Fluctuation-dissipation relation/theorem

Fundamental theory also tells us there is an intimate connection. The fluctuation-dissipation (FD) theorem (7) relates dissipative quantities such as viscosity to the ever present fluctuations in equilibrium systems, such as correlations in local velocities. A significant statement about the viscosity can now be made. The dissipative quantity has the same kind of temperature and pressure discontinuities as does the correlation function. Thus, if we have a first-order diermodynamic transition then we expect the viscosity to display first-order character in its T-P behavior. This is the case. Since we predict an... 

The statistical properties of the random force f(0 are modeled with an extreme economy of assumptions f(t) is assumed to be a stationary and Gaussian stochastic process, with zero mean (f(0 = 0), uncorrelated with the initial value v(t = 0) of the velocity fluctuations, and delta-correlated with itself, f(0f(t ) = f25(t -1 ) (i.e it is a purely random, or white, noise). The stationarity condition is in reality equivalent to the fluctuation-dissipation relation between the random and the dissipative forces in Equation 1.1, which essentially fixes the value of y. In fact, from Equation 1.1 and the assumed properties of f(t), we can derive the expression y(f)v(t) = exp [v(0)v(0) -+ ylM °, where Xg = In equilibrium, the long-time asymptotic value y/M must coincide with the equilibrium average (vv) = (k TIM)t given by the equipartition theorem (with I being the 3 X 3 Cartesian unit tensor), and this fixes the value of y to y=... 

Thus, the requirement that the Brownian particle becomes equilibrated with the surrounding fluid fixes the unknown value of, and provides an expression for it in tenns of the friction coefficient, the thennodynamic temperature of the fluid, and the mass of the Brownian particle. Equation (A3.1.63) is the simplest and best known example of a fluctuation-dissipation theorem, obtained by using an equilibrium condition to relate the strengtii of the fluctuations to the frictional forces acting on the particle [22]. 

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). 

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 ... 

Other spectral densities correspond to memory effects in the generalized Langevin equation, which will be considered in section 5. It is the equivalence between the friction force and the influence of the oscillator bath that allows one to extend (2.21) to the quantum region there the friction coefficient rj and f t) are related by the fluctuation-dissipation theorem (FDT),... 

The Langevin dynamics method simulates the effect of individual solvent molecules through the noise W, which is assumed to be Gaussian. The friction coefficient r is related to the autocorrelation function of W through the fluctuation-dissipation theorem,... 

G. N. Bochkov and Y. E. Kuzovlev, Non-linear fluctuation relations and stochastic models in nonequilibrium thermodynamics. 1. Generalized fluctuation-dissipation theorem. Physica A 106, 443-J79 (1981). 

In equilibrium, or in a stable state, the magnitude of the fluctuations is the outcome of the competition between the jumps and the macroscopic return to equilibrium. Both effects are represented by the second and first term, respectively, on the right of (4.2b). This is the basis of the Einstein relation (VIII.3.9) and of the fluctuation-dissipation theorem. 

Remark. It is easily seen that the second term of (5.2) by itself causes the norm of if/ to change. In order that this is compensated by the fluctuating term the two terms must be linked, as is done by the relation U = V V. This resembles the classical fluctuation-dissipation theorem, which links both terms by the requirement that the fluctuations compensate the energy loss so as to establish the equilibrium. The difference is that the latter requirement involves the temperature T of the environment that makes it possible to suppress the fluctuations by taking T = 0 without losing the damping. This is the reason why in classical theory deterministic equations with damping exist, see XI.5. 

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. 

In general, both (f) and R(t) appearing in Eq. (317) arise from microscopic motions of heat-bath (solvent) modes interacting with reaction coordinate [167-169]. For isomerization reactions in solvents, both of them arise from the microscopic motions of the solvent molecules interacting with the isomerizing moiety. So, they must be related to each other. They are related by the following relation (known as 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]

This is a specific example of the fundamental fluctuation-dissipation theorem that relates the random force / (fluctuation) to the friction constant 7 (dissipation) to ensure that any initial state eventually evolves into a state in thermal equilibrium with the fluid at temperature T. With this requirement, we obtain from Eq. (11.12) the final result... 

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