Landau-Lifshitz Theory Research Articles

Hydrodynamic equations for a U(N) invariant superfluid

Abstract In this paper, we develop an appropriate set of hydrodynamic equations for a U(N) invariant superfluid that couple the dynamics of superflow and magnetization. In the special case when both the superfluid and normal velocities are zero, the hydrodynamic equations reduce to a generalized version of Landau-Lifshitz equation for ferromagnetism with U(N) symmetry. When both velocities are non-zero, there appears couplings between the superflow and magnetization dynamics, and the superfluid velocity no longer satisfies the irrotational condition. On the other hand, the magnitude of magnetization is no longer a constant of motion as was the case for the standard Landau-Lifshitz theory. In comparison with the simple superfluid, the first and second sounds are modified by a non-zero magnetization through various thermodynamic functions. For U(2) invariant superfluid, we get both (zero-) sound wave and a spin wave at zero temperature. It is found that the dispersion of spin wave is always quadratic, which is consistent with microscopic analysis. In the Appendix, we
show that the hydrodynamic equation for a U(N) invariant superfluid can be obtained from the general hydrodynamic equation with arbitrary internal symmetries.

Theory and simulation of electrokinetic fluctuations in electrolyte solutions at the mesoscale

Electrolyte solutions play an important role in energy storage devices, whose performance relies heavily on the electrokinetic processes at sub-micron scales. Although fluctuations and stochastic features become more critical at small scales, the long-range Coulomb interactions pose a particular challenge for both theoretical analysis and simulation of fluid systems with fluctuating hydrodynamic and electrostatic interactions. Here, we present a theoretical framework based on the Landau–Lifshitz theory to derive closed-form expressions for fluctuation correlations in electrolyte solutions, indicating significantly different decorrelation processes of ionic concentration fluctuations from hydrodynamic fluctuations, which provides insights for understanding transport phenomena of coupled fluctuating hydrodynamics and electrokinetics. Furthermore, we simulate fluctuating electrokinetic systems using both molecular dynamics (MD) with explicit ions and mesoscopic charged dissipative particle dynamics (cDPD) with semi-implicit ions, from which we identify that the spatial probability density functions of local charge density follow a gamma distribution at sub-nanometre scale (i.e. $0.3\,{\rm nm}$ ) and converge to a Gaussian distribution above nanometre scales (i.e. $1.55\,{\rm nm}$ ), indicating the existence of a lower limit of length scale for mesoscale models using Gaussian fluctuations. The temporal correlation functions of both hydrodynamic and electrokinetic fluctuations are computed from all-atom MD and mesoscale cDPD simulations, showing good agreement with the theoretical predictions based on the linearized fluctuating hydrodynamics theory.

Fluctuation kinetics in system in the presence of random external field and a generalization of the fluctuation-dissipation theorem

The evolution equation of general form for the parameters of the reduced description (RDP) of system in the presence of a random external field is used to describe the dynamics of the system. The field causes fluctuations of RDP. With taking into account the fluctuations, the system is described along with the average value of the RDP itself by the average values of all RDP products (fluctuations) or their correlation functions. For the generating function of these quantities, a closed time equation of the fluctuation kinetics is derived. The general initial form of the time equation for RDP allows investigating kinetic and hydrodynamic states in a unique way without specifying the spatial dependence of quantities. Compared with the known previous works, this greatly simplified the study. The closed time equation for the generating function (the equation of the fluctuation kinetics) is derived using the generalized Furutzu–Novikov theorem, the proof of which is simplified in the paper. The external field is considered as a Gaussian stationary process with a correlation time much shorter than the characteristic time of system evolution. On this basis, a small parameter is introduced, and the corresponding perturbation theory is built. Cases of the field which is introduced through RDP and directly (additive field) are considered. The definition of a generalized nonlinear fluctuation-dissipation theorem is proposed. To illustrate the developed fluctuation kinetics, the approximations of binary correlations are considered, in which more complex correlations are neglected, as well as the states around equilibrium. Fluctuation hydrodynamics, which is compared with the Landau–Lifshitz theory, is considered as an application.

SCALINGS AND LIMITS OF LANDAU-DE GENNES MODELS FOR LIQUID CRYSTALS: A COMMENT ON SOME RECENT ANALYTICAL PAPERS

Some recent analytical papers have explored limiting behaviors of Landaude Gennes models for liquid crystals in certain extreme ranges of the model parameters: limits of “vanishing elasticity” (in the language of some of these papers) and “low-temperature limits.” We use simple scaling analysis to show that these limits are properly interpreted as limits in which geometric length scales (such as the size of the domain containing the liquid crystal material) become large compared to intrinsic length scales (such as correlation lengths or coherence lengths, which determine defect core sizes). This represents the natural passage from a mesoscopic model to a macroscopic model and is analogous to a “London limit” in the Ginzburg-Landau theory of superconductivity or a “large-body limit” in the Landau-Lifshitz theory of ferromagnetism. Known relevant length scales in these parameter regimes (nematic correlation length, biaxial coherence length) can be seen to emerge via balances in equilibrium Euler-Lagrange equations associated with well-scaled Landau-de Gennes free-energy functionals.

Thermodynamics, phase transitions and Ruppeiner geometry for Einstein\u2013dilaton\u2013Lifshitz black holes in the presence of Maxwell and Born\u2013Infeld electrodynamics

In this paper, we first obtain the higher-dimen-sional dilaton–Lifshitz black hole solutions in the presence of Born–Infeld (BI) electrodynamics. We find that there are two different solutions for the cases of z=n+1 and zne n+1 where z is the dynamical critical exponent and n is the number of spatial dimensions. Calculating the conserved and thermodynamical quantities, we show that the first law of thermodynamics is satisfied for both cases. Then we turn to the study of different phase transitions for our Lifshitz black holes. We start with the Hawking–Page phase transition and explore the effects of different parameters of our model on it for both linearly and BI charged cases. After that, we discuss the phase transitions inside the black holes. We present the improved Davies quantities and prove that the phase transition points shown by them are coincident with the Ruppeiner ones. We show that the zero temperature phase transitions are transitions in the radiance properties of black holes by using the Landau–Lifshitz theory of thermodynamic fluctuations. Next, we turn to the study of the Ruppeiner geometry (thermodynamic geometry) for our solutions. We investigate thermal stability, interaction type of possible black hole molecules and phase transitions of our solutions for linearly and BI charged cases separately. For the linearly charged case, we show that there are no phase transitions at finite temperature for the case zge 2. For z<2, it is found that the number of finite temperature phase transition points depends on the value of the black hole charge and there are not more than two. When we have two finite temperature phase transition points, there is no thermally stable black hole between these two points and we have discontinuous small/large black hole phase transitions. As expected, for small black holes, we observe finite magnitude for the Ruppeiner invariant, which shows the finite correlation between possible black hole molecules, while for large black holes, the correlation is very small. Finally, we study the Ruppeiner geometry and thermal stability of BI charged Lifshtiz black holes for different values of z. We observe that small black holes are thermally unstable in some situations. Also, the behavior of the correlation between possible black hole molecules for large black holes is the same as for the linearly charged case. In both the linearly and the BI charged cases, for some choices of the parameters, the black hole system behaves like a Van der Waals gas near the transition point.

Asymmetric Domain Walls of Small Angle in Soft Ferromagnetic Films

We focus on a special type of domain walls appearing in the Landau-Lifshitz theory for soft ferromagnetic films. These domain walls are divergence-free $S^2$-valued transition layers that connect two directions in $S^2$ (differing by an angle $2\theta$) and minimize the Dirichlet energy. Our main result is the rigorous derivation of the asymptotic structure and energy of such "asymmetric" domain walls in the limit $\theta \to 0$. As an application, we deduce that a supercritical bifurcation causes the transition from symmetric to asymmetric walls in the full micromagnetic model.

Fluctuating hydrodynamics for a rarefied gas based on extended thermodynamics

We develop a theory of fluctuating hydrodynamics based on extended thermodynamics through studying the 13-variable theory for a monatomic rarefied gas as a representative case. After analyzing the relationship between the present theory and the Landau–Lifshitz theory, we discuss the hierarchy structure of the hydrodynamic fluctuations.

Gilbert ferromagnetic damping theory and the fluctuation-dissipation theorem

The phenomenological Gilbert–Landau–Lifshitz theory of magnetic damping in ordered magnetic materials has long served as the central model employed in the analysis of considerable experimental magnetic resonance data. The problem undertaken here is to formulate the microscopic basis of the model in terms of usual theory of irreversible transport embodied in the fluctuation-dissipation-response theorems. As an application of the general method we establish a rigorous relationship between magnetic damping and acoustic damping implicit in magnetoelastic models.

Fluctuating hydrodynamics for dilute granular gases

Starting from the kinetic equations for the fluctuations and correlations of a dilute gas of inelastic hard spheres or disks, a Boltzmann-Langevin equation for the one-particle distribution function of the homogeneous cooling state is constructed. This equation is the linear Boltzmann equation with a fluctuating white-noise term. Balance equations for the fluctuating hydrodynamic fields are derived. New fluctuating forces appear as compared with the elastic limit. The particular case of the transverse velocity field is investigated in detail. Its fluctuations can be described by means of a Langevin equation, but exhibiting two main differences with the Landau-Lifshitz theory: the noise is not white, and its second moment is not determined by the shear viscosity. This shows that the fluctuation-dissipation relations for molecular fluids do not straightforwardly carry over to inelastic gases. The theoretical predictions are shown to be in good agreement with molecular-dynamics simulation results.

Stability and causality in relativistic dissipative hydrodynamics

The stability and causality of the Landau–Lifshitz theory and the Israel–Stewart-type causal dissipative hydrodynamics are discussed. We show that the problems of acausality and instability are correlated in relativistic dissipative hydrodynamics and instability is induced by acausality. We further discuss the stability of the scaling solution. The scaling solution of the causal dissipative hydrodynamics can be unstable against inhomogeneous perturbations.

The Entropy Effects in Binary Mixtures of Polar Mesogenic Solvent/Nonpolar Solute

The paper concerns two aspects of the entropy in mesogenic systems: (i) the entropy jump (Delta S (0) NI) at the phase transition from the isotropic liquid (I) to the nematic liquid crystalline state (N), and (ii) the entropy increment (Delta S) caused by the ordering action of the probing electric field applied to the dipolar system. The system studied are the mixtures of strongly polar mesogenic solvent n-hexylcyanobiphenyl (C 6H 13PhPhCN, 6CB) and the nonpolar nonmesogenic admixture 4-ethylcyclohexyl-4'- n-nonylphenyl (C 2H 5CyHxPhC 9H 19, 2CyPh9). The entropy jump at the I-N phase transition in pure 6CB [Delta S (0) NI= 1.52 J/(mol K)] was evaluated from the analysis of the phase diagram of the mixture 6CB + 2CyPh9 with use of the Landau-Lifshitz theory; the resulting value of the transition enthalpy (Delta H (0) NI = T NIDelta S (0) NI = 0.50 kJ/mol) agrees well to that obtained with the calorimetric methods. The field-induced entropy increment (Delta S) was calculated, at the given temperature, from the static dielectric permittivity derivative value (depsilon s/d T), with use of the Fröhlich theory. The singularities in dependence of the entropy increment on the temperature and on the mixtures composition are discussed in terms of the prenematic molecular self-organization extent in mesogenic liquids of different density of dipoles.

Nonlinear-nonequilibrium wave propagation in sandstones

The transition from linear to nonlinear-nonequilibrium dynamical elasticity in rocks is of considerable interest in seismic wave propagation as well as in understanding the basic dynamical processes in consolidated granular materials. We have carried out a careful experimental investigation of this transition for Berea and Fontainebleau sandstones. These experiments have showed the existence of two strain regimes. At low strain (10-9 up to 10-7) the rocks behave elastically as classical (Landau-Lifshitz theory) nonlinear materials. At higher strains memory effects due to a driven nonequilibrium state complicate the characterization of the nonlinear behavior. The understanding of this second region is not trivial. The main focus of this work is to review the behavior of the low strain, nonlinear region and to report on new experiments meant to understand more about the behavior in the nonequilibrium regime.

Symmetry principles of the formation of ordered phases of crystals

General principles of the formation of low-symmetry ordered phases of crystals, such as the principle of a finite number of low-symmetry phases, the principle of phase diversity, the principle of a finite number of possible structure types of ordered phases, the principle of structural complexity, the compensation principle, and the generalized Fedorov-Groth law, are established in the framework of the Landau-Lifshitz theory of second-order phase transitions. The ranges of their applicability are determined. These principles are derived by extending the results of the structural calculations of the design of new inorganic materials with anomalous physicochemical properties.

Monte Carlo simulations of two-dimensional hard core lattice gases

Monte Carlo simulations are used to study lattice gases of particles with extended hard cores on a two-dimensional square lattice. Exclusions of one and up to five nearest neighbors (NN) are considered. These can be mapped onto hard squares of varying side length, lambda (in lattice units), tilted by some angle with respect to the original lattice. In agreement with earlier studies, the 1NN exclusion undergoes a continuous order-disorder transition in the Ising universality class. Surprisingly, we find that the lattice gas with exclusions of up to second nearest neighbors (2NN) also undergoes a continuous phase transition in the Ising universality class, while the Landau-Lifshitz theory predicts that this transition should be in the universality class of the XY model with cubic anisotropy. The lattice gas of 3NN exclusions is found to undergo a discontinuous order-disorder transition, in agreement with the earlier transfer matrix calculations and the Landau-Lifshitz theory. On the other hand, the gas of 4NN exclusions once again exhibits a continuous phase transition in the Ising universality class-contradicting the predictions of the Landau-Lifshitz theory. Finally, the lattice gas of 5NN exclusions is found to undergo a discontinuous phase transition.

Causal structure of relativistic dissipative hydrodynamics

We present a new formalism for the relativistic dissipative hydrodynamics consistent with causality. We start from the physical analysis of the irreversible currents according to the Landau-Lifshitz theory. Then, the irreversible currents are given by integral expressions which take into account the relaxation time. Only one additional parameter was introduced, the relaxation time, tR. We verified that the linearized equation of motion for small perturbations in the homogeneous, static background coincides with Hiscock-Lindblom

Orbital magnetic dynamics in chiralp-wave superconductors

We present a theory of orbital magnetic dynamics for a chiral $p$-wave superconductor with broken time-reversal symmetry. In contrast to the common Landau-Lifshitz theory for spin ferromagnets, the case of orbital magnetism cannot be described in terms of local magnetization density. Hence, it is impossible to define unambiguously the spontaneous magnetic moment: the latter would depend on conditions of its experimental investigation. As an example of this we consider orbital magnetization waves and the domain-structure energy.

Formula omitted] corrections to semiclassical AdS/CFT states from quantum Landau–Lifshitz model

One way to relate semiclassical string states and dual gauge theory states is to show the equivalence between their low-energy effective 2d actions. The gauge theory effective action, which is represented by an effective Landau–Lifshitz (LL) model, was previously found to match the string theory world-sheet action up to the first two orders in the effective parameter λ ˜ = λ / J 2 , where λ is the 't Hooft coupling and J is the total R-charge. Here we address the question if quantizing the effective LL action reproduces the subleading 1 / J corrections to the spin chain energies as well as the quantum corrections to the string energies. We demonstrate that this is indeed the case provided one chooses an appropriate regularization of the effective LL theory. Expanding near the BPS vacuum, we show that the quantum LL action gives the same 1 / J corrections to energies of BMN states as found previously on the gauge theory and string theory sides. We also compute the subleading 1 / J 2 corrections and show that these too match with corrections computed from the Bethe ansatz. We also compare the results from the LL action with a more direct computation from the spin chain. We repeat the same computation for the β-deformed LL action and find that the quantum LL result is again equal to the 1 / J correction computed from the β-deformed Bethe ansatz equations. We also quantize the LL action near the rotating circular and folded string solutions, generalizing the known gauge/string results for 1 / J corrections to the classical energies. We emphasize the simplicity of this effective field theory approach as compared to the full quantum string computations.

The distribution function for a subsystem experiencing temperature fluctuations

A nonlinear generalization of the Landau-Lifshitz theory of hydrodynamic fluctuations for the simplest case in which only energy flux and temperature fluctuations are observed is used to derive the distribution function for a subsystem with a fluctuating temperature, which coincides with the Levy distribution taken to be one of the main results of the so-called Tsallis’s nonextensive statistics. It is demonstrated that the same distribution function is obtained from the principle of maximum of information entropy if the latter is provided by Renyi’s entropy, which is an extensive quantity. The obtained distribution function is to be used instead of the Gibbs distribution in constructing the thermodynamics of systems with significant temperature fluctuations.

Effect of magnetization ripple at high frequencies in amorphous ferromagnetic thin films

An analytical calculation of the permeability frequency spectra using Landau–Lifshitz theory is presented and compared to experimental results obtained on soft amorphous ferromagnetic thin films with an in-plane uniaxial magnetic anisotropy. The disagreement between theory and experiment is attributed to a small magnetic anisotropy dispersion in the film (magnetization ripple). We have measured the real part of the transverse biased permeability μt′ at different frequencies of the exciting field in order to estimate the ripple effects in our films and to improve our model. We have plotted the value of the bias field Hm, at which μt′ versus exciting field frequency reaches a maximum. Hm is found to be frequency independent at low frequencies (up to 280 MHz). Above a critical frequency it increases rapidly, suggesting that ripple effects are strongly frequency dependent.

Spin dynamics and magnetic anisotropy dispersion in soft ferromagnetic thin films

An analytic calculation of the permeability frequency spectra using the LANDAU-LIFSHITZ theory is compared with experimental results obtained on ferromagnetic thin films with an in-plane uniaxial magnetic anisotropy The disagrement between theory and experiment is attributed to magnetic anisotropy dispersion in the film. This dispersion is studied and seems to be due to magnetic ripples. Measurement of the ripple effect is done through the measurement of the transverse biased susceptibility. A quantitative evaluation of the anisotropy dispersion is suggested in order to improve our model.