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Driven Diffusive Systems An Introduction and Recent Developments

Driven Diffusive Systems An Introduction and Recent Developments
Driven Diffusive Systems An Introduction and Recent Developments

a r X i v :c o n d -m a t /9803392v 1 [c o n d -m a t .s t a t -m e c h ] 31 M a r 1998Driven Di?usive Systems:An Introduction and Recent Developments B.Schmittmann and R.K.P.Zia Center for Stochastic Processes in Science and Engineering Physics Department Virginia Polytechnic Institute and State University Blacksburg,VA,24061-0435USA 1Introduction In nature,there are no true equilibrium phenomena,since all of these require in?nite times and in?nite thermal reservoirs or perfect insulations.Neverthe-less,for a large class of systems,it is possible to set up conditions under which predictions from equilibrium statistical mechanics provide excellent approxi-mations,as many of the inventions of the industrial revolution can attest to.By contrast,non-equilibrium phenomena are not only ubiquitous,but often elude the powers of the Boltzmann-Gibbs framework.Unfortunately,to date,the theoretical development of non-equilibrium statistical mechanics is at a stage comparable to that of its equilibrium counterpart in the days before

Maxwell and https://www.doczj.com/doc/c714629803.html,ing the intuition developed by studying equilib-rium statistical mechanics,we are often “surprised”by the behavior displayed by systems far from equilibrium,even if they appear to be in time-independent states.For example,the well honed arguments,based on the competition be-tween energy and entropy,frequently fail dramatically.In these lectures,we will present some explorations into the intriguing realm of non-equilibrium steady states ,focusing only on a small class,namely,driven di?usive systems.

Motivated by both the theoretical interest in non-equilibrium steady states and the physics of fast ionic conductors[1],Katz,Lebowitz and Spohn[2]intro-duced a deceptively minor modi?cation to a well-known system in equilibrium statistical mechanics:the Ising[3]model with nearest-neighbor interactions. In lattice gas language[4],the time evolution of this model can be speci?ed by particles hopping to nearest vacant sites,with rates which simulate cou-pling to a thermal bath as well as an external?eld,such as gravity.If“brick wall”boundary conditions are imposed in the direction associated with grav-ity(particles re?ected at the boundary,comparable to a?oor or a ceiling), and if the rates obey detailed balance,then this system will eventually settle into an equilibrium state,similar to that of gas molecules in a typical room on earth.Although there is a local bias in the hopping rates(due to gravity), thermal equilibrium is established by an inhomogeneous particle density,at all T<∞.However,if periodic boundary conditions are imposed,then trans-lational invariance is completely restored so that,in the?nal steady state, the particle density is homogeneous,for all T above some?nite critical T c, while a current will be present.Clearly,for gravity,such boundary conditions can be imposed only in art,as by M.C.Escher[5].In physics,it is possible to establish such a situation with an electric?eld,e.g.,by placing the d=2 lattice on the surface of a cylinder and applying a linearly increasing magnetic ?eld down the cylinder xis.If the particles are charged,they will experience a local bias everywhere on the cylinder and a current will persist in the steady state.Echoing the physics of fast ionic conductors,we will therefore use the term“electric”?eld to describe the external drive and imagine our particles to be“charged”,in their response to this drive.This is the prototype of a “driven di?usive system.”Now,such a system constantly gains(loses)en-ergy from(to)the external?eld(thermal bath),so that the time independent state is by no means an equilibrium state,`a la Boltzmann-Gibbs.Instead,it is a non-equilibrium steady state,with an unknown distribution in general.As discovered in the last decade,modifying the Ising model to include a simple local bias leads to a large variety of far-from-simple behavior.The scope of these lectures necessarily limits us to only a bird’s eye view.Since a more extensive review has been published recently[6],we will restrict these notes to a brief“introduction”to this subject.Instead,we choose to include some developments since that review.

For completeness,we summarize,in Section II,the lattice model introduced by KLS and the Langevin equation believed to capture its essence in the long-time large-distance regime.The next section(III)is devoted to some of the surprising behavior displayed by this system,at temperatures far above,near, and well below T c.A collection of recent developments is presented in Section IV.In the?nal section(V),we conclude with a brief summary.

2The Microscopic Model and a Continuum Field Approach

On a square lattice,with L x×L y sites and toroidal boundary conditions,a particle or a hole may occupy each site.A con?guration is speci?ed by the occupation numbers{n i},where i is a site label and n is either1or0.Oc-casionally,we also use spin language,de?ning s≡2n?1=±1.To access the critical point,half-?lled lattices are generally used in Monte Carlo simu-lations: i n i=L x L y/2.The particles are endowed with nearest-neighbor at-traction(ferromagnetic,in spin language),modeled in the usual way through the Hamiltonian:H=?4J n i n j,with J>0.The external drive,with strength E and pointing in the?y direction,will bias in favor of particles hop-ping“downwards”.To simulate coupling to a thermal bath at temperature T, the Metropolis algorithm is typically used,i.e.,the contents of a randomly chosen,nearest-neighbor,particle-hole pair are exchanged,with a probabil-ity min[1,e?(?H??E)/k B T].Here,?H is the change in H after the exchange and?=(?1,0,1),for a particle attempting to hop(against,orthogonal to, along)the drive.Note that these dynamic rules conserve particle number.For E=0,this system will eventually settle into an equilibrium state which is precisely the static Ising model[3,4].In the thermodynamic limit,it is known to undergo a second order phase transition at the Onsager critical tempera-ture T c(0)=(2.2692..)J/k B.When driven(E=0),this system displays the same qualitative properties,i.e.,there is a disordered phase for large T,fol-lowed by a second order transition into a phase-segregated state for low T. However,with more scrutiny,this super?cial similarity gives way to puzzling surprises.In particular,T c(E)appears to be monotonically increasing with E,saturating at about1.41T c(0)[7]for E?J.Why should T c(∞)>T c(0) be surprising?Consider the following“argument”.For very large E,hopping along y becomes like a random walk,in that?H is irrelevant for the rates. Therefore,hops along y might as well be coupled to a thermal bath at in?nite T(apart from the bias).Indeed,recall that our system gains energy from E and loses it to the bath,so that any drive may be thought of as a coupling to a second reservoir with higher temperature.Then,it seems reasonable that T must be lowered to achieve ordering,since this extra reservoir pumps in a higher level of noise,helping to disorder the system!To date,there is neither a convincing argument nor a computation which predicts the correct sign of T c(E)?T c(0),let alone the magnitude.In the next Section,we will brie?y review other puzzling discoveries,only a few of which are understood.

To understand collective behavior in the long-time and large-scale limit,we often rely on continuum descriptions,which are hopefully universal to some extent.Successful examples include hydrodynamics and Landau-Ginzburg the-ories.Certainly,the?4theory,enhanced by renormalization group techniques, o?ers excellent predictions for both the statics and the dynamics near equi-librium of the Ising model.Following these lines,we formulate a continuum

theory for the KLS model,in arbitrary dimension d.In principle,such a de-scription can be obtained by coarse-graining the microscopic dynamics[8]but we will pursue a more phenomenological approach here.Seeking a theory in the long-time,large wavelength limit,we?rst identify the slow variables of the theory.These are typically

?ordering?elds which experience critical slowing down near T c,and ?any conserved densities.

The KLS model is particularly simple since it involves a single ordering?eld, namely the local“magnetization”or excess particle density,?(x,t),which is also the only conserved quantity.Here,x stands for(x1,x2,...,x d?1,x d=y). The last entry denotes the one-dimensional“parallel”subspace selected by E.Thus,we begin with a continuity equation,?t?+?j=0,and postulate an appropriate form for the current j.In the absence of E,it simply takes its Model B[9]form,j(x,t)=?λ?δH

2(??)2+τ

4!

?4}.As usual,τ∝T?T c and u>

0.While the?rst contribution to j is a deterministic term,re?ecting local chemical potential gradients,the second term,η,models the thermal noise. The noise is Gaussian distributed,with zero mean and positive second moment

proportional to the unit matrix,i.e.,<ηiηj>=2σδijδ(x?x′)δ(t?t′),i,j= 1,...,d.In the presence of E,we should expect at least two modi?cations to j,namely?rst,an additional contribution j E modeling the nonvanishing

mass transport through the system and second,the generation of anisotropies, since E singles out a speci?c direction.By virtue of the excluded volume

constraint,the“Ohmic”current j E vanishes at densities1and0,corresponding to?=±1.Writing E for the coarse-grained counterpart of E,pointing along unit vector?y,the simplest form is therefore j E=E(1??2)[1+O(?)]?y,where

the O(?)corrections will turn out to be irrelevant for universal properties. Next,we consider possible anisotropies.Clearly,all?2operators should be split into components parallel(?2)and transverse(?2⊥)to E,accompanied

by di?erent coe?cients.For example,the anisotropic version of the Model B termτ?2?will read(τ ?2+τ⊥?2⊥)?,with two di?erent“di?usion”coe?cients for the parallel and transverse subspaces.Should we expect both of these to vanish as T approaches T c(E)?Or just one of them-but which one?Recalling that the lowering ofτ,in the equilibrium system,is a consequence of the

presence of interparticle interactions,we argue that E,stirring parallel jumps much more e?ectively than transverse ones,should counteract this e?ect in the parallel direction,having less of an impact in the transverse subspace. Thus,we anticipate that genericallyτ >τ⊥,so that criticality,in particular, is marked byτ⊥vanishing at positiveτ .This is borne out by the structure of typical ordered con?gurations,namely,single strips aligned with E,indicating that“antidi?usion”,i.e.,τ⊥<0,dominates in the transverse directions below T c.Finally,the drive also induces anisotropies in the noise terms so that the second moment of their distribution should be taken as<ηiηj>=2σiδijδ(x?

x′)δ(t?t′).Since the transverse subspace is still fully isotropic,we de?ne σ1=...=σd?1≡σ .Generically,however,we should expectσd≡σ⊥=σ . Summarizing,we write down the full Langevin equation:

?t?(x,t)=λ{(τ⊥??2⊥)?2⊥?+(τ ??2)?2??2α×?2?2⊥?+

+

u

δ?.In this case,P?is simply proportional to

exp(?H/σ),irrespective of the choice of N.We will refer to such a dynamics, in an operational sense[6],as“satisfying the?uctuation-dissipation theorem”(FDT)[13].Model B falls into this category,with N=??2.In contrast, for our driven system,F is just the expression in the{...}brackets of(1)

and?≡??η,so that N=?(σ

?2+σ⊥?2⊥).Thus,this F is clearly not

Hamiltonian!In this fashion,our continuum theory re?ects the fact that we are dealing with a generic non-equilibrium steady state.For completeness, we should point out that certain microscopically non-Hamiltonian dynamics may become Hamiltonian if viewed on su?ciently large length scales.The discussion of these subtleties[14–16],while intriguing,is beyond the scope of these lectures.

3Surprising Singular Behavior,near and far from Criticality

For the d=2Ising model in equilibrium,thermodynamic quantities are typ-ically analytic,except at a single point,i.e.,T c.In particular,being a model with only short-range interactions,correlation functions are short-ranged far above T c,decaying exponentially with distance and controlled by a?nite cor-relation length.Far below criticality,a half?lled system in a square geometry will display two strips of equal width,as a result of the co-existence of a particle rich(dense)phase with hole-rich one.Correlations of the?uctuations within each phase are also short-ranged.More interestingly,the interfaces between the phases do represent soft degrees of freedom,being the Goldstone mode of a broken translational invariance[17].One consequence is a divergent struc-ture factor(Fourier transform of the two-point correlation,of deviations from being straight).Although this behavior is singular,it is well understood and can be related to that of a simple random walk[18].Only at criticality does the system possess non-trivial singular properties,the nature of which became transparent only in the70’s[19]despite Onsager’s tour-de-force in1944[20]. When driven into non-equilibrium steady states,this picture changes dramat-ically.In particular,non-trivial singular behavior appears at all T.While the transition itself remains second order,its properties fall into a non-Ising class. Here,we will present some of these surprising features brie?y,referring the reader to[6]for more details.

3.1T far above T c

Focusing on the behavior far above criticality,let us study the two-point cor-relation G(r)(r=(x,y))and its transform,the structure factor,S(k).We?rst point out that the drive clearly induces an asymmetry between x and y,i.e., anisotropy beyond that due to the lattice.Thus,we should not be surprised if the familiar Ornstein Zernike form of S(∝(1+ξ2k2))were to become el-liptical rather than circular,e.g.,similar to Fig.1a.However,simulations[2] showed that there is a discontinuity singularity in S at the origin,as in Fig.1b. To be more precise,we may de?ne

R≡lim k

y→0

S(0,k y)

Fig.1.Ising model in system. Strange the con-text ofτ andτ⊥are positive.a stable min-imum at nonlinearities are the system in the Langevin

?t?(x,t)=λ(τ⊥?2⊥+τ ?2)???η.(3) With the help of a Fourier transform,we can easily?nd?(k,t)for any real-ization ofη,and then compute averages over the Gaussian distributedη,with its second moment given by(1).This forms the starting point for the discus-sion of the disordered phase,quanti?ed by,e.g.,equal-time structure factors or three-point functions[21].

Here,we focus on the former,S(k)≡,which can be easily computed:

S(k)=σ⊥k2x+σ k2y

so that R= σ τ⊥ / σ⊥τ .In order for this S to reduce to the equilibrium Ornstein-Zernike form,the FDT has to be invoked,which constrains R to

unity.For the driven case,R is no longer constrained,so that a discontinuity develops.One consequence of such a singularity in S is that G becomes long-ranged,decaying as r?2at large|r|.The amplitude,in addition to being∝(R?1),has a dipolar angular dependence,so that an appropriate angular average of G is again short-ranged[6].

To end this discussion,let us follow G.Grinstein’s argument[22]that such long-ranged power law decays should be expected.Starting with the dynamic two-point correlation G(r,t),we know that the conservation law leads to the

autocorrelation:G(0,t)→t?d/2.In addition,being a di?usive system,we

should expect r~

L.In our case,if we study an interface aligned along the y axis,then,to a good approximation,we may specify the con?guration by its position along x by the“height”function h(y).Known as capillary waves[24],these?uctuations have a venerable his-tory.Since the interface is a manifestation of broken translational invariance, h(y)are the soft Goldstone modes[17],so that the associated structure fac-

tor S h(q)≡ h(q)h(?q) diverges as q?2.Of course,this property is just the

Fourier version of the

plotting the widths vs.L p with various values of p ,we found that the curves did not straighten out with p as low as 0.05[25].As a phenomenon,roughness suppression is not novel,gravity being the most common example.However,the drive here is parallel to the interface,reminiscent of wind driving across a water surface,which has a destabilizing e?ect by contrast.Subsequently,a more detailed simulation study [26]of the interface with L y ≤600showed that S (q )~q ?2for large q ,crossing over to q ?0.67for small q (Fig.2).Since 1/L q

?0.67dq =O (1),the small q behavior is consistent with p =0.Though 0.67appears temptingly close to a simple rational:2/3,there is no viable theory so far (despite two valiant attempts [27]).Finally,let us note that the crossover from rough to smooth occurs at the scale of q ~E .Although no precise Monte Carlo analysis of this crossover has been performed,it is con-sistent with E having the units of 1/length.In this respect,such a length is comparable to the capillary length which controls the crossover in the gravi-tationally stabilized interface.Of course,in that case,the small q behavior is simply q 0!

q

1/S (q )

h Fig.2.Log-log plot of interface structure factor vs.wavevector.

3.3Critical Properties

Finally,we turn to the critical region,described by τ⊥ 0in (1).In contrast to the situation for T ?T c ,we now need fourth-order terms in ?⊥to stabilize the system against large-wavelength ?uctuations.Similarly,the nonlinear terms are required to ensure a stable ordered phase below T c .However,we still have τ >0,so that fourth-order parallel gradients may safely be neglected.Thus,near criticality,the leading terms on the right hand side of (1)are

(?2⊥)2?and ?2?,implying that parallel and transverse wave vectors,k ⊥and k ,scale naively as |k |~|k ⊥|2,i.e.,parallel gradients are less relevant than transverse ones.More systematically,a naive dimensional analysis [28]reveals

that the nonlinearity associated with E is the most relevant one,having an upper critical dimension d c=5,distinct from the usual value of4for the Ising model.Dropping irrelevant terms,(1)simpli?es to

?t?(x,t)=λ{(τ⊥??2⊥)?2⊥?+?2?+u

Here,μis just a scaling factor.Eqn.(6)can be viewed as a de?nition of the critical exponentsν,z,ηand?.The appearance of the latter is of course consistent with our earlier discussion.Di?erent universality classes are dis-tinguished by the characteristic values of these exponents,expressed,e.g., through their?-expansions.For our model,all exponents,except?,take their

mean-?eld values:ν=1

3.A separate analysis

yields the order parameter exponentβ=1

1+?.Since most

simulations have focused on two-dimensional systems,we set?=3,predicting ?=2andη′

=2/3.Monte Carlo data for the pair correlation along the

drive beautifully display the expected r?2/3

decay[32].Numerical evidence for

the strong anisotropy exponent is somewhat more indirect,extracted from an anisotropic?nite size scaling analysis[7].Convincing data collapse is obtained, using anisotropic systems of size L ×L⊥with?xed“aspect”ratio L /L1+?⊥, for the theoretically predicted values of the exponents.

It is intriguing that the signals of continuous phase transitions in or near equilibrium,namely,a diverging length scale,scale invariance and universal behavior,also mark such transitions in far-from-equilibrium scenarios.

4Some Recent Developments

Beyond the topics discussed in the previous section,we may arbitrarily name three other“levels”of non-equilibrium steady-state systems.The?rst are as-sociated with the KLS model itself,including fascinating results on higher correlations[21],failure of the Cahn-Hilliard approach to coarsening dynam-ics[33],models with shifted periodic and/or open boundary conditions[34], and systems with AC or random drives[35,16].Topics in the next“level”in-volve various generalizations of KLS,such as other interactions[36,37],multi-

layer systems,multi-species models,and systems with quenched impurities [38].Further“a?eld”is a wide range of driven systems,e.g.,surface growth, electrophoresis and sedimentation,granular and tra?c?ow,biological and ge-ological systems,etc.In these brief lecture notes,we present two topics at the “intermediate level”:a driven bi-layer system and a model with two species.

4.1Phase Transitions in a Bi-layer Lattice Gas

The physical motivation for considering multi-layered systems may be traced to intercalated compounds[39].The process of intercalation,where foreign atoms di?use into a layered host material,is well suited for modeling by a driven lattice gas of several layers.In both physical systems and Monte Carlo simulations of a model with realistic parameters[40],?nger formation has been observed.Both are transient phenomena,so that we might ask:are there any novel phenomena in the driven steady states?On the theoretical front,there are two motivations.To study critical behavior of the single layer KLS model, it is necessary to set the overall density at1/2,with the consequence that two interfaces develop below criticality.These interfaces seriously complicate the analysis,since their critical behavior is quite distinct from the bulk.One way to avoid these di?culties is to consider a bi-layer system[41],with no inter-layer interactions.Through particle exchange between the layers,the system may order into a dense and a hole-rich layer,neither of which has interfaces. Certainly,this is expected(and observed)in the equilibrium case,where the steady state is just two Ising models,decoupled except for the overall conser-vation law.The other is using multilayer models as“interpolations”from two-to three-dimensional systems[42].Motivations aside,what the simulations reveal is entirely unexpected.

In the simplest generalization,a bi-layer system is considered as a fully periodic L×L×2model.The?rst simulations were carried out on the case with zero inter-layer interactions[42].Therefore,within each layer,all aspects are identical to the KLS model.Particle exchanges across the layers,una?ected by the drive,are updated according to the local energetics alone,so that these jump rates are identical to a system in equilibrium.The overall particle density is?xed at1/2.In the absence of E,there would be a single second order transition at the Onsager temperature.Since the inter-layer coupling is zero,phase segregation at low temperatures will occur across the layers, i.e.,the ordered phase is characterized by homogeneous layers of di?erent densities(opposite magnetizations,in spin language).Having no interfaces, the free energy of the system is clearly lowest in such a state.At T=0,one layer will be full and the other empty,so that this state will be labeled by FE. Intriguingly,when E is turned on,two transitions appear[42]!As T is lowered,

the homogeneous,disordered(D)phase?rst gives way to a state with strips

in both layers,reminiscent of two entirely unrelated,yet aligned,single layer driven systems.We will refer to this state as the strip phase(S).As T is lowered

further,a?rst order transition takes the S phase into the FE state.Why the S state should interpose between the D and FE states was not understood.

This puzzle,together with the presence of interlayer couplings in intercalated

compounds,motivated our study of the interacting bi-layer driven system[43]. Within this wider context,the presence of two transitions is no longer a total

mystery.On the other hand,this study reveals several unexpected features,

leading to interesting new questions.

Our system consists of two L2Ising models with attractive interparticle in-teractions of strength unity.Arranged in a bi-layer structure,the inter-layer

interaction is speci?ed by J.Thus,the“internal”Hamiltonian is just H≡

?4 nn′?4J nn′′,where the?rst sum is over nearest neighbors within a given layer while n and n′′di?er only by the layer index.The external drive,

E,is imposed through the jump rates,as in Section II.For simulations,we kept the overall density at1/2and chose J∈[?10,10].Note that negative J’s are especially appropriate for intercalated materials[39,40].The eventual goal of such a study is to map out the phase diagram in the T-J-E space.So far,we have data for only three(positive)values of E.Not expecting further surprises,we believe that we have uncovered the main features of the phase diagram.

To begin,let us point out various features in the equilibrium case.Here,the J>0and the J<0systems can be mapped into each other by a gauge transformation.Thus,in the thermodynamic limit,the phase diagram in the T-J plane is exactly symmetric.However,for the lattice gas constrained by a conservation law,the low temperature states of these two systems are not the same,being S or FE,respectively.While the D-S or D-FE transitions are second order(shown as a line in Fig.3),the S-FE line will be?rst order in nature(T axis from0to1in Fig.3).Due to the presence of interfaces,the FE domain includes the J≡0line.For the same reason,it“intrudes”into the J>0half plane by O(1/L),for?nite systems.Of course,on the T axis lies a pair of decoupled Ising systems,so that T c(J=0)assumes the Onsager value,as L→∞.In general,T c(J)is not known exactly,though T c(J=±∞)is precisely2T c(J=0),since every con?guration of the two layers can be mapped into one in the usual Ising model.The lesson from equilibrium is now clear: the S state is as generic as the FE state in this wider context.As we will see below,the presence of two transitions does not represent a qualitative change from the equilibrium phase structure.

Turning to driven systems,we can no longer expect a symmetric phase dia-gram,since the drive breaks the Ising symmetry.The data for E=25are shown in Fig.3,in which the?rst/second order transitions are marked as

Fig.3.second order

shift the phase

remains

lowering of the critical temperature for large|J|.Given that T c(E)>T c(0)in the KLS model,it is quite unexpected that T c(|J|?1,E?1)is less than its equilib-rium counterpart.Perhaps more notable is the presence of a small region in the J<0half plane,in which an S-phase is stable.Since particle-rich strips lie on top of each other,such a phase could not exist if either energy or entropy were to dominate the steady state.Concerned that its presence might be a ?nite size e?ect,we performed simulations using L’s up to100,with J=?0.1 and T∈[1.00,1.20].In all cases,the S-phase prevailed,leading us to conjec-ture that this region exists even in the thermodynamic limit.On the other hand,for lower T,the FE-phase penetrates into the J>0half plane,as in equilibrium cases.Energetics seem to take the upper hand here.Though we believe that,as L→∞,this part of the phase boundary will collapse onto the J=0line,we have not checked the?nite size e?ects explicitly.Of course, these“intrusions”into the positive/negative J region by the FE/S-phase are responsible for the appearance of two transitions in the studies of the J≡0 model[42].

It would clearly be useful to develop some intuitive arguments which can“pre-dict”these qualitative modi?cation of the equilibrium phase structure.Since the usual energy-entropy considerations appear to fail,our attempt[43]is based on the competition between suppression of short-range correlations[44] and enhancement of long-range ones[32],as a result of driving.Let us focus on the two-point correlations in the disordered phase with T being lowered from ∞and see how they are a?ected by the drive.On the one hand,the nearest-

neighbor correlations are found to be suppressed by E,consistent with the picture that the drive acts as an extra noise in breaking bonds.Taken alone, this suppression would lead to the lowering of the critical temperature,as pointed out in Section II.On the other hand,as we showed above,the drive changes signi?cantly the large distance behavior of G,from an exponential to a power law.Further,the amplitude is positive(negative)for correlations parallel(transverse)to E.Both the positive longitudinal correlations and the negative transverse ones should help the process of ordering into strips parallel to the?eld.In other words,the enhanced long-range parts favor the S-phase. Taken alone,we expect this e?ect to increase T c(E).Evidently,for the single layer case,the latter e?ect“wins”.For a bi-layer system,we need to take into account cross-layer correlations,which are necessarily short-ranged. Focusing?rst on the J=0case,where short-range e?ects due to cross-layer interactions are absent,we are led to T c(0,E)/T c(0,0)>1.Indeed,this ratio is comparable to that in the single-layer case.Next,we consider systems with positive J.Without the drive,T c(J,0)is of course enhanced over T c(0,0). For non-vanishing drive,it is not possible to track the competition between the short-and long-range properties of the transverse correlations.Evidently, for small J,the long-range part still dominates,so that T c(J,E)/T c(J,0)> 1.However,for J?J0,the presence of E e?ectively lowers J,since the latter is associated with only short-range correlations.Since a lower e?ective J naturally gives rise to a lower T c,we would“predict”that T c(J,E)/T c(J,0) could decrease considerably as J increases.From Fig.3,we see that T c(J,E)< T c(J,0)for J≥5!The interplay of the competing e?ects is so subtle that either can dominate,in di?erent regions of the phase diagram.Finally,we turn to the J<0case.Here,the two e?ects tend to co-operate rather than compete,since the long-range parts favor the S-phase over the FE-phase.As a result,the domain of FE is smaller everywhere.In particular,note that the J<0branch of T c(J,E)is signi?cantly lower than the J>0branch. The small region of the S-phase can be similarly understood,at least at the qualitative level.In this picture,we may argue that the bicritical point and its trailing?rst order line should be“driven”to the J<0half-plane.

To end this subsection,we point out a few other interesting features.The order parameters of the S and FE phases are conserved and non-conserved, respectively.Based on symmetry arguments,we may expect that the critical properties of the two second order branches will fall into di?erent universality classes.In contrast,the gauge symmetry forces static equilibrium properties to be identical along these two branches[45].Since the drive breaks this sym-metry,we can only speculate that the D-S transitions belong to the same class as the single layer,KLS model[30]while the D-FE ones should remain in the equilibrium Ising class[14].Since these two classes are distinct,we can also expect a rich crossover structure near the bi-critical point.Assuming these conjectures are con?rmed,we can truly marvel at the novelties a driving?eld

can bring.

4.2Biased Di?usion of Two Species

If the particles in the KLS model are considered“charged”,it is natural to

explore the e?ects of having both positive and negative charges.De?ned on a fully periodic L x×L y square lattice,a con?guration of such a“two species”model can be characterized by two local occupation variables,n+r and n?r,

which equal1(0),if a positive or negative particle is present(absent)at site r≡(x,y).In spin language,this corresponds to a spin-1model,and novel behavior is to be expected,since we have altered the internal symmetries of the local order parameter.For simplicity,we will restrict ourselves to zero to-tal charge,Q≡ r(n+r?n?r)=0.Pursuing the analogy to electric charges, positive(negative)charges move preferentially along(against)the drive E which is directed along the?y axis.As a?rst step,we assume that there are no interparticle interactions apart from an excluded volume constraint. Thus,our model can be viewed as the high-temperature,large E limit of a more complicated interacting system.To model biased di?usion,a parti-cle with charge q=±1jumps onto a nearest-neighbor empty site accord-ing to min[1,e?qEδy],whereδy=0,±1is the change in the y-coordinate of the particle.To allow charge exchange between neighboring particles,two nearest-neighbor sites carrying opposite charge may exchange their content with probabilityγmin[1,e?qEδy].Now,δy is the change in the y-coordinate of the positive particle.The parameterγsets the ratio of the characteristic time scales controlling,respectively,charge exchange and di?usion.

Physical motivations for this model come from various directions:fast ionic

conductors with several species of mobile charges[1],electric breakdown of

water-in-oil microemulsions[46],and gel electrophoresis of charged polymers [47].It can also be interpreted as a simple model for some tra?c or granular ?ows[48,49].

Summarizing our simulation data,we?rst map out the phase diagram of the

model,in the space spanned by E,the total mass densityˉm≡ r(n+r+ n?r)/(L x L y),andγ[50].Focusing on smallγ,the di?usive dynamics is limited by the excluded volume constraint.For su?ciently small E andˉm,typical con?gurations are disordered,characterized by homogeneous charge and mass densities and fairly large currents.Nevertheless,this phase is highly nontrivial, supporting anomalous two-point correlations reminiscent of the KLS model:In generic directions,the familiar r?d decay prevails,with the remarkable excep-

tion of the?eld direction,where a novel r?(d+1)/2

dominates[51].With increas-

ing E orˉm,the tendency of the particles to impede one another becomes more pronounced,until a phase transition into a spatially inhomogeneous“ordered”

phase occurs.Beyond the transition line E c (ˉm,γ),typical con?gurations are charge-segregated,consisting of a strip of mostly positive charges “?oating”on a similar strip of negative charges,surrounded by a background of holes.To explore these transitions more quantitatively,we need to de?ne a suitable order parameter.Taking the Fourier transforms of the local hole and charge densities,?φ(k ⊥,k )≡1L x L y r [n +r ?n ?r

]exp {ik ⊥x +ik y },we select the components Φ≡|?φ(0,2πL y )|since these are most sensitive to ordering into a single transverse strip.Naturally,we choose their averages <Φ>and <Ψ>as our

order parameters.To distinguish ?rst from second order transitions,we also measure their ?uctuations and histograms.The resulting phase diagram is shown in Fig.4.Typically,hysteresis loops in <Φ>,<Ψ>and the average current are observed only for ˉm <ˉm o (γ),indicating that the transitions are ?rst order in this regime.On the other hand,?uctuations develop sharp peaks for larger mass densities,signalling continuous https://www.doczj.com/doc/c714629803.html,rger values of γfavor the disordered phase,so that the whole transition line shifts to higher E ,the region between the spinodals narrows,and ˉm o (γ)moves to higher mass densities.Thus,we observe a surface of ?rst order transitions,separated from a surface of continuous ones by a line of multicritical points:E c (ˉm o (γ),γ).

m

E

01

2

3

4

5

Fig.4.Phase diagram for the driven two-species model.The system size is 30×30.The ?lled circles mark the lines of continuous transitions,while the open circles denote the spinodal lines associated with ?rst order transitions.The inset shows a typical ordered con?guration at ˉm =0.40,E =3.00and γ=0.02with the open (?lled)circles representing positive (negative)charges.

While it is not surprising that the disordered phase is stable for large E pro-vided the mass density is su?ciently small,it may appear rather counterintu-itive that it should also dominate theˉm 1region.However,settingˉm≡1 eliminates every hole in the system so that the dynamics is carried entirely by the charge exchange mechanism.Relabelling positive charges as“particles”and negative ones as“holes”,the model becomes equivalent to a noninteract-ing(i.e.,J=0)KLS model.Also termed the asymmetric simple exclusion process(ASEP)in the literature,its steady-state solution is exactly known [52]to be homogeneous.Thus,for allγ>0,our two-species model remains disordered along the entireˉm≡1line.For su?ciently largeγ,the data in-dicate quite clearly that a?nite region of disorder persists,for any E,just below complete?lling.For smallerγ,however,it is less obvious whether such a region exists in the thermodynamic limit,since even a single hole can suf-?ce to induce spatial inhomogeneities in a?nite system:Acting as a catalyst for the charge segregation process,the hole creates a strip of predominantly positive charge,separated by a rather sharp interface from a similar,nega-tively charged domain,located“downstream”.The charge exchange mecha-nism tends to remix the charges but has little e?ect for smallγ.Eventually, the hole ends up trapped in the interfacial region!Finally,we turn to larger values ofγ,speci?cally,γ 0.62:here,the charge exchange mechanism dom-inates over the excluded volume constraint and suppresses the ordered phase completely.

So far,our discussion has mostly drawn upon Monte Carlo results.It is grat-ifying,however,that the same qualitative picture emerges from a mean-?eld theory,based upon a set of equations of motion for the local densities.We brie?y summarize the analytic route.Since the numbers of both positive and negative charges are conserved,we begin with a continuity equation for the coarse-grained densitiesρ±(r,t):?tρ±+?j±=0.For simplicity,we fo-cus on the caseγ=0,i.e.,particle-hole exchanges,?rst.As for the KLS model,j±can be written as the sum of a di?usive piece and an Ohmic term, j±=λ±{??δH

δρ±is taken

at?xedρ?since we are focusing on particle-hole exchanges here.To model the charge exchange mechanism,we simply add a similar term to j±,namely, j±=λ{??δH

δρ±|?±E?y}whereλ′=ρ+ρ?vanishes with bothρ+andρ?,and the functional derivative is taken at?xed hole density. Expressing the resulting equations in the more convenient variables?and charge densityψ≡ρ+?ρ?,we obtain

?t?=?{??+E?ψ?y}

?tψ=?{γ?ψ+(1?γ)[??ψ?ψ??]?E?(1??)?y(7)?γ

1+γ

du .The potential V(u),given

explicitly in[50],exhibits a minimum for a range of J’s,so that inhomoge-neous,periodic solutions of(7)exist.We note in passing that these may even be found explicitly,providedγ=0[53].Otherwise,numerical integration is of course always possible,yielding rather impressive agreement with simula-tion pro?les.Interestingly,our equations of motion predict that E enters only through the scaling variable E L y.If plotted accordingly,our simulation data con?rm this scaling by collapsing rather convincingly,at least within their error bars.

However,the mere existence of two types of steady-state solutions is not suf-?cient to provide evidence for a phase transition.Therefore,we seek insta-bilities of these solutions,as the control parameters E,ˉm andγare var-ied.Performing a linear stability analysis,we?nd that a homogeneous so-lution with massˉm becomes unstable if the drive E exceeds a threshold value E H(ˉm,γ)=2π1?ˉm+γˉm

2?γ<ˉm<1,so that,in particular,the homogeneous phase is always

stable forγ 1.Clearly,we cannot identify this mean-?eld stability boundary with the true transition line:we have not considered the locus of instabilities associated with inhomogeneous solutions,and?uctuations have been neglected throughout.However,it is quite remarkable how well it mirrors the qualitative shape of the phase diagram.

Finally,let us consider the order of the transitions.In principle,two routes

can be pursued here.One of these,namely,the computation of the stability

boundary of the inhomogeneous phase,E I(ˉm,γ),is rather subtle,involving three Goldstone modes[54].Once E I(ˉm,γ)is known,values(ˉm,γ)for which

E I and E H coincide can be identi?ed as loci of continuous transitions;other-wise,E H and E I mark the“spinodals”,near a?rst order transition,where the

homogeneous/inhomogeneous phases become linearly unstable.Alternatively,

the adiabatic elimination of the fast modes results in an e?ective equation of motion for the slow ones which can then be analyzed.Since the technical details of this approach can be found in[50],we need only focus on the result which combines the expected with the surprising.Letting M(q⊥,t)denote the complex amplitude of the unique slow mode,associated with the band of wave vectors(q⊥,2π/L y),its equation of motion takes the Ginzburg-Landau form:?t M=?{(τ+q2⊥)M+gM|M|2+O(M5)}.Here,τis the soft eigenvalue, which vanishes on the stability boundary and g=g(E,ˉm,γ,L y)is a rather complicated function.As in standard Landau theory,the sign of g determines the order of the transition:if positive,the transition is continuous while neg-ative g signals a?rst order one.Sinceτ?0,we can set E=E H and discuss g on the stability boundary itself:being positive forˉm 1,it has a unique zero at a criticalˉm o(γ),below which it becomes negative.Sinceˉm o(γ)increases withγ,we recover the qualitative behavior of the multicritical line observed in the simulations.The surprising aspect of this equation of motion,however, resides in the fact that M is O(2)-symmetric and q⊥spans just a single spatial dimension.Given these symmetries,the Mermin-Wagner theorem[55]should forbid the existence of long-range order!One might hope that a careful analy-sis of?nite-size e?ects in this system would contribute to the disentanglement of this puzzle.

We conclude with two comments.First,we noted earlier that the caseˉm≡1 is exactly soluble.There are,in fact,two other surfaces in the phase dia-gram,corresponding toγ=1andγ=2,for which certain distributions are exactly known.Settingγ=1implies that the rates for particle-hole and particle-particle exchanges become equal,so that any given particle,e.g.,a positive charge,cannot distinguish between negative charges and holes.Thus, it experiences biased di?usion,equivalent to the non-interacting KLS model. Accordingly,the marginal steady-state distribution of the occupation numbers of one species is uniform,i.e.,P[{n±r}]= {n?r}P[{n+r,n?r}]∝1,and observ-ables pertaining to a single species are trivial.For example,the two-point correlation functions for equal charges,and,vanish for r=0.The full distribution,P[{n+r,n?r}],however,is nontrivial,so that,e.g., the two-point function for opposite charges,,remains long-ranged [51].A completely random system,marked by a uniform P[{n+r,n?r}]results only ifγ=2.Finally,we note that the one-dimensional version of our model is exactly soluble by matrix methods[56].

Second,it is natural to wonder about the consequences of having non-zero

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《舌尖上的中国》中提到的所有美食统计 第一集自然的馈赠 1,香格里拉松茸 2,江浙地区冬笋油焖冬笋 3,广西柳州酸笋黄豆酸笋小黄鱼 4,云南大理诺邓山区诺邓盐血肠火腿莴笋炒火腿火腿炒饭 5,湖北嘉鱼藕莲藕炖排骨 6,吉林查干湖湖水大鱼鱼头泡饼(北京) 7,海南香煎马鲛鱼酸菜鱼汤水煮红螺 第二集主食的故事 1,山西襄汾县花馍花卷油卷 2,陕西绥德黄馍馍(糜子面) 3,新疆库车馕饼 4,中原地区馒头(馍馍,古时又叫炊饼,现在想通武大郎为啥喊“炊饼”了) 长期发展形成南稻北麦,2000年前五谷排行顺序:稻,黍shu,稗bai,麦,菽shu;现在以稻,麦,玉米为主。5,贵州黎平米粉汤粉 6,广州沙河河粉干炒牛河 7,陕西西安凉皮汉中米皮(这是本人自己加上的,因为说到米粉,河粉,如果不说凉皮就说不过去了,导 演组失职啊) 8,陕西肉夹馍,牛羊肉泡馍,粉蒸肉夹馍(也叫荷叶饼夹馍,这个也是按自己加的) 9,兰州拉面 10,广州竹升面云吞捞面 11,中原地区手擀面 12,陕西岐山臊子面 13,嘉兴粽肉子蛋黄棕(关于粽子南方有肉粽,蛋黄粽等,北方一般都是蜜枣棕,北方人吃不惯肉粽,咸棕)

15,北方饺子焖面(陕西河南) 第三集转化的灵感 本集介绍三大部分1豆腐,2,酒,3醋(第六集时会讲一下醋所以这集就说了一下),4酱油,5酱油,6大酱 豆腐篇 1,云南红河建木县碳烤豆腐球石屏县老豆腐 2,中原地区石膏豆腐(各种豆制品,相信大家都是很喜欢吃豆腐的吧 ,尤其是嫩的。) 3,内蒙古锡林郭勒旗奶茶奶豆腐奶制品 4,云南白族豆腐皮 5,北京蒙古餐厅烤羊背 6,浙江天台山僧人的素食中豆制品很重要 7,安徽毛豆腐 酒篇 8,绍兴黄酒 9,安徽休林糯米酒 10 ,安昌镇腊肠 11,东北大豆酱酸菜 第四集时间的味道 这一集是介绍腌制品,脱水,酱菜 1,黑龙江绥化市朝鲜族泡菜, 2,广粤地区腊肠各种腊制品煲仔饭荔芋腊鸭煲南安腊鸭 3,湖南靖州县腌鱼腊鸭 4,徽州臭鳜鱼 5,安徽黔县腊八豆腐刀板鱼 黄山火腿咸肉 6,浙江金华火腿蜜汁上方 7,上海的三阳南货店经营各种腌制品腊肉等: 杭州酱鸭 上海腌笃鲜 温州黄鱼鲞xiang 宁波笋干 绍兴梅干菜梅干菜烧肉 8,上海醉虾醉蟹 9,福建霞浦紫菜 10,台湾云朴县乌鱼子 11,香港大奥海盐产地咸鱼虾膏虾酱 12,中原地区各种酱菜(腌萝卜,鬼子姜,辣椒,黄瓜各种蔬菜) 第五集厨房的秘密 这一集最后一句“厨房的秘密就是没有秘密”很是狗血啊。

各种电机种类及应用

电机与拖动基础讨论课课题:各种电机应用现状调研 班级:13级工程机械1班 组员:李昊天张晨阳崔超钰 陈杰董亚飞潘帅 时间:2016年4月5日 指导老师:马云飞

目录 前言 (3) 1、电机产业现状 (4) 2、电机行业格局 (4) 3、电机行业前景预测 (4) 4、电机的分类及应用概述 (5) 5、信号电机 (6) 5.1、位置信号电机 (6) 5.2、感应同步器 (6) 5.3、自整角机 (7) 5.4、速度信号电机 (7) 6、功率电动机 (8) 6.1、直流电动机 (8) 6.2、交流电机 (8) 6.3、同步电动机 (9) 6.4、异步电动机 (10) 7、控制电机 (10) 7.1伺服电机 (10) 7.2、步进电动机 (11) 7.3、力矩电动机 (12) 7.4、开关磁阻电机 (13) 7.5、无刷直流电动机 (13) 参考文献 (15)

前言 从广义上讲,电机是电能的变换装置,包括旋转电机和静止电机。旋转电机是根据电磁感应原理实现电能与机械能之间相互转换的一种能量转换装置;静止电机是根据电磁感应定律和磁势平衡原理实现电压变化的一种电磁装置,也称其为变压器。本文我们主要讨论旋转电机,旋转电机的种类很多,在现代工业领域中应用极其广泛,可以说,有电能应用的场合都会有旋转电机的身影。与内燃机和蒸汽机相比,旋转电机的运行效率要高的多;并且电能比其它能源传输更方便、费用更廉价,此外电能还具有清洁无污、容易控制等特点,所以在实际生活中和工程实践中,旋转电机的应用日益广泛。不同的电机有不同的应用场合,随着电机制造技术的不断发展和对电机工作原理研究的不断深入,目前还出现了许多新型的电机,例如,美国EAD公司研制的无槽无刷直流电动机,日本SERVO公司研制的小功率混合式步进电机,我国自行研制适用于工业机床和电动自行车上的大力矩低转速电机等。本文下面主要讨论部分电机种类及应用情况。

舌尖上的中国中英文介绍

舌尖上的中国中英文介绍 《舌尖上的中国》第二季于2013年1月10日在京正式启动,该片于2014年4月18日至6月6日,每周五在中央电视台综合频道(CCTV-1)21点档、中央电视台纪录频道(CCTV-9)22点档同步开播,同时在爱奇艺,乐视网等网络平台播出。 A bite of China2 was started on January 10th,2013 in Beijing. This documentary will be on from April 18th,2014 to June 6th,2014.Each Friday, audience can enjoy this documentary in CCTV1 at 21.pm, or CCTV9,22pm.In the meanwhile, it shows in IQIYI or Letv and other network platform. 《舌尖上的中国》讲述了从远古时代赖以充饥的自然谷物到如今人们餐桌上丰盛的、让人垂涎欲滴的美食,一个异彩纷呈、变化多端的主食世界呈现在你面前。这部纪录片将着重描绘不同地域、不同民族、不同风貌的有关主食的故事,展现人们对主食的样貌、口感的追求,处理和加工主食的智慧,以及中国人对主食的深厚情感。据导演透露《舌尖上的中国2》将加重川菜的部分。 A bite of China describes food’s transition from ancient to nowadays. In ancient, humans just could solve the hunger , they could not eat delicious food. But now, there are various of food on people’s dinner-table. A colorful and changeable major food world emerges your eyes. This documentary pays most attention on describing the stories about major food in different areas, different races and features. It shows the intelligence about Chinese. They went after food’s features, tastes, solution and process. Of course, it also reflects Chinese feelings. From the director, A bite of China will introduce the food of Sichuan most detail. 《舌尖上的中国2》共分为《脚步》、《心传》、《时节》、《家常》、《秘境》、《相逢》、《三餐》,第八集则为拍摄花絮。每集50分钟 A bite of China covers 8 segments, including 《Steps》,《Heart inheriting》,《Seasons》,《The daily life of a family》,《Nulls》,《Meeting》,《Three meals》. The eighth part is tidbits. Each section lasts 50 minutes. 第一集《脚步》 The first segment:《Steps》. 【汗水中的苦辣酸甜】 Four tastes in sweat, bitter, hot, acid and sweet. “路菜”是先人保存食物的智慧,进而被演化成标志性的中国美食。味觉记忆的强大,往往让人们对故乡食物的迷恋十分牢固,甚至被赋予“乡愁”这样的文学语汇。舌尖第二季分集《脚步》,将跟随那些奔波在路上的人们,品尝辛劳与汗水中的苦辣酸甜。 “Lucai” is wisdom of Chinese ancestor which comes from food’s keeping. It has became a symbol of Chinese delicious food. The strong strength of gustation

电机概述及各类型电机介绍知识培训资料

第一章概述 第一节电机的定义 电机可泛指所有实施电能生产、传输、使用和电能特性变换的机械或装置。然而,由于生产、传输、使用电能和电能特性变换的方式很多,原理各异,如机械摩擦、电磁感应、光电效应、磁光效应、热电效应、压电效应、化学效应等等,内容广泛,不可能由一门课程包括。电机学的主要研究范畴仅限于那些依据电磁感应定律和电磁力定律实现机电能量转换和信号传递与转换的装置。 一、电机的能量交换原理 根据电机学定义,电机是一种将电能转换成机械能或将机械能转换成电能的设备,其中将机械能变为电能的称发电机;将电能变为机械能的称为电动机。电机实现能量转换都基于两个基本定律: 1、法拉第-楞次定律(感应电动势定律) 导体在磁场中运动切割磁力线,在导体两端必然会产生感应电动势,感应电动势e的大小与导体运动速度v、导体的长度L以及磁场的磁感应强度B成正比,即:e=BLV 感应电动势的方向符合右手定律。 2、毕奥-萨戈尔定律(电磁力定律) 载流导体在磁场中必然会受到电磁力的作用。电磁力的大小与导体所载电流I、磁场的磁感应强度B以及导体的长度L成正比,即:F=BIL 电磁力的方向符合左手定律。 所有电机都是根据上述两个定律基本原理设计的。电机类型很多,结构类型很多,结构形式也各不相同,但其能量转换和转矩产生原理是相同的,由于励磁方式不同,各类电机磁场性能和变化不同,才有电机的不同类型和不同特性。 电机内部实现能量转换包括电系统、耦合系统和机械系统三个环节,如图1-1。电动机通常是在其接线端子上输入电能,在轴上输出机械功率。外电路电能通过耦合系统在在机械系统中产生机械能,在机械系统轴头以机械能输出。发电机则是在其轴上输入机械功率,其端子上输出电功率。驱动机械的机械能通过耦合系统,在电系统中产生电能输出。在上述能量转换过程中,三个系统都会产生部分能量损失,最终都以热量形式向周围环境散逸。

电动机种类及其应用教材

哈尔滨工业大学寒假社会调查报告 题目:电机种类及其使用情况调查 指导教师:江善林 院系:电气学院电气工程系 班级:11306108 姓名:况麒麒 学号:1130610823 日期:2013年03月01日

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