Standard Response Theory Research Articles

Generalisation of fluctuation-dissipation theorem to systems with absorbing states

Systems that evolve towards a state from which they cannot depart are common in nature. But the fluctuation-dissipation theorem (FDT), a fundamental result in statistical mechanics, is mainly restricted to systems near-stationarity. In processes with absorbing states, the total probability decays with time, eventually reaching zero and rendering the predictions from the standard response theory invalid. In this article, we investigate how such processes respond to external perturbations and develop a new theory that extends the framework of the FDT. We apply our theory to two paradigmatic examples that span vastly different fields—a birth–death process in forest ecosystems and a targeted search on DNA by proteins. These systems can be affected by perturbations which increase their rate of extinction/absorption, even though the average or the variance of population sizes are left unmodified. These effects, which are not captured by the standard response theory, are exactly predicted by our framework. Our theoretical approach is general and applicable to any system with absorbing states. It can unveil important features of the path to extinction masked by standard approaches.

Nonlinear and divergent responses of fluctuation-driven systems

Most network dynamical systems are out of equilibrium and externally driven by fluctuations. Linear response theory generically characterizes systems responses to such fluctuations for small driving amplitudes yet cannot capture response properties that are either due to strong driving or intrinsically nonlinear. For oscillation-driven systems, we here report average response offsets that scale quadratically with asymptotically small amplitudes. At some critical driving amplitude, responses cease to stay close to a given operating point and may diverge. Standard response theory fails to predict these amplitudes even at arbitrarily high orders. We propose an integral self-consistency condition that captures the full nonlinear system dynamics. We illustrate our approach for a minimal one-dimensional model and capture the nonlinear shift of voltages in the phase, frequency and voltage dynamics of AC power grid networks. Our approach may help to quantitatively predict intrinsically nonlinear response dynamics as well as bifurcations emerging at large driving amplitudes in non-autonomous dynamical systems.

Dissipation Function: Nonequilibrium Physics and Dynamical Systems.

An exact response theory has recently been developed within the field of Nonequilibrium Molecular Dynamics. Its main ingredient is known as the Dissipation Function, . This quantity determines nonequilbrium properties like thermodynamic potentials do with equilibrium states. In particular, can be used to determine the exact response of particle systems obeying classical mechanical laws, subjected to perturbations of arbitrary size. Under certain conditions, it can also be used to express the response of a single system, in contrast to the standard response theory, which concerns ensembles of identical systems. The dimensions of are those of a rate, hence can be associated with the entropy production rate, provided local thermodynamic equilibrium holds. When this is not the case for a particle system, or generic dynamical systems are considered, can equally be defined, and it yields formal, thermodynamic-like, relations. While such relations may have no physical content, they may still constitute interesting characterizations of the relevant dynamics. Moreover, such a formal approach turns physically relevant, because it allows a deeper analysis of and of response theory than possible in case of fully fledged physical models. Here, we investigate the relation between linear and exact response, pointing out conditions for the validity of the response theory, as well as difficulties and opportunities for the physical interpretation of certain formal results.

K- and L-edge X-ray absorption spectrum calculations of closed-shell carbon, silicon, germanium, and sulfur compounds using damped four-component density functional response theory.

X-ray absorption spectra of carbon, silicon, germanium, and sulfur compounds have been investigated by means of damped four-component density functional response theory. It is demonstrated that a reliable description of relativistic effects is obtained at both K- and L-edges. Notably, an excellent agreement with experimental results is obtained for L2,3-spectra-with spin-orbit effects well accounted for-also in cases when the experimental intensity ratio deviates from the statistical one of 2 : 1. The theoretical results are consistent with calculations using standard response theory as well as recently reported real-time propagation methods in time-dependent density functional theory, and the virtues of different approaches are discussed. As compared to silane and silicon tetrachloride, an anomalous error in the absolute energy is reported for the L2,3-spectrum of silicon tetrafluoride, amounting to an additional spectral shift of ∼1 eV. This anomaly is also observed for other exchange-correlation functionals, but it is seen neither at other silicon edges nor at the carbon K-edge of fluorine derivatives of ethene. Considering the series of molecules SiH4-XFX with X = 1, 2, 3, 4, a gradual divergence from interpolated experimental ionization potentials is observed at the level of Kohn-Sham density functional theory (DFT), and to a smaller extent with the use of Hartree-Fock. This anomalous error is thus attributed partly to difficulties in correctly emulating the electronic structure effects imposed by the very electronegative fluorines, and partly due to inconsistencies in the spurious electron self-repulsion in DFT. Substitution with one, or possibly two, fluorine atoms is estimated to yield small enough errors to allow for reliable interpretations and predictions of L2,3-spectra of more complex and extended silicon-based systems.

First-order nonadiabatic coupling matrix elements between excited states: a Lagrangian formulation at the CIS, RPA, TD-HF, and TD-DFT levels.

Analytic expressions for the first-order nonadiabatic coupling matrix elements between electronically excited states are first formulated exactly via both time-independent equation of motion and time-dependent response theory, and are then approximated at the configuration interaction singles, particle-hole/particle-particle random phase approximation, and time-dependent density functional theory/Hartree-Fock levels of theory. Note that, to get the Pulay terms arising from the derivatives of basis functions, the standard response theory designed for electronic perturbations has to be extended to nuclear derivatives. The results are further recast into a Lagrangian form that is similar to that for excited-state energy gradients and allows to use atomic orbital based direct algorithms for large molecules.

A computable evolution equation for the joint response-excitation probability density function of stochastic dynamical systems

By using functional integral methods, we determine a computable evolution equation for the joint response-excitation probability density function of a stochastic dynamical system driven by coloured noise. This equation can be represented in terms of a superimposition of differential constraints, i.e. partial differential equations involving unusual limit partial derivatives, the first one of which was originally proposed by Sapsis & Athanassoulis. A connection with the classical response approach is established in the general case of random noise with arbitrary correlation time, yielding a fully consistent new theory for non-Markovian systems. We also address the question of computability of the joint response-excitation probability density function as a solution to a boundary value problem involving only one differential constraint. By means of a simple analytical example, it is shown that, in general, such a problem is undetermined, in the sense that it admits an infinite number of solutions. This issue can be overcome by completing the system with additional relations yielding a closure problem, which is similar to the one arising in the standard response theory. Numerical verification of the equations for the joint response-excitation density is obtained for a tumour cell growth model under immune response.

Quasienergy formulation of damped response theory

We present a quasienergy-based formulation of damped response theory where a common effective lifetime parameter has been introduced for all excited states in terms of complex excitation energies. The introduction of finite excited state lifetimes leads to a set of (complex) damped response equations, which have the same form to all orders in the perturbation. An algorithm is presented for solving the damped response equations in Hartree-Fock theory and Kohn-Sham density functional theory. The use of the quasienergy formulation allows us to obtain directly the computationally simplest expressions for damped response functions by applying a set of response parameter elimination rules, which minimize the total number of damped response equations to be solved. In standard response theory broadened absorption spectra are obtained by ad hoc superimposing lineshape functions onto the absorption stick spectra, whereas an empirical lineshape function common to all excitations is an integrated part of damped response theory. By superimposing the lineshape functions inherent in damped response theory onto the stick spectra of standard response theory, we show that the absorption spectra obtained in standard and damped response theory calculations are identical. We demonstrate that damped response theory may be applied to obtain absorption spectra in all frequency ranges, also those that are not readily addressed using standard response theory. This makes damped response theory an effective tool, e.g., for determining absorption spectra for large molecules, where the density of the excited states may be very high, and where standard response theory therefore is not applicable in practice. A thorough comparison is given between our formulation of damped response theory and the formulation by Norman et al. [J. Chem. Phys. 123, 194103 (2005)].

Approach to the non-equilibrium time-periodic state in a ‘steady’ shear flow model

The standard non-equilibrium molecular dynamics algorithm for steady shear flow (SLLOD) employs Lees—Edwards periodic boundary conditions. It is not widely known that these boundary conditions make the system non-autonomous. The ‘steady state’ shear stress is in fact time periodic. The standard response theory derivations for steady shear do not take proper account of these non-autonomous terms. In this paper we correct this deficiency. We show that these non-autonomous terms invalidate the Green-Kubo relation for the finite frequency shear viscosity of fluids under Lees-Edwards periodic boundary conditions.