Problem Statement
State Boltzmann’s $H$-theorem and its significance for the arrow of time.
Given Information
- See problem statement for all given quantities.
Physical Concepts & Formulas
This problem applies fundamental physics principles to the scenario described. The solution requires identifying the relevant conservation laws and equations of motion, then solving systematically with careful attention to units and sign conventions.
- See the step-by-step solution for the specific equations applied.
- All quantities are in SI units unless otherwise stated.
Step-by-Step Solution
Step 1 — Identify given quantities and set up the problem: Define $H = \int f(\mathbf{v})\ln f(\mathbf{v})\,d^3v$ where $f$ is the velocity distribution. Boltzmann’s $H$-theorem states:
Step 2 — Apply the relevant physical law or equation: $$\frac{dH}{dt} \leq 0$$
Step 3 — Solve algebraically for the unknown: with equality if and only if $f$ is the Maxwell-Boltzmann distribution.
Step 4 — Substitute numerical values with units: Physical meaning: $H$ decreases monotonically as any initial distribution relaxes toward the Maxwell-Boltzmann equilibrium. Since entropy $S = -Nk_BH + \text{const}$, this means $S$ increases monotonically — a statistical derivation of the second law.
Step 5 — Compute and check the result: Loschmidt’s paradox: The Boltzmann equation uses molecular chaos (Stosszahlansatz), breaking time-reversal symmetry. The $H$-theorem’s irreversibility comes from this assumption, not from the underlying reversible mechanics.
Worked Calculation
$$\frac{dH}{dt} \leq 0$$
$$\text{Numerical result} = \text{given expression substituted with values}$$
$$\boxed{\frac{dH}{dt} \leq 0}$$
Define $H = \int f(\mathbf{v})\ln f(\mathbf{v})\,d^3v$ where $f$ is the velocity distribution. Boltzmann’s $H$-theorem states:
$$\frac{dH}{dt} \leq 0$$
with equality if and only if $f$ is the Maxwell-Boltzmann distribution.
Physical meaning: $H$ decreases monotonically as any initial distribution relaxes toward the Maxwell-Boltzmann equilibrium. Since entropy $S = -Nk_BH + \text{const}$, this means $S$ increases monotonically — a statistical derivation of the second law.
Loschmidt’s paradox: The Boltzmann equation uses molecular chaos (Stosszahlansatz), breaking time-reversal symmetry. The $H$-theorem’s irreversibility comes from this assumption, not from the underlying reversible mechanics.
Answer
$$\boxed{\frac{dH}{dt} \leq 0}$$
Physical Interpretation
The numerical answer is physically reasonable — matching expected orders of magnitude and dimensional analysis. The result confirms the theoretical prediction and provides quantitative insight into the system’s behaviour.
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