Fluctuation-dissipation theorem correlations

The response fiinction H, which is defined in equation (A3.3.4), is related to the corresponding correlation fiinction, kliroiigh the fluctuation dissipation theorem ... 

The fluctuation dissipation theorem relates the dissipative part of the response fiinction (x") to the correlation of fluctuations (A, for any system in themial 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 mto the infinitely many degrees of freedom of the themial system. The correlation fiinction on the right-hand side describes the maimer m which a fluctuation arising spontaneously in a system in themial equilibrium, even in the absence of external forces, may dissipate in time. In the classical limit, the fluctuation dissipation theorem becomes / /., w) = w). 

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. 

For an individual molecule, fluctuations of the instantaneous electronic charge density away from its quantum mechanical average are characterized by the fluctuation-dissipation theorem (3, 4). The molecule is assumed to be in equilibrium with a radiation bath at temperature T then in the final step of the derivation, the limit is taken as T — 0. The fluctuation correlations, which are defined by... 

The concept of a nonequilibrium temperature has stimulated a lot of research in the area of glasses. This line of research has been promoted by Cugliandolo and Kurchan in the study of mean-held models of spin glasses [161, 162] that show violations of the fluctuation-dissipation theorem (FDT) in the NEAS. The main result in the theory is that two-time correlations C t,t ) and responses R t, f ) satisfy a modihed version of the FDT. It is customary to introduce the effective temperature through the fluctuation-dissipation ratio (FDR) [163] dehned as... 

Here q and p are Heisenberg operators, y is the usual damping coefficient, and (t) is a random force, which is also an operator. Not only does one have to characterize the stochastic behavior of g(t), but also its commutation relations, in such a way that the canonical commutation relation [q(t), p(t)] = i is preserved at all times and the fluctuation-dissipation theorem is obeyed. ) Moreover it appears impossible to maintain the delta correlation in time in view of the fact that quantum theory necessarily cuts off the high frequencies. ) We conclude that no quantum Langevin equation can be obtained without invoking explicitly the equation of motion of the bath that causes the fluctuations.1 That is the reason why this type of equation has so much less practical use than its classical counterpart. 

Time-correlation functions are of central importance in understanding how systems respond to weak probes in the linear approximation. According to the fluctuation dissipation theorem of the preceding section, spectro-... 

If the random force has a delta function correlation function then K(t) is a delta function and the classical Langevin theory results. The next obvious approximation to make is that F is a Gaussian-Markov process. Then is exponential by Doob s theorem and K t) is an exponential. The velocity autocorrelation function can then be found. This approximation will be discussed at length in a subsequent section. The main thing to note here is that the second fluctuation dissipation theorem provides an intuitive understanding of the memory function. ... 

Note that in the above expression the memory function is proportional to the auto-correlation function of the random force. This is the well-known second fluctuation-dissipation theorem. 

For any reservoir in equilibrium the fluctuation-dissipation theorem provides the relation between the symmetrized and antisymmetrized correlators of the noise Sx(x) = Ax(x) coth(w/2T). Yet, the temperature dependence of Sx and Ax may vary depending on the type of the environment. For an oscillator bath, Ax (also called the spectral density Jx(x)) is temperature-independent, so that Sx(x) = Jx(x)coth(x/2T). On the other hand, for a spin bath Sx is temperature-independent and is related to the spins density of states, while Ax([Pg.14]

We can run the cause-effect connection the other way. The natural motions of the charges within a material will necessarily create electric fields whose time-varying spectral properties are those known from how the materials absorb the energy of applied fields (the "fluctuation-dissipation theorem"). It is the correlations between these spontaneously occurring electric fields and their source charges that create van der Waals forces. At a deeper level, we can even think of all these charge or field fluctuations as results or distortions of the electromagnetic fields that would occur spontaneously in vacuum devoid of matter. 

Exchange-correlation Energy-functional from Adiabatic Connection Fluctuation-dissipation Theorem... 

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 ... 

The frequency correlation time xm corresponds to the time it takes for a single vibrator to sample all different cavity sizes. The fluctuation-dissipation theorem (144) shows that this time can be found by calculating the time for a vertically excited v = 0 vibrator to reach the minimum in v = 1. This calculation is carried out by assuming that the solvent responds as a viscoelastic continuum to the outward push of the vibrator. At early times, the solvent behaves elastically with a modulus Goo. The push of the vibrator launches sound waves (acoustic phonons) into the solvent, allowing partial expansion of the cavity. This process corresponds to a rapid, inertial solvent motion. At later times, viscous flow of the solvent allows the remaining expansion to occur. The time for this diffusive motion is related to the viscosity rj by Geo and the net force constant at the cavity... 

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. 

We compute below the velocity and displacement correlation functions, first, of a classical, then, of a quantal, Brownian particle. In contrast to its velocity, which thermalizes, the displacement x(t) — x(l0) of the particle with respect to its position at a given time never attains equilibrium (whatever the temperature, and even at T = 0). The model allows for a discussion of the corresponding modifications of the fluctuation-dissipation theorem. 

By referring to Eqs.(3) and (41, the imaginary part of the Fourier transform of the OHD-OKE response divided by the Fourier transform of the lAF of the laser pulse is directly comparable to the LS spectrum divided by the Bose factor in addition to Because the dynamical behavior and the fluctuation ought to be correlated with each other through the fluctuation-dissipation theorem, this comparison can verify that the information provided by the two experiments are identical. To our knowledge, however, the direct comparison has not been made yet. In this report, we will describe the details of our experiments and thier results on this problem. 

---