More is the Same; Phase Transitions and Mean Field Theories

This paper is the first in a series that will look at the theory of phase transitions from the perspectives of physics and the philosophy of science. The series will consider a group of related concepts derived from condensed matter and statistical physics. The key technical ideas go under the names of “singularity”, “order parameter”, “mean field theory”, “variational method”, “correlation length”, “universality class”, “scale changes”, and “renormalization”. The first four of these will be considered here. In a less technical vein, the question here is how can matter, ordinary matter, support a diversity of forms. We see this diversity each time we observe ice in contact with liquid water or see water vapor (steam) come up from a pot of heated water. Different phases can be qualitatively different in that walking on ice is well within human capacity, but walking on liquid water is proverbially forbidden to ordinary humans. These differences have been apparent to humankind for millennia, but only brought within the domain of scientific understanding since the 1880s. A phase transition is a change from one behavior to another. A first order phase transition involves a discontinuous jump in some statistical variable. The discontinuous property is called the order parameter. Each phase transition has its own order parameter. The possible order parameters range over a tremendous variety of physical properties. These properties include the density of a liquid-gas transition, the magnetization in a ferromagnet, the size of a connected cluster in a percolation transition, and a condensate wave function in a superfluid or superconductor. A continuous transition occurs when the discontinuity in the jump approaches zero. This article is about statistical mechanics and the development of mean field theory as a basis for a partial understanding of phase transition phenomena. Much of the material in this review was first prepared for the Royal Netherlands Academy of Arts and Sciences in 2006. It has appeared in draft form on the authors’ web site (http://jfi.uchicago.edu/~leop/) since then. The title of this article is a hommage to Philip Anderson and his essay “More is Different” (Sci. New Ser. 177(4047):393–396, 1972; N.-P. Ong and R. Bhatt (eds.) More is Different: Fifty Years of Condensed Matter Physics, Princeton Series in Physics, Princeton University Press, 2001) which describes how new concepts, not applicable in ordinary classical or quantum mechanics, can arise from the consideration of aggregates of large numbers of particles. Since phase transitions only occur in systems with an infinite number of degrees of freedom, such transitions are a prime example of Anderson’s thesis.     

考虑驾驶员预估效应的交通流格子模型与数值仿真

考虑驾驶员的预估效应对车流的影响,提出了一个改进的一维交通流格子模型.基于线性稳定性理论得到了该模型的线性稳定性判据;运用非线性分析方法导出了描述交通阻塞相变时的mKdV方程.应用数值仿真验证了mKdV方程的解,研究表明适当考虑车流中预估效应的作用能够增强交通流稳定性,从而能有效抑制交通阻塞的形成. 关键词： 预估效应 交通流 格子模型 数值仿真     

Features of sound propagation in the vicinity of “symmetry-dictated” isostructural phase transitions in ferroelastics

Using a proper ferroelastic phase transition of the tension-compression type as an example, it is shown that, if the order parameter characterizing a structural phase transition allows the existence of a third-power invariant in the Landau potential, then there must be “symmetry-dictated” isostructural phase transition lines in the vicinity of the line of that structural phase transition. These isostructural transitions may manifest themselves both directly and as supercritical anomalies in the behavior of elastic moduli and lattice parameters. These effects are discovered and investigated without invoking the perturbation theory in terms of which the second-order phase transitions are commonly described. A hypothesis is made on the basis of the results obtained that the sound velocity anomalies observed in orthoclase and sanidine crystals are due to the super-critical behavior of the lattice parameters in the vicinity of a symmetry-dictated isostructural phase transition in the prototype phase of these crystals.     

Transitions in lyotropic liquid crystals caused by concentration change and γ-radiation

A molecular-statistical theory of phase transitions in lyotropic liquid crystals, which describes the phase transitions between the isotropic (micellar), nematic and lamellar phases was developed. The equations describing the dependence of parameters of orientation and translational long-range order on the concentration were obtained. It was shown that depending on the values of the model microscopic constants, the nematic phase–lamellar phase transition can be both of the first and the second order. The influence of intensive and low intense γ-radiation on the phase transitions mentioned herein was considered. It was shown that the irradiation changes the model constants responsible for the phase transitions. On this basis, it can be assumed that the γ-radiation influences the course of the dependence of the long-range order parameters on concentration as well as it changes the values of the critical concentrations of the phase transitions and even the phase transition order.     

Extraordinary wetting phase diagram for mixtures of Bose-Einstein condensates

The possibility of wetting phase transitions in Bose-Einstein condensed gases is predicted on the basis of Gross-Pitaevskii theory. The surface of a binary mixture of Bose-Einstein condensates can undergo a first-order wetting phase transition upon varying the interparticle interactions, using, e.g., Feshbach resonances. Interesting ultra-low-temperature effects shape the wetting phase diagram. The prewetting transition is, contrary to general expectations, not of first order but critical, and the prewetting line does not meet the bulk phase coexistence line tangentially. Experimental verification of these extraordinary results is called for, especially now that it has become possible, using optical methods, to realize a planar "hard wall" boundary for the condensates.     

Second and first order phase transition analogy in the operation of an organic dye laser

By an extension of the Landau theory of phase transitions for the case of an organic dye laser, it is shown that the threshold behavior of such a laser may be interpreted as an example of a second or first order phase transition. The character of the phase transition primarily depends on the wavelength dependent internal reabsorption of the laser light, which can be controlled by simple wavelength tuning of the dye laser.     

Failure of mean field theory at large N

We study strongly coupled lattice QCD with N colors of staggered fermions in 3+1 dimensions. While mean field theory describes the low temperature behavior of this theory at large N, it fails in the scaling region close to the finite temperature second order chiral phase transition. The universal critical region close to the phase transition belongs to the 3D XY universality class even when N becomes large. This is in contrast to Gross-Neveu models where the critical region shrinks as N (the number of flavors) increases and mean field theory is expected to describe the phase transition exactly in the limit of infinite N. Our work demonstrates that infrared fluctuations can be important close to second order phase transitions even when N is strictly infinite.     

Photon transmission technique for studying multiple phase transitions in a liquid crystal

A photon transmission technique was used to monitor the multiple phase transitions in a 4-butoxyphenyl4(')-declyoxybenzoate (BOPDOB) liquid crystal. Drastic decreases in the transmitted photon intensity (I) were attributed to the sequential phase transitions in BOPDOB upon cooling. In this paper, it is assumed that the order parameter rho is proportional to the transmitted photon intensity. The isotropic-nematic and nematic-smectic-A transitions were observed and found to be of first order. It was observed that the smectic-A-smectic-C and smectic-C-smectic-G transitions are second order. It was found that for the smectic-A-smectic-C transition, critical exponent crosses over from beta=0.513+/-0.006, which is consistent with mean-field theory, to beta=0.35+/-0.009, which is consistent with heliumlike behavior, as the Ginzburg criterion predicts. The critical exponent for the smectic-C-smectic-G transition was found to be beta=0.703+/-0.001. Transition temperatures were established at each phase transitions and found to be 84.92 degrees C, 74.85 degrees C, 52.96 degrees C, and 33.03 degrees C for isotropic-nematic, nematic-smectic-A, smectic-A-smectic-C and, smectic-C-smectic-G transitions, respectively.     

The effect of the negative particle velocity in a soliton gas within Korteweg–de Vries-type equations

The effect of changing the direction of motion of a defect (a soliton of small amplitude) in soliton lattices described by the Korteweg–de Vries and modified Korteweg–de Vries integrable equations (KdV and mKdV) was studied. Manifestation of this effect is possible as a result of the negative phase shift of a small soliton at the moment of nonlinear interaction with large solitons, as noted in [1], within the KdV equation. In the recent paper [2], an expression for the mean soliton velocity in a “cold” KdV-soliton gas has been found using kinetic theory, from which this effect also follows, but this fact has not been mentioned. In the present paper, we will show that the criterion of negative velocity is the same for both the KdV and mKdV equations and it can be obtained using simple kinematic considerations without applying kinetic theory. The averaged dynamics of the “smallest” soliton (defect) in a soliton gas consisting of solitons with random amplitudes has been investigated and the average criterion of changing the sign of the velocity has been derived and confirmed by numerical solutions of the KdV and mKdV equations.     

Dirty, skewed, and backwards: the smectic A-C phase transition in aerogel

We study the smectic A-C transition in anisotropic and uniaxial disordered environments, e.g., uniaxially stretched aerogel. We find very strange behavior of translational correlations: the low-temperature, lower-symmetry smectic C phase is less translationally ordered than the high-temperature, higher-symmetry smectic A phase, with short ranged and algebraic translational correlations, respectively. Specifically, the A and C phases belong to the quasi-long-ranged translationally ordered "XY Bragg glass" and short ranged translationally ordered "m=1 Bragg glass" phase, respectively. The A-C phase transition itself belongs to a new universality class, whose fixed points and exponents we find in a d=5-epsilon expansion.     

一维扩展量子罗盘模型的拓扑序和量子相变

应用矩阵乘积态表示的无限虚时间演化块算法,研究了扩展的量子罗盘模型.为了深入研究该模型的长程拓扑序和量子相变,基于奇数键和偶数键,引入了奇数弦关联和偶数弦关联,计算了保真度、奇数弦关联、偶数弦关联、奇数弦关联饱和性与序参量.弦关联表现出三种截然不同的行为:衰减为零、单调饱和与振荡饱和.基于弦关联的以上特征,给出了量子罗盘模型的基态序参量相图.在临界区,局域磁化强度和单调奇弦序参量的临界指数β=1/8表明:相变的普适类是Ising类型.此外,保真度探测到的相变点、连续性与非连续性和序参量的结果一致.     

Magnetic-Field-Induced Semimetal-Insulator-like Transition in Highly Oriented Pyrolitic Graphite

We report the extraordinarily large positive magnetoresistances （MR, 69400% at 4.5K under a magnetic field of 8.15 T）, de Hass-van Alphen oscillations effect at 10 K and the semimetal-Jnsulator-like transition in a wide range of temperature in highly oriented pyrolitic graphite （HOPG）. Besides a dominating ordinary MR （OMR） mechanism in the free-electron mode, it is realized from qualitative analysis that the Coulomb interacting quasiparticles within graphite layers play some roles. However it is difficult to associate the transition with the simple OMR theory. In order to investigate the possible origins of the transition, further analysis is carried out. It is revealed that the magnetic-field-induced behaviour is responsible for the semimetal-insulator-like transitions in HOPG.     

Unseen Coherences Can Be Felt

Forbidden transitions are not observed in the continuous-wave electron paramagnetic resonance (EPR) spectrum nor in the free induction decay because, unlike allowed transitions, their coherences have no observable magnetic moment and are spectroscopically silent. Yet, the paramagnetic relaxation described by Redfield theory can cause coherence transfer between any types of transitions. Coherence transfer between allowed transitions is now known to cause noticeable changes in EPR spectra, but coherence transfer involving forbidden transitions has long been considered to be negligible because those coherences are silent and unseen. However, our simulations of a simple model system indicate that coherence transfer with silent transitions can introduce new features into EPR spectra. The EPR-silent coherence of a forbidden transition can be transferred to an allowed transition by paramagnetic relaxation. A silent coherence can have consequences felt in the EPR spectrum.     

Coupled m-component fields with cubic anisotropy

The critical behavior at phase transitions of two coupled, m-component systems with cubic anisotropy is studied by a simplified model in which the fluctuations are partially considered. The phase transition could be a new fluctuation-induced first order transition into the anisotropic phase, or a new second order phase transition. Unlike in uncoupled systems, the second order phase transition could be into either the anisotropic or isotropic phase. As expected, upon suppression of fluctuations, all results reduce to those of mean field theory.     

Effects of fluctuations in the orientational order parameter in the cyanobiphenyl (nCB) homologous series

Photopyroelectric measurements of the anisotropy in the thermal conductivity Deltak vs temperature in the nCB (n=5,ellipsis,9) series are reported. The data have been used to deduce the behavior of the orientational order parameter Q close to the nematic-isotropic (N-I) and smectic A-nematic (A-N) phase transitions, respectively. It has been shown that near the N-I transition the data for 5CB and 6CB are consistent with the so-called "tricritical hypothesis," which predicts beta=0.25. This is not true for 7CB and 8CB in which the order parameter exhibits a behavior that could be caused by the presence of fluctuations that become increasingly important when the transition temperature is approached. A very simple model, which takes into account the contribution of fluctuations to the orientational order, has been developed close to the A-N transition and it has been shown that it is in good agreement with the experimental results. A semiquantitative explanation for the observed behavior in compounds with different nematic range has been also given.     

Some features of one-soliton solutions of the Korteweg - de Vries equation

Using the method of inverse scattering problem [1, 2], we study solutions of the Korteweg - de Vries equation under initial conditions in the form of two nonsoliton pulses with not very large amplitudes. It is shown that if the distance between these pulses is not large, then they evolve to one soliton and an oscillating nonlinear tail for t → ∞. As the distance between the pulses or the pulse amplitudes increase, two solitons and an oscillating nonlinear tail are formed. Similar behavior is observed for solutions of the nonlinear Schrödinger equation. The only difference is that three, but not two, solitons are formed if the distance between two initial inphase pulses increases. The results of analytical consideration are illustrated by the numerical solution of the Korteweg - de Vries equation.     

Investigation of the magnetic properties of the Gd1−xErx alloy system in the x < 0.62 composition range

The magnetic properties of the Gd---Er system (with erbium concentrations ranging from 0 to 62 at%) were investigated at temperatures between 4.2 and 350 K under external magnetic fields up to 1 T. The results show that, similar to other rare-earth systems, simple ferromagnetism prevails for concentrations with a de Gennes factor G higher than about 11.5 (≈ 30 at % Er) while complex magnetic structures are present for higher Er concentrations (G ≤ 11.5. The magnetic transition temperature decreases with increasing Er content, showing good agreement with phenomenological expectations. In the paramagnetic region, the Curie constant of the alloy corresponds to the linear combination of the constituent elements' contribution, thus proving that both Gd and Er maintain their individual magnetic moments. From magnetization data below the transition, the change in magnetic entropy for a 1 T field was calculated and this showed distinct differences between ferromagnetic-paramagnetic transitions and antiferromagnetic transitions. The Landau theory of second-order phase transitions can be well applied for evaluating the thermodynamic transition temperature and the specific heat anomaly in the ferromagnetic alloys but it fails for the more complex structures if the applied field is not far higher than the critical field. The magnetic entropy change displays well defined peaks at the transition temperature for both ferromagnetic and antiferromagnetic alloys. There is an indication that the spin-reorientation transition occurring in alloys with higher Er concentrations is also coupled with significant magnetic entropy changes.     

First and second order phase transitions in gauge theories at finite temperature

We consider in general the nature of the phase transition which occurs in 4D gauge theories coupled to scalar and spinor fields at finite temperature. It is shown that the critical behavior can be isolated in an effective 3D theory of the zero frequency mode whose lagrangian may be calculated explicitly in weak coupling perturbation theory. This lagrangian, in turn, may be investigated by means of standard ?-expansion techniques. Theories with an asymptotically free gauge coupling constant possess no stable fixed point in the ?-expansion and are inferred to have weakly first-order phase transitions; theories not satisfying this condition may have second-order transitions.     

Nuclear quadrupole interaction studies of high pressure phase transitions in InSb

Conclusions This study of the behavior of the EFG in the pressure-induced phase transition in InSb shows a high frequency in the high presure metallic phase, that is compatible with the tetragonal, -Sn like structure, but with a certain degree of imperfection. If this imperfection is in fact some disorder in the In and Sb positions, its amount could be promptly clarified by neutron diffraction, since the similarities of the atomic numbers of In and Sb preclude any easy measurement of superlattice lines by X-ray diffraction techniques.The strong frequency distribution that remains after the reversal of the phase transformation must be related with the peculiar kinetics of this reaction, since it depends on temperature and the time interval and since the same is not observed in the T1 transition. So it seems interesting to study other phase transitions in similar systems. A clarification of the precise nature of the generated defects demands the application of other techniques like transmission electron microsconv and neutron diffraction.If the defects observed here are in fact boundary of domains, it might be possible to relate the domain size with the frequency distribution and then to study, by TDPAC, the kinetics of formation and growth of these domains.Work supported in part by Conselho Nacional de Desenvolvimento Cientifico e Tecnológico (CNPq) and Financiadora de Estudos e Projetos (FINEP)     

Solitary waves in van der Waals-like transition in fluidized granular matter

Solitary wave solutions exhibited at the onset of the phase transition in fluidized granular matter are perused. In the quasi-sonic limit the system is modeled by two Korteweg de Vries equations. We study the solitary wave interactions in order to understand the rich dynamics exhibited by the fluidized granular system at the onset of the gas–liquid phase transition.