Nonlinear Response Coefficients Research Articles

Formula omitted], [formula omitted], [formula omitted], [formula omitted]: Nonlinear hydrodynamic response versus LHC data

Higher harmonics of anisotropic flow (vn with n≥4) in heavy-ion collisions can be measured either with respect to their own plane, or with respect to a plane constructed using lower-order harmonics. We explain how such measurements are related to event-plane correlations. We show that CMS data on v4 and v6 are compatible with ATLAS data on event-plane correlations. If one assumes that higher harmonics are the superposition of non-linear and linear responses, then the linear and non-linear parts can be isolated under fairly general assumptions. By combining analyses of higher harmonics with analyses of v2 and v3, one can eliminate the uncertainty from initial conditions and define quantities that only involve nonlinear hydrodynamic response coefficients. Experimental data on v4, v5 and v6 are in good agreement with hydrodynamic calculations. We argue that v7 can be measured with respect to elliptic and triangular flow. We present predictions for v7 versus centrality in Pb–Pb collisions at the LHC.

Multivariate fluctuation relations for currents

This paper is devoted to multivariate fluctuation relations for all the currents flowing across an open system in contact with several reservoirs at different temperatures and chemical potentials, or driven by time-independent external mechanical forces. After some transient behavior, the open system is supposed to reach a nonequilibrium steady state that is controlled by the thermodynamic and mechanical forces, called the affinities. The time-reversal symmetry of the underlying Hamiltonian dynamics implies symmetry relations among the statistical properties of the fluctuating currents, depending on the values of the affinities. These multivariate fluctuation relations are not only compatible with the second law of thermodynamics, but they also imply remarkable relations between the linear or nonlinear response coefficients and the cumulants of the fluctuating currents. These relations include the Onsager and Casimir reciprocity relations, as well as their generalizations beyond linear response. Methods to deduce multivariate fluctuation relations are presented for classical, stochastic and quantum systems. In this way, multivariate fluctuation relations are obtained for energy or particle transport in the effusion of an ideal gas, heat transport in Hamiltonian systems coupled by Langevin stochastic forces to heat reservoirs, driven Brownian motion of an electrically charged particle subjected to an external magnetic field, and quantum electron transport in multi-terminal mesoscopic circuits where the link to the scattering approach is established.

Interplay between relaxation of nonlinear response and coupling coefficient dispersion in the instability spectra of dual core optical fiber

The modulational instability of a dual core optical fiber under the combined effect of coupling coefficient dispersion (CCD) and relaxation of nonlinear response is presented. The effect of wavelength dependence of coupling coefficient is effectively included in the pair of generalized linearly coupled nonlinear Schrödinger equation (CNLSE). In order to account for the finite response time, a time dependent nonlinear response is incorporated in the system of CNLSE using the Debye relaxation model. The results from the linear stability analysis show that in anomalous dispersion regime for any finite relaxation time, the nonlinearity becomes complex and there exist two unstable modes. In the normal dispersion regime, a single unstable band at higher detuning frequency is observed which is identified as the Raman band. In symmetric case the CCD is found to be immaterial, whereas in asymmetric case the CCD is found to be crucial and brings new spectral bands. A critical value of CCD is predicted where the dynamics evolve dramatically in different manners. Thus in this paper, the MI dynamics of a dual core optical fiber is fully explored and the interplay between CCD and relaxation is clearly highlighted.

Modulational instability in a twin-core fiber with the effect of saturable nonlinear response and coupling coefficient dispersion

Modulational instability (MI) in a twin-core optical fiber with the combined effect of saturation of nonlinearity (SNL) and coupling coefficient dispersion (CCD) is presented. The CCD does not dramatically modify the spectrum of the symmetric or antisymmetric continuous wave (CW) case, but the SNL on the other hand behaves in a perceptible manner such that the gain and the unstable region inherently decrease. In the anomalous dispersion region, the gain of the instability spectrum increases (decreases) monotonously with power (coupling coefficient). The effective nonlinearity and the power threshold for the sustained CW solution becomes a function of power for the SNL case. The so-called nonlinear factor behaves in a unique way such that there exist two powers for the same value of nonlinear factor. The interplay between CCD and SNL is emphasized, where any nonzero value of CCD leads to new instability bands and the saturation on the other hand suppresses the gain of the instability band. We identified a pair of power corresponding to the constant value of the nonlinear factor, where the system becomes invariant, such that the number of instability bands, the gain, and the range of the unstable region are all preserved. In the case of the normal dispersion case, the MI is achieved purely by means of the coupling coefficient. In both cases, a critical CCD is predicted, where the system evolves dramatically in a different manner.

Phase-Sensitive Sum-Frequency Spectroscopy

Phase-sensitive sum-frequency spectroscopy (SFS) allows the complete measurement of the complex spectra of surface nonlinear response coefficients. Similar to linear spectroscopy, the spectrum of the imaginary part of a surface response coefficient directly characterizes surface resonances without complication. This newly developed technique has greatly enhanced the capability of surface SFS and provides many new research opportunities for surface science. This article describes the experimental schemes and underlying theory for the technique and briefly reviews works that have clearly demonstrated its power.

Nonlinearities in the harmonic spectrum of heavy ion collisions with ideal and viscous hydrodynamics

We determine the non-linear hydrodynamic response to geometrical fluctuations in heavy ion collisions using ideal and viscous hydrodynamics. This response is characterized with a set of non-linear response coefficients that determine, for example, the $v_5$ that is produced by an $\epsilon_2$ and an $\epsilon_3$. We analyze how viscosity damps both the linear and non-linear response coefficients, and provide an analytical estimate that qualitatively explains most of the trends observed in more complete simulations. Subsequently, we use these nonlinear response coefficients to determine the linear and non-linear contributions to $v_1$, $v_4$ and $v_5$. For viscous hydrodynamics the nonlinear contribution is dominant for $v_4$, $v_5$ and higher harmonics. For $v_1$, the nonlinear response constitutes an important $\sim 25%$ correction in mid-central collisions. The nonlinear response is also analyzed as a function of transverse momentum for $v_1$, $v_4$ and $v_5$. Finally, recent measurements of correlations between event-planes of different harmonic orders are discussed in the context of non-linear response.

Symmetries in fluctuations far from equilibrium

Fluctuations arise universally in nature as a reflection of the discrete microscopic world at the macroscopic level. Despite their apparent noisy origin, fluctuations encode fundamental aspects of the physics of the system at hand, crucial to understand irreversibility and nonequilibrium behavior. To sustain a given fluctuation, a system traverses a precise optimal path in phase space. Here we show that by demanding invariance of optimal paths under symmetry transformations, new and general fluctuation relations valid arbitrarily far from equilibrium are unveiled. This opens an unexplored route toward a deeper understanding of nonequilibrium physics by bringing symmetry principles to the realm of fluctuations. We illustrate this concept studying symmetries of the current distribution out of equilibrium. In particular we derive an isometric fluctuation relation that links in a strikingly simple manner the probabilities of any pair of isometric current fluctuations. This relation, which results from the time-reversibility of the dynamics, includes as a particular instance the Gallavotti-Cohen fluctuation theorem in this context but adds a completely new perspective on the high level of symmetry imposed by time-reversibility on the statistics of nonequilibrium fluctuations. The new symmetry implies remarkable hierarchies of equations for the current cumulants and the nonlinear response coefficients, going far beyond Onsager's reciprocity relations and Green-Kubo formulas. We confirm the validity of the new symmetry relation in extensive numerical simulations, and suggest that the idea of symmetry in fluctuations as invariance of optimal paths has far-reaching consequences in diverse fields.

Nonlinear transport effects in mass separation by effusion

Generalizations of Onsager reciprocity relations are established for the nonlinearresponse coefficients of ballistic transport in the effusion of gaseous mixtures.These generalizations, which have been established on the basis of the fluctuationtheorem for the currents, are here considered for mass separation by effusion. Inthis kinetic process, the mean values of the currents depend nonlinearly on theaffinities or thermodynamic forces controlling the nonequilibrium constraints. Thesenonlinear transport effects are shown to play an important role in the process of massseparation. In particular, the entropy efficiency turns out to be significantly largerthan would be the case if the currents were supposed to depend linearly on theaffinities.

Strong Surface Contribution to the Nonlinear Meissner Effect ind-Wave Superconductors

We demonstrate that in a d-wave superconductor the bulk nonlinear Meissner effect is dominated by a surface effect due to Andreev bound states at low temperatures. The contribution of this surface effect to the nonlinear response coefficient follows a 1/T3 law with the opposite sign compared to the bulk 1/T behavior. The crossover from bulk dominated behavior to surface dominated behavior occurs at a temperature of T/Tc∼1/square root(κ). We present an approximate analytical calculation, which supports our numerical calculations and provides a qualitative understanding of the effect. The effect can be probed by intermodulation distortion experiments.

The fluctuation theorem for currents in open quantum systems

A quantum-mechanical framework is set up to describe the full counting statistics of particles flowing between reservoirs in an open system under time-dependent driving. A symmetry relation is obtained, which is the consequence of microreversibility for the probability of the nonequilibrium work and the transfer of particles and energy between the reservoirs. In some appropriate long-time limit, the symmetry relation leads to a steady-state quantum fluctuation theorem for the currents between the reservoirs. On this basis, relationships are deduced which extend the Onsager–Casimir reciprocity relations to the nonlinear response coefficients.

Temporal disorder and fluctuation theorem in chemical reactions

We report the analytical study of a class of chemical reactions described as birth-and-death stochastic processes ruled by a master equation compatible with the mass action law of chemical kinetics. We solve analytically this master equation to find the generating functions of the fluctuating fluxes and of the Lebowitz-Spohn action functional. These generating functions are explicitly shown to obey fluctuation theorems. In the case of fluxes, we derive relations for the nonlinear response coefficients, extending Onsager's reciprocity relations. Moreover, symmetry relations reminiscent of the fluctuation theorem are obtained for the finite-time probability distributions of the fluxes. The temporal disorder of the stochastic process is also characterized and related to the thermodynamic entropy production.

A fluctuation theorem for currents and non-linear response coefficients

We use a recently proved fluctuation theorem for currents to develop a response theory ofnon-equilibrium phenomena. In this framework, expressions for the response coefficients ofthe currents at arbitrary orders in the thermodynamic forces or affinities are obtained interms of the fluctuations of the cumulative currents and remarkable relations are obtainedwhich are the consequences of microreversibility beyond Onsager reciprocity relations.

Fluctuation theorem for transport in mesoscopic systems

The fluctuation theorem for currents is applied to several mesoscopic systemson the basis of Schnakenberg’s network theory, which allows one to verify itsconditions of validity. A graph is associated with the master equation ruling therandom process and its cycles can be used to obtain the thermodynamic forces oraffinities corresponding to the nonequilibrium constraints. This provides a method ofdefining the independent currents crossing the system in nonequilibrium steadystates and to formulate the fluctuation theorem for the currents. This result isapplied to out-of-equilibrium diffusion in a chain, to a biophysical model of ionchannels in a membrane, and to electronic transport in mesoscopic circuits madeof several tunnel junctions. In this latter, we show that the generalizations ofOnsager’s reciprocity relations to the nonlinear response coefficients also hold.

Hyperpolarizabilities of molecular chains: A real-space approach

First principles calculations can provide important insight into the mechanisms of non-linear molecular response. Chain molecules are of particular interest due to their large and directional response, but calculating their non-linear response coefficients is a computational challenge. We have developed a real-space approach to evaluate the linear polarizability and first and second hyperpolarizability of molecular chains within Kohn–Sham density functional theory. Different approaches to minimize uncertainties in particular in the second hyperpolarizability γ are presented. We tested our scheme by calculating the response of model hydrogen chains using local and non-local density functionals. Differences between explicit density functionals and orbital functionals are discussed.

Calculation of the formation time for microwave magnetic envelope solitons

The theoretical formation and propagation properties of microwave magnetic envelope solitons were modeled from the nonlinear Schrodinger equation without damping. Based on soliton threshold input amplitude criteria from inverse scattering theory, input amplitudes were set midpoint between the one- and two-soliton threshold values. The soliton formation time T/sub s/ was defined as the time required for the pulse amplitude to stabilize after launch. This T/sub s/ was found to be independent of the nonlinear response coefficient N and to vary as the inverse of the dispersion coefficient D. One possible way to model these T/sub s/ dependencies is through the relation T/sub s//spl ap/[T/sub d/ T/sub nl/]/sup 1/2/, where T/sub d/ and T/sub nl/ are characteristic dispersion and nonlinear response times, respectively. This result suggests a simple geometric mean relation between these characteristic times.

Erratum: "Density-functional approach to nonlinear-response coefficients of solids"

Erratum: "Density-functional approach to nonlinear-response coefficients of solids"

Density-functional approach to nonlinear-response coefficients of solids.

We propose a general scheme, within the density-functional theory, for an accurate computation of a large class of nonlinear-response coefficients of solids. The scheme is applicable to all kinds of adiabatic perturbations of the crystal ground state, such as the application of mechanical strains, static electric fields of any wavelength, or individual atomic displacements, and allows the study of coefficients which describe anharmonicities or coupling of simultaneously applied disturbances. Total-energy changes as a function of the perturbation wavelength, as needed to obtain phonon group velocities, can also be considered. The formalism, which avoids the use of supercells and large-matrix inversions, contains as a special case the treatment of linear-response coefficients recently suggested by Baroni, Giannozi, and Testa. The central ingredient in our scheme is an efficient use of the ``2n+1'' theorem of the perturbation theory, which allows us to obtain the third-order derivatives of the total energy by using only byproducts of a first-order perturbation calculation.

Quantum theory of radiation-pressure induced electrical currents: I. Free electrons

The radiation-pressure-induced direct electrical current in an electronic system, arises from the transfer of momentum of photons to the electrons. For free electrons and within classical mechanics, this effect originates from the action of the Lorentz-force. This paper reports the first quantum theory of this effect, where the number conserving relaxation time approximation for the quantum Liouville equation has been used for the calculation of a nonlinear response coefficient.

Electric-Field-Induced Raman Scattering in SrTiO3and KTaO3

In their paraelectric phases, SrTi${\mathrm{O}}_{3}$ and KTa${\mathrm{O}}_{3}$ crystallize in the cubic perovskite structure (space group ${{O}_{h}}^{1}$) in which the first-order Raman effect is forbidden. We discuss here experiments in which first order Raman scattering has been induced by the application of an external electric field that serves to remove the center of inversion symmetry of the crystals. An electric field applied along a $〈001〉$ direction induces a ${C}_{4v}$ symmetry in the crystal and renders all the phonons first-order Raman-active. We have studied induced scattering from all four TO phonons in SrTi${\mathrm{O}}_{3}$ and from three TO phonons in KTa${\mathrm{O}}_{3}$ at temperatures between 8 and 300\ifmmode^\circ\else\textdegree\fi{}K and with electric fields between 0.2 to 15 kV/cm. Most attention was given to the lowest-frequency TO phonon---the "soft" or "ferroelectric" mode---whose striking decrease in frequency as the temperature is lowered signals the lattice instability associated with the ferroelectric phase transition. Detailed investigations were also made of the electric field dependence of the soft-mode frequencies in both materials. At 8\ifmmode^\circ\else\textdegree\fi{}K this frequency in SrTi${\mathrm{O}}_{3}$ increases from 10 ${\mathrm{cm}}^{\ensuremath{-}1}$ at zero field to 45 ${\mathrm{cm}}^{\ensuremath{-}1}$ at 12 kV/cm. Similar behavior is observed in KTa${\mathrm{O}}_{3}$. From this behavior we infer values for the nonlinear dielectric response coefficients of the crystals. Some discussion is given of the large discrepancies between the soft-mode linewidths that we observe and those previously obtained from infrared (IR) reflectivity experiments. At 80\ifmmode^\circ\else\textdegree\fi{}K in SrTi${\mathrm{O}}_{3}$ the Raman and IR measurements yield values of 3 and 61.7 ${\mathrm{cm}}^{\ensuremath{-}1}$, respectively. Additional topics considered include relative electric-field-induced scattering cross sections for the various phonon modes, and the temperature dependence of the soft-mode linewidth. Finally, the question of anomalous first-order Raman scattering in the intrinsic spectrum of SrTi${\mathrm{O}}_{3}$, recently reported by several authors, is considered in the light of the field-induced first-order Raman scattering.