"What does it mean physically that the correlation is extremely brief or the noise is white?"

The white noise term takes into account the random kicking of the colloidal particle (pollen) by fluid molecules. These random kicks are the origin of the particle jiggling. Yet, the kicks do not contribute in average to the velocity of the particle: =0. The delta correlation term simply states that the kicking is memoryless; namely, that the random perturbation to the velocity by the kicking does not "remember" what was the value of the velocity at any other time. In reality the random perturbation has some memory that exponentially decays with time. Still, the decay is so fast that it is usually approximated by a delta function (no memory) and that allow for analytical calculations. I hope it helps. Cheers.

Abdallah,

(I sense that you may be asking this more for commentary and discussion rather than clarification)

My take is that a vanishing correlation function implies a perturbing force that acts momentarily - the agent applying the force has no 'memory' of its previous acts.

[A swarm of perfectly elastic ball-bearings would do the trick.]

If the noise is white then the perturbing agent acts with equal vigour irrespective of the duration over which it is observed.

[here the ball-bearing swarm model would fail - unless you had an infinite number of swarms at work, all with different ball-bearing sizes]

Physically, this isn't a bad model for a particle buffeted by a sea of quanta. But it's also not a bad model for any object affected by similarly brief but frequent interactions.

At least, that's my take on dim recollections of statistical physics from the 1980s.

I do not remember clearly but I remember that I had met with this question when I was undergrad. And the answer were in the Feyman and Hibb's Quantum Mechanics and Path Integrals book. However I am not sure. Somebody should check.

See, for example,

Polymer Chain Dynamics and NMR, Rainer Kimmich, Nail Fatkullin,

12/2004: pages 1-113; , ISBN: ISBN: 978-3-540-20510-4 (Print) 978-3-540-40000-4 and literature sited in it. General discussion can be find in sections 3.3.1.

"What does it mean physically that the correlation is extremely brief or the noise is white?"

The white noise term takes into account the random kicking of the colloidal particle (pollen) by fluid molecules. These random kicks are the origin of the particle jiggling. Yet, the kicks do not contribute in average to the velocity of the particle: =0. The delta correlation term simply states that the kicking is memoryless; namely, that the random perturbation to the velocity by the kicking does not "remember" what was the value of the velocity at any other time. In reality the random perturbation has some memory that exponentially decays with time. Still, the decay is so fast that it is usually approximated by a delta function (no memory) and that allow for analytical calculations. I hope it helps. Cheers.

Thank you James/Gökçe/Nail/Javier for your helpful answers! In fact, I asked this question because I had some ambiguities in the signification of Dirac function in the velocity auto-correlation function. I can barely understand now that the white noise signifies the absence of memory in the collision process between particles.

I found that in other situations, the particle dynamics is governed by the following equations (attachment). Can anyone explain me more the difference between these two cases? Thank you in advance!

Your new equations are nothing but a rewriting of the firsts you wrote. Trivial changes are that the equation of motion is written in term of acceleration instead of force, and that the argument of delta function in the last equation is written explicitely as the difference of time t and t'. I think that the argument of C is inconsistent with the right hand side of the equation (correlation functions should depend on the time difference t-t').

As far as the physical meaning, I would add to previous answers that a rigorous formal derivation of the equation of motion for a particle in a fluid is possible (look for Mori's formalism of classical many body dynamics), justifying the *structure* of Langevin's equation, although the random force memory function, in general is not a delta function. The Langevin's form (delta correlation) can be seen as the limiting case of a memory function decaying over times much shorter than the observation time. This is fully consistent with the expectation that Langevin's equation should be more and more accurate larger is the difference of characteristic times between solute and solvent dynamics.

For a pedagogical and very simple derivation of Langevin's equation I could cite paper by de Grooth : " A simple model for Brownian motion leading to the Langevin equation." American Journal of Physics, 67 (12). pp. 1248-1252 (1999).

Thank you Giorgio for your answer as well as for the paper you referred. I have just rectify the governing equation!

But if the delta correlation term simply states that the kicking is memoryless as Javier has mentioned, how can we model the memory effect in the dispersion of particles? Thank you!

You can add memory effects in the random kicking by substituting the white noise by a colored noise as the Ornstein-Uhlenbeck one that has a memory kernel (correlation) that exponentially decays and that in the limit of zero correlation time (memory) leads to the white noise (http://goo.gl/8lbQ4). We may want to take a look to the "classic" book by Horsthemke and Lefever (http://goo.gl/yK7q9M). Cheers.

Above answers are almost complete on this question I think. No white noise exists, but the time average should be large enough to valid the statistical meaning on the noise. White noise means random kick with a frequency as high as possible.

Just a comment on Lin Zhang response. The term "white" means that if you compute the power spectrum (it is trivial since the correlation function is a delta function) you obtain a flat spectrum: all frequencies are equally weighted. Thus, I guess it is not correct to state that the kicking frequency is "as high as possible" since all frequencies are equally represented in the process.

Thank you all for your clarifications!

There is to be added references on the works by A.O. Caldeira and A.J. Leggett, e.g.

Ann.Phys (N.Y.)149,374 (1983). There, rigorous derivation of the Langevin equation is based on a linear coupling of a system \Phi(x,t) with the reservoir, represented as a system of infinite number of harmonic oscillators with frequencies \omega_k. When solving the coupled system, one arrives to the expression for the oscilltor coordinate in a form which exhibits a memory term \int d\tau \Phi(x,\tau )\sin (\omega_k(t-\tau). Integrating it per partes, one obtains \int d\tau d(\Phi(x,\tau )) d(\tau) \cos(\omega_k(t-\tau).. Providing d(\Phi(x,\tau )) d(\tau) changes very slowly when compared with the \cos (\omega_k(t-\tau), (\omega large)., the slow part can be taken approximately constant and the \cos term be integrated over k. This turns to \sin (\omega_k (t-\tau) )/ \omega_k, which , in the limit tends to the Dirac \delta function . \delta(\t-\tau), which leads to the vanishing of the memory and thus to the diffusion with white noise. The random force in the r.h.s. of the Langevin equation can also be easily evaluated in the process of the above outlined calculation.

Dear Eva, thank you for your reply.

What is the starting point for the derivation of the Langevin equation? I mean what are the initial assumptions that one may made to define \Phi in this approach?

Can you justify please the reason why the mentioned quantity is slowly varying comparing to the other one?

I would appreciate if you possibly could send me the cited paper above. Thank you!

I attach another papers, sorry for not having the electronic version of that cited.

The assumption for slowly varying function \Phi is used to illustrate conditions when the friction term of the Langevin equation and white noise is obtained, if this is not the case, there remains the memory term .

The function \phi represents a dynamical system in a linear coupling with the reservoir

There is not enough place for transfer a longer file, I am going to send it by e-mail.

Abdallah, let me know please, if You received my e-mail with files attached. The files might be too big .

Javier Buceta is right. I am just trying to give a new angle and put an emphasis on the long-time average to valid the statistical meaning. Surely, the Delta correlation function can be expanded by the Fourier components with all frequencies being equally weighted. If we imagine the noises as sequences of kicks, the white noise means no correlation exists between the kicks bursting at different times.

Sorry, neither any Fourier expansion of delta function, nor any additional assumptions are necessary.

The Langevin equation for a time evolving variable X(t) is obtained from the model of reservoir as a system of harmonic oscillators {q_k(t)} of frequencies \omega_k linearly coupled to the system X(t) by setting the solution for q_k(t) into the dynamic equation for X(t). Resulting memory term includes summation over \omega_k (or integration in the interval over \omega) which must be calculated with the density of modes D(\omega). The number of modes in the interval d\omega is D(\omega_k)dk~\omega^2 d\omega for 3D. From this one obtains the Dirac delta function as a limit for large \omega_D. So one obtains in the memory term the Dirac \delta and integration over time then gets the known friction term. So there is not necessary to manipulate with the \delta function by expansion into the Fourier components. The random force in the right hand side is obtained from the initial conditions of the integration of q_k(t). It represents the white noise. For slowly varying X(t) (neglecting its dynamic oscillations), one obtains the Langevin equation.

This is the standard procedure which originated from Einstein and can be found in literature.

The following paper of mine shows that the statistical properties of the Langevin force, presented above by Mr. Moussa Ider, are unnecessary to investigate Brownian motion.

Best regards,

Bolivar.

https://www.researchgate.net/publication/220004561_Non-Markovian_effects_on_the_Brownian_motion_of_a_free_particle_Physica_A_390_(2011)_30953107?fulltextDialog=true

Article Non-Markovian effects on the Brownian motion of a free parti...

Perhaps not quite the answer you wanted:

The Langevin equation is an ad hoc turn of the last century approximation.  It is replaced by the exact Mori equation. While I say "exact", rather little is known about the properties of projected Liouville operators. 

In terms of physical systems, the Langevin equation leads to the notion that particle displacement has, as a probability distribution,  a Gaussian in Delta x, this being the central limit theorem result. The much less well known Doob's theorem then guarantees that the mean-square particle displacement increases linearly in time.

Recently, there has been interest in 'subdiffusion', the notion that mean-square diffusion increases less than linearly in time, leading (I omit detailed constants, because typing equations in ascii is a nuisance) to the alleged equation

P(x,t)  = a \exp(- x^2 / b t^{\alpha} )

where P(x,t) is the probability distribution for x during t, a and b are constants, and \alpha is a power not equal to unity.  The Gaussian in x is said to arise from the central limit theorem, and the non-unitary power \alpha is said to be the subdiffusion.

This equation cannot possibly be correct as described. It is mathematically impossible, because it violates Doob's theorem.  If the central limit theorem is applicable, then \alpha must be unity.

In terms of the Langevin and Mori equations, the Langevin equation becomes erroneous because the noise is no longer white and the friction factor becomes a memory kernel.

How can the central limit theorem fail? Trivial case: Bidisperse suspension, in which P(x,t) is the sum of two exponentials.  Serious case: Diffusion in complex fluids, in which the "random" force has time correlations on experimentally interesting times. Critical case:  Quasielastic Light scattering spectra, for which the form exp(- q^2 /2) is sometimes invoked, is sensitive to this issue.  The form I just quoted is not valid for quasielastic scattering from non-dilute solutions, as first shown 40 years ago.