Ginzburg-Landau Theory Of Phase Transitions Research Articles

Ginzburg-Landau theory of dark energy: A framework to study both temporal and spatial cosmological tensions simultaneously

A dark energy model (DE) is proposed based on Ginzburg-Landau theory of phase transition (GLT). This model, GLTofDE, surprisingly provides a framework to study not only temporal tensions in cosmology e.g. $H_0$ tension but also spatial anomalies of CMB e.g. the hemispherical power asymmetry and quadrupole-octopole alignment. In the mean field (or Landau) approximation of GLTofDE, there is a spontaneously symmetry breaking exactly like the Higgs potential. We modeled this transition, phenomenologically, and showed that GLTofDE can resolve both the $H_0$ tension and Lyman-$\alpha$ anomaly in a non-trivial way. According to $\chi^2$-analysis the transition happens at $z_t=0.738\pm0.028$ while $H_0=71.89\pm0.93$ km/s/Mpc and $\Omega^{like}_{k}=-0.225\pm0.049$ which are consistent with the latest $H(z)$ reconstructions. In addition, the GLTofDE proposes a framework to address the CMB anomalies when it is considered beyond the mean field approximation. In this regime existence of a long wavelength mode is a typical consequence which is named the Goldstone mode in the case of continuous symmetries. This mode, which is an automatic byproduct in GLTofDE, makes cosmological constant, direction dependent. This means one side of the sky should be colder than the other side in agreement with what has been already observed in CMB. In addition between initial stochastic pattern and the final state with one long wavelength mode, we can observe smaller patches or protrusions of the biggest remaining patch in the simulation. Our simulations show these protrusions are few in numbers and will be evolved according to Alan-Cahn mechanism. These protrusions can give an additional effect on CMB which is the existence of aligned quadrupole-octopole mode and its direction should be orthogonal to the dipole direction. We conclude that GLTofDE is a fertile framework both theoretically and phenomenologically.

Size-dependent pressure-induced amorphization: a thermodynamic panorama.

Below a critical particle size, some pressurized compounds (e.g. TiO2, Y2O3, PbTe) undergo a crystal-to-amorphous transformation instead of a polymorphic transition. This effect reflects the greater propensity of nanomaterials for amorphization. In this work, a panorama of thermodynamic interpretations is given: first, a descriptive analysis based on the energy landscape concept gives a general comprehension of the balance between thermodynamics and kinetics to obtain an amorphous state. Then, a formal approach based on Gibbs energy to describe the thermodynamics and phase transitions in nanoparticles gives a basic explanation of size-dependent pressure-induced amorphization. The features of this transformation (amorphization occurs at pressures lower than the polymorphic transition pressure!) and the nanostructuration can be explained in an elaborated model based on the Ginzburg-Landau theory of phase transition and on percolation theory. It is shown that the crossover between polymorphic transition and amorphization is highly dependent on the defect density and interfacial energy, i.e., on the synthesis process. Their behavior at high pressure is a quality control test for the nanoparticles.

Modeling the morphogenesis of brine channels in sea ice

Brine channels are formed in sea ice under certain constraints and represent a habitat of different microorganisms. The complex system depends on a number of various quantities as salinity, density, pH value, or temperature. Each quantity governs the process of brine channel formation. There exists a strong link between bulk salinity and the presence of brine drainage channels in growing ice with respect to both the horizontal and vertical planes. We develop a suitable phenomenological model for the formation of brine channels both referring to the Ginzburg-Landau theory of phase transitions as well as to the chemical basis of morphogenesis according to Turing. It is possible to conclude from the critical wave number on the size of the structure and the critical parameters. The theoretically deduced transition rates have the same magnitude as the experimental values. The model creates channels of similar size as observed experimentally. An extension of the model toward channels with different sizes is possible. The microstructure of ice determines the albedo feedback and plays therefore an important role for large-scale global circulation models.

A phase-field model with two-component order parameter for spontaneous crystallization in a melt of pure metal

A phase-field model with the two-component order parameter is proposed for describing the crystallization of a pure metal. The model is constructed on the basis of the Ginzburg-Landau theory of phase transitions using the principle of the positivity of entropy production in an open weakly nonequilibrium system. This makes it possible to use the proposed model for describing nonisothermal processes. A number of one-dimensional and two-dimensional problems associated with the simulation of a solidified structure under different temperature conditions are solved.

Description of thermal conductivity in the Ginzburg-Landau theory of phase transitions

A generalization of the Ginzburg-Landau theory of phase transitions is presented which allows one to describe states with variable temperatures. The approach is based on an expression for the entropy in the form of a functional which depends on the temperature gradient and the order parameter. It is shown that the theory is compatible with the zero-th law of thermodynamics (constancy of the temperature in equilibrium). For equilibrium thermodynamically stable states the results of the theory agree with the results of the isothermal approach based on the free energy functional. General limitations on the possible form of the nonlinear dynamic equations are given, in particular for the heat flux vector, and possible particular versions are given. The dynamics of linear fluctuations has been included, including temperature fluctuations.

Phase-Transition Theory of Instabilities. I. Second-Harmonic Instability and Bifurcation Points

view Abstract Citations (22) References (36) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Phase-Transition Theory of Instabilities. I. Second-Harmonic Instability and Bifurcation Points Christodoulou, Dimitris M. ; Kazanas, Demosthenes ; Shlosman, Isaac ; Tohline, Joel E. Abstract We use a free-energy minimization approach to describe in simple and clear physical terms the secular and dynamical instabilities as well as the bifurcations along well-known sequences of rotating, self-gravitating fluid and stellar systems such as the Maclaurin spheroids, the Jacobi, Dedekind, and Riemann ellipsoids, and the fluid/stellar disks. Our approach stems from the Ginzburg-Landau theory of phase transitions. In this paper, we focus on the Maclaurin sequence of oblate spheroidal equilibria and on the effects of nonaxisymmetric, second-harmonic disturbances. The free-energy approach has been pioneered in astrophysics by Bertin & Radicati (1976) who showed that the secular instability beyond the Maclaurin-Jacobi bifurcation can be interpreted as a second-order phase transition. We show that second-order phase transitions appear on the Maclaurin sequence also at the points of dynamical instability (bifurcation of the x = +1 self-adjoint Riemann sequence) and of bifurcation of the Dedekind sequence. The distinguishing characteristic of each second-order phase transition is the conservation/nonconservation of an integral of motion (a "conserved/nonconserved current") which, in effect, determines uniquely whether the transition appears or not. The secular instability beyond the Jacobi bifurcation appears only if circulation is not conserved. The secular instability at the Dedekind bifurcation appears only if angular momentum is not conserved. We show by an explicit calculation that, in the presence of dissipation agents that violate one or the other conservation law, the global minimum of the free-energy function beyond the onset of secular instability belongs to the Jacobi and to the Dedekind sequence, respectively. In the case of a "perfect" fluid which conserves both circulation and angular momentum, the "secular" phase transitions are no longer realized and the Jacobi/Dedekind bifurcation point becomes irrelevant. The Maclaurin spheroid remains at the global minimum of the free-energy function up to the bifurcation point of the x = +1 Riemann sequence. The x = +1 equilibria have lower free energy than the corresponding Maclaurin spheroids for the same values of angular momentum and circulation. Thus, a "dynamical" second- order phase transition is allowed to take place beyond this bifurcation point. This phase transition brings the spheroid, now sitting at a saddle point of the free-energy function, to the new global minimum on the x = +1 Riemann sequence. Circulation is not conserved in stellar systems because the stress-tensor gradient terms that appear in the Jeans equations of motion include "viscosity-like" off-diagonal terms of the same order of magnitude as the conventional "pressure" gradient terms. For this reason, globally unstable axisymmetric stellar systems evolve toward the "stellar" Jacobi sequence on dynamical timescales. This explains why the Jacobi bifurcation is a point of dynamical instability in stellar systems but only a point of secular instability in viscous fluids. The second-order phase transitions on the Maclaurin sequence are discussed in relation to the dynamical instability of stellar systems, the λ-transition of liquid 4He, the second-order phase transition in superconductivity, and the mechanism of spontaneous symmetry breaking. Publication: The Astrophysical Journal Pub Date: June 1995 DOI: 10.1086/175806 arXiv: arXiv:astro-ph/9409039 Bibcode: 1995ApJ...446..472C Keywords: GALAXIES: FORMATION; HYDRODYNAMICS; INSTABILITIES; STARS: BINARIES: GENERAL; STARS: FORMATION; Astrophysics E-Print: Postscript, compressed, uuencoded. Submitted to ApJ full text sources arXiv | ADS | Related Materials (3) Part 2: 1995ApJ...446..485C Part 3: 1995ApJ...446..500C Part 4: 1995ApJ...446..510C

Phase-Transition Theory of Instabilities. II. Fourth-Harmonic Bifurcations and lambda -Transitions

view Abstract Citations (8) References (55) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Phase-Transition Theory of Instabilities. II. Fourth-Harmonic Bifurcations and lambda -Transitions Christodoulou, Dimitris M. ; Kazanas, Demosthenes ; Shlosman, Isaac ; Tohline, Joel E. Abstract We use a free-energy minimization approach to describe in simple and clear physical terms the secular and dynamical instabilities as well as the bifurcations along equilibrium sequences of rotating, self-gravitating fluid systems. Our approach is fully nonlinear and stems from the Ginzburg-Landau theory of phase transitions. In this paper, we examine fourth-harmonic axisymmetric disturbances in Maclaurin spheroids and fourth-harmonic nonaxisymmetric disturbances in Jacobi ellipsoids. These two cases are very similar in the framework of phase transitions. It has been conjectured (Hachisu & Eriguchi 1983) that third-order phase transitions, manifested as smooth bifurcations in the angular momentum-rotation frequency plane, may occur on the Maclaurin sequence at the bifurcation point of the axisymmetric one-ring sequence and on the Jacobi sequence at the bifurcation point of the dumbbell-binary sequence. We show that these transitions are forbidden when viscosity maintains uniform rotation. The uniformly rotating one-ring/dumbbell equilibria close to each bifurcation point and their neighboring uniformly rotating nonequilibrium states have higher free energies than the Maclaurin/Jacobi equilibria of the same mass and angular momentum. These high-energy states act as free-energy barriers preventing the transition of spheroids/ellipsoids from their local minima to the free-energy minima that exist on the low rotation frequency branch of the one-ring/binary sequence. At a critical point, the two minima of the free-energy function are equal, signaling the appearance of a first-order phase transition. This transition can take place beyond the critical point only nonlinearly if the applied perturbations contribute enough energy to send the system over the top of the barrier (and if, in addition, viscosity maintains uniform rotation). In the angular momentum-rotation frequency plane, the one-ring and dumbbell-binary sequences have the shape of an "inverted S" and two corresponding turning points each. Because of this shape, the free-energy barrier disappears suddenly past the higher turning point, leaving the spheroid/ellipsoid on a saddle point but also causing a "catastrophe" by permitting a "secular" transition toward a one-ring/binary minimum energy state. This transition appears as a typical second-order phase transition, although there is no associated sequence bifurcating at the transition point (cf. Christodoulou et al. 1995a). Irrespective of whether a nonlinear first-order phase transition occurs between the critical point and the higher turning point or an apparent second-order phase transition occurs beyond the higher turning point, the result is fission (i.e., "spontaneous breaking" of the topology) of the original object on a secular timescale: the Maclaurin spheroid becomes a uniformly rotating axisymmetric torus, and the Jacobi ellipsoid becomes a binary. The presence of viscosity is crucial since angular momentum needs to be redistributed for uniform rotation to be maintained. We strongly suspect that the "secular catastrophe" is the dynamical analog of the notorious λ-transition of liquid 4He because it appears as a "second-order" phase transition with infinite "specific heat" at the point where the free-energy barrier disappears suddenly. This transition is not an elementary catastrophe. In contrast to this case, a "dynamical catastrophe" takes place from the bifurcation point to the lower branch of the Maclaurin toroid sequence because all conservation laws are automatically satisfied between the two equilibrium states. Furthermore, the free-energy barrier disappears gradually, and this transition is part of the elementary cusp catastrophe. This type of "λ-transition" is the dynamical analog of the Bose-Einstein condensation of an ideal Bose gas. The phase transitions of the dynamical systems are briefly discussed in relation to previous numerical simulations of the formation and evolution of protostellar systems. Some technical discussions concerning related results obtained from linear stability analyses, the breaking of topology, and the nonlinear theories of structural stability and catastrophic morphogenesis are included in an appendix. Publication: The Astrophysical Journal Pub Date: June 1995 DOI: 10.1086/175807 arXiv: arXiv:astro-ph/9504062 Bibcode: 1995ApJ...446..485C Keywords: GALAXIES: KINEMATICS AND DYNAMICS; GALAXIES: STRUCTURE; INSTABILITIES; STARS: FORMATION; STARS: ROTATION; Astrophysics E-Print: 34 pages, postscript, compressed,uuencoded. 7 figures available in postscript, compressed form by anonymous ftp from asta.pa.uky.edu (cd /shlosman/paper2 mget *.ps.Z). To appear in ApJ full text sources arXiv | ADS | Related Materials (3) Part 1: 1995ApJ...446..472C Part 3: 1995ApJ...446..500C Part 4: 1995ApJ...446..510C

Phase-Transition Theory of Instabilities. IV. Critical Points on the Maclaurin Sequence and Nonlinear Fission Processes

view Abstract Citations (9) References (34) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Phase-Transition Theory of Instabilities. IV. Critical Points on the Maclaurin Sequence and Nonlinear Fission Processes Christodoulou, Dimitris M. ; Kazanas, Demosthenes ; Shlosman, Isaac ; Tohline, Joel E. Abstract We use a free-energy minimization approach to describe in simple and clear physical terms the secular and dynamical instabilities as well as the bifurcations along equilibrium sequences of rotating, self-gravitating fluid systems. Our approach is fully nonlinear and stems from the Ginzburg-Landau theory of phase transitions. In the final paper of this series, we examine higher than second-harmonic disturbances applied to Maclaurin spheroids, the corresponding bifurcating sequences, and their relation to nonlinear fission processes. The triangle and ammonite sequences bifurcate from the two third-harmonic neutral points on the Maclaurin sequence, while the square and one-ring sequences bifurcate from two of the three known fourth-harmonic neutral points. The one-ring sequence has been analyzed in Christodoulou et al. (1995b). In the other three cases, secular instability does not set in at the corresponding bifurcation points because the sequences stand and terminate at higher energies relative to the Maclaurin sequence. Consequently, an anticipated (numerically unresolved) third-order phase transition at the ammonite bifurcation and numerically resolved second-order phase transitions at the triangle and square bifurcations are strictly forbidden. Furthermore, the ammonite sequence exists at higher rotation frequencies as well and is similar in every respect to the pear-shaped sequence that has been analyzed in Christodoulou et al. (1995c). There is no known bifurcating sequence at the point of third-harmonic dynamical instability. This point represents a discontinuous λ-transition of type 3 that brings a Maclaurin spheroid on a dynamical timescale directly to the binary sequence while the original symmetry and topology are broken in series. The remaining fourth-harmonic neutral point also appears to be related to a type-3 λ-transition which however takes place from the lower turning point of the one-ring sequence toward the starting point and then on toward the stable branch of the three-fluid-body (triple) sequence. A third type-3 λ-transition, taking place from the one-ring sequence toward the starting point and then on toward the stable branch of the four-fluid-body (quadruple) sequence, is also discussed. The two-ring sequence bifurcates from the axisymmetric sixth-harmonic neutral point on the Maclaurin sequence also toward higher energies initially but eventually turns around and proceeds to lower energies relative to the Maclaurin sequence. The point where the two sequences have equal energies represents a fourth type of λ-transition which is not preceded by a first-order phase transition. This type-4 λ-transition results in double fission on a secular timescale: a Maclaurin spheroid breaks into two coaxial axisymmetric tori that rotate uniformly and with the same frequency. Finally, our nonlinear approach easily identifies resonances between the Maclaurin sequence and various multi-fluid-body sequences that cannot be detected by linear stability analyses. Resonances appear as first-order phase transitions at points where the energies of the two sequences are nearly equal but the lower energy state belongs to one of the multi-fluid-body sequences. Three nonlinear resonances leading to the turning points of the binary, triple, and quadruple sequences are described. Publication: The Astrophysical Journal Pub Date: June 1995 DOI: 10.1086/175809 arXiv: arXiv:astro-ph/9505101 Bibcode: 1995ApJ...446..510C Keywords: GALAXIES: FORMATION; STARS: BINARIES: GENERAL; HYDRODYNAMICS; INSTABILITIES; STARS: FORMATION; Astrophysics E-Print: 23 pages, postscript, compressed, uuencoded. Figs. (6) available by anonymous ftp from ftp://asta.pa.uky.edu/shlosman/paper4/ , get *.ps.Z). To appear in ApJ full text sources arXiv | ADS | Related Materials (3) Part 1: 1995ApJ...446..472C Part 2: 1995ApJ...446..485C Part 3: 1995ApJ...446..500C

Ginzburg-Landau theory of phase transitions in quasi-one-dimensional systems.

A wide range of quasi-one-dimensional materials, consisting of weakly coupled chains, undergo three-dimensional phase transitions that can be described by a complex order parameter. A Ginzburg-Landau theory is derived for such a transition. It is shown that intrachain fluctuations in the order parameter play a crucial role and must be treated exactly. The effect of these fluctuations is determined by a single dimensionless parameter. The three-dimensional transition temperature, the associated specific heat jump, coherence lengths, and width of the critical region, are computed assuming that the single chain Ginzburg-Landau coefficients are independent of temperature. The width of the critical region, estimated from the Ginzburg criterion, is virtually parameter independent, being about 5-8 per cent of the transition temperature. To appear in {\it Physical Review B,} March 1, 1995.

Theory of pattern formation in gels: Surface folding in highly compressible elastic bodies.

Gels are highly compressible elastic bodies near the volume phase transition. They undergo large elastic deformations that cannot be described within the usual elastic theory. A theoretical framework is proposed to incorporate a nonlinear elastic theory into the Ginzburg-Landau theory of phase transition. Then we analyze patterns emerging on the upper surface of uniaxially deformed gels whose lower surface is clamped to a substrate. On the upper surface a two-dimensional spinodal decomposition is shown to occur when the elongation ratio exceeds a critical value. If the gel is uniaxially stretched and sufficiently compressible, the surface eventually folds itself, yielding periodic cusps in the late stage of the coarsening process. The folded parts are singular surfaces like cracks in solids accompanied by large strains. Developing some mathematical techniques, we properly take into account a drastic change of the boundary conditions at the folded parts, but elastic deformations inside the gel are treated within a linear theory around uniaxial homogeneous states. We also find that the cusp pattern exists only in a limited region of the parameter space, outside which it is still an open question as to what kinds of patterns emerge in final stages of the spinodal decomposition. We expect corrugated surfaces without folding when the gel is uniaxially stretched and rather incompressible or when it is uniaxially compressed near the critical point.

Ginzburg-Landau theory of phase transitions in pseudo-one-dimensional systems

Abstract The static and dynamic properties of weakly coupled chains undergoing a phase transition are reviewed. The discussion is based on the functional generalization of the Ginzburg-Landau theory, including systems with real and complex order parameter. Various predictions of the theory, such as static correlations, renormalized phonon frequencies and a central resonance in the dynamic form factor near structural instabilities are discussed and compared with recent experiments on linear conductors that undergo a Peierls transition. New results are obtained for the thermodynamic anomalies near the onset of 3-d ordering and for the dynamic form factor of systems with an incommensurate Peierls distortion.

A Statistical Theory of Particle Production

A statistical theory of particle production is formulated in analogy with the generalized Ginzburg-Landau theory of phase transitions in superconductors and intensity fluctuations in lasers. (auth)