Fluctuation-dissipation theorem in a surface

[paloureiro]

Hi,

my question regards the application of the fluctuation-dissipation theorem (Kubo - 1966 Rep. Prog. Phys. 29 255) to a collection of particles (molecules in liquid state) in a plane.
Being Na, the number of particles contained in a macroscopic region of volume Va, one has:

<(Na - <Na>)**2> = <Na>{1 + n ∫{g(R) -1}dR}

, where, n is the average number density and g(R) is the radial distribution function.

My question is, can I make a direct use of this relation to a plane? In this case, Na would be number of particles in a given area and so on.
If so, can I make a transformation such as instead of using the fluctuation of number of particles in a certain area, I would use the fluctuation of the areas occupied by each molecule?

Regards,

Pedro L. Loureiro

 

[eljose79]

Hello Pedro,

Thank you for your question. The fluctuation-dissipation theorem is a powerful tool in statistical mechanics that relates the fluctuations of a system to its response to external perturbations. In the case of a collection of particles in a plane, the theorem can indeed be applied, but some modifications need to be made.

Firstly, the relation you have provided is for a three-dimensional system, so it cannot be directly applied to a two-dimensional system like a plane. However, it can be modified to account for the different dimensions. Instead of using the number of particles, Na, you can use the number of particles per unit area, which we can denote as nA.

Additionally, the radial distribution function, g(R), needs to be modified for a two-dimensional system. In three dimensions, it represents the probability of finding a particle at a distance R from another particle. In two dimensions, it represents the probability of finding a particle at a distance R from a line of particles. This can be calculated using the two-dimensional analog of the radial distribution function, which is defined as:

g(R) = nA/2πR

where nA is the number of particles per unit area and R is the distance from the line of particles.

With these modifications, the fluctuation-dissipation theorem can be applied to a collection of particles in a plane, with the relation becoming:

<(nA - <nA>)**2> = <nA>{1 + nA ∫{g(R) -1}dR}

This relation can then be used to study the fluctuations of the number of particles per unit area in a given region of the plane. However, it is important to note that this is a simplified version and may not take into account all the complexities of a real system. It is always recommended to carefully consider the assumptions and limitations of any theoretical model before applying it to a real system.

I hope this helps answer your question. Best of luck with your research.

Regards,