Nonlocal scalar electrodynamics from Chern-Simons theory

有限元模拟固态电解质锂离子传输路径
固态电解质在锂离子电池中扮演着重要的角色，它们具有高离子导电性能，能够有效地传输锂离子。

了解固态电解质中锂离子的传输路径对于提高锂离子电池的性能至关重要。

固态电解质中锂离子的传输路径可以分为两个方面：晶格传导和界面传导。

晶格传导是指锂离子在固态电解质的晶格中通过空位或离子空穴的传输。

当锂离子在固态电解质中发生扩散时，离子会通过晶格中的空位或离子空穴进行传输。

晶格传导是固态电解质中锂离子传输的主要路径之一，它直接影响着固态电解质的离子导电性能。

界面传导是指锂离子在固态电解质与电极材料之间的界面上进行传输。

在锂离子电池中，固态电解质与正极和负极之间形成了界面。

锂离子通过界面传导到达电极材料，并在充放电过程中与电极材料发生反应。

界面传导是固态电解质中锂离子传输的另一个重要路径，它影响着锂离子电池的充放电性能和循环稳定性。

固态电解质中锂离子的传输路径的研究对于优化固态电解质的结构和性能具有重要意义。

通过改变固态电解质的晶格结构和优化界面结构，可以提高锂离子的传输速率和电池的功率密度。

此外，了解锂离子在固态电解质中的传输路径还可以帮助我们理解固态电解质的失活机制，并设计出更加稳定和安全的锂离子电池。

固态电解质中锂离子的传输路径对于锂离子电池的性能至关重要。

通过研究锂离子在固态电解质中的晶格传导和界面传导，可以优化固态电解质的结构和性能，提高锂离子电池的性能和循环稳定性。

这将对于推动锂离子电池的发展和应用具有重要的意义。

用VASP计算Ｈ原子的能量氢原子的能量为。

在这一节中，我们用VASP计算H原子的能量。

对于原子计算，我们可以采用如下的INCAR文件PREC=ACCURATENELMDL = 5 make five delays till charge mixingISMEAR = 0; SIGMA=0.05 use smearing method采用如下的KPOINTS文件。

由于增加K点的数目只能改进描述原子间的相互作用，而在单原子计算中并不需要。

所以我们只需要一个K点。

Monkhorst Pack 0 Monkhorst Pack1 1 10 0 0采用如下的POSCAR文件atom 115.00000 .00000 .00000.00000 15.00000 .00000.00000 .00000 15.000001cart0 0 0采用标准的H的POTCAR得到结果如下：k-point 1 : 0.0000 0.0000 0.0000band No. band energies occupation1 -6.3145 1.000002 -0.0527 0.000003 0.4829 0.000004 0.4829 0.00000我们可以看到，电子的能级不为。

Free energy of the ion-electron system (eV)---------------------------------------------------alpha Z PSCENC = 0.00060791Ewald energy TEWEN = -1.36188267-1/2 Hartree DENC = -6.27429270-V(xc)+E(xc) XCENC = 1.90099128PAW double counting = 0.00000000 0.00000000entropy T*S EENTRO = -0.02820948eigenvalues EBANDS = -6.31447362atomic energy EATOM = 12.04670449---------------------------------------------------free energy TOTEN = -0.03055478 eVenergy without entropy = -0.00234530 energy(sigma->0) = -0.01645004我们可以看到也不等于。

A nonlinear dynamics method for signal identificationT. L. CarrollCitation: Chaos 17, 023109 (2007); doi: 10.1063/1.2722870View online: /10.1063/1.2722870View Table of Contents: /resource/1/CHAOEH/v17/i2Published by the American Institute of Physics.Related ArticlesExperimental verification of photon angular momentum and vorticity with radio techniquesAppl. Phys. Lett. 99, 204102 (2011)A new low-cost 10 ns pulsed Ka-band radarRev. Sci. Instrum. 82, 074706 (2011)A new technique for the characterization of chaff elementsRev. Sci. Instrum. 82, 074702 (2011)Thermophotonic radar imaging: An emissivity-normalized modality with advantages over phase lock-in thermographyAppl. Phys. Lett. 98, 163706 (2011)Microcontroller-based binary integrator for millimeter-wave radar experimentsRev. Sci. Instrum. 81, 054704 (2010)Additional information on ChaosJournal Homepage: /Journal Information: /about/about_the_journalTop downloads: /features/most_downloadedInformation for Authors: /authorsA nonlinear dynamics method for signal identificationT.L.Carroll a͒U.S.Naval Research Lab,Washington,D.C.20375͑Received22February2007;accepted15March2007;published online9May2007͒When a radio frequency signal is radiated by a transmitter,the properties of the transmitter itself affect the properties of the signal.These transmitter-induced changes are known as unintentional modulation,to differentiate them from intentional modulation used to add information to the signal.The unintentional modulation can be used to identify which transmitter produced a signal.This paper shows how phase space analysis based on nonlinear dynamics ideas can be used to determine which of several amplifiers produced a signal.©2007American Institute of Physics.͓DOI:10.1063/1.2722870͔Often in the military,it is useful to know how many re-sources,such as radars,that an adversary has.Each ra-dar may be tagged based on a signature derived from its signal.In order to extract these signatures,there is afield known as specific emitter identification.Traditionally, specific emitter identification has depended on using lin-ear signal processing to extract signatures from the tran-sient parts of pulsed radar signals.I take a different ap-proach in this paper;I consider the power amplifier in the radar transmitter to be a driven nonlinear system, and then apply methods from nonlinear dynamics to ex-tract a signature based on the different nonlinearities in the different amplifiers.Because traditional analysis has considered the transient parts of signals,I consider non-transient signals in this work so that the techniques stud-ied here will be complementary to traditional analysis.I.INTRODUCTIONThefield of nonlinear dynamics has led to new tech-niques for analyzing signals,and has seen existing tech-niques͑singular value decomposition,for example͒applied in new ways.1The study of chaos has led to signal analysis methods that do not depend on linear techniques such as Fourier transforms.Some of these new approaches to analyz-ing signals that were developed with chaos in mind might also be useful for existing signal analysis problems where linear analysis provides only limited information.In this pa-per,phase space analysis methods are applied to the problem of specific emitter identification,in which a radar transmitter is uniquely identified based on the signal that it transmits. II.PHASE SPACE ANALYSISMany common analysis methods in nonlinear dynamics begin by reconstructing the phase space trajectory of an ex-perimental system by embedding a scalar time series signal from the system in a phase space.2,3Given a signal s͑t͒,a vector w͑t͒=͑s͑t͒,s͑t+␶͒,s͑t+2␶͒,...͒is constructed.There are several methods for determining the values of the time delays␶and the phase space dimension D required so that the series of vectors w͑t͒forms an accurate representation,or an embedding,of the phase space trajectory of the system that generated the signal s͑t͒.4This delay embedding method is very general,and does not depend on the system that gen-erated s͑t͒being linear or nonlinear.III.SPECIFIC EMITTER IDENTIFICATIONThe problem of specific emitter identification͑SEI͒was chosen because it may be an easy application for nonlinear dynamics methods.The particular SEI application consid-ered in this paper is that an airplane is being illuminated by a radar signal,and the pilot wants to identify the particular radar transmitter that is sending the signal.Because of the nature of radar,the signal to noise ratio for the radar signal that the pilot sees will be large.A radar signal has to travel a distance R from the transmitter to the target,be scattered by the target,and return to the transmitter.The power of the scattered radar signal measured at the receiver is decreased by a factor of1/R4from the transmitted signal,so the am-plitude of the radar signal at the target͑the airplane͒must be large compared to the background noise to insure that a large enough signal is scattered back to the transmitter to allow for detection.The radar transmitter uses a power amplifier to create this large signal,and the power amplifier is normally run at as high a power as practical,so the amplifier is usually operated in a range where its nonlinear properties will affect the signal.Radar signals are usually sent as pulses,so radar ampli-fiers are normally pulsed on and mon SEI tech-niques use linear signal analysis to identify unique transients generated by pulsing the amplifier.5–7I would like to concen-trate on applications where linear methods are not sufficient, so in this paper,I will analyze continuous signals that are not pulsed.For a pulsed signal,this would involve analyzing the middle part of the pulse,where no transients are present. Linear signal analysis is not currently used to analyze the nontransient part of the pulse.a͒Electronic mail:Thomas.L.Carroll@CHAOS17,023109͑2007͒1054-1500/2007/17͑2͒/023109/7/$23.00©2007American Institute of Physics17,023109-1IV.ANALYSIS METHODAmplifiers which depend on semiconductors are inher-ently nonlinear,so all such amplifiers contain unavoidable nonlinearities.The algorithm developed here depends on treating the amplifier as a driven nonlinear dynamical sys-tem.Two identical dynamical systems,driven by the same signal and sampled at the same point in phase space,should have the same derivatives.The idea that derivatives mea-sured at the same point in phase space should be identical for a deterministic dynamical system was previously used to de-tect determinism in a single dynamical system.8In the present paper,it is assumed that the two signals come from deterministic dynamical systems.The algorithm in this pa-per,called the phase space difference algorithm,involvesembedding two signals in phase space,measuring the deriva-tives of signals at the same point in the phase space,and taking the difference.The concept of comparing derivatives at the same point in phase space is essentially the same as the idea of cross-prediction,which was used to determine if a time series was stationary,9but the execution of the idea in this paper is different.A similar method for comparing time series from the same dynamical system compared the prob-ability distributions in a D-dimensional phase space to deter-mine if a dynamical system was stationary.10The algorithm in this paper does depend on the two sig-nals staying close enough to each other in the phase space so that the derivative difference estimates are accurate,so in this paper,the algorithm will only be applied to systems driven with the same type of driving signal,i.e.,both ampli-fiers are driven with a pure sine wave,or both are driven with a frequency modulated signal.Thefirst step in the algorithm is to obtain output signals from a known set of amplifiers,to be used as reference sig-nals r␣͑t͒,where the subscript␣indicates the particular am-plifier.Each of the digitized r␣͑t͒signals is embedded in aphase space.An embedded point from the reference signal r␣͑t͒is designated v i=r␣͑i+␶1,i+␶2,...,i+␶d͒,where the subscript␣has been dropped from v i to avoid overly clut-tered notation.In this paper,it will be understood that v i comes from one of the r␣͑t͒time series.The next step is to measure a signal u͑t͒from an un-known amplifier.The unknown signal u͑t͒is embedded with the same embedding parameters as the reference signals.In the simplest form of the phase space difference algorithm,a search would be made for embedded points v i from the ref-erence signal that were near u j from the unknown signal.The time derivative u j+1−u j could then be compared to the time derivative of the reference signal at the same point in phase space,v i+1−v i,to get a phase space difference.In some situ-ations,however,this approach can lead to large errors.In one example,used in this paper,a frequency modulated sine wave was randomly switched between two different frequen-cies at the zero phase of the sine.Sometimes the sine wave switched from one frequency to the other,but sometimes the frequencies before and after switching were the same.This switching leads to two different values for the derivative of the sine wave at its0phase.If the point v i is at the point where the frequency switches,then nearby neighboring points may have two different values for the derivative v i+1−v i,and the derivative of the unknown signal u j+1−u j may also take on two different values.The possibility of different derivatives leads to large errors in the phase space difference statistic at these switching points.Figure1shows a sche-matic plot of a point in phase space where such an error can occur.To avoid errors at switching points,the point͑u j,u j+1͒from the unknown signal is used as an index point.The reference signal is then searched for a single point of the form͑v i,v i+1͒on the reference signal that is the closest such point to the index point.This amounts to searching the ref-erence signal for the derivative that is closest to a derivative on the unknown signal.The absolute value of the difference between derivatives͑v i+1−v i͒−͑u j+1−u j͒is calculated and averaged over the unknown signal to get the average phase space difference͗␦͘.All the example signals in this paper are based on sine waves,so only one-dimensional embeddings are used.While a sine wave requires two dimensions for an embedding,if only a small region of phase space is considered at any one time,the sine wave appears to be close to one dimension for a small enough region.The algorithm as applied in this paper may be summarized as follows:first,obtain reference signals from known amplifiers.Several signals obtained at different times from the same amplifier may be combined to create more reference points.Next,record a signal from an un-known amplifier.Pick an index point͑u j,u j+1͒on the un-known signal,and search for the closest pair of points ͑v i,v i+1͒on the reference signal.This procedure is essen-tially the same as searching for the closest strand.11Calculate the difference in derivatives␦=͉͑v i+1−v i͒−͑u j+1−u j͉͒and average over the attractor to get͗␦͘.plications caused by real dataReal data adds an additional complication to the calcu-lation of the phase space statistic͗␦͘.When different signals are digitized at different times,they may not be phase coher-ent with each other;that is,the digitized points may come at different phases of the waveform.As a result,when the sig-nals are embedded,the nearest reference neighbor͑v i,v i+1͒to a point͑u j,u j+1͒from the unknown signal may not be close,so the resulting derivative estimates will be takenat FIG.1.Ambiguous point in phase space.Points v i or v k may be the nearest neighbor of u j,but the derivative differences͑v i+1−v i͒−͑u j+1−u j͒or͑v k+1−v k͒−͑u j+1−u j͒will be very different.023109-2Thomas Carroll Chaos17,023109͑2007͒points that are not close in phase space,resulting in an error.A schematic of this situation is shown in Fig.2.In order to get a better estimate of the derivative,a line is drawn between v i and v i +1.The point at which this line comes closest to the unknown point u j is designated z j ,and is used to replace v i .In some cases,z j may be between v i −1and v i .The next point,z j +1,is found by searching for the nearest neighbor of ͑u j +1,u j +2͒and repeating the procedure.It was observed from the data that the difference be-tween derivatives was roughly proportional to the distance between ͑u j ,u j +1͒and ͑z j ,z j +1͒,meaning that the derivative difference had two parts;one part caused by the actual dif-ference between derivatives at the same location in phase space,and one part which caused the measured difference to become larger when the two points were separated by some distance in the phase space.In order to correct for the second part,the derivative difference is divided by the distance be-tween the phase space points at which the derivatives are measured,yielding the normalized phase space difference statistic⌬=͚k =1d͉͑z j +1k −z j k ͒−͑u j +1k −u j k͉͒ͱ͚k =1d͓͑z j +1k −u j +1k ͒2+͑z j k −u j k ͒2͔͑1͒where the superscript k indicates the k th component.The statistic ⌬is averaged over the entire unknown signal to produce the average phase space difference ͗⌬͘.The numera-tor of Eq.͑1͒is a difference between two derivatives,each of which has a sign,so the linear difference was used and ab-solute value was taken before averaging.The difference could also be squared before averaging;a similar result would be expected.As a final correction,the absolute amplitude of the un-known signal u ͑t ͒is not known,so both the unknown signal and the reference signal v ͑t ͒are normalized to have rms values of 1.The object of the normalization is to make sure that the unknown signal and the reference signal are close together in phase space when they are embedded.It is pos-sible that the amplifier that produced u ͑t ͒was driven with avery different amplitude signal than the amplifier that pro-duced v ͑t ͒,so it may be that u ͑t ͒and v ͑t ͒should not actually be close in phase space,and the statistic ͗⌬͘will be in error.The size of this error will depend on the particular amplifier nonlinearity.For now,only signals that are truly close in phase space will be considered.B.The algorithmSummarizing the entire algorithm:͑1͒Accumulate signals from several amplifiers known to be different to serve as reference signals.Normalize the sig-nals to have a rms amplitude of 1.͑2͒Record an unknown signal u ͑t ͒and normalize so that u ͑t ͒has a rms amplitude of 1.For each reference signal v ͑t ͒.͑3͒Embed both the unknown signal u ͑t ͒and the refer-ence signal v ͑t ͒in phase spaces with identical dimensions and delays.The proper dimension and delay may be deter-mined by known methods.1,4͑4͒From the embedded unknown vector u j ,create pairs of points ͑u j ,u j +1͒and search the reference signal for the closest pair ͑v i ,v i +1͒.In this work,a “slice”search was used,12but other search algorithms could also work.͑5͒Correct for phase errors in sampling:Calculate the equations for the line from ͑v i −1,v i ͒to ͑v i ,v i +1͒,and the line from ͑v i ,v i +1͒to ͑v i +1,v i +2͒.For an embedding dimension D ,there will be D lines,of the form y l k =m l k v k +b l k,where the subscript l =1refers to the line from ͑v i −1,v i ͒to ͑v i ,v i +1͒,and l =2refers to the line from ͑v i ,v i +1͒to ͑v i +1,v i +2͒.The superscript k refers to the particular dimension.͑6͒Find the closest point on the lines y 1k or y 2kto thepoint ͑u j ,u j +1͒from the relation d l k =͉͑m l k u j k −u j +1k +b l k͉͒/ͱ͑m l ͒2+1,where d l k is the distance.For a phase space ofdimension D Ͼ1,the distances for the different dimensions are added together for a total distance d l .͑7͒For whichever line ͑l =1or l =2͒gives the smaller distance d l ,find the line through ͑u j ,u j +1͒that is perpendicu-lar to the line l =1or l =2.The slope of this line is m p k =−1/m lk .Knowing this line,find the point z j where the per-pendicular intercepts the l =1or l =2line.Store this point z j ,increment the index j ,and return to step ͑5͒.͑8͒Calculate the phase space difference statistic ⌬from Eq.͑1͒,and average over the unknown signal to get ͗⌬͘.V.NUMERICAL EXAMPLEA simple numerical example is first used to illustrate the phase space difference method.The amplifier model is de-scribed by␰͑t ͒=sin ͑␻t ͒,dx ␣dt=␥␣͑g ͑␰͒−x ␣͒,͑2͒dy␣dt =␥␣ͩdx i dt −y ␣ͪ,g ͑x ͒=x +␳␣x 3.FIG.2.For real data,the unknown signal u ͑t ͒may not be sampled at the same phase as the reference signal v ͑t ͒,so that the distance between nearest neighbors u j and v i may be large.To improve on the derivative estimate,the interpolated point z j is used instead of u j .023109-3NLD for signal ID Chaos 17,023109͑2007͒Equation ͑2͒is a model for a bandpass filter driven by a nonlinear function ͓g ͑x ͔͒of a sinusoidal signal.The param-eter ␳␣controls the size of the nonlinearity,while ␥␣sets the time constant for the bandpass filter.In order to model two different amplifiers,there are two different versions of Eq.͑2͒,with different parameters ␥␣and ␳␣.The frequency ␻is fixed to produce 20points/cycle of the sine wave.First,two different linear amplifiers are modeled,so ␳1and ␳2=0.Figure 3shows plots of the averaged phase space difference ͗␦͘for ␥1=1and different values of ␥2͑solid line ͒.The signal from amplifier 1was used as the reference signal,with the amplifier 2signal as the unknown.There is some variation in the phase space statistic ͗␦͘when both amplifiers are linear,but the variation is small,indicating that the statistic is not very sensitive to differences in linear amplifiers.The statistic ͗␦͘is much more sensitive to differences in nonlinear amplifiers.Figure 3also shows the value of the statistic when the nonlinearity parameters ␳1=␳2=0.1,so that both simulated amplifiers are nonlinear ͑dashed line ͒.The statistic ͗␦͘now varies by an amount that is roughly propor-tional to the difference between the two simulated amplifiers.For the next test,the two time constants ␥1and ␥2were set equal to 1.0,and the nonlinear parameter ␳1=0,and ␳2was varied.Figure 4shows that the statistic ͗␦͘is sensitive to am-plifiers that have the same time constants but different non-linearities.VI.EXPERIMENTSFor an experimental test of the phase space statistic,3OP-07operational amplifiers were driven with a common signal.The three amplifiers were nominally identical,butbecause of unavoidable variations in the semiconductors,in practice they were well matched but not identical.Figure 5is a schematic of the experiment.The amplifiers were all set to have a gain of −1,although there was some variation due to the 1%tolerance of the resistors.For experimental tests involving an unmodulated sine wave signal,the three amplifiers were driven with a sine wave with a 1V amplitude and a frequency of 25kHz.The bandwidth of these amplifiers for unity gain is 1MHz,so the amplifiers were not being driven out of their normal range of operation.Figure 6shows that only weak nonlinearity was present for these driving parameters.Figure 6shows that the sine wave signal amplified by amplifier A does contain some small harmonics at 50,75,and 100kHz,indicating weak nonlinearity.The power spec-trum does show other sources of interference that are larger in amplitude than the harmonics,so the nonlinear effects from this amplifier are not large.It will be shown below that this nonlinearity is still large enough to allow identification of the amplifiers.Figure 7shows the results of measurements of the aver-age of the phase space difference ͗⌬͘for digitizedsineFIG.3.͑Solid line ͒Phase space statistic ͗␦͘as a function of time constant ␥2when both simulated amplifiers are linear.͑Dashed line ͒Phase space statistic ͗␦͘as a function of time constant ␥2when both simulated amplifiers contain a cubicnonlinearity.FIG.4.Phase space statistic ͗␦͘as a function of nonlinear parameter ␳2when ␳1=0.1and ␥1=␥2=1.0for the simulatedamplifiers.FIG.5.Schematic of the experiment.The amplifiers were all nominally identical OP-07operational amplifiers,driven by a common signal.The resistors R were all 1k ⍀±1%.FIG.6.Power spectrum of a 25kHz sine wave signal with an amplitude of 1V amplified by amplifier A.The harmonics at 50,75,and 100kHz indi-cate the presence of weak nonlinearity in the amplifier,while the other peaks in the power spectrum are caused by other interference sources.023109-4Thomas Carroll Chaos 17,023109͑2007͒waves amplified by amplifiers A,B,or C.The reference signal used to produce Fig.7was a 400000point signal from amplifier A.The unknown signal was a different signal from amplifier A,or a signal from amplifier B or C.In order to get good statistics on how well the phase space difference statistic ͗⌬͘differentiated between different amplifiers,a time series of 100000points from the unknown amplifier was used.The phase space difference statistic was computed for each point in this time series,and ͗⌬͘was computed as a running average of the previous 500points.All calculated values for ͗⌬͘were accumulated into a histogram.The his-togram shows the probability of measuring a particular value of ͗⌬͘for a 500point signal from the unknown amplifier.500points was about 25cycles of the sine wave.Figure 7shows that the unknown signal gives smaller values of ͗⌬͘when compared to a reference signal from amplifier A ͑solid line ͒,then when compared to reference signals from amplifiers B or C ͑dashed and dotted lines ͒,the unknown is correctly identified as coming from A.The over-lap between the histogram for A and the other histograms is very small,indicating that there is a very low probability of misidentifying the amplifier based on these signals.VII.FM SIGNALSBecause all the signals in the previous section were simple sine waves,one could probably distinguish which amplifier each signal came from by taking a simple linear difference between the unknown signal and each reference signal.In this section,sine waves with a random frequency modulation are considered.Each signal has a different modu-lation,so a simple linear difference will not reveal the origin of the signal.To generate the FM ͑frequency modulated ͒signals,a sine wave was randomly switched between frequencies of 22.5and 27.5kHz with equal probabilities.The average switching time was 2cycles.As before,amplifiers A,B,or C were driven with FM signals with an amplitude of 1V,and a 400000point time series from each amplifier was stored to be used as a reference signal.A different FM signal from amplifier A was used as the unknown signal.Figure 8shows histograms of the statistic ͗⌬͘from the unknown signal when compared to each of the reference signals.In Fig.8,there is some overlap between the histogram when the reference comes from A ͑solid line ͒and when the reference comes from B ͑dashed line ͒,so there is some prob-ability of falsely identifying which amplifier generated the unknown signal.The histogram when the reference signal comes from C does not overlap with the histogram for A,so there is very little probability of misidentifying the unknown signal as coming from amplifier rger amplitude signalsWhen the amplifiers are driven with larger signals,the effects of their inherent nonlinearities should be more pro-nounced,which may make it easier to identify which ampli-fier an unknown signal came from.Figure 9is the power spectrum of a 25kHz sine wave with an amplitude of 2V amplified by amplifier A.The harmonics of the 2.0V sine wave in Fig.9are larger than the harmonics of the 1.0V sine wave in Fig.6by 20–30dB,indicating that the amplifier nonlinearities have a larger effect on the larger signal.Figure 10shows histograms of the phase space statistic ͗⌬͘for unmodulated sine waves with an amplitude of 2.0V.The statistic worked well for sine waves of 1.0V amplitude ͑Fig.7͒,so it is not surprising that amplifier A is easily identified as the source of the unknown signal for these larger sine waves.The phase space statistic ͗⌬͘could misidentify which amplifier had generated a 1.0V random FM signal,as seen in Fig.8,but this possibility is less likely when theamplitudeFIG.7.Experimental histograms of the phase space difference ͗⌬͘for am-plifiers driven by unmodulated sine waves with an amplitude of 1.0V.p ͑͗⌬͒͘is the probability of measuring a particular value of ͗⌬͘for a 500point ͑25cycle ͒signal from the unknown amplifier.The unknown signal u ͑t ͒came from amplifier A.The solid line is the histogram when the refer-ence signal comes from amplifier A,the dotted line is for a reference from B,and the dashed line is for a reference fromC.FIG.8.Experimental histograms of the phase space difference ͗⌬͘for am-plifiers driven by randomly frequency modulated sine waves with an ampli-tude of 1.0V.p ͑͗⌬͒͘is the probability of measuring a particular value of ͗⌬͘for a 500point ͑25cycle ͒signal from the unknown amplifier.The unknown signal u ͑t ͒came from amplifier A.The solid line is the histogram when the reference signal comes from amplifier A,the dotted line is for a reference from B,and the dashed line is for a reference fromC.FIG.9.Power spectrum of a 25kHz sine wave with an amplitude of 2.0V amplified by amplifier A.Note the larger harmonics at 50,75,and 100kHz compared to Fig.6.023109-5NLD for signal ID Chaos 17,023109͑2007͒of the FM signal is 2.0V.Figure 11shows histograms of ͗⌬͘for random FM signals with an amplitude of 2.0V.Compared to Fig.8,Fig.11shows a much larger sepa-ration between histograms when signals from the different amplifiers are used as reference signals.Unlike Fig.8,when the signal amplitude was 1.0V,there is no overlap between the histogram when the reference signal comes from ampli-fier A and the histograms when amplifiers B or C are used as references.Figure 11correctly identifies the unknown ran-dom FM signal as coming from amplifier A,with a very small probability of error.When the nonlinear properties of the amplifier have a larger effect,signal identification using the phase space statistic ͗⌬͘becomes easier.B.Higher dimensional embeddingThe calculation of ͗⌬͘by searching for points of the form ͑v i ,v i +1͒may be extended to higher dimensions.The disadvantage of calculating ͗⌬͘in higher dimensions is that the actual dimension of the phase space to be searched in-creases by 2on adding a dimension,greatly increasing the computational time;the advantage is that fewer reference points are required.Figure 12shows the result of calculating ͗⌬͘using only 40000reference points ͑in the preceding examples,400000reference points were used ͒.Figure 12͑a ͒shows the result of calculating ͗⌬͘for a 1D embedding of the unknown signal using 40000reference points from amplifier A,amplifier B,or amplifier C.As be-fore,the histogram when a signal from amplifier A is used s a reference is shown by a solid line,the histogram for a reference from B is a dotted line,and C is a dashed line.Figure 12͑a ͒shows the same calculation as in Fig.8,but with one tenth the number of points in the reference signal.Figure 12͑a ͒shows that it is not possible to distinguish which amplifier produced the 1V random FM signal using a 1D embedding with only 40000reference points.When fewer reference points are used,then on the average,the nearest reference point ͑v i ,v i +1͒to the unknown point ͑u j ,u j +1͒is farther away,and the interpolated reference point ͑z j ,z j +1͒is also farther from the true reference point ͑v i ,v i +1͒.When 400000reference points are used,the aver-age of the distance between ͑u j ,u j +1͒and ͑z j ,z j +1͒is 3.6ϫ10−4,but when only 40000reference points are used,this average distance increases to 8.1ϫ10−2.All signals were normalized before the calculation,so the distances are unit-less.The greater distance means that derivatives for the un-known and reference signals are being measured at points that are farther apart in the phase space,so the measurement of the difference of derivatives is less accurate.Figure 12͑b ͒shows the same result with a 2D embed-ding ͑because the algorithm is searching for derivatives,this is effectively a 4D embedding ͒.While Fig.12͑b ͒shows con-siderable overlap between the histograms of ͗⌬͘when the reference signal comes from amplifier A or B,the 2D em-bedding still distinguishes the different signals better than when a 1D embedding was used.Figure 12shows that even for simple signals,there is some advantage to using higher dimensional embeddings;the higher dimensionalembeddingFIG.10.Experimental histograms of the phase space difference ͗⌬͘for amplifiers driven by unmodulated sine waves with an amplitude of 2.0V.p ͑͗⌬͒͘is the probability of measuring a particular value of ͗⌬͘for a 500point ͑25cycle ͒signal from the unknown amplifier.The unknown signal u ͑t ͒came from amplifier A.The solid line is the histogram when the refer-ence signal comes from amplifier A,the dotted line is for a reference from B,and the dashed line is for a reference fromC.FIG.11.Experimental histograms of the phase space difference ͗⌬͘for amplifiers driven by randomly frequency modulated sine waves with an amplitude of 2.0V.p ͑͗⌬͒͘is the probability of measuring a particular value of ͗⌬͘for a 500point ͑25cycle ͒signal from the unknown amplifier.The unknown signal u ͑t ͒came from amplifier A.The solid line is the histogram when the reference signal comes from amplifier A,the dotted line is for a reference from B,and the dashed line is for a reference fromC.FIG.12.Experimental histograms of the phase space difference ͗⌬͘for amplifiers driven by randomly frequency modulated sine waves with an amplitude of 1.0V,when the reference signal contains only 40000points,one tenth the number used in Fig.8.͑a ͒Shows the histograms when all signals were embedded in a 1D phase space,and ͑b ͒shows the same results when all signals were embedded in a 2D phase space.p ͑͗⌬͒͘is the prob-ability of measuring a particular value of ͗⌬͘for a 500point ͑25cycle ͒signal from the unknown amplifier.The unknown signal u ͑t ͒came from amplifier A.The solid line is the histogram when the reference signal comes from amplifier A,the dotted line is for a reference from B,and the dashed line is for a reference from C.023109-6Thomas Carroll Chaos 17,023109͑2007͒。

华中师范大学物理学院物理学专业英语仅供内部学习参考！2014一、课程的任务和教学目的通过学习《物理学专业英语》，学生将掌握物理学领域使用频率较高的专业词汇和表达方法，进而具备基本的阅读理解物理学专业文献的能力。

通过分析《物理学专业英语》课程教材中的范文，学生还将从英语角度理解物理学中个学科的研究内容和主要思想，提高学生的专业英语能力和了解物理学研究前沿的能力。

培养专业英语阅读能力，了解科技英语的特点，提高专业外语的阅读质量和阅读速度；掌握一定量的本专业英文词汇，基本达到能够独立完成一般性本专业外文资料的阅读；达到一定的笔译水平。

要求译文通顺、准确和专业化。

要求译文通顺、准确和专业化。

二、课程内容课程内容包括以下章节：物理学、经典力学、热力学、电磁学、光学、原子物理、统计力学、量子力学和狭义相对论三、基本要求1.充分利用课内时间保证充足的阅读量（约1200~1500词/学时），要求正确理解原文。

2.泛读适量课外相关英文读物，要求基本理解原文主要内容。

3.掌握基本专业词汇（不少于200词）。

4.应具有流利阅读、翻译及赏析专业英语文献，并能简单地进行写作的能力。

四、参考书目录1 Physics 物理学 (1)Introduction to physics (1)Classical and modern physics (2)Research fields (4)V ocabulary (7)2 Classical mechanics 经典力学 (10)Introduction (10)Description of classical mechanics (10)Momentum and collisions (14)Angular momentum (15)V ocabulary (16)3 Thermodynamics 热力学 (18)Introduction (18)Laws of thermodynamics (21)System models (22)Thermodynamic processes (27)Scope of thermodynamics (29)V ocabulary (30)4 Electromagnetism 电磁学 (33)Introduction (33)Electrostatics (33)Magnetostatics (35)Electromagnetic induction (40)V ocabulary (43)5 Optics 光学 (45)Introduction (45)Geometrical optics (45)Physical optics (47)Polarization (50)V ocabulary (51)6 Atomic physics 原子物理 (52)Introduction (52)Electronic configuration (52)Excitation and ionization (56)V ocabulary (59)7 Statistical mechanics 统计力学 (60)Overview (60)Fundamentals (60)Statistical ensembles (63)V ocabulary (65)8 Quantum mechanics 量子力学 (67)Introduction (67)Mathematical formulations (68)Quantization (71)Wave-particle duality (72)Quantum entanglement (75)V ocabulary (77)9 Special relativity 狭义相对论 (79)Introduction (79)Relativity of simultaneity (80)Lorentz transformations (80)Time dilation and length contraction (81)Mass-energy equivalence (82)Relativistic energy-momentum relation (86)V ocabulary (89)正文标记说明：蓝色Arial字体（例如energy）：已知的专业词汇蓝色Arial字体加下划线（例如electromagnetism）：新学的专业词汇黑色Times New Roman字体加下划线（例如postulate）：新学的普通词汇1 Physics 物理学1 Physics 物理学Introduction to physicsPhysics is a part of natural philosophy and a natural science that involves the study of matter and its motion through space and time, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic disciplines, perhaps the oldest through its inclusion of astronomy. Over the last two millennia, physics was a part of natural philosophy along with chemistry, certain branches of mathematics, and biology, but during the Scientific Revolution in the 17th century, the natural sciences emerged as unique research programs in their own right. Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry,and the boundaries of physics are not rigidly defined. New ideas in physics often explain the fundamental mechanisms of other sciences, while opening new avenues of research in areas such as mathematics and philosophy.Physics also makes significant contributions through advances in new technologies that arise from theoretical breakthroughs. For example, advances in the understanding of electromagnetism or nuclear physics led directly to the development of new products which have dramatically transformed modern-day society, such as television, computers, domestic appliances, and nuclear weapons; advances in thermodynamics led to the development of industrialization; and advances in mechanics inspired the development of calculus.Core theoriesThough physics deals with a wide variety of systems, certain theories are used by all physicists. Each of these theories were experimentally tested numerous times and found correct as an approximation of nature (within a certain domain of validity).For instance, the theory of classical mechanics accurately describes the motion of objects, provided they are much larger than atoms and moving at much less than the speed of light. These theories continue to be areas of active research, and a remarkable aspect of classical mechanics known as chaos was discovered in the 20th century, three centuries after the original formulation of classical mechanics by Isaac Newton (1642–1727) 【艾萨克·牛顿】.University PhysicsThese central theories are important tools for research into more specialized topics, and any physicist, regardless of his or her specialization, is expected to be literate in them. These include classical mechanics, quantum mechanics, thermodynamics and statistical mechanics, electromagnetism, and special relativity.Classical and modern physicsClassical mechanicsClassical physics includes the traditional branches and topics that were recognized and well-developed before the beginning of the 20th century—classical mechanics, acoustics, optics, thermodynamics, and electromagnetism.Classical mechanics is concerned with bodies acted on by forces and bodies in motion and may be divided into statics (study of the forces on a body or bodies at rest), kinematics (study of motion without regard to its causes), and dynamics (study of motion and the forces that affect it); mechanics may also be divided into solid mechanics and fluid mechanics (known together as continuum mechanics), the latter including such branches as hydrostatics, hydrodynamics, aerodynamics, and pneumatics.Acoustics is the study of how sound is produced, controlled, transmitted and received. Important modern branches of acoustics include ultrasonics, the study of sound waves of very high frequency beyond the range of human hearing; bioacoustics the physics of animal calls and hearing, and electroacoustics, the manipulation of audible sound waves using electronics.Optics, the study of light, is concerned not only with visible light but also with infrared and ultraviolet radiation, which exhibit all of the phenomena of visible light except visibility, e.g., reflection, refraction, interference, diffraction, dispersion, and polarization of light.Heat is a form of energy, the internal energy possessed by the particles of which a substance is composed; thermodynamics deals with the relationships between heat and other forms of energy.Electricity and magnetism have been studied as a single branch of physics since the intimate connection between them was discovered in the early 19th century; an electric current gives rise to a magnetic field and a changing magnetic field induces an electric current. Electrostatics deals with electric charges at rest, electrodynamics with moving charges, and magnetostatics with magnetic poles at rest.Modern PhysicsClassical physics is generally concerned with matter and energy on the normal scale of1 Physics 物理学observation, while much of modern physics is concerned with the behavior of matter and energy under extreme conditions or on the very large or very small scale.For example, atomic and nuclear physics studies matter on the smallest scale at which chemical elements can be identified.The physics of elementary particles is on an even smaller scale, as it is concerned with the most basic units of matter; this branch of physics is also known as high-energy physics because of the extremely high energies necessary to produce many types of particles in large particle accelerators. On this scale, ordinary, commonsense notions of space, time, matter, and energy are no longer valid.The two chief theories of modern physics present a different picture of the concepts of space, time, and matter from that presented by classical physics.Quantum theory is concerned with the discrete, rather than continuous, nature of many phenomena at the atomic and subatomic level, and with the complementary aspects of particles and waves in the description of such phenomena.The theory of relativity is concerned with the description of phenomena that take place in a frame of reference that is in motion with respect to an observer; the special theory of relativity is concerned with relative uniform motion in a straight line and the general theory of relativity with accelerated motion and its connection with gravitation.Both quantum theory and the theory of relativity find applications in all areas of modern physics.Difference between classical and modern physicsWhile physics aims to discover universal laws, its theories lie in explicit domains of applicability. Loosely speaking, the laws of classical physics accurately describe systems whose important length scales are greater than the atomic scale and whose motions are much slower than the speed of light. Outside of this domain, observations do not match their predictions.Albert Einstein【阿尔伯特·爱因斯坦】contributed the framework of special relativity, which replaced notions of absolute time and space with space-time and allowed an accurate description of systems whose components have speeds approaching the speed of light.Max Planck【普朗克】, Erwin Schrödinger【薛定谔】, and others introduced quantum mechanics, a probabilistic notion of particles and interactions that allowed an accurate description of atomic and subatomic scales.Later, quantum field theory unified quantum mechanics and special relativity.General relativity allowed for a dynamical, curved space-time, with which highly massiveUniversity Physicssystems and the large-scale structure of the universe can be well-described. General relativity has not yet been unified with the other fundamental descriptions; several candidate theories of quantum gravity are being developed.Research fieldsContemporary research in physics can be broadly divided into condensed matter physics; atomic, molecular, and optical physics; particle physics; astrophysics; geophysics and biophysics. Some physics departments also support research in Physics education.Since the 20th century, the individual fields of physics have become increasingly specialized, and today most physicists work in a single field for their entire careers. "Universalists" such as Albert Einstein (1879–1955) and Lev Landau (1908–1968)【列夫·朗道】, who worked in multiple fields of physics, are now very rare.Condensed matter physicsCondensed matter physics is the field of physics that deals with the macroscopic physical properties of matter. In particular, it is concerned with the "condensed" phases that appear whenever the number of particles in a system is extremely large and the interactions between them are strong.The most familiar examples of condensed phases are solids and liquids, which arise from the bonding by way of the electromagnetic force between atoms. More exotic condensed phases include the super-fluid and the Bose–Einstein condensate found in certain atomic systems at very low temperature, the superconducting phase exhibited by conduction electrons in certain materials,and the ferromagnetic and antiferromagnetic phases of spins on atomic lattices.Condensed matter physics is by far the largest field of contemporary physics.Historically, condensed matter physics grew out of solid-state physics, which is now considered one of its main subfields. The term condensed matter physics was apparently coined by Philip Anderson when he renamed his research group—previously solid-state theory—in 1967. In 1978, the Division of Solid State Physics of the American Physical Society was renamed as the Division of Condensed Matter Physics.Condensed matter physics has a large overlap with chemistry, materials science, nanotechnology and engineering.Atomic, molecular and optical physicsAtomic, molecular, and optical physics (AMO) is the study of matter–matter and light–matter interactions on the scale of single atoms and molecules.1 Physics 物理学The three areas are grouped together because of their interrelationships, the similarity of methods used, and the commonality of the energy scales that are relevant. All three areas include both classical, semi-classical and quantum treatments; they can treat their subject from a microscopic view (in contrast to a macroscopic view).Atomic physics studies the electron shells of atoms. Current research focuses on activities in quantum control, cooling and trapping of atoms and ions, low-temperature collision dynamics and the effects of electron correlation on structure and dynamics. Atomic physics is influenced by the nucleus (see, e.g., hyperfine splitting), but intra-nuclear phenomena such as fission and fusion are considered part of high-energy physics.Molecular physics focuses on multi-atomic structures and their internal and external interactions with matter and light.Optical physics is distinct from optics in that it tends to focus not on the control of classical light fields by macroscopic objects, but on the fundamental properties of optical fields and their interactions with matter in the microscopic realm.High-energy physics (particle physics) and nuclear physicsParticle physics is the study of the elementary constituents of matter and energy, and the interactions between them.In addition, particle physicists design and develop the high energy accelerators,detectors, and computer programs necessary for this research. The field is also called "high-energy physics" because many elementary particles do not occur naturally, but are created only during high-energy collisions of other particles.Currently, the interactions of elementary particles and fields are described by the Standard Model.●The model accounts for the 12 known particles of matter (quarks and leptons) thatinteract via the strong, weak, and electromagnetic fundamental forces.●Dynamics are described in terms of matter particles exchanging gauge bosons (gluons,W and Z bosons, and photons, respectively).●The Standard Model also predicts a particle known as the Higgs boson. In July 2012CERN, the European laboratory for particle physics, announced the detection of a particle consistent with the Higgs boson.Nuclear Physics is the field of physics that studies the constituents and interactions of atomic nuclei. The most commonly known applications of nuclear physics are nuclear power generation and nuclear weapons technology, but the research has provided application in many fields, including those in nuclear medicine and magnetic resonance imaging, ion implantation in materials engineering, and radiocarbon dating in geology and archaeology.University PhysicsAstrophysics and Physical CosmologyAstrophysics and astronomy are the application of the theories and methods of physics to the study of stellar structure, stellar evolution, the origin of the solar system, and related problems of cosmology. Because astrophysics is a broad subject, astrophysicists typically apply many disciplines of physics, including mechanics, electromagnetism, statistical mechanics, thermodynamics, quantum mechanics, relativity, nuclear and particle physics, and atomic and molecular physics.The discovery by Karl Jansky in 1931 that radio signals were emitted by celestial bodies initiated the science of radio astronomy. Most recently, the frontiers of astronomy have been expanded by space exploration. Perturbations and interference from the earth's atmosphere make space-based observations necessary for infrared, ultraviolet, gamma-ray, and X-ray astronomy.Physical cosmology is the study of the formation and evolution of the universe on its largest scales. Albert Einstein's theory of relativity plays a central role in all modern cosmological theories. In the early 20th century, Hubble's discovery that the universe was expanding, as shown by the Hubble diagram, prompted rival explanations known as the steady state universe and the Big Bang.The Big Bang was confirmed by the success of Big Bang nucleo-synthesis and the discovery of the cosmic microwave background in 1964. The Big Bang model rests on two theoretical pillars: Albert Einstein's general relativity and the cosmological principle (On a sufficiently large scale, the properties of the Universe are the same for all observers). Cosmologists have recently established the ΛCDM model (the standard model of Big Bang cosmology) of the evolution of the universe, which includes cosmic inflation, dark energy and dark matter.Current research frontiersIn condensed matter physics, an important unsolved theoretical problem is that of high-temperature superconductivity. Many condensed matter experiments are aiming to fabricate workable spintronics and quantum computers.In particle physics, the first pieces of experimental evidence for physics beyond the Standard Model have begun to appear. Foremost among these are indications that neutrinos have non-zero mass. These experimental results appear to have solved the long-standing solar neutrino problem, and the physics of massive neutrinos remains an area of active theoretical and experimental research. Particle accelerators have begun probing energy scales in the TeV range, in which experimentalists are hoping to find evidence for the super-symmetric particles, after discovery of the Higgs boson.Theoretical attempts to unify quantum mechanics and general relativity into a single theory1 Physics 物理学of quantum gravity, a program ongoing for over half a century, have not yet been decisively resolved. The current leading candidates are M-theory, superstring theory and loop quantum gravity.Many astronomical and cosmological phenomena have yet to be satisfactorily explained, including the existence of ultra-high energy cosmic rays, the baryon asymmetry, the acceleration of the universe and the anomalous rotation rates of galaxies.Although much progress has been made in high-energy, quantum, and astronomical physics, many everyday phenomena involving complexity, chaos, or turbulence are still poorly understood. Complex problems that seem like they could be solved by a clever application of dynamics and mechanics remain unsolved; examples include the formation of sand-piles, nodes in trickling water, the shape of water droplets, mechanisms of surface tension catastrophes, and self-sorting in shaken heterogeneous collections.These complex phenomena have received growing attention since the 1970s for several reasons, including the availability of modern mathematical methods and computers, which enabled complex systems to be modeled in new ways. Complex physics has become part of increasingly interdisciplinary research, as exemplified by the study of turbulence in aerodynamics and the observation of pattern formation in biological systems.Vocabulary★natural science 自然科学academic disciplines 学科astronomy 天文学in their own right 凭他们本身的实力intersects相交，交叉interdisciplinary交叉学科的，跨学科的★quantum 量子的theoretical breakthroughs 理论突破★electromagnetism 电磁学dramatically显著地★thermodynamics热力学★calculus微积分validity★classical mechanics 经典力学chaos 混沌literate 学者★quantum mechanics量子力学★thermodynamics and statistical mechanics热力学与统计物理★special relativity狭义相对论is concerned with 关注，讨论，考虑acoustics 声学★optics 光学statics静力学at rest 静息kinematics运动学★dynamics动力学ultrasonics超声学manipulation 操作，处理，使用University Physicsinfrared红外ultraviolet紫外radiation辐射reflection 反射refraction 折射★interference 干涉★diffraction 衍射dispersion散射★polarization 极化，偏振internal energy 内能Electricity电性Magnetism 磁性intimate 亲密的induces 诱导，感应scale尺度★elementary particles基本粒子★high-energy physics 高能物理particle accelerators 粒子加速器valid 有效的，正当的★discrete离散的continuous 连续的complementary 互补的★frame of reference 参照系★the special theory of relativity 狭义相对论★general theory of relativity 广义相对论gravitation 重力，万有引力explicit 详细的，清楚的★quantum field theory 量子场论★condensed matter physics凝聚态物理astrophysics天体物理geophysics地球物理Universalist博学多才者★Macroscopic宏观Exotic奇异的★Superconducting 超导Ferromagnetic铁磁质Antiferromagnetic 反铁磁质★Spin自旋Lattice 晶格，点阵，网格★Society社会，学会★microscopic微观的hyperfine splitting超精细分裂fission分裂，裂变fusion熔合，聚变constituents成分，组分accelerators加速器detectors 检测器★quarks夸克lepton 轻子gauge bosons规范玻色子gluons胶子★Higgs boson希格斯玻色子CERN欧洲核子研究中心★Magnetic Resonance Imaging磁共振成像，核磁共振ion implantation 离子注入radiocarbon dating放射性碳年代测定法geology地质学archaeology考古学stellar 恒星cosmology宇宙论celestial bodies 天体Hubble diagram 哈勃图Rival竞争的★Big Bang大爆炸nucleo-synthesis核聚合，核合成pillar支柱cosmological principle宇宙学原理ΛCDM modelΛ-冷暗物质模型cosmic inflation宇宙膨胀1 Physics 物理学fabricate制造，建造spintronics自旋电子元件，自旋电子学★neutrinos 中微子superstring 超弦baryon重子turbulence湍流，扰动，骚动catastrophes突变，灾变，灾难heterogeneous collections异质性集合pattern formation模式形成University Physics2 Classical mechanics 经典力学IntroductionIn physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces. The study of the motion of bodies is an ancient one, making classical mechanics one of the oldest and largest subjects in science, engineering and technology.Classical mechanics describes the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. Besides this, many specializations within the subject deal with gases, liquids, solids, and other specific sub-topics.Classical mechanics provides extremely accurate results as long as the domain of study is restricted to large objects and the speeds involved do not approach the speed of light. When the objects being dealt with become sufficiently small, it becomes necessary to introduce the other major sub-field of mechanics, quantum mechanics, which reconciles the macroscopic laws of physics with the atomic nature of matter and handles the wave–particle duality of atoms and molecules. In the case of high velocity objects approaching the speed of light, classical mechanics is enhanced by special relativity. General relativity unifies special relativity with Newton's law of universal gravitation, allowing physicists to handle gravitation at a deeper level.The initial stage in the development of classical mechanics is often referred to as Newtonian mechanics, and is associated with the physical concepts employed by and the mathematical methods invented by Newton himself, in parallel with Leibniz【莱布尼兹】, and others.Later, more abstract and general methods were developed, leading to reformulations of classical mechanics known as Lagrangian mechanics and Hamiltonian mechanics. These advances were largely made in the 18th and 19th centuries, and they extend substantially beyond Newton's work, particularly through their use of analytical mechanics. Ultimately, the mathematics developed for these were central to the creation of quantum mechanics.Description of classical mechanicsThe following introduces the basic concepts of classical mechanics. For simplicity, it often2 Classical mechanics 经典力学models real-world objects as point particles, objects with negligible size. The motion of a point particle is characterized by a small number of parameters: its position, mass, and the forces applied to it.In reality, the kind of objects that classical mechanics can describe always have a non-zero size. (The physics of very small particles, such as the electron, is more accurately described by quantum mechanics). Objects with non-zero size have more complicated behavior than hypothetical point particles, because of the additional degrees of freedom—for example, a baseball can spin while it is moving. However, the results for point particles can be used to study such objects by treating them as composite objects, made up of a large number of interacting point particles. The center of mass of a composite object behaves like a point particle.Classical mechanics uses common-sense notions of how matter and forces exist and interact. It assumes that matter and energy have definite, knowable attributes such as where an object is in space and its speed. It also assumes that objects may be directly influenced only by their immediate surroundings, known as the principle of locality.In quantum mechanics objects may have unknowable position or velocity, or instantaneously interact with other objects at a distance.Position and its derivativesThe position of a point particle is defined with respect to an arbitrary fixed reference point, O, in space, usually accompanied by a coordinate system, with the reference point located at the origin of the coordinate system. It is defined as the vector r from O to the particle.In general, the point particle need not be stationary relative to O, so r is a function of t, the time elapsed since an arbitrary initial time.In pre-Einstein relativity (known as Galilean relativity), time is considered an absolute, i.e., the time interval between any given pair of events is the same for all observers. In addition to relying on absolute time, classical mechanics assumes Euclidean geometry for the structure of space.Velocity and speedThe velocity, or the rate of change of position with time, is defined as the derivative of the position with respect to time. In classical mechanics, velocities are directly additive and subtractive as vector quantities; they must be dealt with using vector analysis.When both objects are moving in the same direction, the difference can be given in terms of speed only by ignoring direction.University PhysicsAccelerationThe acceleration , or rate of change of velocity, is the derivative of the velocity with respect to time (the second derivative of the position with respect to time).Acceleration can arise from a change with time of the magnitude of the velocity or of the direction of the velocity or both . If only the magnitude v of the velocity decreases, this is sometimes referred to as deceleration , but generally any change in the velocity with time, including deceleration, is simply referred to as acceleration.Inertial frames of referenceWhile the position and velocity and acceleration of a particle can be referred to any observer in any state of motion, classical mechanics assumes the existence of a special family of reference frames in terms of which the mechanical laws of nature take a comparatively simple form. These special reference frames are called inertial frames .An inertial frame is such that when an object without any force interactions (an idealized situation) is viewed from it, it appears either to be at rest or in a state of uniform motion in a straight line. This is the fundamental definition of an inertial frame. They are characterized by the requirement that all forces entering the observer's physical laws originate in identifiable sources (charges, gravitational bodies, and so forth).A non-inertial reference frame is one accelerating with respect to an inertial one, and in such a non-inertial frame a particle is subject to acceleration by fictitious forces that enter the equations of motion solely as a result of its accelerated motion, and do not originate in identifiable sources. These fictitious forces are in addition to the real forces recognized in an inertial frame.A key concept of inertial frames is the method for identifying them. For practical purposes, reference frames that are un-accelerated with respect to the distant stars are regarded as good approximations to inertial frames.Forces; Newton's second lawNewton was the first to mathematically express the relationship between force and momentum . Some physicists interpret Newton's second law of motion as a definition of force and mass, while others consider it a fundamental postulate, a law of nature. Either interpretation has the same mathematical consequences, historically known as "Newton's Second Law":a m t v m t p F ===d )(d d dThe quantity m v is called the (canonical ) momentum . The net force on a particle is thus equal to rate of change of momentum of the particle with time.So long as the force acting on a particle is known, Newton's second law is sufficient to。

文章标题：探索三维非均匀介质波动方程有限差分python开源代码1. 简介在地质勘探、医学成像和地震监测等领域，对三维非均匀介质波动方程的研究与应用日益重要。

而有限差分方法在数值求解波动方程中具有广泛的应用。

在本文中，我们将探讨如何利用Python编程语言实现三维非均匀介质波动方程的有限差分方法，并开源共享相应的代码，以便更多人能够深入理解和应用这一重要领域。

2. 三维非均匀介质波动方程简介三维非均匀介质波动方程描述了波在非均匀介质中的传播规律，是地震勘探、医学成像等领域中常见的数学模型之一。

该方程的数值求解通常采用有限差分方法，通过离散网格化空间和时间来逼近连续的微分方程，从而得到数值解。

3. 有限差分方法有限差分方法是数值求解微分方程的一种常见方法，其基本思想是将微分方程中的导数用差分近似代替，从而将连续的问题转化为离散的问题。

在三维非均匀介质波动方程中，有限差分方法可以有效地模拟波的传播过程，并得到波场的数值解。

4. Python编程实现利用Python编程语言实现三维非均匀介质波动方程的有限差分方法具有许多优势，如简洁易读的代码、丰富的科学计算库等。

在实现过程中，我们可以利用NumPy库进行数组操作，使用Matplotlib库进行波场可视化，并通过SciPy库进行数值求解等。

5. 开源代码共享在本文中，我们将共享我们编写的三维非均匀介质波动方程有限差分Python开源代码，包括空间离散化、时间离散化、边界条件处理、波场更新等关键部分。

我们也会附上详细的注释和使用说明，以便感兴趣的读者能够下载并运行我们的代码，深入理解和学习有限差分方法在波动方程中的应用。

6. 个人观点和理解通过编写三维非均匀介质波动方程的有限差分Python开源代码，我深刻体会到数值模拟在地质勘探、医学成像等领域中的重要作用。

Python作为一种强大的科学计算语言，为我们提供了丰富的工具和库，使得数值模拟变得更加高效和灵活。

a rXiv:h ep-ph/96238v114Fe b1996BI-TP 96/09February 1996MAGNETIC MASS IN HOT SCALAR ELECTRODYNAMICS 1O.K.Kalashnikov 2Fakult¨a t f¨u r Physik Universit¨a t Bielefeld D-33501Bielefeld,Germany Abstract Using the Slavnov-Taylor identities we prove that the so-called ”magnetic mass”is exactly equal to zero within hot scalar electrodynamics.The same result is valid for hot QED and seems for any abelian theory but this is not the case for hot QCD where one expects that m 2mag =0.At present for hot QCD and many other gauge theories it is very essen-tial to calculate the so-called”magnetic mass,”which is an infrared cutofffor gluomagnetic forces and in many cases it can protect this theory from infrared divergencies.This question has a very long history[1,2,3]but till now it is open to discussions.There are only the estimates made perturba-tively for this parameter[4,5,6]although another possibility,which considers a nonanalytical behaviour[7,8],is also not excluded.Nevertheless this pa-rameter(when m2mag=0)is widely used today for many applications[9], especially when the next-to-leading order term is calculated[10]within hot QCD.Moreover it is often stated(starting from paper[11])that for hot scalar electrodynamics and for any hot abelian theory this parameter is equal to zero although this fact has not been proven.The goal of this paper is to calculate exactly the magnetic mass for hot scalar electrodynamics using the Slavnov-Taylor identities.Here we exploit the exact graph representation for the photon self-energy tensor and de-mostrate that,indeed,this parameter is equal to zero after the simple algebra being performed.Moreover we also see arguments that this result is valid for hot QED and it is correct for any abelian theory.For hot QCD m2mag=0 although the analogous calculations are also valid.On the formal level,the graphs with other numerical coefficients define the QCD self-energy tensor but,of course,the real reason is connected with the essential different nature of hot QCD infrared divergencies.Scalar electrodynamics is determined through the LagrangianL A=−14(φ+φ)2(1)where Aµis an abelian gaugefield andφ+(φ)are the complex scalar ones. Here Fµνis the standard electromagneticfield strength tensor and the last term in Eq.(1)is necessary to make the model(1)renormalizable.The quantum Lagrangian for the theory under consideration is built as usual and has the formL=L A+L g.f.L g.f=1The set of equations for the temperature Green functions can be eas-ily obtained via the stationary-action principle[12]and has the standard Schwinger-Dyson formD−1(k4,k)=D−10(k4,k)+Π(k4,k),G(k4,k)=G−10(k4,k)+Σ(k4,k)(3)whereΠandΣare the self-energy part of the photon Green function and the Green function of scalarfields,respectively.The explicit form ofΠcan be represented by the four nonperturbative graphs(4)where all lines and the bold points should be identified with the exact Green and vertex functions.All the bare vertices are found to beΓ0Aφφ+(k|p+k,p)µ=e(2p+k)µΓ0A2φφ+|µν=−2eδµν,Γ0(φφ+)2=−λ(5)and they are independent from the gauge chosen.The last two functions are independent from momenta as well.For the Feynman gauge(whereα=1)the photon self-energy tensor is transversalkµΠµν(k)=0(6)and can be represented with the aid of two scalar functions in the formΠij(k4,k)=(δij−k i k jk2k24k2Π44(k4,k),i,j=1,2,3(7)The magnetic mass is determined to bem2mag=A(k4=0,k→0)(8) where the limit is defined in the infrared manner.However,sinceΠ44(k4= 0,k→0)=0for this theory,it is more convenient to use for calculating m2magthe relationm2mag =1which directly follows from Eq.(7)when the Feynman gauge is used.Our tool for transforming Eq.(4)is the exact Slavnov-Taylor identitiesΓAφφ+(0|p,p)i=e∂G−1(p)∂p j(10) which can be found by using the known prescription[12].They are valid if one momentum is equal to zero in the infrared manner and for indices i,j=4.One-loop nonperturbative graphs and two-loop ones in Eq.(4)are can-celed independently.The twofirst(one-loop)graphs are easily put in the formΠ(1)ii(0)=6e2(2π)3G(p)−e2(2π)3(2p i)G(p)∂G−1(p)β p4d3pβ p4 d3p∂p i(12)which is exactly equal to zero when one calculates the last integral by parts (the surface term being zero is ommited[13]).On this level we have that the nonperturbative one-loopΠ(1)ii(0)=0.The same situation with the one-loop nonperturbative graphs takes place also in hot QCD which is possible to see,for example,in axial temporal gauge[13].For hot QED the analogous calculations prove that m2mag=0at once since the exact graph representation for the photon self-energy part in QED does not contain the nonperturbative two-loop graphs[12].But there is a problem when the nonperturbative two-loop graphs are considered.For the model(1),however,we demostrate that the two last nonperturbative graphs in Eq.(4)seem to be equal to zero as well.Here we take the third graph(below called G3)from(4)which(after thefirst formula from Eq.(10)being used)has the form(G3)=2e3(2π)3d3p∂p i(13)4and we perform the intergation by parts within Eq.(13).If the integral(below K-term)K=2e3(2π)3d3p∂p iG(p+k)Γj(k|p+k,p)G(p) (14)is equal to zero,the expression Eq.(13)becomes(G3)=−2e3(2π)3d3p∂p iG(p+k)Γj(k|p+k,p) G(p)(15)and this representation for G3is enough to prove within the model(1)that m2mag=0exactly.Now one should explicitly perform a differentiation within Eq.(15)andfind the simple identity for the exact graphs within Eq.(4)(16) which shows that the magnetic mass for this model is indeed equal to zerom2mag=0(17) However we should prove else that from Eq.(14)K=0.In the lowest perturbative order(here this means the e4-term)one can demonstrate that K(0)=0,transforming Eq.(14)to the formK(0)=2e4(2π)3d3pk2 6(p+k)2p4(18)and then calculates it in the usual manner.For example,using the infrared manner of calculation,onefinds at once thatK(0)=2e4(2π)3d3pk2 −2(p+k)2p4=0(19)So there is not any problem with the leading g4-term calculated for m2mag and it being zero strongly indicates that m2mag=0within all perturbative orders.For hot QCD m2mag=0already within the g4-order[4]although the analogous calculations are also possible,for example,in the axial temporal gauge.On the formal level the graphs with other numerical coefficients define the QCD self-energy tensor but,of course,the real reason is connected with the essential different nature of the QCD infrared divergencies.5AcknowledgementsI would like to thank Rudolf Baier for useful discussions and all the col-leagues from the Department of Theoretical Physics of the Bielefeld Univer-sity for the kind hospitality.References1.)A.D.Linde,Phys.Lett.B96(1980)289.2.)D.J.Gross,R.D.Pisarski and L.G.Yaffe,Rev.Mod.Phys.53(1981)43.3.)O.K.Kalashnikov,JETP Lett.33(1981)165.4.)O.K.Kalashnikov and E.Kh.Veliev,Sov.Phys.Lebedev Inst.Rep.(1986)No.3,39;for review see:O.K.Kalashnikov,Phys.Lett.B279 (1992)367.5.)J.Blaizot,E.Iancu and R.R.Parwani,Phys.Rev.D52(1995)2543.6.)R.Jackiw,So-Young Pi,Phys.Lett.B368(1996)131.7.)R.Jackiw and S.Templeton,Phys.Rev.D23(1981)2291.8.)O.K.Kalashnikov,JETP Lett.54(1991)181.9.)R.D.Pisarski,Phys.Rev.D47(1993)5589;F.Flechsig,A.K.Rebhanand H.Schulz,Phys.Rev.D52(1995)2994.10.)A.K.Rebhan,Phys.Rev.D48(1993)R3967;E.Braaten and A.Nieto,Phys.Rev.Lett.73(1994)2402.11.)O.K.Kalashnikov and V.V.Klimov,Phys.Lett.B95(1980)423.12.)E.S.Fradkin,Proc.Lebedev Inst.29(1965)6.13.)O.K.Kalashnikov,JETP Lett.39(1984)405.6。

a rXiv:h ep-th/94618v115J un1994IITM-TH-94-03June 1994Quantum Field Theory without divergences:Quantum Spacetime G.H.Gadiyar Department of Mathematics,Indian Institute of Technology,Madras 600036,INDIA.Abstract:A fundamental length is introduced into physics in a way which respects the principles of relativity and quantum field theory.This improves the properties of quantum field theory:divergences are removed.How to quantize gravity is also indicated.When the fundamental length tends to zero the present version of quantum field theory is recovered.PACS 11.General Theory of fields and particles.1In this paper the problem of divergences in quantumfield theory is solved. This is done by a simple modification of the concept of spacetime.Math-ematically what is being done is called a deformation[1].The idea is to introduce a fundamental length into physics.There are compelling reasons to do this.Masters like Heisenberg,Dirac,Landau and Schwinger[2]have on various occasions talked about the logical inconsistencies of quantumfield theory.Further the problem of quantizing gravity remains[3].Wheeler[4] talks of the possibility offluctuations at the Planck length.The origin of this work is as follows.Weinberg[5]has shown that the equivalence principle and the General Theory of Relativity can be derived from special relativity and quantumfield theory.However the problem of renormalization remains.Thus the author was tempted to play with the axioms underlying the concepts of spacetime and pursue the consequences. The article contains no exhaustive references to the literature.This is because the results were arrived at without reference to the literature and attempts to give references would be misleading.Mathematicians call the process of introducing a fundamental constant into a structure as deformation.Thus relativity which introduces the velocity of light c and quantum mechanics which introduces the Planck constant¯h into the older structures,namely Galilean relativity and classical mechanics, are called deformations.In each case the two structures will be formally and mathematically similar to the older structures though conceptually severe2changes are needed.Let us attempt to deform quantumfield theory.It is immediate that the Planck length L is the only parameter available.To introduce the parameter let us introduce a pair of operatorsˆxµandˆx#µsatisfying the commutation relations[ˆxµ,ˆx#ν]=iηµνL2.Further the interval is now defined as1ˆs2=2L2for these states.This corresponds to taking a subset of the overcomplete set.Tofix ideas take only one coordinate x.Then[ˆx,ˆx#]=iL23andˆs 2=1√√2(dd †+d †d ).Now comes the halving of coordinates.We cannot let ˆx equal ˆx #as operators.So we set x =x #on the coherent states which we assume as the state of spacetime.Then|x >=e −x 2√L ]nn !|n ><x |x ′>=e−(x −x ′)2√2L 2.This goes over to δ(x −x ′)as L →0.The reason for doing this will become clear shortly.We now go over to quantum field theory.We have so far introduced two ideas.We have doubled the number of coordinates to introduce a fundamen-tal length and changed the definition of interval.This leads to quantization4of the interval.We have next introduced the idea that spacetime is in a coher-ent state and halved the number of coordinates whichfinally appear.These two ideas leave us with afine balance between discreteness and continuity as the following analysis will show.Thus we see that functions of x will now be functions of annihilation operators acting on states|x>.This is just like in optics.To make contact with quantumfield theory is our next task.Since there are spacetimefluctuations,a little thought shows that we should take the action and average over the spacetimefluctuations.Thus we are led to replace the action for the scalarfield ∂µφ∂µφd4x by d4x d4y∂µφ(x)K(x−y)∂µφ(y)whereK(x−y)= 12πL 4e−(x−y)2∂φ∂φ+Jφ2where∂φ∂φ= d4x∂µφ(x)∂µφ(x)andJφ= d4x J(x)φ(x)5byZ= Dφe−1p2are replaced bye−p2L2p2.Thiscorresponds to a regularization of the theory.It is remarkable that this means something very simple in the parametric representation[6]of the propagator.e−p2L2p2=e−p2L22)p2dα= ∞L22byαand changing limits of integration.Thus the theory is automatically regularized.To do quantum electrodynamics is now simple. It turns out that the propagator and vertices will be modified.However the gauge principle survives.Nonlocalfield theory thus arises naturally from our assumptions.The parametric representation of Feynman amplitudes can be utilized with a natural cutoff.Thus no new techniques need be invented for calculations.Further the results of QED will be reproduced but now with a natural cutoff.6We recapitulate what has happened.We assumed that spacetime is in a coherent state and averaged over thefluctuations and this caused the propa-gators to be regulated.Thus a natural regularization comes about.Further the parametric representation makes the calculations very simple.This is because we have deliberately deformed the original theory and such results are bound to appear.To quantize gravity is now fairly simple.The work of Weinberg[5]shows the way.However with the structure of spacetime modified the infinities which plague the older theory will go away.The spin-two graviton will couple to a modified energy momentum tensor which will have nonlocal structure. Physically one can say that point particles are replaced by objects which are roughly of the order of L in size due to the new structure of spacetime.Notice that modifying the action leads to modified classical equations of motion as well.Thus the hope that Dirac expressed of modifying the classical and quantum theory together is fulfilled in an unexpected way.There are several conceptual issues involved.The reason is as follows. Whenever one deforms an older theory the newer theory will have mathe-matically similar structure.Witness the fact that the Lie algebra structure survives in both classical and quantum mechanics.Thus the language of the original theory and its deformation will be similar but conceptual issues will be thorny.The author refrains from any discussion of conceptual issues as this is a luxury he cannot afford at present.7Acknowledgements:Thanks to Prof.E.C.G.Sudarshan for frequently em-phasizing the structural similarity between classical and quantum mechanics and the beauty of coherent states.Thanks to Dr.H.S.Sharatchandra for short,illuminating discussions in the initial stages of the work.Thanks to L.Kannan of PPST Foundation and Shambu Prasad of Dastkar for encour-aging new modes of thinking.The author wishes to dedicate this work to Paramahamsa Yogananda whose centenary is being celebrated.8References1.On the Relationship between Mathematics and Physics,L.D.Faddeev,Asia Pacific Physics News vol.3,June-July,1988,pp.21-22.2.Niels Bohr and the Development of Physics,W.Pauli Ed.Theoretical Physics in the Twentieth century:A memorial volume to Wolfgang Pauli,M.Fierz and V.Weisskopf Eds.Inward Bound,A.Pais.Selected Papers on Quantum Electrodynamics,J.Schwinger Ed.3.Quantum Theory of Gravity.Essays in honour of the sixtieth birthdayof Bryce S.DeWitt,S.M.Christensen Ed.4.Gravitation,C.W.Misner,K.S.Thorne and J.A.Wheeler.Magic without Magic.John Archibald Wheeler,J.R.Klauder Ed.5.Photons and Gravitons in S-Matrix Theory:Derivation of Charge Con-servation and Equality of Gravitational and Inertial Mass,S.Weinberg, Phys.Rev B1049-1056vol.135,4B,1964.Photons and Gravitons in Perturbation Theory:Derivation of Maxwell’s and Einstein’s Equations,S.Weinberg,Phys.Rev B988-1002,vol.138, 4B,1965.6.Quantum Field Theory,C.Itzykson and J.B.Zuber.9。

《几种非对称结构二维材料中的电-声子相互作用》篇一一、引言近年来，二维材料因其独特的物理和化学性质，在电子器件、光电器件、传感器等领域得到了广泛的应用。

其中，非对称结构二维材料因其具有特殊的电子能带结构和声子模式，使得其电-声子相互作用具有独特的性质。

本文将探讨几种典型的非对称结构二维材料中的电-声子相互作用，并对其在电子器件中的应用进行探讨。

二、非对称结构二维材料的电-声子相互作用1. 过渡金属硫化物过渡金属硫化物（TMDs）是一种典型的非对称结构二维材料，其具有独特的电子能带结构和声子模式。

在TMDs中，电子和声子之间的相互作用主要发生在导带和价带之间的跃迁过程中。

由于TMDs的电子态和声子模式的不对称性，使得电子在跃迁过程中能够与特定的声子模式发生耦合，从而产生独特的电-声子相互作用。

这种电-声子相互作用对于电子输运具有重要影响，如可调节的半导体性能和负泊松效应等。

在TMDs基的电子器件中，这种电-声子相互作用可用于设计高效的电子输运和热管理策略。

2. 拓扑绝缘体拓扑绝缘体是一种具有非平凡能带结构的二维材料，其表面具有特殊的电子态和声子模式。

在拓扑绝缘体中，由于电子态的特殊性质，使得其与声子的相互作用也具有独特的性质。

在拓扑绝缘体中，由于表面态的存在，使得电子和声子之间的相互作用更加复杂。

这种相互作用不仅与材料的电子能带结构和声子模式有关，还与表面态的分布和能量有关。

因此，在拓扑绝缘体基的电子器件中，电-声子相互作用可用于实现特殊的电子输运和热管理策略。

3. 石墨烯及其衍生物石墨烯及其衍生物是一种具有蜂窝状结构的二维材料，其具有独特的电子能带结构和声子模式。

在石墨烯及其衍生物中，由于结构的非对称性，使得其电子和声子之间的相互作用也具有特殊的性质。

在石墨烯及其衍生物中，电子和声子的相互作用受到结构的特殊调制。

由于蜂窝状结构的非对称性，使得电子在跃迁过程中能够与特定的声子模式发生耦合。

这种电-声子相互作用对于石墨烯基的电子器件具有重要的应用价值，如高性能的晶体管、传感器等。