Question

Small differences in large numbers can lead to nonsense. Using the results from Problem 8, show that the propagated error is larger than the difference itself for \(f(x, y)=x-y\), with \(x=20 \pm 2\) and \(y=19 \pm 2\).

Short answer

Error (2.83) is larger than the difference (1).

Step-by-step solution

Understand the Problem

We need to find the propagated error for the function \(f(x, y) = x - y\). The given values are \(x = 20 \pm 2\) and \(y = 19 \pm 2\).

Calculate the Difference

First, calculate the difference \(f(x, y) = x - y\). For the given values, this is \(f(20, 19) = 20 - 19 = 1\).

Determine Partial Derivatives

Find the partial derivatives of \(f(x, y)\) with respect to \(x\) and \(y\). \(\frac{\partial f}{\partial x} = 1\) and \(\frac{\partial f}{\partial y} = -1\).

Calculate the Propagated Error

Use the formula for propagated error: \[ \sigma_f = \sqrt{ \left( \frac{\partial f}{\partial x} \sigma_x \right)^2 + \left( \frac{\partial f}{\partial y} \sigma_y \right)^2 } \]Substitute the given values: \[ \sigma_f = \sqrt{(1 \cdot 2)^2 + (-1 \cdot 2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \approx 2.83 \]

Compare the Propagated Error with the Difference

The difference \(f(20, 19) = 1\). The propagated error is approximately \(2.83\), which is larger than the difference itself.

Conclusion

Therefore, the propagated error is larger than the difference itself for \(f(x, y) = x - y\) with the given values and uncertainties.

Key concepts

Partial Derivatives

In scientific computations, especially when dealing with functions of multiple variables, understanding partial derivatives is crucial. Partial derivatives measure how a function changes as each variable changes, independently of the others. For example, consider the function given in the exercise, \(f(x, y) = x - y\).

To understand this concept better:

• The partial derivative of \(f\) with respect to \(x\), noted as \(\frac{\partial f}{\partial x}\), tells us how the function \(f\) changes when only \(x\) is varied.
• The partial derivative of \(f\) with respect to \(y\), noted as \(\frac{\partial f}{\partial y}\), tells us how the function \(f\) changes when only \(y\) is varied.

In the given problem,
\(\frac{\partial f}{\partial x} = 1\)
and
\(\frac{\partial f}{\partial y} = -1\).
These values help us understand how the overall function value changes with small changes in \(x\) and \(y\).

Propagated Error

When performing calculations with measurements that have uncertainties, the inaccuracies can propagate through the computations. Propagated error gives us an understanding of how the uncertainties in the input variables affect the result of the function.

To quantify this, we use the following formula:
\[ \sigma_f = \sqrt{ \left( \frac{\partial f}{\partial x} \sigma_x \right)^2 + \left( \frac{\partial f}{\partial y} \sigma_y \right)^2 } \] Here,

• \(\sigma_x\) and \(\sigma_y\) are the uncertainties in the variables \(x\) and \(y\).
• The partial derivatives \(\frac{\partial f}{\partial x}\) and \(\frac{\partial f}{\partial y}\) show how sensitive the function is to changes in \(x\) and \(y\).

By substituting the given values:
\(\sigma_x = 2\)
and
\(\sigma_y = 2\)
we calculated:
\( \sigma_f = \sqrt{ (1 \cdot 2)^2 + (-1 \cdot 2)^2 } = \sqrt{8} \approx 2.83 \).
This example shows that the uncertainty in the result (\(\sigma_f\)) is larger than the calculated difference, highlighting the significance of propagated error in scientific computations.

Statistical Thermodynamics

Statistical thermodynamics often involves complex functions of multiple variables where errors and uncertainties can immensely affect the results. Statistical mechanics provides a bridge between microscopic properties of individual atoms and molecules and macroscopic properties of materials.

Concepts in statistical thermodynamics include:

• Ensemble averages, where we calculate average properties over a large collection of microstates.
• Partition functions, which summarize the statistical properties of a system in thermodynamic equilibrium.
• Fluctuations and response functions, where uncertainties and their propagation play a crucial role.

The calculations of propagated errors (like in our example) ensure that we understand how small measurement errors can impact predictions of macroscopic properties. This helps in developing more accurate models which are vital for advancements in areas like material science and thermodynamic processes.