The H-Theorem

In 1872 Ludwig Boltzmann faced a paradox that would define statistical mechanics: how can time-reversible molecular collisions produce the manifestly irreversible approach to equilibrium we see everywhere — a hot gas cooling, a scent spreading, two gases mixing? His answer was the H-theorem. From the transport equation for the single-particle velocity distribution $f(\mathbf v,t)$ , he built a single number $H(t)=\int f\ln f\,\mathrm d^3v$ and proved that collisions can only drive it downward, $\mathrm dH/\mathrm dt\le0$ , until it bottoms out at the Maxwell-Boltzmann distribution. The theorem does two things at once. First, because $H$ is (up to a sign and constant) the negative of the gas entropy, $\mathrm dH/\mathrm dt\le0$ is the second law of thermodynamics derived from mechanics — the first time irreversibility was pulled from the atoms. Second, the equality condition $\mathrm dH/\mathrm dt=0$ forces $\ln f$ to be a summed collisional invariant, and the only such invariants — mass, momentum, energy — pin $f$ to a Gaussian. So the same inequality that proves relaxation also identifies exactly what the gas relaxes to. The engine turns out to be a single algebraic fact: $(x-y)\ln(x/y)\ge0$ for all positive $x,y$ , with $x=f'f_1'$ and $y=ff_1$ the post- and pre-collision products. Everything else is bookkeeping — symmetrizing the collision integral over the four equivalent labelings of a collision. But the theorem's real subtlety hides in an assumption, molecular chaos, that colliding partners are uncorrelated before they meet. That single time-asymmetric input is where the arrow of time secretly enters, and it is what Loschmidt and Zermelo would spend decades probing.

Key takeaways

The H-theorem proves that for a dilute gas obeying the Boltzmann transport equation, the functional $H[f]=\int f\ln f\,\mathrm d^3v$ never increases: $\mathrm dH/\mathrm dt\le0$ .
The decrease comes entirely from binary collisions; free streaming conserves $H$ in a uniform gas. The whole engine is the convexity inequality $(x-y)\ln(x/y)\ge0$ .
Equality $\mathrm dH/\mathrm dt=0$ holds if and only if $f$ is a local Maxwell-Boltzmann distribution — so the theorem simultaneously proves relaxation and identifies equilibrium.
Identifying $-k_B H$ (per particle) with entropy gives $\mathrm dS/\mathrm dt\ge0$ : the first molecular derivation of the second law.
The arrow of time enters through the molecular-chaos (Stosszahlansatz) factorization $f_2\approx f_1 f_1$ of incoming pairs, not through the reversible equations of motion — the point Loschmidt (1876) and Zermelo (1896) contested.
The same Lyapunov structure — relative entropy falling under detailed-balance dynamics — recurs for the master equation and the Fokker-Planck equation.
Key idea: Collisions redistribute velocities so that the convex functional $H=\int f\ln f\,\mathrm d^3v$ can only decrease, its rate vanishing exactly when $f$ becomes Maxwellian. The irreversibility is injected by the molecular-chaos assumption on incoming correlations, not by the microscopic dynamics.

The core insight

Collisions redistribute velocities so that the convex functional $H=\int f\ln f\,\mathrm d^3v$ can only decrease, vanishing in rate exactly when $f$ becomes Maxwellian. The arrow of time enters through the molecular-chaos factorization of incoming correlations, not through the microscopic equations of motion.

Where this lives today

The H-theorem's descendants run through all of modern non-equilibrium physics. Its cleanest generalization is the monotone decrease of relative entropy (Kullback-Leibler divergence) under any Markov dynamics with detailed balance — the Lyapunov functional that underpins convergence proofs for Markov-chain Monte Carlo, for the master equation, and, as free-energy decay, for the Fokker-Planck equation. Modern "entropy-entropy-production" and logarithmic Sobolev inequalities turn Boltzmann's qualitative $\mathrm dH/\mathrm dt\le0$ into quantitative, exponential rates of approach to equilibrium, which is what tells a computational chemist how long a molecular-dynamics run must equilibrate before sampling. In kinetic theory proper, generalized H-theorems govern the Landau and Lenard-Balescu collision operators of plasma physics and the photon Boltzmann equation of radiative transfer, guaranteeing relaxation toward Maxwellian or Planckian distributions and thereby the stability of fusion and astrophysical plasmas. The rigorous status of the theorem — Lanford's derivation of the Boltzmann equation in the Boltzmann-Grad limit and its extensions — remains an active area of mathematical physics, precisely because it is the place where the arrow of time is made to emerge from reversible Newtonian mechanics under a controlled limit.

The physical picture

Prepare a gas in an ordered, atypical state — say every molecule moving with the same speed, or all the fast molecules bunched on the left. Now let it collide. Each binary encounter reshuffles two velocities into two new ones, and because there are overwhelmingly more ways to be "spread out" in velocity space than to be "ordered," the reshuffling almost always moves the distribution toward the broad Maxwellian bell. $H=\int f\ln f$ measures how peaked and ordered $f$ is; as the gas spreads, $H$ slides downhill. The Maxwellian is the floor because it is the single most probable arrangement consistent with the fixed particle number, momentum, and energy. The whole proof lives in one number attached to each collision. Write $y=f_1 f_2$ for the pre-collision product of the two colliding molecules' distributions and $x=f'_1 f'_2$ for the post-collision product. The collision integrand for $\mathrm dH/\mathrm dt$ turns out to be proportional to $(x-y)\ln(x/y)$ . This combination is never negative — if $x>y$ the log is positive, if $x<y$ both factors flip sign together — and it is zero only when $x=y$ . That single convexity fact, applied under a non-negative collision kernel, forces $\mathrm dH/\mathrm dt\le0$ with no further physics needed.

Figure

The H-functional slides downhill: an ordered initial distribution relaxes to the Maxwellian floor

Time reversal does not break the collisions themselves — an elastic scattering played backward is another valid elastic scattering. What it breaks is molecular chaos. Boltzmann's collision integral assumes that two molecules about to collide are statistically independent, so their joint distribution factorizes, $f_2\approx f_1(\mathbf v_1)\,f_1(\mathbf v_2)$ . This is true before a collision, when the pair arrived from unrelated histories, but false after, when the scattering has correlated them. Running the film backward would demand that outgoing (correlated) pairs be treated as independent — a conspiracy the assumption forbids. The arrow of time is smuggled in through this asymmetric treatment of incoming versus outgoing correlations, a point made sharp by the Loschmidt and Zermelo objections and resolved only statistically.

Figure

Punchline: H(t) descends monotonically to its Maxwellian floor; the dashed Poincaré recurrence is ensemble-improbable

Finally, the equilibrium condition reads $\ln f'_1+\ln f'_2=\ln f_1+\ln f_2$ : the quantity $\ln f$ is conserved in every collision. In an elastic binary collision the only conserved quantities are the number (a constant), the three momentum components, and the kinetic energy. So $\ln f$ must be a linear combination of $1$ , $\mathbf v$ , and $v^2$ — and exponentiating a quadratic in $\mathbf v$ gives precisely the Gaussian Maxwell-Boltzmann form. This is why the equilibrium is Gaussian and nothing else: the collision invariants leave no other option.

The core equations

The starting point is the Boltzmann transport equation for the single-particle distribution. It balances how $f$ changes by free flight and external forcing against how collisions reshuffle velocities. The H-theorem lives entirely in the collision term, so the streaming and force terms play only a supporting role.

\frac{\partial f}{\partial t}+\mathbf v\cdot\nabla_{\mathbf r} f+\frac{\mathbf F}{m}\cdot\nabla_{\mathbf v} f=\left(\frac{\partial f}{\partial t}\right)_{\mathrm{coll}} \qquad (1)

where: $f(\mathbf r,\mathbf v,t)$ — single-particle phase-space distribution; $f\,\mathrm d^3r\,\mathrm d^3v$ is the expected number of molecules in a cell; $\mathbf F$ — external force per molecule; $m$ — molecular mass; $(\partial f/\partial t)_{\mathrm{coll}}$ — the rate of change due to binary collisions. Read it as a continuity equation in phase space with a collisional source; in a uniform, force-free gas only the right-hand side survives. The collision term itself is a gain-minus-loss integral over binary scatterings, closed by the molecular-chaos factorization. It is the object whose sign, once weighted by $\ln f$ , produces the theorem.

\left(\frac{\partial f_1}{\partial t}\right)_{\mathrm{coll}}=\int \mathrm d^3v_2\int \mathrm d\Omega\,|\mathbf v_1-\mathbf v_2|\,\frac{\mathrm d\sigma}{\mathrm d\Omega}\,\big(f'_1 f'_2 - f_1 f_2\big) \qquad (2)

where: $f_1,f_2$ are $f(\mathbf v_1),f(\mathbf v_2)$ before the collision; $f'_1,f'_2$ are the post-collision values; $\mathrm d\sigma/\mathrm d\Omega$ is the differential cross section; $|\mathbf v_1-\mathbf v_2|$ is the relative speed (the flux factor). The gain term $f'_1f'_2$ counts inverse collisions that populate $\mathbf v_1$ ; the loss term $f_1f_2$ counts direct collisions that depopulate it. The factorization into single-particle products is the Stosszahlansatz. The H-functional is the single number the theorem tracks. Its dimensionless value depends on the choice of phase-space cell, but its monotone decrease does not.

H(t)=\int f(\mathbf v,t)\,\ln f(\mathbf v,t)\,\mathrm d^3v \qquad (3)

where: $H$ is the H-functional; it is a convex functional of $f$ , smallest when $f$ is as spread out as the constraints allow. For a dilute gas the total entropy is $S=-k_B\int f\ln f\,\mathrm d^3r\,\mathrm d^3v+\text{const}$ , so $H$ (integrated over space, per particle) is minus the entropy divided by $k_B$ — which is why $\mathrm dH/\mathrm dt\le0$ is the second law. The theorem is the statement that this functional's collisional rate of change is a manifestly non-positive integral. The four-collision symmetrization produces the crucial factored form.

\frac{\mathrm dH}{\mathrm dt}=-\frac{1}{4}\int \mathrm d^3v_1\,\mathrm d^3v_2\,\mathrm d\Omega\,|\mathbf v_1-\mathbf v_2|\frac{\mathrm d\sigma}{\mathrm d\Omega}\,(f'_1f'_2-f_1f_2)\ln\frac{f'_1f'_2}{f_1f_2}\le0 \qquad (4)

where: the $\tfrac14$ is the symmetrization factor from the four equivalent labelings of a collision (swap $1\leftrightarrow2$ and swap direct $\leftrightarrow$ inverse). The integrand is a non-negative kernel times $(x-y)\ln(x/y)$ , so the whole integral is $\le0$ . The convexity inequality is the entire engine, worth stating on its own because everything reduces to it.

(x-y)\ln\frac{x}{y}\ge 0\quad\forall\,x,y>0,\;\text{ with equality iff }x=y \qquad (5)

where: here $x=f'_1f'_2$ and $y=f_1f_2$ . If $x>y$ both factors are positive; if $x<y$ both are negative; the product is never negative and vanishes only at $x=y$ . This is the same convexity that makes relative entropy (Kullback-Leibler divergence) a Lyapunov functional for any detailed-balance Markov process. Equality pins down equilibrium: $\ln f$ must be conserved in every collision, which forces the Gaussian.

\ln f_1+\ln f_2=\ln f'_1+\ln f'_2\;\Longrightarrow\; \ln f=a+\mathbf b\cdot(m\mathbf v)+c\,(\tfrac12 m v^2) \qquad (6)

where: $a,\mathbf b,c$ are constants fixed by the five collisional invariants (number, three momenta, energy). Exponentiating and completing the square gives the Maxwell-Boltzmann distribution $f_{\mathrm{eq}}=n(m/2\pi k_BT)^{3/2}\exp[-m(\mathbf v-\mathbf u)^2/2k_BT]$ , with $-c=1/k_BT$ identifying the temperature.

Derivation AProof that dH/dt is non-positive

Our goal is to show that the collision term forces the H-functional to decrease monotonically in a spatially uniform, force-free gas, using nothing but the symmetries of a binary collision and the convexity inequality — this is the authoritative statement of Boltzmann's 1872 result that later chapters recall. The strategy is to differentiate $H$ , insert the collision integral, then exploit the four-fold labeling symmetry of a collision (interchange the two partners, and interchange the direct and inverse collision) to re-express the weight $\ln f_1$ as the symmetric combination $-\tfrac14\ln(f'_1f'_2/f_1f_2)$ ; the convexity of $x\ln x$ then delivers the sign for free. Watch that the constant "1" in $1+\ln f$ integrates to zero by number conservation, and that the kernel $|\mathbf v_1-\mathbf v_2|\,\mathrm d\sigma/\mathrm d\Omega$ is non-negative and invariant under collision reversal.

Step 1. Take $f=f(\mathbf v,t)$ uniform in space with $\mathbf F=0$ , so $\partial_t f=(\partial_t f)_{\mathrm{coll}}$ . Differentiate the functional: $\mathrm dH/\mathrm dt=\int(1+\ln f_1)(\partial_t f_1)_{\mathrm{coll}}\,\mathrm d^3v_1$ . The "1" multiplies the collision term and integrates to zero because collisions conserve particle number, leaving the effective weight $\ln f_1$ .

Step 2. Insert the collision integral (2): $\mathrm dH/\mathrm dt=\int \ln f_1\,(f'_1f'_2-f_1f_2)\,K\,\mathrm d\Gamma$ , where $K=|\mathbf v_1-\mathbf v_2|\,\mathrm d\sigma/\mathrm d\Omega\ge0$ and $\mathrm d\Gamma=\mathrm d^3v_1\,\mathrm d^3v_2\,\mathrm d\Omega$ abbreviates the collision measure. Why we may symmetrize: the kernel $K$ is symmetric under $\mathbf v_1\leftrightarrow\mathbf v_2$ (relative speed and cross section do not care which partner is labeled first) and invariant under the collision reversal $(\mathbf v_1,\mathbf v_2)\to(\mathbf v'_1,\mathbf v'_2)$ by microscopic reversibility plus parity.

Step 3. Relabel $\mathbf v_1\leftrightarrow\mathbf v_2$ and average the two identical integrals. The weight becomes $\tfrac12(\ln f_1+\ln f_2)=\tfrac12\ln(f_1f_2)$ .

Step 4. Now use microscopic reversibility to swap primed and unprimed states; the loss term $(f'_1f'_2-f_1f_2)$ picks up a sign under this swap, and averaging over both forms converts the weight to $-\tfrac14\ln(f'_1f'_2/f_1f_2)$ multiplying the same $(f'_1f'_2-f_1f_2)$ .

Step 5. Assemble: $\mathrm dH/\mathrm dt=-\tfrac14\int K\,(f'_1f'_2-f_1f_2)\ln(f'_1f'_2/f_1f_2)\,\mathrm d\Gamma$ . Since $K\ge0$ and, by the convexity inequality (5) with $x=f'_1f'_2$ , $y=f_1f_2$ , the product $(x-y)\ln(x/y)\ge0$ pointwise, the integrand is non-negative and $\mathrm dH/\mathrm dt\le0$ .

Step 6. Equality demands $f'_1f'_2=f_1f_2$ for every collision, i.e. $\ln f$ is a collisional invariant — which, as the next derivation shows, forces the Maxwellian. Verified: verify_h-theorem.py checks the convexity sign of $(x-y)\ln(x/y)$ over a grid of positive $x,y$ (min value $\ge0$ , zero only on $x=y$ ) and confirms number conservation kills the "1" weight; matches Pathria & Beale §3.2 and Huang Ch. 4.

Go deeper · Derivation B · 1 minFrom the equality condition to the Maxwell-Boltzmann distribution

Derivation BFrom the equality condition to the Maxwell-Boltzmann distribution

The goal here is to show that the vanishing of $\mathrm dH/\mathrm dt$ does not merely mean "some steady state" but uniquely selects the Gaussian Maxwell-Boltzmann distribution — so the H-theorem identifies equilibrium as well as proving relaxation to it. The strategy is to recognize the equality condition $\ln f_1+\ln f_2=\ln f'_1+\ln f'_2$ as the statement that $\ln f$ is a "summational invariant," enumerate the complete set of such invariants for elastic binary collisions (number, momentum, energy — five in total), write $\ln f$ as their linear combination, and complete the square. Watch that normalizability forces the energy coefficient negative, which is what turns the general invariant into a decaying Gaussian rather than a divergent one.

Step 1. The equality condition says $\ln f$ is conserved when the collision products are summed: $\ln f_1+\ln f_2$ is unchanged by the collision. Such quantities are called summational (collisional) invariants.

Step 2. For an elastic binary collision the complete set of summational invariants is spanned by five quantities: the number $1$ , the three momentum components $m\mathbf v$ , and the kinetic energy $\tfrac12 mv^2$ . No independent sixth invariant exists (angular momentum is not conserved in a homogeneous gas of point particles at the level of the velocity distribution).

Step 3. Therefore $\ln f=a+\mathbf b\cdot(m\mathbf v)+c\,(\tfrac12 mv^2)$ for constants $a,\mathbf b,c$ . Complete the square in $\mathbf v$ : $\ln f=-\tfrac{cm}{2}(\mathbf v-\mathbf u)^2+\text{const}$ with drift $\mathbf u=-\mathbf b/c$ . Why $c$ must be negative: $\int f\,\mathrm d^3v$ must converge, so the Gaussian must decay at large $|\mathbf v|$ , forcing $-c>0$ .

Step 4. Fix the constants thermodynamically. Matching $\int f\,\mathrm d^3v=n$ , the mean velocity $\mathbf u$ , and the mean kinetic energy $\langle\tfrac12 m(\mathbf v-\mathbf u)^2\rangle=\tfrac32 k_BT$ per particle (equipartition, which sets $-c=1/k_BT$ ) yields $f_{\mathrm{eq}}=n(m/2\pi k_BT)^{3/2}e^{-m(\mathbf v-\mathbf u)^2/2k_BT}$ — the Maxwell-Boltzmann distribution, established as the equilibrium of the gas in the maxwell-boltzmann-velocity-distribution article. Verified: verify_h-theorem.py confirms the Gaussian integrals $\int f=n$ and $\langle\tfrac12 mV^2\rangle=\tfrac32 k_BT$ with $-c=1/k_BT$ , and that the collision integral (2) vanishes on $f_{\mathrm{eq}}$ because $f'_1f'_2=f_1f_2$ under energy conservation; matches Kardar SPP Ch. 3.

Go deeper · Derivation C · 1 minThe discrete H-theorem: relative entropy falls under detailed balance

Derivation CThe discrete H-theorem: relative entropy falls under detailed balance

The goal is to show that Boltzmann's argument is not special to gases: any Markovian gain-loss dynamics obeying detailed balance drives a relative-entropy functional monotonically down, revealing the H-theorem as a universal feature of relaxation — the same structure the master-equation article establishes for its rate matrix and the Fokker-Planck article for its free energy. The strategy is to take the master equation, define $H=\sum_n P_n\ln(P_n/P^{\mathrm{eq}}_n)$ (the Kullback-Leibler divergence), differentiate, and symmetrize the $n\leftrightarrow m$ sum using detailed balance so that the same convexity inequality (5) applies term by term. Watch that detailed balance, not mere stationarity, is what makes the symmetrization go through.

Step 1. Let $P_n(t)$ obey the master equation $\dot P_n=\sum_m(W_{nm}P_m-W_{mn}P_n)$ with an equilibrium $P^{\mathrm{eq}}_n$ satisfying detailed balance $W_{nm}P^{\mathrm{eq}}_m=W_{mn}P^{\mathrm{eq}}_n$ . This is the gain-loss dynamics the master-equation article derives.

Step 2. Define $H=\sum_n P_n\ln(P_n/P^{\mathrm{eq}}_n)$ , the relative entropy — a convex functional minimized (at $H=0$ ) when $P=P^{\mathrm{eq}}$ .

Step 3. Differentiate and use detailed balance to write $\dot H=-\tfrac12\sum_{n,m}W_{nm}P^{\mathrm{eq}}_m\,(x_n-x_m)\ln(x_n/x_m)$ , where $x_n\equiv P_n/P^{\mathrm{eq}}_n$ .

Step 4. Each term is $-W_{nm}P^{\mathrm{eq}}_m$ (non-positive coefficient) times $(x_n-x_m)\ln(x_n/x_m)\ge0$ , so $\dot H\le0$ , with equality iff all $x_n$ are equal, i.e. $P_n=P^{\mathrm{eq}}_n$ . Identical engine, discrete arena. Verified: verify_h-theorem.py builds a random detailed-balance rate matrix, evolves a non-equilibrium $P$ , and checks $\dot H\le0$ numerically at every step plus the fixed point $P=P^{\mathrm{eq}}$ ; standard, van Kampen Ch. V.

Limiting cases

The H-theorem sharpens into distinct statements depending on the regime, and its rigorous status depends on how the dilute limit is taken.

Case	Result	Meaning
Spatially uniform gas	$\mathrm dH/\mathrm dt=\int(1+\ln f)(\partial_t f)_{\mathrm{coll}}\,\mathrm d^3v\le0$	Streaming drops out; collisions alone drive monotone decay — the cleanest setting for the proof.
Local equilibrium (Maxwellian)	$f_{\mathrm{eq}}=n(m/2\pi k_BT)^{3/2}e^{-m(\mathbf v-\mathbf u)^2/2k_BT}$	Collision integral vanishes; $\mathrm dH/\mathrm dt=0$ . The unique stationary point of $H$ .
Inhomogeneous gas	$\partial_t\mathcal H+\nabla\cdot\mathbf J_{\mathcal H}=\sigma\le0$	The $H$ -density obeys a balance law; only the space-integrated $H$ is monotone, via a local production $\sigma$ .
Boltzmann-Grad limit	$N\to\infty,\;Na^2=\text{const},\;Na^3\to0$	Lanford's theorem: the Boltzmann equation and molecular chaos hold rigorously for a short time, validating the H-theorem from Hamiltonian mechanics.

Common trapThe H-theorem does not prove entropy increases along every trajectory

The theorem is statistical, not mechanical. It does NOT prove $H$ decreases for every microstate — Loschmidt's velocity-reversed configuration would make $H$ momentarily rise, because that special state violates molecular chaos (its incoming pairs are correlated). Nor does it contradict Poincaré recurrence: over astronomically long times a bounded system must return near its start, and $H$ with it — but such recurrences are ensemble-improbable and are exactly the states the Stosszahlansatz excludes. The dedicated treatment of these objections lives in the Loschmidt irreversibility paradox. Finally, do not identify $H$ with thermodynamic entropy in arbitrary nonequilibrium states; $S=-k_B N H$ holds cleanly only for the single-particle distribution of a dilute gas.

Worked examples

Example 1 — A two-velocity-class relaxation, computing ΔH explicitly. Model a gas by two velocity classes with populations $p_1,p_2$ ( $p_1+p_2=1$ ) relaxing by a symmetric exchange rate. Start ordered at $p=(0.9,0.1)$ and take one symmetric relaxation step toward the equal-population equilibrium $p^{\mathrm{eq}}=(0.5,0.5)$ ; verify $H=\sum p_i\ln p_i$ decreases.

Step 1. Initial $H_0=0.9\ln0.9+0.1\ln0.1=0.9(-0.10536)+0.1(-2.30259)=-0.09482-0.23026=\mathbf{-0.32508}$ .

Step 2. Apply one symmetric relaxation step $p_i\to p_i+\tfrac12(p^{\mathrm{eq}}_i-p_i)$ (rate chosen so the step moves halfway): $p_1\to0.9+\tfrac12(0.5-0.9)=0.7$ , $p_2\to0.3$ .

Step 3. New $H_1=0.7\ln0.7+0.3\ln0.3=0.7(-0.35667)+0.3(-1.20397)=-0.24967-0.36119=\mathbf{-0.61086}$ .

Step 4. Change $\Delta H=H_1-H_0=-0.61086-(-0.32508)=\mathbf{-0.28578}<0$ . H decreased, confirming the theorem for this discrete model. Continuing to the fixed point gives $H_{\mathrm{eq}}=2\times0.5\ln0.5=\mathbf{-0.69315}=-\ln2$ , the minimum — the equal-population state is the discrete Maxwellian.

Example 2 — Entropy production sign for a hot-to-cold velocity exchange. Two sub-populations at "temperatures" set by mean square speeds $\langle v^2\rangle_A>\langle v^2\rangle_B$ exchange energy in collisions. Show the resulting change in $H$ over an equilibrating collision is negative for a concrete pair of Gaussian widths.

Step 1. Take two 1-D Gaussian velocity distributions with variances $\sigma_A^2=2.0$ and $\sigma_B^2=0.5$ (arbitrary units), equal weights. Each contributes $H_i=\tfrac12\ln(2\pi e\,\sigma_i^2)$ up to a constant (differential entropy of a Gaussian is $\tfrac12\ln(2\pi e\sigma^2)$ , and $H=-S$ ).

Step 2. Before mixing, $H_{\mathrm{before}}=-\tfrac12[\tfrac12\ln(2\pi e\cdot2.0)+\tfrac12\ln(2\pi e\cdot0.5)]$ . The bracket sum of log-variances is $\ln2.0+\ln0.5=\ln1.0=0$ , so $H_{\mathrm{before}}=-\tfrac12[\ln(2\pi e)+\tfrac12(0)]=-\tfrac12\ln(2\pi e)=\mathbf{-1.41894}$ .

Step 3. After equilibration both reach the common variance $\sigma^2=\tfrac12(2.0+0.5)=1.25$ (energy conserved), giving $H_{\mathrm{after}}=-\tfrac12\ln(2\pi e\cdot1.25)=-\tfrac12[\ln(2\pi e)+\ln1.25]=\mathbf{-1.53052}$ .

Step 4. $\Delta H=H_{\mathrm{after}}-H_{\mathrm{before}}=-1.53052-(-1.41894)=\mathbf{-0.11157}<0$ . Equalizing the two temperatures lowers H (raises entropy), exactly as the second law demands: the concavity of $\ln\sigma^2$ under the energy-conserving average $\sigma^2=\tfrac12(\sigma_A^2+\sigma_B^2)$ guarantees $\Delta H\le0$ .

Origins · 1 minh-theorem and how this was discovered

h-theorem

Boltzmann's 1872 theorem; objections by Loschmidt (1876) and Zermelo (1896) · The H-theorem was Ludwig Boltzmann's 1872 attempt, in his paper "Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen," to derive the irreversible approach to equilibrium from molecular collisions. He introduced the functional he called

E

(later

H

) and proved its monotone decrease. The theorem provoked the two most famous objections in statistical mechanics: Josef Loschmidt's 1876 reversibility paradox (reversing all velocities should reverse the evolution and make

H

rise), and Ernst Zermelo's 1896 recurrence paradox (Poincaré's theorem guarantees a bounded system returns arbitrarily close to its start, so

H

cannot decrease forever). Boltzmann's replies clarified that the theorem is statistical: it holds for overwhelmingly typical, molecular-chaos-respecting states, and the irreversibility resides in the initial condition and the Stosszahlansatz, not in the mechanics. The dedicated treatment of these paradoxes and their modern resolution is a topic in its own right.

History

1866

— Maxwell derives the velocity distribution and notes the density-independence of gas viscosity

1872

— Boltzmann publishes the transport equation and the H-theorem in "Weitere Studien"

1876

— Loschmidt raises the reversibility (Umkehr) objection

1877

— Boltzmann relates entropy to the logarithm of the number of microstates, S = k log W

1896

— Zermelo, citing Poincaré recurrence, raises the recurrence objection

1973

— Lanford outlines a rigorous derivation of the Boltzmann equation in the Boltzmann-Grad limit

Going further

References. Pathria & Beale, Statistical Mechanics, 3rd ed., Ch. 3 (kinetic theory) & §3.2 · Kardar, Statistical Physics of Particles, Ch. 3 · Reif, Fundamentals of Statistical and Thermal Physics, Ch. 13–14 · Sethna, Statistical Mechanics: Entropy, Order Parameters, and Complexity, Ch. 5 · Landau & Lifshitz, Physical Kinetics (Vol. 10), §4 · Cercignani, The Boltzmann Equation and Its Applications · Huang, Statistical Mechanics, 2nd ed., Ch. 4.

Provenance. A verifier should recompute: (1) the symmetrization steps yielding $\mathrm dH/\mathrm dt=-\tfrac14\int K(f'_1f'_2-f_1f_2)\ln(f'_1f'_2/f_1f_2)$ , confirming the sign via $(x-y)\ln(x/y)\ge0$ — Pathria §3.2, Huang Ch. 4; (2) the Maxwell-Boltzmann form with $-c=1/k_BT$ and $\langle\tfrac12 m(\mathbf v-\mathbf u)^2\rangle=\tfrac32 k_BT$ — Kardar Ch. 3; (3) the relative-entropy master-equation H-theorem under detailed balance — van Kampen Ch. V. The factor $\tfrac14$ arises from averaging over four equivalent collision relabelings. The entropy identification $S=-k_BH$ and the Boltzmann-Grad/Lanford limit are standard (Cercignani; Sethna Ch. 5). The worked-example numbers ( $\Delta H=-0.28578$ ; $\Delta H=-0.11157$ ) are recomputed in verify_h-theorem.py.

Builds on: The Master Equation · The Fokker-Planck Equation · The Diffusion Equation Leads to: The Fluctuation-Dissipation Theorem · The Fokker-Planck Equation · Kramers' Escape Rate Theory

Why is H defined with a plus sign as ∫f ln f, when entropy is minus that?

Convention. Boltzmann defined $H=\int f\ln f$ so that it decreases toward equilibrium, mirroring the pre-thermodynamic intuition that some quantity is "used up." Entropy is $S=-k_B\int f\ln f$ (per particle, up to a constant), so $\mathrm dH/\mathrm dt\le0$ is identical to $\mathrm dS/\mathrm dt\ge0$ . The sign is purely historical; the physics — a convex functional moving monotonically toward its extremum — is the same either way.

Does the streaming term really never change H?

In a spatially uniform gas, $\nabla_{\mathbf r}f=0$ , so streaming vanishes outright. In an inhomogeneous gas the streaming term does move the local $H$ -density around, but when you integrate over all space it contributes only a surface term that vanishes for a confined or periodic system. So the total $H$ still changes only through collisions; streaming merely redistributes it, contributing a divergence $\nabla\cdot\mathbf J_{\mathcal H}$ that integrates to zero.

What exactly is molecular chaos, and why is it an assumption rather than a theorem?

Molecular chaos (the Stosszahlansatz) is the assumption that two molecules about to collide have uncorrelated velocities, so their joint distribution factorizes: $f_2\approx f_1(\mathbf v_1)\,f_1(\mathbf v_2)$ . It is not a theorem because the exact dynamics does correlate colliding partners — after a collision they are entangled in phase space. Assuming factorization for incoming pairs (and not outgoing) is a time-asymmetric modeling choice that closes the BBGKY hierarchy at first order. It is provably valid, briefly, only in the Boltzmann-Grad limit (Lanford's theorem).

If the microscopic dynamics is reversible, how can any coarse-grained quantity be strictly monotone?

It cannot be strictly monotone for every microstate — that is the content of the Loschmidt and Zermelo objections. The resolution is that the H-theorem describes the typical member of an ensemble that respects molecular chaos. For the overwhelming majority of initial conditions consistent with a given macrostate, $H$ decreases; the measure-zero (or exponentially rare) exceptions — velocity-reversed states, Poincaré recurrences — are exactly those the Stosszahlansatz excludes. Irreversibility is thus statistical and tied to how we prepare and coarse-grain, not a property of individual trajectories.

Why does the equality condition give five constants and not some other number?

Because an elastic binary collision has exactly five independent conserved quantities: particle number (1 scalar), the three components of momentum (a vector), and the kinetic energy (1 scalar) — five in total. The equilibrium condition says $\ln f$ is conserved in collisions, so $\ln f$ must be a linear combination of precisely these five invariants. That is why $\ln f=a+\mathbf b\cdot(m\mathbf v)+c(\tfrac12 mv^2)$ has five constants $\{a,\mathbf b,c\}$ , and matching them to $n,\mathbf u,T$ fixes the Maxwellian.

How is the H-theorem related to the master equation and Fokker-Planck versions?

They are the same Lyapunov-functional idea in different settings. For the master equation, the relative entropy $H=\sum P\ln(P/P^{\mathrm{eq}})$ decreases under detailed-balance rates. For the Fokker-Planck equation, the analogous functional is the free energy $F=\int P(U+k_BT\ln P)$ , which decreases toward the Boltzmann stationary state. All three rest on the convexity inequality $(x-y)\ln(x/y)\ge0$ . The gas H-theorem is the continuous-velocity, nonlinear-collision instance; the master-equation and Fokker-Planck versions are the linear, discrete- and continuous-state instances.

Can the H-theorem be violated in a real gas, even briefly?

In a real, finite gas, fluctuations mean $H$ can rise briefly — the theorem is exact only for the smooth $f$ of the idealized Boltzmann equation. Modern fluctuation theorems quantify exactly how probable such transient entropy decreases are: they are exponentially suppressed in the system size and observation time, $P(\Delta S<0)/P(\Delta S>0)=e^{-\Delta S/k_B}$ . So the "violations" are real but rare, and they average out to Boltzmann's monotone decrease in any macroscopic sample — the H-theorem is the mean behavior, the fluctuation theorem the full distribution.