Linear response theory diffusion

The observation that the rate constant may be expressed in terms of an auto-time-correlation function of the flux, averaged over an equilibrium ensemble, has a parallel in statistical mechanics. There it is shown, within the frame of linear response theory, that any transport coefficients, like diffusion constants, viscosities, conductivities, etc., may also be expressed in terms of auto-time-correlation functions of proper chosen quantities, averaged over an equilibrium ensemble. 

Let us now come back to the specific problem of the diffusion of a particle in an out-of-equilibrium environment. In a quasi-stationary regime, the particle velocity obeys the generalized Langevin equation (22). The generalized susceptibilities of interest are the particle mobility p(co) = Xvxi03) and the generalized friction coefficient y(co) = — (l/mm)x ( ) [the latter formula deriving from the relation (170) between y(f) and Xj> (f))- The results of linear response theory as applied to the particle velocity, namely the Kubo formula (156) and the Einstein relation (159), are not valid out-of-equilibrium. The same... 

As a step toward the study of thermodynamic equilibrium in the case of anomalous statistical physics, in Section VII we study how the generators of anomalous diffusion respond to external perturbation. The ordinary linear response theory is violated and, in some conditions, is replaced by a different kind of linear response. In Section VIII we review the results of an ambitious attempt at deriving thermodynamics from dynamics for the main purpose of exploring a dynamic approach to the still unsettled issue of the thermodynamics of Levy statistics. The Levy walk perspective seems to be the only possible way to establish a satisfactory connection between dynamics and thermodynamics in... 

In the present section, it is demonstrated how the linear response of an assembly of noninteracting polar Brownian particles to a small external field F applied parallel and perpendicular to the bias field Fo may be calculated in the context of the fractional noninertial rotational diffusion in the same manner as normal rotational diffusion [8]. In order to carry out the calculation, it is assumed that the rotational Brownian motion of a particle may be described by a fractional noninertial Fokker-Planck (Smoluchowski) equation, in which the inertial effects are neglected. Both exact and approximate solutions of this equation are presented. We shall demonstrate that the characteristic times of the normal diffusion process, namely, the integral and effective relaxation times obtained in Refs. 8, 65, and 67, allow one to evaluate the dielectric response for anomalous diffusion. Moreover, these characteristic times yield a simple analytical equation for the complex dielectric susceptibility tensor describing the anomalous relaxation of the system. The exact solution of the problem reduces to the solution of the infinite hierarchies of differential-recurrence equations for the corresponding relaxation functions. The longitudinal and transverse components of the susceptibility tensor may be calculated exactly from the Laplace transform of these relaxation functions using linear response theory [72]. 

The basic concepts of linear response theory are best illustrated by considering the rotational diffusion model of an assembly of electric dipoles constrained to rotate in two dimensions due to Debye [14] which is governed by the Smoluchowski equation... 

We compute effective thermal transport coefficients for proteins using linear response theory and beginning in the harmonic approximation, with anharmonic contributions included as a correction. The correction can in fact be rather large, as we compute anharmonicity to nearly double the magnitude of the thermal conductivity and thermal diffusivity of myoglobin. We expect that anharmonicity will generally enhance thermal transport in proteins, in contrast, for example, to crystals, where anharmonicity leads to thermal resistance, since most of the harmonic modes of the protein are spatially localized and transport heat only inefficiently. 

Several transport properties can be evaluated from equilibrium simulations with use of linear response theory, which relates correlation fimctions of spontaneously fluctuating molecular properties to phenomenological transport coefficients. These relations can be used to evaluate diffusion coefficients, thermal conductivities, viscosities, IR spectra, and so on. However, most of these properties are evaluated more directly using appropriately devised techniques of nonequilibrium molecular dynamics. Particularly challenging for polymers is the direct... 

One of the most important principles of linear response theory relates the system s response to an externally imposed perturbation, which causes it to depart from equilibrium, to its equilibrium fluctuations. Indeed, the system response to a small perturbation should not depend on whether this perturbation is a result of some external force, or whether it is just a random thermal fluctuation. Spontaneous concentration fluctuations, for instance, occur all the time in equilibrium systems at finite temperatures. If the concentration c at some point of a liquid at time zero is (c) + 3c(r, t), where (c) is an average concentration, concentration values at time t + 8t dXt and other points in its vicinity will be affected by this. The relaxation of the spontaneous concentration fluctuation is governed by the same diffusion equation that describes the evolution of concentration in response to the external imposition of a compositional heterogeneity. The relationship between kinetic coefficients and correlations of the fluctuations is derived in the framework of linear response theory. In general, a kinetic coefficient is related to the integral of the time correlation function of some relevant microscopic quantity. 

Zwanzig (65) considered a lattice model of dipoles reorienting by rotational diffusion and coupled by electrostatic dipole-dipole interaction energy. He was able to obtain results by a high temperature expansion of the Boltzmann factor in the linear response theory correlation... 

The challenge created by the fluctuation-dissipation theorem and linear response theory is that the modes of motion and mechanisms of dissipation that are driven by the weakest linear perturbations are required to be the same as the modes of motion and mechanisms of dissipation that appear in the corresponding diffusive process(24). If teuthidic motion represents the linear translational response to a weak applied field, then the fluctuation-dissipation theorem and linear response theory guarantee that teuthidic motion must also characterize diffusion of star polymers through a polymer solution matrix. 

The photomicrographic measurements refer directly to polymer motion under the influence of an external force. However, measurements of migration velocity v as a function of applied electrical field E show that some of these electrophoretic measurements were made in a low-field linear regime, in which the electrophoretic mobility jx is independent of E. Linear response theory and the fluctuation-dissipation theorem are then applicable they provide that the modes of motion used by a polymer undergoing electrophoresis in the linear regime, and the modes of motion used by the same polymer as it diffuses, must be the same. This requirement on the equality of drag coefficients for driven and diffusive motion was first seen in Einstein s derivation of the Stokes-Einstein equation(16), namely thermal equilibrium requires that the drag coefficients / that determine the sedimentation rate v = mg/f and the diffusion coefficient D = kBT/f must be the same. 

The calculation of the thermal conductivity of gas hydrate using EMD and the Green-Kubo linear response theory was repeated recently. In that work, convergences of the relevant quantities were monitored carefully as a function of the model size. Subtleties in the numerical procedures were also carefully considered. The thermal conductivity of methane hydrate was found to converge within numerical accuracy for 3 x 3 x 3 and 4x4x4 supercells. In the calculation of the heat flux vector there is an interactive term that is a pairwise summation over the forces exerted by atomic sites on one another. The species (i.e., water and methane) enthalpy correction term requires that the total enthalpy of the system is decomposed into contributions from each species. Because of the partial transformation from pairwise, real-space treatment to a reciprocal space form in Ewald electrostatics, it is necessary to recast the diffusive and interactive terms in this expression in a form amenable for use with the Ewald method using the formulation of Petravic. ... 

This diffusion constant T> q) can be calculated within the framework of linear response theory by using a Kubo formula ... 

The three-pulse experiments contain more information than two-pulse methods when the direction and timing of all three pulses is controlled. We have seen that this additional information cannot be interpreted within a Bloch picture. We will therefore outline in the following a more detailed theory, which includes spectral diffusion and which simultaneously explains the linear response (absorption spectrum) and the nonlinear response (four wave mixing, photon echo, transient grating, pump-probe) of vibrational transitions. 

This is very important, because the interface is stable in the presence of the diffusion-limited current, that is, the maximum current available, flowing across the interface. In other words, the flow of current or ions across the interface is not directly responsible for the interfacial turbulences, which in fact makes the strong contrast of the electrochemical instability with the instability associated with interfacial chemical reactions of the type treated within the framework of the linear stability theory [9,10,30]. 

We turn first to computation of thermal transport coefficients, which provides a description of heat flow in the linear response regime. We compute the coefficient of thermal conductivity, from which we obtain the thermal diffusivity that appears in Fourier s heat law. Starting with the kinetic theory of gases, the main focus of the computation of the thermal conductivity is the frequency-dependent energy diffusion coefficient, or mode diffusivity. In previous woik, we computed this quantity by propagating wave packets filtered to contain only vibrational modes around a particular mode frequency [26]. This approach has the advantage that one can place the wave packets in a particular region of interest, for instance the core of the protein to avoid surface effects. Another approach, which we apply in this chapter, is via the heat current operator [27], and this method is detailed in Section 11.2. 

Following the developments outlined in [8,9], we now stress the fact that anomalous diffusion in the scahng form of Eqs. 119 and 120 is closely connected to descriptions based on fractional derivatives, given that they allow us to invert, in a simple way, the integral expressions which follow from the theory of Hnear response, when the anomalous behavior has a power-law character going as Eq. 119, with y < 1. For technical reasons and because of an intimate relation to linear response we prefer, as in [8,9], to extend the lower integration Emit in Eq. 121 to - oo in this way we obtain the Weyl-form. 

The field and frequency dependence of the drift velocity Vd(P> n) in a 1-d system has been analysed in great detail in references [5,9] using mean field theory and scaling theory. The disorder was described by allowing a subset of the diffusion or jump rates W to differ and follow a distribution function p(W). We shall not repeat all the details here but only recall the main conclusions [9] the linear response relationship between and the diffusivity D in the low field limit n l given by... 

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