Theorem fluctuation-dissipation

This theorem is called the fluctuation dissipation theorem.  

To prove eqn (3.54), we consider the situation that a constant field h is applied for a long time until the system reaches equilibrium, and that the 

The function a(t), called the relaxation function, is expressed by the response function ju(r) as 

Note that in the above proof the explicit form of the time evolution equation for is not used. Therefore the proof applies to a pure dynamical system which is described by the Liouville equation. The fluctuation dissipation theorem holds quite generedly in physiced systems near equilibrium. 

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)  

Consider a system at equilibrium with a phase space probability distribution function Dg. If an external field F t) that couples to a mechanical property A p, q) is applied at t = 0, the system will move away from its equilibrium state. The phase space probability distribution will now become a function of time, D = )(p, q,t), changing according to Liouville s equation, Eq. 12.1. 

Generally, the observable response to an external perturbation is a mechanical property B p, q). Properties A and B may be identical, although they need not be. 

We can define the response function of a classical mechanical system to an external perturbation as follows  

Generally, the ensemble average response B(t)) can be determined with 

In the linear response limit of a small external field, we may assume that the probability distribution function is determined as 

---

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). 

These are the two components of the Navier-Stokes equation including fluctuations s., which obey the fluctuation dissipation theorem, valid for incompressible, classical fluids ... 

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),... 

MD runs for polymers typically exceed the stability Umits of a micro-canonical simulation, so using the fluctuation-dissipation theorem one can define a canonical ensemble and stabilize the runs. For the noise term one can use equally distributed random numbers which have the mean value and the second moment required by Eq. (13). In most cases the equations of motion are then solved using a third- or fifth-order predictor-corrector or Verlet s algorithms. 

R. Kubo, Statistical-mechanical theory of irreversible processes. 1. General theory and simple applications to magnetic and conduction problems, J. Phys. Soc. Japan 12, 570 (1957) R. Kubo, The fluctuation-dissipation theorem, Rep. Prog. Phys. 29, 255 (1966). 

R. Kubo, The fluctuation-dissipation theorem, Rep. Prog. Phys. 29, 255 (1966). 

Fluctuation-dissipation theorem, transition state trajectory, white noise, 203—207 Fluctuation theorem, nonequilibrium thermodynamics, 6—7... 

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,... 

Here, 7 is the friction coefficient and Si is a Gaussian random force uncorrelated in time satisfying the fluctuation dissipation theorem, (Si(0)S (t)) = 2mrykBT6(t) [21], where 6(t) is the Dirac delta function. The random force is thought to stem from fast and uncorrelated collisions of the particle with solvent atoms. The above equation of motion, often used to describe the dynamics of particles immersed in a solvent, can be solved numerically in small time steps, a procedure called Brownian dynamics [22], Each Brownian dynamics step consists of a deterministic part depending on the force derived from the potential energy and a random displacement SqR caused by the integrated effect of the random force... 

As mentioned, this equivalence is a consequence of the fluctuation-dissipation theorem (the general basis of linear response theory [51]). In (12.68), we have dropped nonlinear terms and we have not indicated for which state Variance (rj) is computed (because the reactant and product state results only differ by nonlinear terms). We see that A A, AAstat, and AAr x are all linked and are all sensitive to the model parameters, with different computational routes giving a different sensitivity for AArtx. 

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. 

Kubo, R. The fluctuation-dissipation theorem, Benjamin,Inc., New York, 1969... 

A Nonlocal Energy Functional Derived from the Fluctuation-Dissipation Theorem... 

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... 

---