe is generic;it does not depend on speci?c characteristics of the system.When the local rules are suf?ciently simple,it produces explicit formulas that depend on exactly 4parameters:the temperatures and rates of tracer injection at the left and right ends of the chain.
For demonstration purposes,we carry out this proposed program for a few examples.Our main stochastic example,dubbed the “random-halves model”,is particularly sim-ple:A clock rings with rate proportional to √x,where x is the (kinetic)energy of the tracer;at the clock,energy exchange between tracer and tank takes place;and the rule of exchange consists simply of pooling the two energies together and randomly dividing –in an unbiased way –the total energy into two parts.Our main Hamiltonian example
Nonequilibrium Energy Pro?les for a Class of1-D Models3 is a variant of the model studied in[17,12].Here the role of the“tank”is played by a rotating disk nailed down at its center,and stored energy isω2,whereωis the angular velocity of the disk.Explicit formulas for the pro?les in question are correctly predicted in all examples.
In terms of methodology,this paper has a theory part and a simulations part.The the-ory part is rigorous in the sense that all points that are not proven are isolated and stated explicitly as“assumptions”(see the next paragraph).It also serves to elucidate the rela-tion between various concepts regardless of the extent to which the assumed properties hold.Simulations are used to verify these properties for the models considered.
Our main assumption is in the direction of local thermodynamic equilibrium.For our stochastic models,a proof of this property seems within reach though technically involved(see e.g.[5,22,10]and[11]);no known techniques are available for Hamilto-nian systems.Extra assumptions pertaining to ergodicity and mixing issues are needed for our Hamiltonian models.It is easy to“improve ergodicity”via model design,harder to mathematically eliminate the possibility of all(small)invariant regions.In the absence of perfect mixing within cells,actual pro?les show small deviations from those predicted for the ideal case.
In summary,we introduce in this paper a relatively tractable class of models that can be seen as paradigms for heat conduction,and put forth a program which–under natural assumptions–takes one from the microscopic dynamics of a system to its phe-nomenological laws of conduction.
2.Main Ideas
2.1.General setup.The models considered in this paper–both stochastic and Hamil-tonian–have in common a basic set of characteristics which we now describe.
There is a?nite,linearly ordered collection of sites or cells labeled1,2,...,N.In isolation,i.e.,when the chain is not in contact with any external heat source,the system is driven by the interaction between two distinct types of energy-carrying objects:?Objects of the?rst kind are?xed in place,and there is exactly one at each site.These objects play the role of storage facility,and serve at the same time to mark the energy level at?xed locations.For brevity and for lack of a better word,let us call them energy tanks.Each tank holds a?nite amount of energy at any one point in time;it is not to be confused with an in?nite reservoir.We will refer to the energy in the tank at site i as the stored energy at site i.
?The second type of objects are moving particles called tracers.Each tracer carries with it a?nite amount of energy,and moves from site to site.For de?niteness,we assume that from site i,it can go only to sites i±1.
With regard to microscopic dynamics,the following is assumed:When a tracer is at site i,it may interact–possibly multiple times–with the tank at that site.In each interaction, the two energies are pooled together and redistributed,so that energy is conserved in each interaction.The times of interaction and manner of redistribution are determined by the microscopic laws of the system,which depend solely on conditions within that site.These laws determine also the exit times of the tracers and their next locations.A priori there is no limit to how many tracers are allowed at each site.We stress that this tracer-tank interaction is the only type of interaction permitted:the tanks at different sites can communicate with each other only via the tracers,and the tracers do not“see”each other directly.
4J.-P.Eckmann,L.-S.Young All stochastic models considered in this paper are Markovian.Typically in stochastic rules of interaction,energy exchanges occur when exponential clocks ring,and energy is redistributed according to probability distributions.In Hamiltonian models,tracers are usually embodied by real-life moving particles,and energy exchanges usually involve some types of collisions.
The two ends of the chain above are coupled to two heat baths,which are in?nite reservoirs emitting tracers at characteristic temperature(and also absorbing them).It is sometimes convenient to think of them as located at sites0and N+1.The two baths inject tracers into the system according to certain rules(to be described).Tracers at site1 or N can exit the system;when they do so,they are absorbed by the baths.The actions of the two baths are assumed to be independent of each other and independent of the state of the chain.The left bath is set at temperature T L;the energies of the tracers it injects into the system are iid with a law depending on the model.These tracers are injected at exponential rates,with mean L.Similarly the bath on the right is set at temperature T R and injects tracers into the system at rate R.
To allow for a broad spectrum of possibilities,we have deliberately left unspeci?ed (i)the rules of interaction between tracers and tanks,and(ii)the coupling to heat baths, i.e.,the energy distribution of the injected tracers.(Readers who wish to see concrete examples immediately can skip ahead with no dif?culty to Sects.3.1and4.1,where two examples are presented.)We stress that once(i)and(ii)are chosen,and the4parameters T L,T R, L and R are set,then all is determined:the system will evolve on its own,and there is to be no other intervention of any kind.
Remark2.1.Our approach can be viewed as that of a grand-canonical ensemble,since we?x the rates at which tracers are injected into the system(which indirectly determine the density and energy?ux at steady state).An alternate setup would be one in which the density of tracers is given,with particles being replaced upon exit.In this alternate setup,the4natural intensive variables would be the temperatures T L and T R,the density of tracers(mean number of tracers per cell)and the mean energy?ux.For de?niteness, we will adhere to our original formulation.
We now introduce the quantities of interest.For?xed N,letμN denote the invariant measure corresponding to the unique steady state of the N-chain(assuming there is a unique steady state).The word“mean”below refers to averages with respect toμN.The main quantities of interest in this paper are
?s i=mean stored energy at site i;
?e i=mean energy of individual tracers at site i;
?k i=mean number of tracers at site i;
?E i=mean total energy at site i,including stored energy and the energies of all tracers present.
For simplicity,we will refer to e i as tracer energy and E i as total-cell energy.
We are primarily interested in the pro?les of these quantities,i.e.,in the functions i→s i,e i,k i and E i as N→∞with the temperatures and injection rates of the baths held?xed.More precisely,we?x T L,T R, L and R.Then spacing{1,2,...,N} evenly along the unit interval[0,1]and letting N→∞,the?nite-volume pro?les i→s i,e i,k i,E i give rise to functionsξ→s(ξ),e(ξ),k(ξ),E(ξ),ξ∈[0,1].It is these functions that we seek to predict given the microscopic rules that de?ne a system.
Nonequilibrium Energy Pro?les for a Class of1-D Models5 2.2.Proposed program:from local rules to global pro?les.We?x N,T L,T R, L and R,and consider an N-chain with these parameters.To determine the pro?les in Sect.2.1, we distinguish between the following two kinds of information:
(a)cell-to-cell traf?c,and
(b)statistical information pertaining to the dynamics within individual cells.
In(a),we regard the cells as black boxes,and observe only what goes in and what comes out.Where left-right exit distributions are known,standard arguments balancing energy and tracer?uxes give easily the mean number of tracers and energy transported from site to site.While these numbers are indicative of the internal states of the cells(for exam-ple,high-energy tracers emerging from a cell suggest higher temperatures inside),the pro?les we seek depend on more intricate relations than these numbers alone would tell us.
We turn therefore to(b).Our very na¨?ve idea is to study a single cell,and to bring to bear on chains of arbitrary length the information so obtained.We propose the following plan of action:
(i)Consider a single cell plugged to two heat baths(one on its left,the other on its
right),both of which are at temperature T and have injection rate ,T and being arbitrary.Finding the invariant measureμT, describing the state of the cell in this equilibrium situation is,in general,relatively simple compared to?ndingμN. (ii)Suppose the measureμT, has been found.We then look at an N-chain with T L= T R=T and L= R= ,and verify that the marginals at site i of the invariant measureμN are equal toμT, .(By the marginal at site i,we refer to the measure obtained by integrating out all variables pertaining to all sites=i.)
(iii)Once the family{μT, }is found and(ii)veri?ed,we assume that the structure com-mon to theμT, passes to all marginals ofμN as N→∞even when(T L, L)= (T R, R).More precisely,for allξ∈(0,1),we assume that all limit points of the marginals ofμN at site[ξN](where[x]denotes the integer part of x)inherit,as N→∞,the structure common toμT, .
We observe that(i)–(iii)alone are inadequate for determining the sought-after pro-?les,for they give no information on which T and are relevant at any given site. The main point of this program is that(a)and(b)together is suf?cient for uniquely determining the pro?les in question.
Remark2.2.Our rationale for(iii)is as follows:Fix an integer, .As N→∞,the gradients of temperature and injection rate on the sites centered at[ξN]tend to0,so that the subsystem consisting of these sites resembles more and more the situation in (ii).Though rather natural from the point of view of physics[4],this argument does not constitute a proof.Indeed our program is in the direction of proving the existence of well de?ned Gibbs measures and then assuming,when the system is taken out of equilibrium, that thermodynamic equilibrium is attained locally;in particular,local temperatures are well de?ned.The full force of local thermal equilibrium is not needed for our purposes; however.The assumption in(iii)pertains only to marginals at single sites.
The rest of this paper is devoted to illustrating the program outlined above in concrete examples.
6J.-P.Eckmann,L.-S.Young
3.Stochastic Models
3.1.The “random-halves”model.This is perhaps the simplest stochastic model of the general type described in Sect.2.1.The microscopic laws that govern the dynamics in each cell are as follows:Let δ>0be a ?xed number.Each tracer is equipped with two independent exponential clocks.Clock 1,which signals the times of energy exchanges with the tanks,rings at rate 1δ√x,where x is the (current)energy of the tracer.Clock 2,which signals the times of site-to-site movements,rings at rate √x .The stored energy at site i is denoted by y i .In the description below,we assume the tracer is at site i .
(i)When Clock 1rings,the energy carried by the tracer and the stored energy at site
i are pooled together and split randomly.That is to say,the tracer gets p(x +y i )units of energy and the tank gets (1?p)(x +y i ),where p ∈[0,1]is uniformly distributed and independent of all other random variables.
(ii)When Clock 2rings,the tracer leaves site i .It jumps with equal probability to sites
i ±1.If i =1or N ,going to sites 0or N +1means the tracer exits the system.It remains to specify the coupling to the heat baths.Here it is natural to assume that the energies of the emitted tracers are exponentially distributed with means T L and T R .
This completes the formal description of the model.
Remark 3.1.The rates of the two clocks are to be understood as follows:We assume the energy carried by the tracer is purely kinetic,so that its speed is √x .We assume also that a tracer travels,on average,a distance δbetween successive interactions with the tank,and a distance 1before exiting each site.
Remark 3.2.As we will show,the invariant measure does not depend on the value of δ,which can be large or small.The size of δdoes affect the rate of convergence to equilibrium,however.
Remark 3.3.While the tracers do not “see”each other in the sense that there is no direct interaction,their evolutions cannot be decoupled.The number of tracers present at a site varies with time.When two or more tracers are present,they interact with the tank whenever their clocks go off,thereby sharing information about their energies.A new tracer may enter at some random moment,bringing its energy to the pool;just as randomly,a tracer leaves,taking with it the energy it happens to be carrying at that time.
3.2.Single-cell analysis.
3.2.1.Single cell in equilibrium with 2identical heat baths.We consider ?rst the fol-lowing special case of the model described in Sect.3.1:N =1,T L =T R =T ,and L = R = .Each state of the cell in this model is represented by a point in
=∞
k =0 k (disjoint union ),
where k ={({x 1,...,x k },y):x ,y ∈[0,∞)}.Here {x 1,...,x k },is an unordered k -tuple representing the energies of the k tracers,y denotes the stored energy,and a point in k represents a state of the cell when exactly k tracers are present.
Remark 3.4.We motivate our choice of .During a time interval when there are exactly k tracers in the cell –with no tracers entering or exiting –it makes little difference
Nonequilibrium Energy Pro?les for a Class of1-D Models7 whether we think of the tracers as named,and represent the state of the cell by a point in[0,∞)k+1,or if we think of them as indistinguishable,and represent the state by a point in k.With tracers entering and exiting,however,thinking of tracers as named will require that all exiting tracers return later,otherwise the system is transient and has no invariant measure.Since any rule that assigns to each departing tracer a new tracer to carry its name is necessarily arti?cial,and for present purposes exact identities of tracers play no role,we have opted to regard the tracers as indistinguishable.
We clarify the relationship between[0,∞)k+1and k and set some notation:Let πk:[0,∞)k+1→ k be the mapπk(x1,...,x k,y)=({x1,...x k},y),i.e.,πk is the (k!)-to-1map that forgets the order in the ordered k-tuple(x1,...,x k).For a measure?μon[0,∞)k+1that is symmetric with respect to the x coordinates,ifμ=(πk)??μ,and ?σandσare the densities of?μandμrespectively,then?σandσare related by
σ({x1,...,x k},y)=k!?σ(x1,...,x k,y).
We also write d{x1,...,x k}dy=(πk)?(dx1...dx k dy),and use I to denote the char-acteristic function.
Proposition3.5.The model in Sect.3.1with N=1,T L=T R=T,and L= R= has a unique invariant probability measureμ=μT, on .This measure has the
following properties:
?the number of tracers present is a Poisson random variable with meanκ≡2 √
π/T,
i.e.,
μ( k)=κk
k!
e?κ,k=0,1,2,...;(1)
?the conditional density ofμon k is c kσk d{x1,...,x k}dy,where
σk({x1,...,x k},y)=I{x1,...,x k,y≥0}
1
√
x1·...·x k
e?β(x1+···+x k+y);(2)
hereβ=1/T,and c k=βk!(β/π)k/2is the normalizing constant.
Proof.Uniqueness is straightforward,since one can go from a neighborhood of any point in to a neighborhood of any other point via positive measure sets of sample paths.We focus on checking the invariance ofμas de?ned above.
For z,z ∈ ,let P h(dz |z)denote the transition probabilities for time h≥0starting from z.We?x a small cube A? ˉk for someˉk,and seek to prove that
d dh
I A(z )P h(dz |z)
μ(dz)
h=0
=0.
On the time interval(0,h),the following three types of events may occur: Event E1:Entrance of a new tracer
Event E2:Exit of a tracer from the cell
Event E3:Exchange of energy between a tracer and the tank
We claim that with initial distributionμ,the probability of more than one of these events occurring before time h is o(h)as h→0.This assertion applies to events both of the
8J.-P.Eckmann,L.-S.Young same type and of distinct types.It follows primarily from the fact that these events are independent and occur at exponential rates.Of relevance also are the exponential tails of σk and the Poisson distribution of p k :=μ( k )in the de?nition of μ.To illustrate the arguments involved,we will verify at the end of the proof that the probability of two or more tracers exiting on the time interval (0,h)is o (h),but let us accept the above assertion for now and go on with the main argument.
Starting from the initial distribution μ,we let P (E i ,A)denote the probability that E i occurs before time h resulting in a state in A ,and let P (E c 1∩E c 2∩E c 3,A)denote the probability of starting from a state in A and having none of the E i occur before time h .We will prove
P (E 1,A)+P (E 2,A)+P (E 3,A)+P (E c 1∩E c 2∩E c 3,A)?μ(A)=o (h)μ(A).(3)
Notice that A can be represented as the union of disjoint sets ∪i A i ,where each A i is of the form
A ε(ˉz )={({x 1,...,x ˉk },y):x ∈[ˉx
,ˉx +ε], =1,...,ˉk,y ∈[ˉy,ˉy +ε]}for some ˉz =({ˉx 1,...,ˉx ˉk },ˉy)∈ ,ε 1,and with the intervals [ˉx ,ˉx +ε]pairwise disjoint for =1,2,...,ˉk.To prove (3)for A ,it suf?ces to prove it for each A i provided o (h)in (3)is uniformly small for all i .We consider from here on A =A ε(ˉz )with the properties above.Let σdenote the density of μ.With εsuf?ciently small,we have μ(A)≈σ(ˉz )εˉk +1=p ˉk c ˉk σˉk (ˉz )εˉk +1,where c k and σk are as in the proposition.The other terms in (3)are estimated as follows:P (E 1,A):E 1is in fact the union of 2ˉk subevents,corresponding to a new tracer coming from the left or right bath and the ˉk approximate values of energy of the new tracer.For de?niteness,we assume the new tracer arrives from the left bath,and has energy in
[ˉx 1,ˉx 1+ε].That is to say,the initial state of the cell is described by
B ={({x 2,...,x ˉk },y):x ∈[ˉx ,ˉx +ε],y ∈[ˉy,ˉy +ε]}? ˉk ?1.
The contribution to P (E 1,A)of this subevent is μ(B)h ˉx 1+εˉx 1
βe ?βx dx ≈h μ(B)e ?βˉx 1βε≈h p ˉk ?1c ˉk ?1σˉk (ˉz ) ˉx 1βεˉ
k +1.Here,h is the probability that a tracer is injected,and the integral above is the proba-bility that the injected tracer lies in the speci?ed range.Summing over all 2ˉk subevents,we obtain
P (E 1,A)≈2h β(ˉk =1 ˉx )p ˉk ?1c ˉk ?1σˉk (ˉz
)εˉk +1.(4)
P (E 2,A):In order to result in a state in A ,the initial state must be in
C 1={({x 1,...,x ˉk ,x },y):x ∈[ˉx ,ˉx +ε],x ∈[0,∞),y ∈[ˉy,ˉy +ε]}? ˉk +1.We assume here that the tracer with energy x exits between time 0and time h .This gives P (E 2,A)≈p ˉk +1c ˉk +1σˉk (ˉz )εˉk +1 ∞0
min {h √x,1}1√x e ?βx dx =p ˉk +1c ˉk +1σˉk (ˉz )εˉk +1β?1(h +o (h)).(5)
Nonequilibrium Energy Pro?les for a Class of 1-D Models 9P (E 3,A):For de?niteness,we assume it is the tracer with energy near ˉx 1that is the product of the interaction with the tank.To arrive in a state in A ε,one must start from
D ={({x 1,...,x ˉk },y):x ∈[ˉx
,ˉx +ε]for ≥2,x 1+y ∈[ˉx 1+ˉy,ˉx 1+ˉy +2ε]}.A simple integration using the rule of interaction in Sect.3.1gives
P (E 3,A)≈h ˉk =1√ˉx δp ˉk c ˉk σˉk (ˉz )εˉk +1.(6)
P (E c 1∩E c 2∩E c 3,A):We ?rst note that starting from A ,the probability of the tracer with energy ≈ˉx 1exiting is
p ˉk c ˉk σˉk (ˉz ) ˉx 1e βˉx 1εˉk ˉx 1+εˉx 1
h √x 1√x e ?βx dx =h ˉx 1p ˉk c ˉk σˉk (ˉz )εˉk +1;the probability of the tracer with energy ≈ˉx 1interacting with the tank is
h √ˉx 1δ
p ˉk c ˉk σˉk (ˉz )εˉk +1;and the probability of a new tracer entering the cell from the left (resp.right)bath is h μ(A ε).Thus P (E c 1∩E c 2∩E c 3,A)≈p ˉk c ˉk σˉk (ˉz )εˉk +1· (1?h ˉx )·(1?h )2· (1?h ˉx /δ)≈p ˉk c ˉk σˉk (ˉz )εˉk +1· 1?h ˉk =1 ˉx +2 +1δ
ˉk =1 ˉx .(7)
Summing Eqs.(4)–(7),we obtain (3)provided
2 βp ˉk ?1c ˉk ?1=p ˉk c ˉk and 2 p ˉk c ˉk =p ˉk +1c ˉk +1T .
Note that these two equations represent the same relation for different k .We write this relation as c k p k c k +1p k +1=T 2 ,(8)
and verify that it is compatible with assertion (1):Since
c k 1k !
k =1 1√x e ?βx dx e ?βy dy =1,and ∞0x ?1/2e ?βx dx =√πT ,we have
p k +1=2 T c k c k +1p k =2 T 1k +1 ∞01√x e ?βx dx p k =1k +12√π √T
p k .To complete the proof,we estimate the probability of two or more tracers exiting before time h .For n =2,3,...,let E 2,n be the event that the initial state is in
C n ={({x 1,...,x ˉk ,x (1),...,x (n)},y):x ∈[ˉx ,ˉx +ε],x (
)∈[0,∞),y ∈[ˉy,ˉy +ε]},
10J.-P.Eckmann,L.-S.Young and during the time interval(0,h),all n of the tracers carrying energies x( ), = 1,2,...,n,exit the system.The probability of∪n≥2E2,n is
n≥2
pˉk+n cˉk+nσˉk(ˉz)εˉk+1(O(h))n,
which,from Eq.(8),is bounded by
pˉk cˉkσˉk(ˉz)εˉk+1
n≥2
2
T
n
(O(h))n=pˉk cˉkσˉk(ˉz)εˉk+1o(h).
Remark3.6.In the setting of Proposition3.5,since the cell is in equilibrium with the two heat baths,it is obvious that it ejects,on average,2 tracers per unit time,and the energies of the tracers ejected have mean T.We observe that the cell in fact reciprocates the action of the bath more strongly than this:the distribution of the energies of the ejected tracers is also exponential.To see this,?x k and consider one tracer at a time. The probability of the tracer exiting with energy>u is
～ ∞
u
√
x·
1
√
x
e?βx dx=β?1e?βu.
3.2.2.Chain of N cells in equilibrium with2identical heat baths.We treat next the case of arbitrary N.That is to say,the system is as de?ned in Sect.3.1,but with T L=T R=T and L= R= .Letμbe as in Proposition3.5.
Proposition3.7.The N-fold productμ×···×μis invariant.
Remark3.8.That the invariant measure is a product tells us that at steady state,there are no spatial correlations.We do not?nd this to be entirely obvious on the intuitive level: one might think that above-average energy levels on the left half of the chain may cause the right half to be below average;that is evidently not the case.This result should not be confused with the absence of space-time correlations.
Proof.Proceeding as before,we consider a small time interval(0,h),and treat sepa-rately the individual events that may occur during this period.One of the new events (not relevant in the case of a single cell)is the jumping of a tracer from site i±1to site i.We?x a phase point
ˉz=(ˉz(1),...,ˉz(N))=({ˉx(1)1,...,ˉx(1)k
1},y(1);...;{ˉx(N)1,...,ˉx(N)k
N
},y(N)),
and let A= N i=1A(i)? N,where A(i)=Aε(ˉz(i))is as in Proposition3.5.Let σ(i)=σk i(ˉz(i)).For de?niteness,we?x also an integer n,1 (i)at site n+1,there are k n+1+1tracers the energies of which lie in disjoint intervals [ˉx(n+1) 1,ˉx(n+1) 1 +ε],...,[ˉx(n+1) k n+1 ,ˉx(n+1) k n+1 +ε]and[ˉx(n)1,ˉx(n)1+ε], (ii)at site n,there are k n?1tracers whose energies lie in [ˉx(n)2,ˉx(n)2+ε],...,[ˉx(n)k n ,ˉx(n)k n +ε]. Nonequilibrium Energy Pro?les for a Class of1-D Models11 The probability of this event occurring and the tracer with energy≈ˉx(n)1jumping from site n+1into site n is then given by1 2 I·II·II I,where I= i=n,n+1(p k i c k i σ(i)εk i+1), I I=p k n?1c k n?1 σ(n) ˉx(n)1eβˉx(n)1εk n, I II=p k n+1+1c k n+1+1 σ(n+1)εk n+1+1 ˉx(n) 1 +ε ˉx(n)1 h √ x 1 √ x e?βx dx. This product can be written as h 2 N i=1p k i c k i σ(i)εk i+1 ·p k n?1 p k n c k n?1 c k n ·p k n+1+1 p k n+1 c k n+1+1 c k n+1 · ˉx(n)1, which,by Eq.(8),is equal to h 2 N i=1p k i c k i σ(i)εk i+1 · ˉx(n)1.(9) There are many terms of this kind that contribute to I A(z )P h(dz |z)μ(dz),two forˉx(n) for each pair(n, ).We claim that the system has detailed balance,i.e.,the term associated with the scenario above is balanced by the probability of starting from a state in A and having the tracer at site n carrying energy≈ˉx(n)1jump to site n+1.The probability of the latter is 1 2( N i=1p k i c k i σ(i)εk i+1)· ˉx(n)1eβˉx(n)11 ε · ˉx(n) 1 +ε ˉx(n)1 h √ x 1 √ x e?βx dx, which balances exactly(9)as claimed. An argument combining the one above with that in Proposition3.5regarding the injection of new tracers holds at sites1and N. Propositions3.5and3.7are steps(i)and(ii)in the proposed scheme in Sect.2.2. 3.3.Derivation of equations of macroscopic pro?les.Having found a candidate family of equilibrium measures{μT, },we now complete the rest of the program outlined in Sect.2.2.The next step,according to this program,is to assume that for N 1,the marginals of the invariant measureμN at site i are approximately equal toμT, for some T=T i and = i.We identify those parts of our proposed theory that are not proved in this paper and state them precisely as“Assumptions”. Assumption1.Given T L,T R>0, L, R≥0,and N∈Z+,the N-chain de?ned in Sect.3.1with these parameters has an invariant probability measureμN. We do not believe this existence result is hard to prove but prefer not to depart from the main line of reasoning in this paper.Once existence is established,uniqueness(or ergodicity)follows easily since any invariant measure clearly has strictly positive density everywhere.A proof of the statement in Assumption2below is more challenging.For ξ∈(0,1),letμN,[ξN]denote the marginal ofμN at the site[ξN]. 12J.-P.Eckmann,L.-S.Young Assumption 2.For every ξ∈(0,1),every limit point as N →∞of μN,[ξN ]is a member of the family {μT , ,T >0, ≥0}.1 In Sect.2.1,we introduced four quantities of interest.There is one that was somewhat ambiguously de?ned,namely e i .Its precise meaning is as follows:e i := ∞k =1p i,k e i,k ,where p i,k is the probability that the number of tracers at site i is equal to k and e i,k is 1k of the mean total tracer energy conditioned on the number of tracers present being equal to k . Theorem 3.9is about the pro?les of certain observables as N →∞.We refer the reader to the end of Sect.2.1for the precise meaning of the word “pro?le”in the theorem.Theorem 3.9.The following hold for the “random-halves”model de?ned in Sect.3.1with arbitrary T L ,T R , L , R .Under Assumptions 1and 2above: ?the pro?le for the mean number of jumps out of a site per unit time is j (ξ)=2 L +( R ? L )ξ . ?the pro?le for the mean total energy transported out of a site per unit time is Q(ξ)=2 L T L +( R T R ? L T L )ξ ; ?the pro?le for the mean stored energy at a site is s(ξ)=Q(ξ)j (ξ) = L T L +( R T R ? L T L )ξ L +( R ? L )ξ;in the case L = R ,this simpli?es to s(ξ)=T L +(T R ?T L )ξ;?the pro?le for mean tracer energy is e(ξ)=12s(ξ); ?the pro?le for mean number of tracers is κ(ξ)= πs(ξ) j (ξ);?the pro?le for mean total-cell energy is E(ξ)=s(ξ)+κ(ξ)e(ξ)=s(ξ)+12 j (ξ).Proof of Theorem 3.9.We divide the proof into the following three steps: https://www.doczj.com/doc/db14616804.html,rmation on single cells.Items (i)–(iv)are strictly in the domain of internal cell dynamics .The setting is that of Proposition 3.5,and the results below are deduced (in straightforward computations)from the invariant density given by that proposition.The parameters are,as usual,T and .(Note that this means the rate at which tracers enter the site is 2 .) (i)stored energy has density βe ?βy and mean T ;(ii)tracer energy has density √β√πx e ?βx and mean T 2;(iii)mean number of tracers,κ=2 πT ;(iv)mean total-cell energy,E =T ·(1+κ2). 1As stated in Sect.2,we assume all marginals μN,[ξN ],N ≥1,have uniform tail bounds in energy and in number of tracers of the type suggested in the single-cell analysis. Nonequilibrium Energy Pro?les for a Class of 1-D Models 13Items (v)and (vi)are in preparation for the analysis of cell-to-cell traf?c : (v)mean number of jumps out of the cell per unit time,j =2 ; (vi)mean total energy transported out of the cell per unit time,Q =Tj . II.Global phenomenological equations.Consider now a chain with N cells with settings T L ,T R , L and R at the two ends.The following results use only standard conservation laws together with the local rule that when a tracer exits a cell,it has equal probability of going left and right. (A)Balancing tracer ?uxes.Let j i denote the number of jumps per unit time out of site i .Then j i =2 L +i N +1 ( R ? L ) .(10)Proof.Consider an (imaginary)partition between site i and site i +1.We let ? j i denote the ?ux across this partition.Then j i =12(j i +1?j i )for i =0,1,...,N ,where j 0and j N +1are de?ned to be 2 L and 2 R respectively.For i =0,N ,the 12is there because only half of the tracers out of site i +1jump left,and half of those out of site i jump right.The ?uxes across partitions between all consecutive sites must be equal,or there would be a pile-up of tracers somewhere.This together with i j i = R ? L gives the asserted formula. (B)Balancing energy ?uxes.Let Q i denote the mean total energy transported out of site i per unit time.Then Q i =2 L T L +i N +1( R T R ? L T L ) . Proof.The argument is identical to that in (A),with Q i =12(Q i +1?Q i )and Q 0and Q N +1de?ned to be 2 L T L and 2 R T R respectively. https://www.doczj.com/doc/db14616804.html,bining the results from I and II.Fix ξ∈(0,1).Passing to a subsequence if necessary,we have,by Assumption 2,μN,[ξN ]→μT (ξ), (ξ)for some T (ξ)and (ξ).We identify the two numbers T (ξ)and (ξ)as follows: Let j N (ξ)denote the mean number of jumps out of site [ξN ]per unit time in the N -chain.Then by (A)in Part II,j (ξ):=lim N →∞j N (ξ)=2 L +( R ? L )ξ .This is,therefore,the mean number of jumps out of a cell with invariant measure μT (ξ), (ξ).Similarly,we deduce that the mean total energy transported out of a cell with the same invariant measure is Q(ξ):=lim N →∞Q N (ξ)=2 L T L +( R T R ? L T L )ξ .Appeal- ing now to the information in Part I,we deduce from (v)and (vi)that (ξ)=12j (ξ)and T (ξ)=Q(ξ)/j (ξ).The rest of the pro?les follow readily:we read off s =T from (i),and deduce the relation between s and e by comparing (i)and (ii).The pro?les for κand E follow from (iii)and (iv)together with our knowledge of and T .The proof of Theorem 3.9is complete. We remark that in the terminology of [7]our random halves model is of gradient type . Remark 3.10.It is instructive to see what Theorem 3.9says in the special case when L =0.Since no particles are injected at the left end,clearly,T L cannot matter.But 14J.-P.Eckmann,L.-S.Young since tracers do exit from the left,one expects an energy?ux across the system.Upon substituting L=0into the formulas above,one gets j(ξ)=2 Rξ,e(ξ)=12s(ξ)=12T R,κ(ξ)=2 Rξ√ π/T R, and an energy?ux of?1 2 Q (ξ)=? R T R. 3.4.Simulations.Numerical simulations are used to validate Assumptions1and2in Theorem3.9. We mention here only those details of the simulations that differ from the theoretical study.Needless to say,we work with a?nite number of sites,usually20.Simulations start in a random initial state,and are?rst run for a period of time to let the system reach its steady state.All times are in number of events(E1,E2,and E3).Up to half the simulation time is used to reach stationarity;statistics are then gathered during the remaining simulation time.Since total simulation time is?nite,we?nd it necessary to take measures to deal with tracers of exceptionally low energy:these tracers appear to remain in a cell inde?nitely,skewing the statistics on the number of tracers.We opted to terminate events involving a single tracer at a single site after about0.001of total simulation time.This was done about10times in the course of109events. Simulations are performed both to verify directly the properties of the marginals at individual sites and to plot empirically the various pro?les of interest.Excellent agree-ment with predicted values is observed in all runs.A sample of the results is shown in Fig.1. 3.5.Interpretation of results.We gather here our main observations,discuss their phys-ical implications,and provide explanations for the reasons behind the derived formulas. 1.Linearity of pro?les.We distinguish between the following3types of pro?les: a.Transport of energy and tracers:j(ξ)and Q(ξ)are always linear by simple conserva-tion laws and by the imposed left-right symmetry in out?ow of tracers and energy from each site. b.Mean stored and(individual)tracer energies:s(ξ)and e(ξ)are linear if and only if there is no tracer?ux across the system.(See item4below.) c.Tracer densities and total-cell energy:κ(ξ)is never linear(unless T L=T R and L= R);in addition to the obvious bias brought about by different injection rates, tracers have a tendency to accumulate at the cold end(see item3for elaboration).As a result,E(ξ)is also never linear. 2.Heat?ux and the Fourier Law.Heat?ux from left to right is given by =T L L?T R R.Thinking of the temperature of the system as given by T(ξ),Theorem 3.9says that thermal conductivity is constant and proportional to T R?T L if and only if there is no tracer?ux across the system,i.e.,if and only if L= R. 3.Distribution of tracers along the chain.In the case L= R,more tracers are con-gregated at the cold end than at the hot.This is because the only way to balance the tracer equation is to have the number of jumps out of a site be constant along the chain. Inside the cells,however,tracers move more slowly at the cold end,hence they jump less frequently,and the only way to maintain the required number of jumps is to have more tracers.When L= R,the idea above continues to be valid,except that one needs also to take into consideration the bias in favor of more tracers at the end where the injection rate is higher. Nonequilibrium Energy Pro?les for a Class of1-D Models15 Tank energy tracers @ site Total energy Fig.1.Random-halves model with20sites,temperatures T L=10,T R=100and injection rates L=10, R=5.Top left:Mean tank energies s i.Top right:Mean number of tracersκi.Bottom:Mean total energy E i.Simulations in perfect agreement with predictions from theory 4.Tracer?ux and concavity of stored energy.One of the interesting facts that have emerged is that s(ξ)is linear if and only if L= R,and when L= R,their rel-ative strength is re?ected in the concavity of s(ξ).This may be a little perplexing at ?rst because no mechanism is built into the microscopic rules for the tanks to recognize the directions of travel of the tracers with which they come into contact.The reason behind this phenomenon is,in fact,quite simple:If there is a tracer?ux across the system,say from right to left,then the tank at site i hears from site i+1more fre-quently than it hears from site i?1(because j i+1>j i?1).It therefore has a greater tendency to equilibrate with the energy level on the right than on the left,causing s i to be>1 2(s i+1+s i?1).Since this happens at every site,a curvature for the pro?le of s i is created.The reader should further note that tracer?ux and heat?ux go in opposite directions if( L? R)·( L T L? R T R)<0. 16J.-P.Eckmann,L.-S.Young 5.Individual cells mimicking heat baths.The cells in our models are clearly not in?nite heat reservoirs,yet for large N ,they acquire some of the characteristics of the heat baths with which they are in contact.More precisely,the i th cell injects each of its two neighbors with i tracers per unit time.These tracers,which have mean energy T i ,are distributed according to a law of the same type as that with which tracers are emitted from the baths (exponential in the case of the random halves model);see Remark 3. 6.Unlike the conditions at the two ends,however,the numbers i and T i are self-selected . 3.6.A second example.We consider here a model of the same type as the “random-halves”model but with different microscopic rules .The purpose of this exercise is to highlight the role played by these rules and to make transparent which part of our scheme is generic. The rules for energy exchange in this model simulate a Hamiltonian model in which both tracers and tanks have one degree of freedom.Write x =v 2and y =ω2,v,ω∈(?∞,∞),and think of energy as uniformly distributed on the circle {(v,ω):v 2+ω2=c },so that when the tracer interacts with stored energy,the redistribution is such that a random point on this circle is chosen with weight |v |(this is the measure induced on a cross-section by the invariant measure of the ?ow).That is to say,if (x,y)are the stored energy and tracer energy before an interaction,and (x ,y )afterwards,then for a ∈[0,x +y ], P {y >a }=P {|ω |>√a }=1? √a 0v 1+ dv dω dω √x +y 0 v 1+ dv dω 2dω=1?√a √x +y (11)or,equivalently,the density of y is 121 y 1√x +y for y ∈[0,x +y ]. Assume now that all is as in Sect.3.1,except that when Clock 1of a tracer rings,it exchanges energy with the tank according to the rule in (11)and not the random-halves rule.Following the computation in Sect.3.2(details of which are left to the reader),we see that Propositions 3.5and 3.7hold for the present model provided σk is replaced by σk ({x 1,...,x k },y)=I {x 1,...,x k ,y ≥0}1√x 1·...·x k y e ?β(x 1+...+x k +y).This de?nes a new family o f {?μT , }for this model.With ?μT , in hand,we make the assumption as before that for N 1,the marginals of individual sites have the same form.Proceeding as in Sect.3.3,we read off the following information on single cells:(i)stored energy has density const .e ?βy /√y and mean T /2;(ii)tracer energy has density const .e ?βx /√x and mean T /2;(iii)mean number of tracers,κ=2 πT ; (iv)mean total-cell energy,E =T 2(1+κ). The rest of the analysis,including (iv),(v),(A)and (B),do not depend on the local rules (aside from the fact that tracers exiting a cell have equal chance of going left and right).Thus they remain unchanged.Reasoning as in Theorem 3.9,we obtain the following: Nonequilibrium Energy Pro?les for a Class of 1-D Models 17Proposition 3.11.Under Assumptions 1and 2,the pro?les for the model with energy exchange rule (11)are ?j (ξ)=2( L +( R ? L )ξ);?Q(ξ)=2( L T L +( R T R ? L T L )ξ);?s(ξ)=e(ξ)=12Q(ξ)/j (ξ);?κ(ξ)=√2π/s(ξ)j (ξ)/2;?E(ξ)=s(ξ)+κ(ξ)e(ξ)=s(ξ)+√2πs(ξ)j (ξ)/2. Numerical simulations give results in excellent agreement with these theoretical pre-dictions. 4.Hamiltonian Models In Sect.4.1,we introduce a family of Hamiltonian models generalizing those studied numerically in [17,12].A single-cell analysis similar to that in Sect.3is carried out for this family in Sect.4.2,and predictions of energy and tracer density pro?les are made in Sect.4.3.We again use the Assumptions in Sect.3,but the predictions here are made under an additional ergodicity assumption,ergodicity being a property that is easy to arrange in stochastic models but not in Hamiltonian ones.Results of simulations for a speci?c model are shown in Sect.4.5.A brief discussion of related models is given in Sect.4.6. 4.1.Rotating disks models.We describe in this subsection a family of models quite close to those studied numerically in [17,12].The rules of interaction (though not the coupling to heat baths)are,in fact,used earlier in [20]. 4.1.1.Dynamics in a closed cell.We treat ?rst the dynamics within individual cells assuming the cell or box is sealed ,i.e.,it is not connected to its neighbors or to external heat sources. Let 0?R 2be a bounded domain with piecewise C 3boundary.In the interior of 0lies a (circular)disk D ,which we think of as nailed down at its center.This disk rotates freely,carrying with it a ?nite amount of kinetic energy derived from its angular velocity;it will play the role of the “energy tank”in Sect.2.1.The system below describes the free motion of k point particles (i.e.,tracers)in = 0\D .When a tracer runs into ? 0,the boundary of 0,the re?ection is specular.When it hits the rotating disk,the energy exchange is according to the rules introduced in [20,17,12].A more precise description of the system follows. The phase space of this dynamical system is ˉ k =( k ×?D ×R 2k +1)/～,where x =(x 1,...,x k )∈ k denotes the positions of the k tracers,?∈?D denotes the angular position of a (marked)point on the boundary of the turning disk,v =(v 1,...,v k )∈R 2k denotes the velocities of the k tracers,ω∈R denotes the angular velocity of the turning disk,and ～is a relation that identi?es pairs of points in the collision manifold M k ={(x ,?,v ,ω):x ∈? for some }.The rule of identi?cation is given below. 18J.-P.Eckmann,L.-S.Young The?ow onˉ k is denoted byˉ s.As long as no collisions are involved,we have ˉ s(x,?,v,ω)=(x+s v,?+sω,v,ω). We assume at most one tracer collides with? at any one point in time.(ˉ s is not de?ned at multiple collisions,which occur on a set of measure zero.)At the point of impact,i.e.,when x ∈? for one of the ,let v =(v t ,v n )be the tangential and normal components of v .What happens subsequent to impact depends on whether x ∈? 0 or?D.In the case of a collision with? 0,the tracer bounces off? 0with angle of re?ection equal to angle of incidence,i.e., (v n ) =?v n ,(v t ) =v t ,(12) and the other variables are unchanged.In the case of a collision with the disk,the following energy exchange takes place between the disk and the tracer: (v n ) =?v n ,(v t ) =ω,ω =v t .(13) Here we have,for simplicity,taken the radius of the disk,the moment of inertia of the disk,and the mass of tracer in such a way that the coef?cients in Eq.(13)are equal to1. The identi?cation in the de?nition ofˉ k is z～z where z,z ∈M k are such that all of their coordinates are equal except that v andωin z are replaced by the corresponding quantities with primes in Eqs.(12)and(13)for z .We also write F(z)=z . Note that in both(12)and(13),total energy is conserved,i.e.,|v|2+ω2=|v |2+(ω )2. The energy surfaces in this model are therefore ˉ k,E=( k×?D×S2k E )/～, where S2k E={(v1,...,v k,ω)∈R2k+1: |v |2+ω2=E}. We claim that the natural invariant measure,or Liouville measure,of the(discontin-uous)?owˉ s onˉ k is ˉm k=(λ2| )k×(ν1|?D)×λ2k+1, whereλd is d-dimensional Lebesgue measure andνd is surface area on the relevant d-sphere.Once the invariance ofˉm k is checked,it will follow immediately that the induced measuresˉm k,E=(λ2| )k×(ν1|?D)×ν2k onˉ k,E areˉ s-invariant,as are all measures onˉ k of the formψ(E)ˉm k,E for someψ:[0,∞)→[0,∞). The invariance ofˉm k is obvious away from collisions and at collisions with? 0. Because collisions occur one at a time,it suf?ces to consider a single collision between a single tracer and the disk.The problem,therefore,is reduced to the following:Consider ˉ s onˉ 1,and let M1,D denote the part of the collision manifold involving D.To prove thatˉm1is preserved in a collision with D,it suf?ces to check that for A?M1,D and ε>0arbitrarily small, ˉm1 ∪?ε =ˉm1 ∪0 . We leave this as a calculus exercise. Nonequilibrium Energy Pro?les for a Class of1-D Models19 4.1.2.Coupling to neighbors and heat baths.We now consider a chain of N identical copies of the dynamical system described in Sect.4.1.1,and de?ne couplings between nearest neighbors and between end cells and heat baths. LetγL andγR be two marked subsegments of? 0of equal length;these segments will serve as openings to allow tracers to pass between cells.For now it is best to think of 0 as having a left-right symmetry,and to think ofγL andγR as vertical and symmetrically placed(as in Fig.2),although as we will see,these geometric details are not relevant for the derivation of mean energy and tracer pro?les.We call the segmentsγL andγR in the i th cellγ(i)L andγ(i)R.For i=1,...,N?1,we identifyγ(i)R withγ(i+1) L ,that is to say,we think of the domains of the i th cell and the(i+1)st cell as having a wall in common,namelyγ(i)R andγ(i+1) L ,and remove this wall,so that tracers that would have collided with it simply continue in a straight line into the adjacent cell.(See Fig.2.) Tracers are injected into the system as follows.Consider,for example,the bath on the left.We say the injection rate is if at the ring of an exponential clock of rate , a single tracer enters cell1viaγ(1) L .(Note that the rate is not the injection rate per unit length of the openingγ(1) L but per unit time.)The points of entry and velocities of entering tracers are iid,the law being the one governing the collisions of tracers with γ(1)L.That is to say,the point of entry is uniformly distributed onγ(1)L,and the velocity v has density c e?β|v|2|v||sin(?)|dv,c=2β3/2 √ π ,(14) where v∈R2points intoγ(1) L and?∈(0,π)is the angle v makes withγ(1) L at the point of entry.(This is the distribution of v at collisions for particles with velocity distribution βπexp(?β|v|2)dv.)Hereβ=1/T,where T is said to be the temperature of the bath. Observe that the mean energy of the tracers injected into the system by a bath at tem-perature T is not T but3T/2.Injection from the right is done similarly,via the opening γ(N) R .When a tracer in the chain reachesγ(1) L orγ(N) R ,it vanishes into the baths. This completes the description of our models.We remark that the process above is a Markov process in which the only randomness comes from the action of the baths.Once a tracer is in the system,its motion is governed by rules that are entirely deterministic. Fig.2.A row of diamond-shaped boxes with small lateral holes(made by removing vertical walls corre- sponding toγ(i)R=γ(i+1) L )to allow the tracers to go from one box to the next.The shapes of the“boxes” can be quite general for much of our theory.The con?guration shown is the one used in the simulations discussed in Sect.4.5,but with larger holes for better visibility 20J.-P.Eckmann,L.-S.Young 4.2.Single-cell analysis.In analogy with Sect.3.2,we investigate in this subsection the invariant measure for a single cell coupled to two heat baths with parameters T and .Let ˉ k and ˉ k,E be as in Sect.4.1.1.As before,a state of this system is represented by a point in =∪∞k =0 k =∪∞k =0∪E ≥0 k,E , where k and k,E are quotients of ˉ k and ˉ k,E respectively obtained by identifying permutations of the k tracers.With {···}representing unordered sets as before,points in are denoted by z =({x 1,...,x k },?;{v 1,...,v k },ω)or simply ({x },?;{v },ω),with v understood to be attached to x .The quotient measures of ˉm k and ˉm k,E are respectively m k and m k,E .Abusing notation slightly,we continue to use ˉ s to denote the semi-?ow on ˉ ,and let s denote the induced semi-?ow on .Then s is as in Sect.4.1.1except where tracers exit or enter the system.More precisely,if s (z)∈ k for all 0≤s Let |γ|denote the length of the segment γL or γR . Proposition 4.1.There is an invariant probability measure μwith the following prop-erties: (a)the number of tracers present is a Poisson random variable with mean κwhere κ=2√πλ2( )|γ| √T ;(b)the conditional density of μon k is c k σk dm k ,where σk ({x },?;{v },ω)=e ?β(ω 2+ k =1|v |2)and c k is the normalizing constant. We observe as before that the Poisson parameter κis proportional to (the higher the injection rate,the more tracers in the cell)and inversely proportional to √T ,i.e.,the speed of the tracers (the faster the tracers,the sooner they leave).Unlike the models considered in Sect.3,where the tracers are assumed to leave the cell at a rate equal to their speed,here the ratio λ2( )/|γ|appears,as it should:the smaller the passage way,the longer it takes for the tracers to leave. Notice that we have not claimed that μis unique. We introduce some notation in preparation for the proof.For A ? k and h >0,we let ?h (A)denote the set of all initial states in that in time h evolve into A assuming no new tracers are injected into the system between times 0and h .2Then ?h (A)=∪n ≥0 (n)?h (A),where (n)?h (A)= ?h (A)∩ k +n ,i.e., (n)?h (A)is the set of states where initially k +n tracers are present,and by the end of time h exactly n of these tracers have exited and the remaining k are described by a state in A . Lemma 4.2.Let μbe as in Proposition 4.1,and let A εbe a cube of sides εin k ,εsmall enough that μ(A ε)≈p k c k σk (ˉz )ε4k +2for some ˉz .We assume the following holds for all small h >0: 2Notice that (1){ h ,h ≥0}is a semi-?ow,and ?h is not de?ned;(2) ?h (A)as de?ned is =( h )?1(A). Mathematica函数大全--运算符及特殊符号一、运算符及特殊符号 Line1;执行Line，不显示结果 Line1,line2顺次执行Line1，2，并显示结果 ?name关于系统变量name的信息 ??name关于系统变量name的全部信息 !command执行Dos命令 n! N的阶乘 !!filename显示文件内容 a-b减 a*b或a b 乘 a/b除 a^b 乘方 base^^num以base为进位的数 lhs&&rhs且 lhs||rhs或 !lha非 ++,-- 自加1，自减1 +=,-=,*=,/= 同C语言 >,<,>=,<=,==,!=逻辑判断（同c） lhs=rhs立即赋值 lhs:=rhs建立动态赋值 lhs:>rhs建立替换规则 expr//funname相当于filename[expr] expr/.rule将规则rule应用于expr expr//.rule 将规则rule不断应用于expr知道不变为止param_ 名为param的一个任意表达式（形式变量）param__名为param的任意多个任意表达式（形式变量） 二、系统常数 Pi 3.1415....的无限精度数值 E 2.17828...的无限精度数值 Catalan 0.915966..卡塔兰常数 EulerGamma 0.5772....高斯常数 GoldenRatio 1.61803...黄金分割数 Degree Pi/180角度弧度换算 I复数单位 Infinity无穷大 Word中MathType公式调整的一些技巧 一、批量修改公式的字号和大小 数学试卷编辑中，由于排版等要求往往需要修改公式的大小，一个一个的修改不仅费时费力，还容易产生各种错误。如果采用下面介绍的方法，就可以达到批量修改公式大小的效果。 （1）双击一个公式，打开MathType，进入编辑状态； （2）点击size（尺寸）菜单→define（自定义）→字号对应的pt（磅）值，一般五号对应10pt（磅），小四对应12pt（磅）； （3）根据具体要求调节pt（磅）值，然后点击OK（确定）按钮； （4）然后点击preference（选项）→equation preference （公式选项）→save to file（保存到文档），保存一个与默认配置文件不同的名字，然后关闭MathType 回到Word文档； （5）点击Word界面上的菜单MathType→format equations（公式格式）→load equation preferrence（加载公式选项），点击选项下面的browse（浏览）按钮，选中刚才保存的配置文件，并点选whole document（整个文档）选项，最后单击OK（确定）按钮。 到此，就安心等着公式一个个自动改过来吧…… 但这样处理后，下次使用Word文档进行MathType公式编辑时，将以上述选定的格式作为默认设置。如果需要恢复初始状态，可以按以下步骤操作：（1）双击一个公式，打开MathType，进入编辑状态； （2）然后点击preference（选项）→equation preference （公式选项）→Load factory settings（加载出厂设置），然后关闭MathType回到Word文档。 二、调整被公式撑大的Word行距 点击“文件”菜单下的“页面设置”项。在“文档网格”标签页中的“网格”一栏，勾选“无网格”项。 但此时也存在一个问题，就是此时的行间距一般比预期的行间距要小。这时 Word中使用MathType使公式自动编号，并可左对齐，右对齐 2011-05-31 13:09:12| 分类：杂七杂八| 标签：|字号大中小订阅 MathType是一款功能强大的数学公式编辑器，与常见的文字处理软件和演示程序配合使用，能够在各种 文档中加入复杂的数学公式和符号。它的另一优势就是可以插入带编号公式，建立编号与公式的关联，并 能自动更新编号，避免了人工改编号的麻烦和错误。 一、安装及界面（以 6.5中文版为例） 安装完MathType之后，在word软件中，会生成一个工具栏（如下），在word菜单栏也会有个MathType的菜单项，列出了MathType的一般功能。 Mathtype编辑器的界面如下： 只要通过工具栏就可以很方便的进行公式编辑，如希腊字母、上标下标、积分号、求和、分数、矩阵等。相信大家浏览一下工具栏的选项就能明白它们的用法，这里不做赘述。这里要讲的是一些细节性 的东西。在此要记住一句话，凡是你在书上见过的公式格式，mathtype都能做出来。 二、在word软件中使用MathType。 在word中，MathType工具栏各按钮的功能如下图： 1．插入公式说明 前四项都是在word中插入公式，点任何一项都会打开MathType编辑器。 用于在正文段落中插入小公式或变量符号等； 1）“Insert Inline Equation” 2）“Insert Display Equation” 用于插入无编号公式，第三、四项顾名思义就是插入带左、右编号的公式，这三项都是用于插入单独占一行的公式，如下图： 2．首次参数配置 在写论文时，用到较多的是第一、四项。首次使用带右编号的公式“Insert Display Equation, Number on Right”时，会弹出一个提示框，如下图： 的是输入公式的章、节号，一般学位论文中公式用“章号（chapter）+公式序号”格式，如（1.1），不写节号（section），如（1.1.1）。Mathtype默认公式编号为“节号+公式序号”，通过菜单栏“MathType---Format Equation Numbers…”打开下图窗口，可以更改公式编号的数字格式、括号形式、分隔符等。（下图选项是章号+公式序号，小括号，圆点隔开）。 目录 1 详解MathType中如何批量修改公式字体和大小 (2) 2 如何在等号上插入容 (3) 3 详解MathType快捷键使用技巧 (3) 4 数学上的恒不等于符号怎么打 (3) 5 MathType表示分类的大括号怎么打 (3) 6 编辑公式时如何让括号的容居中 (3) 7 MathType怎么编辑叉符号 (3) 8 怎样用MathType编辑竖式加减法 (3) 9 如何用MathType编辑除法竖式 (3) 10 如何用MathType编辑短除法 (3) 如何调整MathType矩阵行列间距 (3) 12在MathType中怎样表示将公式叉掉 (3) 13 MathType怎么输入字母上方的黑点 (3) 14 MathType如何编辑大于或约等于符号 (3) 1 详解MathType中如何批量修改公式字体和大小 MathType应用在论文中时，有时会因为排版问题批量修改公式字体和大小，一个一个的修改不仅费时费力，还容易出现错误，本教程将详解如何在MathType公式编辑器中批量修改公式字体和大小。批量修改公式字体和大小的操作步骤： 步骤一双击论文中的任意一个公式，打开MathType公式编辑器软件。 步骤二单击菜单栏中的大小——定义命令，打开“定义尺寸”对话框。如果使用的是英文版MathType，点击size——define即可。 步骤三在“定义尺寸”对话框中，通过更改pt值的大小可以达到修改MathType字体的效果。英文版下为“Full”。 一般情况下，五号字对应的pt值为10，小四号字对应的pt值为12。因为“磅”是大家比较熟悉的单位，用户也可以将pt值换成“磅”来衡量。 步骤四菜单栏中的选项——公式选项——保存到文件，选择保存路径。英文版的MathType点击preference——equation preference —— save to file 步骤六关闭MathType软件后，点击word文档中的MathType——Insert Number——format equation，打开format equation对话框。 Math Studio—— Math Studio1M Catalog Catalog Math Studio https://www.doczj.com/doc/db14616804.html,/Manual Manual Wolfram mathematica Det det diff Diff ALGEBRA Apart, Coefficient, Degree , Denominator, Divisors , DivisorSigma, Eval, Expand , Factor, GCD, LCM , PolyDivide, PolyFit, PolyGCD , PolyLCM, PowerExpand, Quotient , Remainder, Sequence, SimplifyPoly , Solve, SolveSystem, Together BASIC Abs, Arg, Conj, Exp, Hyperbolic Functions, Im , Imag, Ln, Log, Re, Real, Trigonometric Functions CALCULUS D, Diff, DSolve , fDiff, FourierCos, FourierSeries , FourierSin, iDiff, iLaplace , Integrate, Laplace, Limit , NIntegrate, pDiff, Product , Series, Sum CAS Append Call,Caps , Char , Choose, Clear, Command, Date , Delete, Extract , Function, Insert, IsList , IsMatrix, IsNumber , IsPoly, Left, Length , List, Matrix , Part, Replace, Reshape , Reverse, Right , Size, Sort, String , Value Converts a string to a value, Variables DATA Constant, Finance , HRStoHMS , LoadList , LoadMatrix, Table ELEMENTARY Binomial, Ceil, Eulerian, Factorial, Floor , fPart, iPart , Mod, Multinomial, nCr , nPr, nRoot, Pochhammer, Round, Sign, Sqrt GRAPHING clip, FullRectSineWave, HalfRectSineWave( ), SawToothWave(), SquareWave, StaircaseWave, TriangleWave MANUAL Code Files, Commands, Creating Scripts, Entering Expressions, Graphing Equations, Include Folder, Lists, Matrices, Strings , Symbols, Time Graphing MATRIX MathType是“公式编辑器”的功能强大而全面的版本。如果要经常在文档中编排各种复杂的数学、化学公式，则MathType是非常合适的选择。MathType用法与“公式编辑器”一样简单易学，而且其额外的功能使你的工作更快捷，文档更美观。 MathType包括： (1)Euclid字体设置了几百个数学符号。 (2)具有应用于几何、化学及其他方面的新样板和符号。 (3)专业的颜色支持。 (4)为全球广域网创建公式。 (5)将输出公式译成其他语言(例如：TeX、AMS-TeX、LaTeX、MathML及自定义语言)的翻译器。 (6)用于公式编号、格式设置及转换Microsoft Word文档的专用命令。 (7)可自定义的工具栏，可容纳最近使用过的几百个符号、表达式和公式。 (8)可自定义的键盘快捷键。 在编辑word文档时，如果需要录入公式将是一件非常痛苦的事情。利用M athtype作为辅助工具，会为文档的公式编辑和修改提供很多方便。下面介绍几种mathtype中比较重要的技巧 一、批量修改公式的字号和大小 论文中，由于排版要求往往需要修改公式的大小，一个一个修改不仅费时 费力还容易使word产生非法操作。 解决办法，批量修改：双击一个公式，打开mathtype，进入编辑状态， 点击size菜单->define->字号对应的pt值，一般五号对应10pt，小四对应12pt 其他可以自己按照具体要求自行调节。其他默认大小设置不推荐改动。 然后点击preference->equation preference -> save to file ->存一个与默认配置文件不同的名字，然后关闭mathtype回到word文档。 点击word界面上的mathtype ->format equation -> load equation pr eferrence选项下面的browse按钮，选中刚才存的配置文件，点选whole doc ument选项，确定，就安心等着公式一个个改过来。 word中被行距被撑大的解决方法 －－－－－－－－－－－－－－－－－－－－－－－－－－ 在Word文档中插入公式后，行距便会变得很大，简单的调整段落的行距是行不通的。逐个点选公式，然后拖动下角的箭头倒可以将它任意放大缩小以调整行距，但是如果在一篇文档中使用了大量的公式，这种操作显然太麻烦，手工操作也容易使得公式大小不一，一些小的公式还会影响到显示的效果。下面介绍两种调整公式行距的方法： ·全部一次调整 依次单击菜单命令“文件→页面设置”。单击“文档网格”选项卡，如图1所示。选中“无网格”，单击“确定”按钮就可以了。 一、运算符及特殊符号 Line1; 执行Line，不显示结果 Line1,line2 顺次执行Line1，2，并显示结果?name 关于系统变量name的信息 ??name 关于系统变量name的全部信息 !command 执行Dos命令 n! N的阶乘 !!filename 显示文件内容 < Expr>> filename 打开文件写 Expr>>>filename 打开文件从文件末写 () 结合率 [] 函数 {} 一个表 <*Math Fun*> 在c语言中使用math的函数(*Note*) 程序的注释 #n 第n个参数 ## 所有参数 rule& 把rule作用于后面的式子 % 前一次的输出 %% 倒数第二次的输出 %n 第n个输出 var::note 变量var的注释 "Astring " 字符串 Context ` 上下文 a+b 加 a-b 减 a*b或a b 乘 a/b 除 a^b 乘方 base^^num 以base为进位的数 lhs&&rhs 且 lhs||rhs 或 !lha 非 ++,-- 自加1，自减1 +=,-=,*=,/= 同C语言 >,<,>=,<=,==,!= 逻辑判断（同c） lhs=rhs 立即赋值 lhs:=rhs 建立动态赋值 lhs:>rhs 建立替换规则 lhs->rhs 建立替换规则 expr//funname 相当于filename[expr] expr/.rule 将规则rule应用于expr expr//.rule 将规则rule不断应用于expr知道不变为止 param_ 名为param的一个任意表达式（形式变量） param__ 名为param的任意多个任意表达式（形式变量） ————————————————————————————————————— 二、系统常数 Pi 3.1415....的无限精度数值 E 2.17828...的无限精度数值 Catalan 0.915966..卡塔兰常数 EulerGamma 0.5772....高斯常数 GoldenRatio 1.61803...黄金分割数 Degree Pi/180角度弧度换算 I 复数单位 Infinity 无穷大 -Infinity 负无穷大 ComplexInfinity 复无穷大 Indeterminate 不定式 ————————————————————————————————————— 三、代数计算 Expand[expr] 展开表达式 Factor[expr] 展开表达式 Simplify[expr] 化简表达式 word公式编辑器MathType使用技巧 当你在用mathtype编辑公式的时候，是否因频繁的鼠标点击而对这个软件颇有抱怨，其实mathtype 的研发人员早就替你想到了这一点并给出了解决的方案。 1. 常见的数学符号的快捷键（Ctrl是王道） (1) 分式 Ctrl+F（分式） Ctrl+/（斜杠分式） (2) 根式 Ctrl+R（根式） 先按“Ctrl+T”，放开后，再按N（n次根式）。 例如，先按“Ctrl+T”，放开后，再按N，然后在空格中分别填入2，3就得到2的3次方根。 (3) 上、下标 Ctrl+H（上标）例如，按y Ctrl+H+2 就得到 Ctrl+L（下标）例如，按y Ctrl+L+2就得到 Ctrl+J（上下标）例如，按y Ctrl+J 然后在空格中分别填入2，3就得到 (4) 不等式 先按“Ctrl+K”，放开后，再按逗号，就得到小于等于符号≤ 先按“Ctrl+K”，放开后，再按句号，就得到大于等于符号≥ (5) 导数、积分 Ctrl+Alt+'（单撇（导数符号）） Ctrl+Shift+”（双撇（二阶导数符号）） Ctrl+I（定积分记号） Ctrl+Shift+I, ! （不定积分记号） (6)上横线、矢量箭头 Ctrl+Shift+连字符（上横线） Ctrl+Alt+连字符（矢量箭头） （7）括号快捷键（效率提高50% o(∩_∩)o ） Ctrl+9或Ctrl+0（小括号） Ctrl+[或Ctrl+]（中括号） Ctrl+{ 或Ctrl+}（大括号） （8）放大或缩小尺寸,只是显示，并不改变字号 Ctrl+1（100%） Ctrl+2（200%） 1、Aurora 方程的类型 在Aurora中有三种类型方程，分别为：inline, display和numbered display。inline 是指该方程与其它的文字混在一起，构成文档的一行，可以理解为行内方程。display类型是指该方程单独构成一个段落，并且居中显示，可以理解为行间显示的方程。numbered display类型就是带数字标号的行间显示。 2、快捷键 Microsoft Word 为了避免跟其他应用程序冲突, 在word里使用Aurora快捷键大多数都是以 函数以及其他一些库函数 函数名称: abs 函数原型: int abs(int x); 函数功能: 求整数x的绝对值 函数返回: 计算结果 参数说明: 所属文件: <>,<> 使用范例： #include <> #include <> int main() { int number=-1234; printf("number: %d absolute value: %d",number,abs(number)); return 0; } @函数名称: fabs 函数原型: double fabs(double x); 函数功能: 求x的绝对值. 函数返回: 计算结果 参数说明: 所属文件: <> 使用范例： #include <> #include <> int main() { float number=; printf("number: %f absolute value: %f",number,fabs(number)); return 0; } @函数名称: cabs 函数原型: double cabs(struct complex znum) 函数功能: 求复数的绝对值 函数返回: 复数的绝对值 参数说明: zuum为用结构struct complex表示的复数，定义如下：struct complex{ double m; double n; } 所属文件: <> #include <> #include <> int main() { struct complex z; double val; =; =; val=cabs(z); printf("The absolute value of %.2lfi %.2lfj is %.2lf",,,val); return 0; } @函数名称: ceil 函数原型: double ceil(double num) 函数功能: 得到不小于num的最小整数 函数返回: 用双精度表示的最小整数 参数说明: num-实数 所属文件: <> #include <> #include <> int main() { double number=; double down,up; down=floor(number); up=ceil(number); printf("original number %",number); printf("number rounded down %",down); printf("number rounded up %",up); return 0; } @函数名称: sin 函数原型: double sin(double x); 函数功能: 计算sinx的值.正弦函数 函数返回: 计算结果 参数说明: 单位为弧度 所属文件: <> 使用范例： 当你在用mathtype编辑公式的时候，是否因频繁的鼠标点击而对这个软件颇有抱怨，其实mathtype的研发人员早就替你想到了这一点并给出了解决的方案，只是我们往往“满足”于“鼠标流”而没有去进一步的探究，但是，现在还不晚.... 1. 常见的数学符号的快捷键（Ctrl是王道） (1) 分式 Ctrl+F（分式） Ctrl+/（斜杠分式） (2) 根式 Ctrl+R（根式） 先按“Ctrl+T”，放开后，再按N（n次根式）。 例如，先按“Ctrl+T”，放开后，再按N，然后在空格中分别填入2，3就得到2的3次方根。 (3) 上、下标 Ctrl+H（上标）例如，按y Ctrl+H+2 就得到 Ctrl+L（下标）例如，按y Ctrl+L+2就得到 Ctrl+J（上下标）例如，按y Ctrl+J 然后在空格中分别填入2，3就得到 (4) 不等式 先按“Ctrl+K”，放开后，再按逗号，就得到小于等于符号≤ 先按“Ctrl+K”，放开后，再按句号，就得到大于等于符号≥ (5) 导数、积分 Ctrl+Alt+'（单撇（导数符号）） Ctrl+Shift+”（双撇（二阶导数符号）） Ctrl+I（定积分记号） Ctrl+Shift+I, ! （不定积分记号） (6)上横线、矢量箭头 Ctrl+Shift+连字符（上横线） Ctrl+Alt+连字符（矢量箭头） （7）括号快捷键（效率提高50% o(∩_∩)o ） Ctrl+9或Ctrl+0（小括号） Ctrl+[或Ctrl+]（中括号） Ctrl+{ 或Ctrl+}（大括号） （8）放大或缩小尺寸,只是显示，并不改变字号 Ctrl+1（100%） Ctrl+2（200%） Ctrl+4（400%） Ctrl+8（800%） （9）空格和加粗 Ctrl＋Shift＋Space 空格 Ctrl＋Shift＋B 加粗 （如果你要问如何记下这些快捷键，其实只要注意把那些字母和英文对应就很好记忆了。比如，R代表Root，F代表Fraction，I代表Integate，H代表Higher等等） 2.希腊字母 先按“Ctrl+G”，放开后，再按英语字母字母得到相应的小写希腊字母；如果再按“Shift+字母”，得到相应的大写希腊字母。 例如，先按“Ctrl+G”，放开后，再按字母A得到小写希腊字母α。 又如，先按“Ctrl+G”，放开后，再按“Shift+S”，得到大写希腊字母Σ。 3.添加常用公式 MathType的一大特色就是可以自己添加或删除一些常用公式，添加的办法是：先输入我们要添加的公式，然后选中该公式，用鼠标左键拖到工具栏中适当位置即可。删除的方式是右击工具图标，选择“删除”命令即可。 4.元素间跳转 每一步完成后转向下一步（如输入分子后转向分母的输入等）可用Tab键，换行用Enter 键 cmath是c++语言中的库函数,其中的c表示函数是来自c标准库的函数，math为数学常用库函数。cmath 库函数列表: using ::abs; //绝对值 using ::acos; //反余弦 using ::acosf; //反余弦 using ::acosl; //反余弦 using ::asin; //反正弦 using ::asinf; //反正弦 using ::asinl; //反正弦 using ::atan; //反正切 using ::atan2; //y/x的反正切 using ::atan2f; //y/x的反正切 using ::atan2l; //y/x的反正切 using ::atanf; //反正切 using ::atanl; //反正切 using ::ceil; //上取整 using ::ceilf; //上取整 using ::ceill; //上取整 using ::cos; //余弦 using ::cosf; //余弦 using ::cosh; //双曲余弦 using ::coshf; //双曲余弦 using ::coshl; //双曲余弦 using ::cosl; //余弦 using ::exp; //指数值 using ::expf; //指数值 using ::expl; //指数值 using ::fabs; //绝对值 using ::fabsf; //绝对值 using ::fabsl; //绝对值 using ::floor; //下取整 using ::floorf; //下取整 using ::floorl; //下取整 using ::fmod; //求余 using ::fmodf; //求余 using ::fmodl; //求余 using ::frexp; //返回value=x*2n中x的值,n存贮在eptr中 using ::frexpf; //返回value=x*2n中x的值,n存贮在eptr中 MathType使用技巧 MathType是“公式编辑器”的功能强大而全面的版本。如果要经常在文档中编排各种复杂的数学、化学公式，则MathType是非常合适的选择。MathType用法与“公式编辑器”一样简单易学，而且其额外的功能使你的工作更快捷，文档更美观。 MathType包括： (1)Euclid字体设置了几百个数学符号。 (2)具有应用于几何、化学及其他方面的新样板和符号。 (3)专业的颜色支持。 (4)为全球广域网创建公式。 (5)将输出公式译成其他语言(例如：TeX、AMS-TeX、LaTeX、MathML及自定义语言)的翻译器。 (6)用于公式编号、格式设置及转换Microsoft Word文档的专用命令。 (7)可自定义的工具栏，可容纳最近使用过的几百个符号、表达式和公式。 (8)可自定义的键盘快捷键。 图1 MathType公式编辑器 在编辑word文档时，如果需要录入公式将是一件非常痛苦的事情。利用Mathtype作为辅助工具，会为文档的公式编辑和修改提供很多方便。下面介绍几种mathtype中比较重要的技巧 1． 放大或缩小尺寸 Ctrl+1（100%）；Ctrl+2（200%）； 2．在数学公式中插入一些符号 Ctrl+9或Ctrl+0（小括号）；Ctrl+[ 或Ctrl+]（中括号）；Ctrl+{ 或Ctrl+}（大括号）； Ctrl+F（分式）；Ctrl+/（斜杠分式）；Ctrl+H（上标）；Ctrl+L（下标）；Ctrl+I（积分号）； Ctrl+R（根式）；Ctrl+Shift+连字符（上横线）；Ctrl+Alt+连字符（矢量箭头）；Ctrl+Alt+'（单撇）；Ctrl+Alt+"（双撇）；先按“Ctrl+T”放开后，再按N（n次根式）、S（求和符号）、P（乘积符号）等。 3．微移间隔 /* #include int abs( int num ); double fabs( double arg ); long labs( long num ); 函数返回num的绝对值 #include double acos( double arg ); 函数返回arg的反余弦值，arg的值应该在-1到1之间 #include double asin( double arg ); 函数返回arg的反正弦值，arg的值应该在-1到1之间 #include double atan( double arg ); 函数返回arg的反正切值 #include double atan2( double y, double x ); 函数返回y/x的反正切值，并且它可以通过x，y的符号判断 （x，y）所表示的象限，其返回的也是对应象限的角度值 #include double ceil( double num ); double floor( double arg ); ceil函数返回不小于num的最小整数，如num = 6.04, 则返回7.0 floor函数返回不大于num的最大的数，如num = 6.04, 则返回6.0 #include double cos( double arg ); double sin( double arg ); double tan( double arg ); 函数分别返回arg的余弦，正弦，正切值，arg都是用弧度表示 #include double cosh( double arg ); double sinh( double arg ); double tanh( double arg ); 函数分别返回arg的双曲余弦，双曲正弦，双曲正切，arg都是用弧度表示的 #include double fmod( double x, double y ); Mathtype常用快捷键(一) 1. 打开/关闭MathType窗口 Alt+M:打开Word工具栏中的MathType菜单，然后用上下键选择想要的操作，打开MathType窗口。 默认的插入inline公式的快捷键是alt+ctrl+q Alt+F4:保存并关闭MathType窗口，返回Word。 2. 公式输入 Ctrl+G+希腊字母英文名的首字母:小写希腊字母（先按Ctrl+G，再按相应的希腊字母英文名的首字母） Ctrl+G+Shift+希腊字母英文名的首字母:大写希腊字母 Ctrl+F: 分式 Ctrl+I:积分 Ctrl+T+S:求和 Ctrl+shift+space:空格 Ctrl+B:输入向量格式字符 Ctrl+H:上角标 Ctrl+J:上、下角标 Ctrl+L:下角标 Ctrl+(:左右圆括弧 Ctrl+[:左右方括弧 Ctrl+{:左右话括弧 Ctrl+K+<:小于等于号 Ctrl+K+>:大于等于号 3．微移间隔 先选取要移动的公式（选取办法是用“Shift+箭头键”），再用“Ctrl+箭头键”配合操作即可实现上、下、左、右的平移；用“Ctrl+Alt+空格”键可适当增加空格。 4．元素间的跳转 每一步完成后转向下一步（如输入分子后转向分母的输入等）可用Tab 键，换行用Enter键。 二、用键盘选取菜单或工具条 按Alt键与箭头键或F10与箭头键可进入菜单；分别按F2、F6、F7、F8、F9键可分别进入工具条的第一至第第五行，再配合箭头键可选取适合的符号进行输入（参见上面图示）。 三、贴加常用公式 公式编辑器MathType 5.0的一大特色就是可以自己贴加或删除一些常用公式，如图中的工具栏的最后两行就是为贴加或删除用的，它还为我们分门别类（名称也可自己改，图中我就把第一类改为“代数符号”）。贴加的办法是：先输入我们要贴加的公式，然后选中用鼠标左键拖到工具栏中适当位置就行，删除则右击工具图标，选删除即可。图中我就添加了两个集合符号“ ”和“ ”。 Mathtype常用快捷键(二) 1．放大或缩小尺寸 Ctrl+1（100%）；Ctrl+2（200%）；Ctrl+4（400%）；Ctrl+8（800%）。2．在数学公式中插入一些符号 公式编辑器快捷输入方法 使用alt+ctrl+q 可直接打开公式编辑器窗口 一、常见的数学符号的快捷键(Ctrl 是王道) 1、根式、分式及上下标 ()()()()Ctrl+t n :Ctrl+f; :Ctrl *()Ctrl Shif +/ :Ctrl+h; :Ctrl+j;:Ctrl+l; :Ctrl+i t ;I,! ++??** * ** * 、*（分式，大）* ** 上标上下标下标不定积分积分模板 2、括号快捷键 ()[]{}:Ctrl+9Ctrl+0; :Ctrl+[Ctrl+];:Ctrl+{Ctrl+}; *或者*或者*或者 3、希腊字母 ()()()()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()() Ctrl+g #$%^&*_~|\[]a b c d e f g h i j k l m n o p q r s t u v w x y z A B C D E F G H I J K L M N O P Q R S T U V W X Y Z αβχδεφγηι?κλμνοπθρστυ?ωξψζ??A B X ?E ΦΓH I K ΛM N O ∏ΘP ∑T Y ΩΞψZ ?#?%⊥&*?~|∴[]后输入以下字母对应出现希腊字母@4、数学符号 ()()()()()()()()()()()( )()()()()()()()()()()() Ctrl+Shift+k 78\/.*|Ctrl+m 24n a b d e l n o p t x A O ????Ω⊥∴?∠∧?=±∨÷?*+?后输入以下字母对应出现 后输入到为对应距阵，输入为距阵模板对话框 5、不等式 ()()()()()()()()()()()()() ()()()()( )()()()()()() ()()()() ()()()()()( )()()() Ctrl+k 1~4~.,2468Shift+Shift+Shift+Shift+Alt+Alt+Alt+Shift+Tab Enter a c d e h i l o p s t u x C E I R U X ???∈∞?∝????????≈=≡+≠≥≤↓←→↑←?↓?→?↑?←?↓↑?后输入以下字母对应出现 为从小到大的空格按箭头键或输入小键盘的数字可得到：6、数学公式 函数 公式编辑器快捷输入方法 一、常见的数学符号的快捷键(Ctrl 是王道) 1、根式、分式及上下标 ()()()()Ctrl+t n :Ctrl+f; :Ctrl *()Ctrl Shif +/ :Ctrl+h;:Ctrl+j; :Ctrl+l; :Ctrl+i t ;I,! ++??*** ***、*（分式，大）* ** 上标上下标下标不定积分积分模板 2、括号快捷键 ()[]{}:Ctrl+9Ctrl+0; :Ctrl+[Ctrl+];:Ctrl+{Ctrl+}; *或者*或者*或者 3、希腊字母 ()()()()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()() Ctrl+g #$%^&*_~|\[]a b c d e f g h i j k l m n o p q r s t u v w x y z A B C D E F G H I J K L M N O P Q R S T U V W X Y Z αβχδεφγηι?κλμνοπθρστυ?ωξψζ??A B X ?E ΦΓH I K ΛM N O ∏ΘP ∑T Y ΩΞψZ ?#?%⊥&*?~|∴[]后输入以下字母对应出现希腊字母 @ 4、数学符号 ()()()()()()()()()()()()()()()()()()()()()()()Ctrl+Shift+k 78\/.*|Ctrl+m 24n a b d e l n o p t x A O ????Ω⊥∴?∠∧?=±∨÷?*+?Q l J m P 后输入以下字母对应出现 后输入到为对应距阵，输入为距阵模板对话框 5、不等式 ()()()()()()()()()()()()() ()()()()()()()()()()() ()()()() ()()()() ()()()()()Ctrl+k 1~4~.,2468Shift+Shift+Shift+Shift+Alt+Alt+Alt+Shift+Tab Enter a c d e h i l o p s t u x C E I R U X ???∈∞?∝????????≈=≡+≠≥≤↓←→↑←?↓?→?↑?←?↓↑?h D U I b c a 后输入以下字母对应出现 为从小到大的空格按箭头键或输入小键盘的数字可得到：6、数学公式 ( ) ()()()()()()()()()()()(){()Ctrl+t |[*]*{*}c d f i l n o p r s u C D F I L O P S U ???????????????? ? ? ? ? ? ? ? ? ? ? ? ? ??????? ???????????????? ? ? ? ? ? ????? ????→←??←?→????∑∏∑∏→←↑C I U C I U 后输入以下字母对应出现 ******************************************** }()()()()() ()()()()***?/*Alt+/**Alt+s *Alt+p *Alt+l lim n +*****Shift+Shift+Shift+Alt+Alt+Alt+***?????? ? ? ? ? ??????? ????????→←??←?→ ? ? ?? ?? ?? ? ??→←??←?→<>∑∏→∞→←↑→←↑**Mathematica函数大全(内置)
Word中MathType公式调整的一些技巧
(完整版)Word中使用MathType使公式自动编号
MathType使用技巧
MathStudio函数说明
MathType使用技巧大全
math 函数大全
word公式编辑器MathType使用技巧
MathType_使用技巧教程
math中函数以及其他一些库函数
mathtype使用方法
math库函数中文对照表
MathType使用技巧
C++math库函数
mathtype使用技巧总结——特别方便
MathType数学公式编辑器使用技巧
math.h中的常用函数
abs(arg)
fabs(arg) ceil（ arg) floor(arg) exp(arg) log(arg) log10(arg) pow(arg1,ar g2)
-camth-的数值函数
说明
返 回 arg 的 绝 对 值 ，其 类 型 与 arg 相 同 ，arg 可 以 是 任 意 浮 点 类 型 。在
-cmath-的三角函数
函数
cos(angle) sin(angle) tan(angle) cosh(angle)
sinh(angle)
tanh(angle)
acos(arg)
asin(arg)
atan(arg)
atan2(arg1,ar g2)
说明
返 回 angle 的 余 弦， angle 变 元 以 弧度 为单位 。 返 回 angle 的 正 弦， angle 变 元 以 弧度 为单位 。 返 回 angle 的 正 切， angle 变 元 以 弧度 为单位 。 返 回 angle 的 双 曲余 弦 { [e^x - e^(-x)]/2 }， angle 变 元 以 弧度 为单位。
返 回 angle 的 双 曲正 弦 { [e^x + e^(-x)]/2 }， angle 变 元 以 弧度 为单位。
返 回 angle 的 双 曲正 切 { [e^x + e^(-x)]/[e^x - e^(-x)] }， angle 变元以弧度为单位。
返 回 arg 的 反 余 弦 。变 元 在 1 到 -1 之 间 。结 果 以 弧 度 为 单 位 ， 其范围是 0 到π 。 返 回 arg 的 反 正 弦 。 变 元 在 1 到 -1 之 间 。 结 果 以 弧 度 为 单 位 ， 其 范 围是 -π /2 到 π /2。 返 回 arg 的 反 正 切 。 结 果 以 弧 度 为 单 位 ， 其 范 围 是 -π /2 到 π /2。
这 个 函 数 需 要 两 个 浮 点 类 型 的 变 元 ，返 回 arg1/arg2 的 反 正 切 。 结果以弧度为单位，范围是 0 到π ，类型与变元相同。MathType数学公式编辑器使用技巧及常用快捷键
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