# Velocity correlation functions

• thrillhouse86
Then in the fundamental theorem of calculus, the derivative of the integral with respect to a is the integrand evaluated at a, so in this case v(a).In summary, The conversation is discussing non-equilibrium statistical mechanics and how to show that the partial derivative of <x^{2}> with respect to t is equal to 2 times the integral of <v(s)v(t)> from 0 to t. The solution involves using the fundamental theorem of calculus and the derivatives of the integrals.
thrillhouse86
Hi, I am going through Non Equilibrium Statistical Mechanics by Zwanzig and I can't follow, the step below:

I have the equation:
$$<x^{2}> = \int^{t}_{0}ds_{1}\int^{t}_{0}ds_{2}<v(s_{1})v(s_{2})>$$

I can't show that:
$$\frac{\partial <x^{2}>}{\partial t} = 2 \int^{t}_{0}ds<v(s)v(t)>$$

I'm sure that the answer lies with that fundamental theorem of calculus, but I can't show it. For one thing, do I apply the product rule to the two integrals above ?

Thanks

call the upper indices (which are now both called t) a(t)=t and b(t)=t. Then use that the total derivative (well, total at least with respect to t) is d/dt=da/dt d/da+db/dt d/db.

## 1. What are velocity correlation functions?

Velocity correlation functions are mathematical tools used in the study of fluid dynamics to describe the relationship between the velocity of a fluid particle at one point and the velocity of another fluid particle at a different point. They provide information about the spatial and temporal correlations of fluid flow.

## 2. How are velocity correlation functions used in fluid dynamics?

Velocity correlation functions are used to study the behavior of fluids, such as turbulence, diffusion, and mixing. They can also provide insight into the transport properties of fluids, such as viscosity and diffusion coefficients.

## 3. How are velocity correlation functions calculated?

Velocity correlation functions are typically calculated using experimental data or numerical simulations. This involves measuring the velocities of fluid particles at different points in space and time and then using mathematical formulas to calculate the correlations between them.

## 4. What is the significance of velocity correlation functions?

Velocity correlation functions are important in understanding the dynamics of fluids and their transport properties. They can also be used to validate and improve theoretical models of fluid flow and to design more efficient processes in various industries, such as aerospace, automotive, and chemical engineering.

## 5. How do velocity correlation functions relate to other correlation functions?

Velocity correlation functions are a type of correlation function that specifically describes the relationship between fluid velocities. Other types of correlation functions may describe the relationships between other physical quantities, such as temperature or density, and can be used in conjunction with velocity correlation functions to provide a more complete understanding of fluid behavior.

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