Fluctuation-dissipation theorem in a surface

In summary, the Fluctuation-Dissipation Theorem (FDT) is a fundamental principle in statistical mechanics that describes the relationship between fluctuations and dissipation in a physical system in thermal equilibrium. In a surface, it is used to study surface properties and processes, such as roughness and diffusion, by measuring the thermal energy and understanding its impact on dynamics. While primarily applied to equilibrium systems, the FDT can also be used for non-equilibrium systems under specific conditions. It is closely related to other thermodynamic principles and violations of the FDT can provide valuable insights into the behavior of surfaces and their response to external forces.
  • #1
paloureiro
1
0
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
 
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  • #2


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,

 

1. What is the Fluctuation-Dissipation Theorem (FDT) in a surface?

The Fluctuation-Dissipation Theorem is a fundamental principle in statistical mechanics that describes the relationship between the fluctuation and dissipation of a physical system in thermal equilibrium. In a surface, the FDT relates the fluctuations in the surface height or displacement to the dissipation of energy due to thermal motion.

2. How is the FDT applied in surface science and materials research?

The FDT is commonly used in surface science and materials research to study the properties of surfaces, such as surface roughness and surface diffusion. It provides a way to measure the thermal energy of a surface and to understand how it affects the dynamics of surface processes.

3. Can the FDT be applied to non-equilibrium systems?

While the FDT is primarily used for systems in thermal equilibrium, it can also be applied to non-equilibrium systems under certain conditions. However, the relationship between fluctuations and dissipation may not be as straightforward as in equilibrium systems.

4. How does the FDT relate to other thermodynamic principles?

The FDT is closely related to other thermodynamic principles, such as the fluctuation theorem and the principle of detailed balance. These principles all describe the behavior of a system in thermal equilibrium, with the FDT specifically relating fluctuations and dissipation.

5. What are the implications of violating the FDT in a surface?

If the FDT is violated in a surface, it suggests that the system is not in thermal equilibrium. This could be due to external factors, such as applied stress or temperature gradients, or it could indicate the presence of non-equilibrium processes. Violations of the FDT can provide valuable insights into the behavior of surfaces and their response to external forces.

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