Linear response theory particles

The idea of the magnetic stochastic resonance has emerged in a natural way first as a theoretical issue [25-27] and shortly afterward was supported by some experimental evidence [28,31]. A consistent theoretical treatment of magnetic SR in a superparamagnetic particle in the framework of the linear response theory was developed in Refs. 29 and 30. 

Linear response theory, applied to the particle velocity, considered as a dynamic variable of the isolated particle-plus-bath system, allows to express the mobility in terms of the equilibrium velocity correlation function. Since the mobility p(co) is simply the generalized susceptibility %vx(o ), one has the Kubo formula... 

Equations (161) and (162) are two equivalent formulations of the second FDT [30,31]. The Kubo formula (162) for the generalized friction coefficient can also be established directly by applying linear response theory to the force exerted by the bath on the particle, this force being considered as a dynamical variable of the isolated particle-plus-bath system. We will come back to this point in Section VI.B. 

An equation of motion is derived in Section VI.B for the case of a bilinear coupling between the particle and its non equilibrated environment. As for the out-of-equilibrium extension of the linear response theory, it will be discussed in Section VI.C. 

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... 

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 familiar shear modulus of linear response theory describes thermodynamic stress fluctuations in equilibrium, and is obtained from (5b, lid) by setting y = 0 [1, 3, 57], While (5b) then gives the exact Green-Kubo relation, the approximation (lid) turns into the well-studied MCT formula (see (17)). For finite shear rates, (lid) describes how afflne particle motion causes stress fluctuations to explore shorter and shorter length scales. There the effective forces, as measured by the gradient of the direct correlation function, = nc = ndck/dk, become smaller, and vanish asympotically, 0 the direct correlation function is connected... 

However, these corrections have been shown to be less important when the charged particle moves outside the solid [56]. Hence, in the case of a bounded three-dimensional electron gas we restrict the calculations to linear-response theory. Assuming translational invariance in two directions, which we take to be normal to the z-axis, to first order in the external perturbation (linear-response theory) the energy loss of equation (6) may be expressed as follows [57]... 

If carrier-carrier interactions are not disregarded we cannot obtain the above transport coefficients from the single particle response (11.63). Linear response theory should now be used for collective variables. Starting from Eqs (11.58) and (11.59) we seek the response in the current density... 

The appendices contain an account of those parts of the theory of Brownian motion and linear response theory which are essential for the reader in order to achieve an understanding of relaxational phenomena in magnetic domains and in ferrofluid particles. The analogy with dielectric relaxation is emphasized throughout these appendices. Appendix D contains the rigorous derivation of Brown s equation. 

---