Jensen’s Inequality in Statistical Thermodynamics of Ideal Gases

Problem: Jensen's Inequality in Statistical Thermodynamics of Ideal Gases

In the study of kinetic theory and statistical mechanics, the velocities of gas molecules are not uniform. Instead, they follow a probability distribution, such as the Maxwell-Boltzmann distribution. Because the velocities are distributed, we characterize the gas using various statistical moments. Two critical measures of molecular speed are the average speed ##v_{\text{avg}}## and the root-mean-square speed ##v_{\text{rms}}##.

Jensen's Inequality provides a powerful framework for relating the mean of a function of a random variable to the function of the mean of that variable.

**Statement of the Problem:**

Jensen's inequality states that for any convex function ##\phi(x)## and a random variable ##X##:

where ##E[\cdot]## denotes the expected value (mean).

(a) Identify why the function ##\phi(v) = v^2## is convex for ##v \ge 0##.

(b) Use Jensen's inequality to prove that the root-mean-square speed ##v_{\text{rms}} = \sqrt{\langle v^2 \rangle}## is always greater than or equal to the average speed ##v_{\text{avg}} = \langle v \rangle##.

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Worked Solution & Step-by-Step Explanation

Part (a): Establishing Convexity

A function ##\phi(v)## is classified as convex on an interval if its second derivative is non-negative throughout that interval. This geometric property ensures that the chord joining any two points on the graph of the function lies above or on the graph itself.

1. **Define the function:**

##\phi(v) = v^2##

2. **First Derivative:**

Applying the power rule, we find the rate of change:

##\phi'(v) = \dfrac{d}{dv}(v^2) = 2v##

3. **Second Derivative:**

Differentiating once more gives the concavity:

##\phi''(v) = \dfrac{d}{dv}(2v) = 2##

4. **Conclusion:**

Since ##\phi''(v) = 2##, and ##2 > 0## for all ##v \in \mathbb{R}##, the function is strictly convex. This satisfies the condition required to apply Jensen's Inequality.

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Part (b): Proving the Speed Relationship

We apply the inequality to the molecular speed ##v##, treating it as our random variable.

1. **Substitution into Jensen's Inequality:**

Substitute ##\phi(v) = v^2## into the inequality ##\phi(E[v]) \le E[\phi(v)]##:

2. **Mapping to Physics Notation:**

In statistical mechanics, the expected value ##E[v]## corresponds to the average speed, denoted as ##\langle v \rangle##. Similarly, the expected value of the squared speed ##E[v^2]## is the mean square speed, denoted as ##\langle v^2 \rangle##.

3. **Algebraic Manipulation:**

Since speed is a non-negative scalar quantity (##v \ge 0##), the average speed ##\langle v \rangle## is also non-negative. We may take the square root of both sides without reversing the inequality:

4. **Definition Application:**

By definition, ##v_{\text{avg}} = \langle v \rangle## and ##v_{\text{rms}} = \sqrt{\langle v^2 \rangle}##. Substituting these yields:

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Summary Table of Statistical Definitions

### Key Takeaways for JEE/NEET

* **Mathematical Rigor:** The inequality ##v_{\text{rms}} \ge v_{\text{avg}}## is a direct consequence of the convexity of the square function.

* **Equality Condition:** Equality (##v_{\text{rms}} = v_{\text{avg}}##) holds if and only if the variance of the speed is zero, meaning every molecule in the gas has the exact same velocity.

* **Physical Meaning:** In any real gas distribution, molecules possess a spread of velocities, which guarantees that ##v_{\text{rms}} > v_{\text{avg}}##.