Macroscopic fluctuations, linear response

Using the fluctuation-dissipation theorem [361, which relates microscopic fluctuations at equilibrium to macroscopic behaviour in the limit of linear responses, the time-dependent shear modulus can be evaluated [371 ... 

For systems close to equilibrium the non-equilibrium behaviour of macroscopic systems is described by linear response theory, which is based on the fluctuation-dissipation theorem. This theorem defines a relationship between rates of relaxation and absorption and the correlation of fluctuations that occur spontaneously at different times in equilibrium systems. 

The world surrounding us is mostly out of equihbrium, equilibrium being just an idealization that requires specific conditions to be met in the laboratory. Even today we do not have a general theory about nonequilibrium macroscopic systems as we have for equilibrium ones. Onsager theory is probably the most successful attempt, albeit its domain of validity is restricted to the linear response regime. In small systems the situation seems to be the opposite. Over the past years, a set of theoretical results that go under the name of fluctuation theorems have been unveiled. These theorems make specific predictions about energy processes in small systems that can be scrutinized in the laboratory. 

The extraction of a homogeneous process from a stationary Markov process is a familiar procedure in the theory of linear response. As an example take a sample of a paramagnetic material placed in a constant external magnetic field B. The magnetization Y in the direction of the field is a stationary stochastic process with a macroscopic average value and small fluctuations around it. For the moment we assume that it is a Markov process. The function Px (y) is given by the canonical distribution... 

An alternative approach to DS study is to examine the dynamic molecular properties of a substance directly in the time domain. In the linear response approximation, the fluctuations of polarization caused by thermal motion are the same as for the macroscopic rearrangements induced by the electric field [27,28], Thus, one can equate the relaxation function < )(t) and the macroscopic dipole correlation function (DCF) V(t) as follows ... 

Kubo s linear-response theory provides the full, quantum-mechanical relation between the response of a system to external perturbations and the spontaneous decay of fluctuations in the unperturbed system. Of course, the paper had important predecessors Nyquist s [1] paper on thermal noise in resistors and Onsager s [2] seminal paper on the relation between decay of macroscopic and microscopic fluctuations, to name but the earliest. 

Before we come to these models, we will first introduce a basic law of statistical thermodynamics which we require for the subsequent treatments and this is the fluctuation-dissipation theorem . We learned in the previous chapter that the relaxation times showing up in time- or frequency dependent response functions equal certain characteristic times of the molecular dynamics in thermal equilibrium. This is true in the range of linear responses, where interactions with applied fields are always weak compared to the internal interaction potentials and therefore leave the times of motion unchanged. The fluctuation-dissipation theorem concerns this situation and describes explicitly the relation between the microscopic dynamics in thermal equilibration and macroscopic response functions. 

The left-hand side involves a correlation function associated with the spontaneous fluctuations in thermal equilibrium, as they arise fi om the molecular dynamics. The response function on the right-hand side incorporates the reaction of the sample to the imposition of an external field. The fluctuation-dissipation theorem states that linear responses of macroscopic systems are related to and can, indeed, be calculated from equilibrium fluctuations. More... 

P(co) is an internal field factor and A t) is a time-correlation function which represents the fluctuations of the macroscopic dipole moment of the volume V in time in the absence of an applied electric field. Equations (44) and (45) are a consequence of applying linear-response theory (Kubo-Callen-Green) to the case of dielectric relaxation, as was first described by Glarum in connexion with dipolar liquids. For the special case of flexible polymer chains of high molecular weight having intramolecular correlations between dipoles but no intermolecular correlations between dipoles of different chains we may write... 

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