# Looking for materials about ensemble average

• I
• Haorong Wu
In summary, the conversation involves the request for materials related to calculating the average of the derivative of a function A(x) satisfying a Gaussian white noise. The formula for A(x) is not provided, making it difficult to calculate A'(x). The person has searched for resources, but has not found any helpful results.
Haorong Wu
Hello. I am looking for some materials related to the ensemble average.

Specifically, suppose there is a function ##A(x)## satisfying a Gaussian white noise $$\left < A(x)A(x') \right > =A_0^2\exp \left ( -\frac 1 {L^2}(x-x')^2\right )$$ where the average is taken over an ensemble.

Now I need to calculate the average ##\left < \frac {d A(x)}{dx} \frac {d A(x')}{ dx'} \right >##. I am not sure how to do this. My guess is ##\left < \frac {d A(x)}{dx} \frac {d A(x')}{ dx'} \right >=\frac {d^2}{dxdx'}\left < A(x)A(x') \right >##.

I have look it up in An Introduction to Thermal Physics by Schroeder without results. I am not sure what kind of materials is related to this kind of calculation.

Thanks!

Could you show us formula of A(x) so that we know A'(x) ?

## What is ensemble average?

Ensemble average is a statistical concept used in physics and other fields to describe the average behavior of a large number of individual systems or particles. It is calculated by taking the average of a property or variable across all members of an ensemble, or collection, of systems.

## Why is ensemble average important?

Ensemble average is important because it allows us to make predictions and draw conclusions about the behavior of a system as a whole, rather than just individual components. It also helps to smooth out any random fluctuations or errors that may occur in individual measurements.

## How is ensemble average calculated?

Ensemble average is calculated by taking the sum of the property or variable of interest for all members of the ensemble, and dividing it by the total number of members in the ensemble. This is represented mathematically as:

⟨A⟩ = (1/N) ∑Ai

where A is the property or variable, N is the total number of members, and ∑Ai represents the sum of A for all members i in the ensemble.

## What is the difference between ensemble average and time average?

Ensemble average and time average are two different methods of calculating the average behavior of a system. Ensemble average is based on the behavior of a large number of systems at a single point in time, while time average is based on the behavior of a single system over a period of time. Ensemble average is used to describe the overall behavior of a system, while time average is used to analyze the fluctuations and changes in a system over time.

## What are some applications of ensemble average?

Ensemble average is used in a variety of fields, including physics, chemistry, biology, and economics. It is commonly used to study the behavior of gases, liquids, and solids, as well as the behavior of particles at the atomic and molecular level. It is also used in statistical mechanics to describe the properties of large systems, and in economics to analyze the behavior of markets and financial systems.

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