Fluctuation-dissipation theorems dissipative response function

Here T is the local-equilibrium temperature. In extended irreversible thermodynamics fluxes are independent variables. The kinetic temperature associated to the three spatial directions of along the flow, along the velocity gradient, and perpendicular to the previous to directions may be different from each other. To define temperature from the entropy is the most fundamental definition, and the nonequilibrium temperature may come from the derivative of a nonequilibrium entropy du/dS) -p. Effective nonequilibrium temperature may be defined from the fluctuation-dissipation theorem relating response function and correlation function. 

The classical linear response function can be written using the fluctuation-dissipation theorem as a single term,... 

Adiabatic-connection fluctuation-dissipation theorem allows one to express the exchange-correlation energy-functional by means of imaginary-frequency density response function ( A) of the system with the scaled Coulomb potential (A/ r — r )11,13 ... 

This chapter relates to some recent developments concerning the physics of out-of-equilibrium, slowly relaxing systems. In many complex systems such as glasses, polymers, proteins, and so on, temporal evolutions differ from standard laws and are often much slower. Very slowly relaxing systems display aging effects [1]. This means in particular that the time scale of the response to an external perturbation, and/or of the associated correlation function, increases with the age of the system (i.e., the waiting time, which is the time elapsed since the preparation). In such situations, time-invariance properties are lost, and the fluctuation-dissipation theorem (FDT) does not hold. 

The response function Xij (f 1, which is defined in equation (A3.3.4), is related to the corresponding correlation function, Sij (j . / (through the fluctuation dissipation theorem ... 

The fluctuation dissipation theorem relates the dissipative part of the response function (x") to the correlation of fluctuations A, for any system in thermal equilibrium. The left-hand side describes the dissipative behaviour of a many-body system all or part of the work done by the external forces is irreversibly distributed into the infinitely many degrees of freedom of the thermal system. The correlation function on the right-hand side describes the manner in which a fluctuation arising spontaneously in a system in thermal equilibrium, even in the absence of external forces, may dissipate in time. In the classical limit, the fluctuation dissipation theorem becomes >). 

Key words Linear-response theory - Correlation function - Fluctuation-dissipation theorem - Reaction rates... 

Let us first consider spectroscopy. Linear-response theory, in particular the fluctuation dissipation theorem - which relates the absorption of an incident monochromatic field to the correlation function of (e.g. dipole) fluctuations in equilibrium - has changed our perspective on spectroscopy of dense media. It has moved away from a static Schrodinger picture -phrased in terms of transitions between immutable (but usually incomputable) quantum levels - to a dynamic Heisenberg picture, in which the spectral line shape is related by Fourier transform to a correlation function that describes the decay of fluctuations. Of course, any property that cannot be computed in the Schrodinger picture, cannot be computed in the Heisenberg picture either however, correlation functions, unlike wave-functions, have a clear meaning in the classical limit. This makes it much easier to come up with simple (semi) classical interpretations and approximations. 

In the case of A = B, the fluctuation dissipation theorem can be stated in a more convenient form. Let us define the growth function j8(r) as the response to the sudden application of a step field (see Fig. 3.5) ... 

The properties of polymer crystals may be computed using the same relations as presented earlier for lattice dynamics. However, it is generally more expedient to take advantage of the fluctuation-dissipation theorem [64] to relate response functions to the magnitude of fluctuations at equi-hbrium. For example, the fluctuation formula for the constant volume heat capacity computed in the canonical NVT) ensemble is... 

In liquids and dense gases where collisions, intramolecular molecular motions and energy relaxation occur on the picosecond timescales, spectroscopic lineshape studies in the frequency domain were for a long time the principle source of dynamical information on the equilibrium state of manybody systems. These interpretations were based on the scattering of incident radiation as a consequence of molecular motion such as vibration, rotation and translation. Spectroscopic lineshape analyses were intepreted through arguments based on the fluctuation-dissipation theorem and linear response theory (9,10). In generating details of the dynamics of molecules, this approach relies on FT techniques, but the statistical physics depends on the fact that the radiation probe is only weakly coupled to the system. If the pertubation does not disturb the system from its equilibrium properties, then linear response theory allows one to evaluate the response in terms of the time correlation functions (TCF) of the equilibrium state. Since each spectroscopic technique probes the expectation value... 

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. 

Imagine that we select within a sample a subsystem contained in a volume Vj which is small but still macroscopic in the sense that statistical thermodynamics can be applied. If we could measure the properties of this subsystem we would observe time dependent fluctuations, for example in the shape of the volume, i.e. the local strain, the internal energy, the total dipole moment, or the local stress. The fluctuation-dissipation theorem relates these spontaneous, thermally driven fluctuations to the response functions of the system. We formulate the relationship for two cases of interest, the fluctuations of the dipole moments in a polar sample and the fluctuations of stress in a melt. 

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... 

Most important is the role played by the fluctuation-dissipation theorem in theoretical treatments, as it may be regarded as an interface between the microscopic and the macroscopic properties of a sample. It provides us with a precise prescription of how to proceed when these two are to be related. On the microscopic side, theoretical analysis of dynamical models usually enables us to make a calculation of equilibrium correlation functions for all properties of interest. The fluctuation-dissipation theorem then relates these correlation functions with the results of measurement, as described by the various response functions. 

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 thermal conductivity of a material can be calculated directly from equilibrium molecular dynamics (EMD) simulation based on the linear response theory Green-Kubo relationship. " The fluctuation-dissipation theorem provides a connection between the energy dissipation in irreversible processes and the thermal fluctuations in equilibrium. The thermal conductivity tensor. A, can be expressed in terms of heat current autocorrelation correlation functions (HCACFs),/, ... 

Going back to the general expressions [4.3.19] and [4.3.20], linear response theory relates non-equilibrium relaxation close to equilibrium to the dynamics of equilibrium fluctuations The first fluctuation dissipation theorem states that following a step function change inF ... 

The terms Xa(q t ) are sometimes called the response functions [212]. It is well known in the literature that the fluctuation-dissipation theorem results in the power spectrum /a(q,w) being given by the relation [38, 212]... 

The fluctuation-dissipation theorem dictates that the response of a system to a weak perturbation is entirely determined by the equilibrium correlation of the fluctuating dynamic variable that couples with the field, i.e., ili p (f) = cmm (0> where cmm is the correlation function of the fluctuations of the total dipole moment in equilibrium ... 

The fluctuation-dissipation theorem (FDT) of Callen and Welton states a general relationship between the response of a given system to an external disturbance and the internal fluctuation of the system in the absence of the dismrbance. Such a response is characterized by a response function or equivalently by an admittance or an impedance. For dielectric relaxation, the complex dielectric function, e ( u), is related to the dipole moment correlation function < >( ) via Fourier transformation ... 

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