Classical Nonlinear Response Research Articles

Nonclassical nonlinear elasticity of crystalline structures.

Hysteretic elastic nonlinearity has been shown to result in a dynamic nonlinear response which deviates from the known classical nonlinear response; hence this phenomenon was termed nonclassical nonlinearity. Metallic structures, which typically exhibit weak nonlinearity, are typically categorized as classical nonlinear materials. This article presents a material model which derives stress amplitude dependent nonlinearity and damping from the mesoscale dislocation pinning and breakaway to show that the lattice defects in crystalline structures can give rise to nonclassical nonlinearity. The dynamic nonlinearity arising from dislocations was evaluated using resonant frequency shift and higher order harmonic scaling. The results show that the model can capture the nonlinear dynamic response across the three stress ranges: linear, classical nonlinear, and nonclassical nonlinear. Additionally, the model also predicts that the amplitude dependent damping can introduce a softening-hardening nonlinear response. The present model can be generalized to accommodate a wide range of lattice defects to further explain nonclassical nonlinearity of crystalline structures.

Classical coherent two-dimensional vibrational spectroscopy.

Two-dimensional (2D) ultrafast spectroscopy is a powerful tool for studying the electronic and vibrational structures of complex systems. Unfortunately, the physical interpretation of these experiments is obscured by conceptual problems in classical response theory, i.e., the divergence of classical nonlinear response functions. We demonstrate that these difficulties are avoided by modeling classical 2D experiments nonperturbatively, illustrating that nonlinear spectroscopy and nonlinear response are not synonymous. Numerical simulations allow a direct comparison between classical and quantum 2D spectra for simple, weakly anharmonic systems relevant to vibrational spectroscopy. We find that nonperturbative classical theory-although differing in quantitative details-accurately captures the key qualitative features of the quantum 2D spectrum, including the separation of the signal into wavevector-selected pathways, formation of cross peaks between coupled vibrational modes, and coherent beating in the signal as a function of waiting time (so-called "quantum beats"). These results are discussed in terms of a simple analytical model which captures the key physical features of classical 2D spectroscopy and provides a link between classical and quantum descriptions. One interesting conclusion from this comparison is that the "coherence" observed in ultrafast spectroscopy may (at least in vibrational experiments) be understood as a purely classical phenomenon, without reference to quantum mechanics.

Classical nonlinear response of a chaotic system. II. Langevin dynamics and spectral decomposition

The spectrum of a strongly chaotic system consists of discrete complex Ruelle-Pollicott (RP) resonances. We interpret the RP resonances as eigenstates and eigenvalues of the Fokker-Planck operator obtained by adding an infinitesimal diffusion term to the first-order Liouville operator. We demonstrate how the deterministic expression for the linear response is reproduced in the limit of vanishing noise. For the second-order response function we establish an equivalence of the spectral decomposition in the limit of vanishing noise and the long-time asymptotic expansion in the deterministic case.

Classical nonlinear response of a chaotic system. I. Collective resonances

We develop a general semiquantitative picture of nonlinear classical response in strongly chaotic systems. In contrast to behavior in integrable or almost integrable systems, the nonlinear classical response in chaotic systems vanishes at long times. The exponential decay of the response functions in the case of strong chaos is attributed to both exponentially decaying and growing elements in the stability matrices. We calculate the linear and second-order response in one of the simplest chaotic systems: free classical motion on a compact surface of constant negative curvature. The response reveals certain features of collective resonances which do not correspond to any periodic classical trajectories. We demonstrate the relevance of the model for the interpretation of spectroscopic experiments.

Collective oscillations in the classical nonlinear response of a chaotic system

We establish a general semiquantitative phase-space picture of the classical nonlinear response in a strongly chaotic system. As opposed to the case of stable dynamics, the response functions decay exponentially at long times. Damped oscillations in response functions are attributed to collective resonances which do not correspond to any periodic classical motions. We calculate analytically the second-order response in a simple chaotic system and demonstrate the relevance of the concept for the interpretation of spectroscopic data.

Classical Divergence of Nonlinear Response Functions

The time divergence of classical nonlinear response functions reveals the fundamental difficulty of dynamic perturbation based on classical mechanics. The nature of the divergence is established for systems in regular motions using asymptotic decomposition of Fourier integrals. The asymptotic analysis shows that the divergence cannot be removed by phase-space averaging such as the Boltzmann distribution function. The implications of this study are discussed in the context of the conceptual development of quantum-classical correspondence in dynamic response.

Simulation strategies and signatures of chaos in classical nonlinear response.

Algorithms are presented for overcoming the computational challenge of nonlinear response functions which describe the response of a classical system to a sequence of n pulses and depend on nth order multipoint stability matrices containing signatures of chaos. Simulations for the Lorentz gas demonstrate that finite field algorithms can be effectively used for the robust, long time calculation of nonlinear response functions. These offer the possibility to characterize chaos beyond the commonly used Lyapunov exponents and suggest new experimentally accessible measures of chaos.

Generalized Langevin equation approach to higher-order classical response: second-order-response time-resolved Raman experiment in CS2.

A simple, systematic generalized Langevin equation approach for calculating classical nonlinear response functions is formulated and discussed. The two-time Poisson brackets appearing at second and higher order are rendered tractable by a physically motivated approximation. The method is used to calculate the fifth order (second order response) Raman response of liquid CS2. Agreement with simulation is good, but the simplicity of the theoretical expression suggests that the path to obtaining qualitatively new information about liquids with the fifth order experiment is uncertain. Further applications of the basic approach are suggested.

Second-order classical nonlinear response theory

Second-order classical nonlinear response theory