Effective Equations Of Motion Research Articles

Dynamic asymptotic homogenization for wave propagation in magneto-electro-elastic laminated composite periodic structure

In the present work, anti-plane wave propagation with oblique incidence in a magneto-electro-elastic multi-laminated composite periodic structure with perfect contact is studied. A general asymptotic homogenization method is applied to analyze dispersion phenomena and dynamic process. On assuming the solution having a single-frequency dependency, the higher-order terms for the displacement, electric potential and magnetic potential in asymptotic expansions are studied. Higher order local problems up to a finite order and a generalization of the local problem of infinite order are derived through rigorous calculations satisfying the necessary and sufficient condition for the existence of 1-periodic solutions. The effective equation of motion also known as “Good” Boussinesq equation has been achieved directly which is one of the highlight of the present work. Moreover, closed-form expression of dispersion equation and closed-form solutions of first and second local problems have been obtained. The analytical results have been validated for particular elastic case. Graphical illustrations have been done for tri-laminated magneto-electro-elastic composite periodic structure and the effect of size of unit cell, angle of incidence of wave, volume fraction of the composite and magneto-electric properties on the dispersion curve and also slowness curves have been examined. The numerical results have been validated with previous work and compared with results obtained by other approximation method. The present dynamic homogenization method has been proven to provide a more dispersive system and better approximation.

The constrained state of minimum energy and the effective equation of motion

In this paper, we define the state of minimum energy while the expectation values of the field operators and their time derivatives in a determined moment in such a state are constrained. As an axiom, we consider such a state as the background of the quantum field theory. As an example, we consider the scalar field with [Formula: see text] interaction. To the third-order of perturbation, we obtain the equation of motion of the dynamic expectation value of the scalar field in the defined state.

Computing first-passage times with the functional renormalisation group

We use Functional Renormalisation Group (FRG) techniques to analyse the behaviour of a spectator field, σ, during inflation that obeys an overdamped Langevin equation. We briefly review how a derivative expansion of the FRG can be used to obtain Effective Equations of Motion (EEOM) for the one- and two-point function and derive the EEOM for the three-point function. We show how to compute quantities like the amplitude of the power spectrum and the spectral tilt from the FRG. We do this explicitly for a potential with multiple barriers and show that in general many different potentials will give identical predictions for the spectral tilt suggesting that observations are agnostic to localised features in the potential. Finally we use the EEOM to compute first-passage time (FPT) quantities for the spectator field. The EEOM for the one- and two-point function are enough to accurately predict the average time taken 〈\U0001d4a9〉 to travel between two field values with a barrier in between and the variation in that time δ\U0001d4a9 2. It can also accurately resolve the full PDF for time taken ρ(\U0001d4a9), predicting the correct exponential tail. This suggests that an extension of this analysis to the inflaton can correctly capture the exponential tail that is expected in models producing Primordial Black Holes.

Emergent non-Abelian gauge theory in coupled spin-electron dynamics

A clear separation of the time scales governing the dynamics of "slow" and "fast" degrees of freedom often serves as a prerequisite for the emergence of an independent low-energy theory. Here, we consider (slow) classical spins exchange coupled to a tight-binding system of (fast) conduction electrons. The effective equations of motion are derived under the constraint that the quantum state of the electron system at any instant of time $t$ lies in the $n$-dimensional low-energy subspace for the corresponding spin configuration at $t$. The effective low-energy theory unfolds itself straightforwardly and takes the form of a non-abelian gauge theory with the gauge freedom given by the arbitrariness of the basis spanning the instantaneous low-energy sector. The holonomic constraint generates a gauge covariant spin-Berry curvature tensor in the equations of motion for the classical spins. In the non-abelian theory for $n>1$, opposed to the $n=1$ adiabatic spin dynamics theory, the spin-Berry curvature is generically nonzero, even for time-reversal symmetric systems. Its expectation value with the representation of the electron state is gauge invariant and gives rise to an additional {\em geometrical} spin torque. Besides anomalous precession, the $n\ge 2$ theory also captures the spin nutational motion, which is usually considered as a retardation effect. This is demonstrated by proof-of-principle numerical calculations for a minimal model with a single classical spin. Already for $n=2$ and in parameter regimes where the $n=1$ adiabatic theory breaks down, we find good agreement with results obtained from the full (unconstrained) theory.

Scalar field damping at high temperatures

The motion of a scalar field that interacts with a hot plasma, like the inflaton during reheating, is damped, which is a dissipative process. At high temperatures the damping can be described by a local term in the effective equation of motion. The damping coefficient is sensitive to multiple scattering. In the loop expansion its computation would require an all-order resummation. Instead we solve an effective Boltzmann equation, similarly to the computation of transport coefficients. For an interaction with another scalar field we obtain a simple relation between the damping coefficient and the bulk viscosity, so that one can make use of known results for the latter. The numerical prefactor of the damping coefficient turns out to be rather large, of order $ 10 ^ 4 $.

Effective picture of bubble expansion

Recently the thermal friction on an expanding bubble from the cosmic first-order phase transition has been calculated to all orders of the interactions between the bubble wall and thermal plasma, leading to a γ2-scaling instead of the previously estimated γ1-scaling for the thermal friction exerted on a fast-moving bubble wall with a Lorentz factor γ. We propose for the first time the effective equation of motion (EOM) for an expanding bubble wall in the presence of an arbitrary γ-scaling friction to compute the efficiency factor from bubble collisions, which, in the case of γ2-scaling friction, is found to be larger than the recently updated estimation when the bubble walls collide after starting to approach a constant terminal velocity, leading to a slightly larger signal of the gravitational waves background from bubble collisions due to its quadratic dependence on the bubble collision efficiency factor, although the γ2-scaling friction itself has already suppressed the contribution from bubble collisions compared to that with γ1-scaling friction. We also suggest a phenomenological parameterization for the out-of-equilibrium term in the Boltzmann equation that could reproduce the recently found (γ2-1)-scaling of the friction term in the effective EOM of an expanding bubble wall, which merits further study in future numerical simulations of bubble expansion and collisions.

Invisible neutrino decay in precision cosmology

We revisit the topic of invisible neutrino decay in the precision cosmological context, via a first-principles approach to understanding the cosmic microwave background and large-scale structure phenomenology of such a non-standard physics scenario. Assuming an effective Lagrangian in which a heavier standard-model neutrino νH couples to a lighter one νl and a massless scalar particle ϕ via a Yukawa interaction, we derive from first principles the complete set of Boltzmann equations, at both the spatially homogeneous and the first-order inhomogeneous levels, for the phase space densities of νH, νl, and ϕ in the presence of the relevant decay and inverse decay processes. With this set of equations in hand, we perform a critical survey of recent works on cosmological invisible neutrino decay in both limits of decay while νH is ultra-relativistic and non-relativistic. Our two main findings are: (i) in the non-relativistic limit, the effective equations of motion used to describe perturbations in the neutrino-scalar system in the existing literature formally violate momentum conservation and gauge invariance, and (ii) in the ultra-relativistic limit, exponential damping of the anisotropic stress does not occur at the commonly-used rate ΓT =(1/τ0) (mνH/EνH)3, but at a rate ∼ (1/0) (mνH/EνH)5. Both results are model-independent. The impact of the former finding on the cosmology of invisible neutrino decay is likely small.The latter, however, implies a significant revision of the cosmological limit on the neutrino lifetime τ0 from τ0 old ≳ 1.2 × 109 s (mνH/50 meV)3 to τ0 ≳ (4 × 105 → 4 × 106) s (mνH/50 meV)5.

Improved μ¯ -scheme effective dynamics of full loop quantum gravity

We propose a new derivation from the full Loop Quantum Gravity (LQG) to the Loop Quantum Cosmology (LQC) improved $\bar{\mu}$-scheme effective dynamics, based on the reduced phase space formulation of LQG and a proposal of effective Hamiltonian/action in the full LQG. A key step of our program is an improved regularization of the full LQG Hamiltonian on a cubic lattice. The improved Hamiltonian uses a set of "dressed holonomies" $h_\Delta(\mathfrak{s})$ which not only depend on the connection $A$ but also depend on the length of the curve $\mathfrak{s}$. With the improved Hamiltonian, we propose a quantum effective action and derive a new set effective equations of motion (EOMs) for the full LQG. Then we show that these new EOMs imply the $\bar{\mu}$-scheme effective dynamics for both the homogeneous-isotropic and Bianchi-I cosmology, and predict bounce and Planckian critical density. As a byproduct, although the model is defined on a cubic lattice, we find that the improved effective Hamiltonian of cosmology is invariant under the lattice refinement. The cosmological effective dynamics, predictions of bounce and critical density are results at the continuum limit.

Domain wall diode based on functionally graded Dzyaloshinskii–Moriya interaction

We present a general approach for studying the dynamics of domain walls in biaxial ferromagnetic stripes with functionally graded Dzyaloshinskii–Moriya interaction (DMI). By engineering the spatial profile of the DMI parameter, we propose the concept of a diode, which implements the filtering of domain walls of a certain topological charge and helicity. We base our study on the phenomenological Landau–Lifshitz–Gilbert equations with additional Zhang–Li spin-transfer terms using a collective variable approach. In the effective equations of motion, the gradients of DMI play the role of a driving force, which competes with the current driving. All analytical predictions are confirmed by numerical simulations.

Effective dynamics from coherent state path integral of full loop quantum gravity

A new routine is proposed to relate Loop Quant Cosmology (LQC) to Loop Quantum Gravity (LQG) from the perspective of effective dynamics. We derive the big-bang singularity resolution and big bounce from the first principle of full canonical LQG. Our results are obtained in the framework of the reduced phase space quantization of LQG. As a key step in our work, we derive with coherent states a new discrete path integral formula of the transition amplitude generated by the physical Hamiltonian. The semiclassical approximation of the path integral formula gives an interesting set of effective equations of motion (EOMs) for full LQG. When solving the EOMs with homogeneous and isotropic ansatz, we reproduce the LQC effective dynamics in $\mu_0$-scheme. The solution replaces the big-bang singularity by a big bounce. In the end, we comment on the possible relation between the $\bar{\mu}$-scheme of effective dynamics and the continuum limit of the path integral formula.

Morphology of Dune-Like Relief in Rivers

Linearized equations of two-dimensional hydraulics have been analyzed by the method of small perturbations within a wide range of alluvial features sizes at large values of Froude numbers and hydraulic resistance. The preservation of three-dimensional effects in depth-averaged equations of motion, continuity, and deformation enabled the authors to identify domains with combinations of flow hydraulic characteristics that correspond to alluvial features of different size and morphology. The study confirmed the results of the earlier analysis for subcritical flows with small hydraulic resistance and showed new types of relationships between alluvial features morphology and the hydraulic characteristics of flow for flows with large Froude numbers and high hydraulic resistance. In addition, a relationship between the length of two-dimensional ultramicroforms and hydraulic resistance has been established. A new class of macroforms—two-dimensional macroforms—have been identified. The verification of the results of theoretical analysis by measurements data on the morphology of channel formations and hydraulic characteristics of the flow has shown that the analysis of linearized equations of two-dimensional hydrodynamics by small-perturbation method can be used to determine the morphology and dimensions of channel forms in both subcritical and supercritical streams. The step–pool systems in torrential streams in mountain rivers are analogues of antidunes (in supercritical flows) and ripples (in subcritical flows), obtained in large flumes with sand alluvium. Three-dimensional macroforms are most common in rivers. If such macroforms are well developed in wide channels (at flow width greater than the half-width of the macroform), their length is determined by channel depth. Macroforms in narrower channels cannot develop completely, and their lengths are limited by channel width and can be calculated only through this width.

Multiband effects in equations of motion of observables beyond the semiclassical approach

The equations of motion (EOM) for the position and gauge invariant crystal momentum are considered for multiband wave packets of Bloch electrons. For a localized packet in a subset of bands well-separated from the rest of the band structure of the crystal, one can construct an effective electromagnetic Hamiltonian with respect to the center of the packet. We show that the EOM can be obtained via a projected operator procedure, which is derived from the adiabatic approximation within perturbation theory. These relations explicitly contain information from each band captured in the expansion coefficients and energy band structure of the Bloch states as well as non-Abelian features originating from interband Berry phase properties. This general and transparent Hamiltonian-based approach is applied to a wave packet spread over a single band, a set of degenerate bands, and two linear crossing bands. The generalized EOM hold promise for novel effects in transport currents and Hall effect phenomena.

Can all the infrared secular growth really be understood as increase of classical statistical variance?

It is known that in the theory of light scalar fields during inflation, correlation functions suffer from infrared (IR) divergences or large IR loop corrections, leading to the breakdown of perturbation theory. In order to understand the physical meaning of such IR enhancement, we investigate the stochastic properties of an effective equation of motion (EoM) for long-wavelength modes of a canonically normalized light scalar field ϕ with a general sufficiently flat interaction potential on de Sitter background. Firstly, we provide an alternative refined derivation of the effective action for long-wavelength modes which leads to the effective EoM that correctly reproduces all the IR correlation functions in a good approximation at a late time, by integrating out short-wavelength modes. Next, under the assumption that one can neglect non-local correlations in the influence functional exceeding the coarse-graining scale, we show that the effective EoM for IR modes of the “average field” in Schwinger-Keldysh formalism ϕc< can be interpreted as a classical stochastic process in the present model.

Stationary solutions from the large D membrane paradigm

It has recently been shown that the dynamics of black holes in large number of dimensions D can be recast as the dynamics of a probe membrane propagating in the background spacetime which solves Einstein equations without matter. The equations of motion of this membrane are simply the statement of conservation of the stress tensor and charge current defined on this membrane. In this paper we obtain the effective equations of motion for stationary membranes in any empty background both in presence and absence of charge. It turns out that the thermodynamic quantities associated with the stationary membranes that satisfy these effective equations also satisfy the first law of black hole thermodynamics. These stationary membrane equations have some interesting solutions such as charged rotating black holes in flat and AdS backgrounds as well as black ring solutions in large D.

Expansion of the effective action around non-Gaussian theories

This paper derives the Feynman rules for the diagrammatic perturbation expansion of the effective action around an arbitrary solvable problem. The perturbation expansion around a Gaussian theory is well-known and composed of one-line irreducible diagrams only. For the expansions around an arbitrary, non-Gaussian problem, we show that a more general class of irreducible diagrams remains in addition to a second set of diagrams that has no analogue in the Gaussian case. The effective action is central to field theory, in particular to the study of phase transitions, symmetry breaking, effective equations of motion, and renormalization. We exemplify the method on the Ising model, where the effective action amounts to the Gibbs free energy, recovering the Thouless–Anderson–Palmer mean-field theory in a fully diagrammatic derivation. Higher order corrections follow with only minimal effort compared to existing techniques. Our results show further that the Plefka expansion and the high-temperature expansion are special cases of the general formalism presented here.

Statistical nature of infrared dynamics on de Sitter background

In this study, we formulate a systematic way of deriving an effective equation of motion(EoM) for long wavelength modes of a massless scalar field with a general potential V(ϕ) on de Sitter background, and investigate whether or not the effective EoM can be described as a classical stochastic process. Our formulation gives an extension of the usual stochastic formalism to including sub-leading secular growth coming from the nonlinearity of short wavelength modes. Applying our formalism to λ ϕ4 theory, we explicitly derive an effective EoM which correctly recovers the next-to-leading secularly growing part at a late time, and show that this effective EoM can be seen as a classical stochastic process. Our extended stochastic formalism can describe all secularly growing terms which appear in all correlation functions with a specific operator ordering. The restriction of the operator ordering will not be a big drawback because the commutator of a light scalar field becomes negligible at large scales owing to the squeezing.

Space-time models based on random fields with local interactions

The analysis of space-time data from complex, real-life phenomena requires the use of flexible and physically motivated covariance functions. In most cases, it is not possible to explicitly solve the equations of motion for the fields or the respective covariance functions. In the statistical literature, covariance functions are often based on mathematical constructions. In this paper, we propose deriving space-time covariance functions by solving “effective equations of motion”, which can be used as statistical representations of systems with diffusive behavior. In particular, we propose to formulate space-time covariance functions based on an equilibrium effective Hamiltonian using the linear response theory. The effective space-time dynamics is then generated by a stochastic perturbation around the equilibrium point of the classical field Hamiltonian leading to an associated Langevin equation. We employ a Hamiltonian which extends the classical Gaussian field theory by including a curvature term and leads to a diffusive Langevin equation. Finally, we derive new forms of space-time covariance functions.

Theory of electronic structure and some related properties of ZnO

We derive an expression for the effective mass of an electron for a multi-band system, starting from a thermodynamic potential expressed in terms of a single-particle Green’s function and using an effective equation of motion in the effective mass representation (EMR). We consider a four-band model around point-one conduction band and three valence bands, for the w-ZnO. We diagonalize the Hamiltonian, using double-group basis functions for the energy levels, by considering the conduction band and each of the valence bands separately, and obtain the energy dispersions in terms of Luttinger–Kohn parameters. Effects of other bands on each of the valence band energy is considered by going beyond parabolic approximation and adding fourth degree terms in the wave vector to the energy dispersion. The resulting bands’ structures agree well with other reported electronic structures. The formalism is then used to calculate the effective masses and effective g-factors at the point, and as functions of the wave vector. Comparisons with other calculated values and experimental values show reasonably good agreement.

Profiling the overdamped dynamics of a nonadiabatic system

Rapidly oscillating time-periodic potentials with a vanishing time average have been exploited to investigate the dynamics of an overdamped particle. Using the multiple scale perturbation theory, it has been shown that the dynamics can be adequately characterized by an explicitly time-independent effective potential. The resulting “effective equation of motion” can offer various avenues to handle the dynamics of the system driven by a high-frequency field. We study the effects of the field parameters on the mobility of the overdamped particle moving in the effective potential. The variation of the mobility with the field parameters is associated with the interplay of spatially periodic gradients, time periodic modulation and thermal noise in the overdamped region. Good agreement between the simulations and theoretical estimates validates our methodology that captures the constitutional features ruling the dynamics in the overdamped limit. The results observed here can also be extended to the quantum system.

4D topological mass by gauging spin

We propose a spin gauge field theory in which the curl of a Dirac fermion current density plays the role of the pseudovector charge density. In this field-theoretic model, spin interactions are mediated by a single scalar gauge boson in its antisymmetric tensor formulation. We show that these long range spin interactions induce a gauge invariant photon mass in the one-loop effective action. The fermion loop generates a coupling between photons and the spin gauge boson, which acquires thus charge. This coupling represents also an induced, gauge invariant, topological mass for the photons, leading to the Meissner effect. The one-loop effective equations of motion for the charged spin gauge boson are the London equations. We propose thus spin gauge interactions as an alternative, topological mechanism for superconductivity in which no spontaneous symmetry breaking is involved.