The Fluctuation-Dissipation Theorem

The Fluctuation-dissipation Theorem Relates Equilibrium Fluctuations to the Rate of Approach to Equilibrium 

An important theorem of statistical mechanics relates a property of kinetics to a property of equilibrium. At equilibrium, a system undergoes thermal fluctuations. Remarkably, the magnitudes of these equilibrium fluctuations are related to how fast the system approaches equilibrium. This theorem is quite general, and applies to many different physical and chemical processes. It allows you to determine the diffusion constant, viscosity, and other transport properties from knowledge of the equilibrium fluctuations. 

We use this model to illustrate the idea of a time correlation function. 

The Time Correlation Function Describes How Quickly Browmian Motion Erases the Memory of the Initial PcU ticle Velocity 

Here s how to construct a time-correlation function. Take the velocity (0) of the particle at time t = 0. Multiply by the velocity of the particle v(t) at a later time t. Take the equilibrium ensemble average of this product over many different collisions to get (if(O)v(t)). This is the velocity autocorrelation 

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. 

It describes the correlation between the results of two measurements of carried out at and t +1, whereby t is arbitrary since systems in thermal equilibrium are homogeneous in time. Fluctuations occur independently along rc, y and 2 . The correlation function for one of the components, denoted py, is therefore 

In the previous chapter we dealt with the dielectric response. Application of an electric field produces a polarization. If the field is imposed at zero time the polarization develops as described by Eq. (5.16) 

If the electric field then is switched off, the polarization returns back to zero, in a time dependent process described by 

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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). 

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... 

The Onsager coefficient is given by the fluctuation-dissipation theorem ... 

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... 

The averaged potential energy (V) includes contributions from fluctuations in the charge density at all real frequencies. The fluctuation-dissipation theorem restricts the contributing frequencies to co = -o), but allows for all real co. The effects on the energy are contained in the term defined by... 

Sometimes R is called the fluctuation-dissipation ratio, not to be confused with the identically called but different quantity introduced in glassy systems (see Section VLB) that quantifies deviations from the fluctuation-dissipation theorem that is valid in equilibrium. 

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