The Second Law from Statistics

Drop a hot stone into cold water and the heat always flows one way. Let a gas out of a bottle and it always fills the room; it never crawls back in. These everyday certainties are the second law of thermodynamics — the entropy of an isolated system never decreases — and they are so reliable that Eddington called the law's status "supreme among the laws of Nature." Yet the microscopic equations of motion, Newtonian or quantum, are perfectly symmetric in time. Reverse every velocity and they run backward just as happily. So where does the arrow come from? Nothing in the dynamics prefers the future. The answer, Boltzmann's, is that the second law is not a law of dynamics at all — it is a law of counting. A macrostate (gas uniform, or gas crammed in the left half) corresponds to a region of phase space, and the volume of that region is the multiplicity $\Omega$ : the number of microscopic arrangements realizing it. The catch is the sheer scale of the inequality. Because $\Omega\sim e^{S/k_{\mathrm B}}$ with $S\propto N$ , two macrostates differing by an ordinary amount of entropy differ in size by a factor like $e^{10^{23}}$ . Release a constraint and the system, wandering blindly among equally-weighted microstates, is overwhelmingly likely to end up in the vastly larger region. Entropy increase is nothing but a system falling into the biggest available bucket. This article proves that mechanism cleanly: removing an internal constraint can only enlarge the accessible phase space, so $\Omega_{\text{new}}\ge\Omega_{\text{old}}$ exactly, with equality only if the constraint was already slack — hence $\Delta S\ge0$ . We compute the canonical case, a free expansion, and get the textbook $\Delta S=Nk_{\mathrm B}\ln 2$ . The famous objections of Loschmidt and Zermelo — that reversed and recurrent trajectories must exist — are real and are answered in full by Statistical Origin of Irreversibility; here we establish the positive statistical content: the second law is the overwhelmingly probable behaviour of a system with astronomically many degrees of freedom.

Key takeaways

The second law $\Delta S\ge0$ for an isolated system is a statistical statement, not a dynamical theorem — the microscopic laws are time-symmetric.
A macrostate has multiplicity $\Omega\sim e^{S/k_{\mathrm B}}$ ; two macrostates differing by extensive entropy differ in phase-space volume by $\sim e^{N}$ .
Under equal a priori probabilities a system is found in the largest accessible macrostate with effective certainty, so releasing a constraint drives it to larger $\Omega$ .
Removing an internal constraint can only add accessible microstates, so $\Omega_{\text{new}}\ge\Omega_{\text{old}}$ exactly, giving $\Delta S\ge0$ with equality iff the constraint was already inactive.
Free expansion of $N$ particles into double the volume gives $\Delta S=Nk_{\mathrm B}\ln 2$ — a concrete instance computed here.
Key idea: entropy increases because a system flows from a macrostate of small $\Omega$ to one of overwhelmingly larger $\Omega$ ; the arrow of time is the arrow of counting, and $\Delta S\ge0$ is a statement of probability, not of mechanics.

The core insight

The second law is a counting statement. Each macrostate is a region of phase space of volume $\Omega\sim e^{S/k_{\mathrm B}}$ , and the equilibrium macrostate is exponentially larger than any other. A system released from a constraint wanders into the biggest accessible region simply because it holds almost all the microstates — so entropy increases with overwhelming probability, not by any asymmetry in the equations of motion.

Where this lives today

The statistical reading of the second law is now a working tool, not just a philosophical resolution. Modern fluctuation theorems (Jarzynski 1997, Crooks 1999) sharpen Eq. (4) into exact equalities: the ratio of probabilities for entropy-producing and entropy-consuming trajectories is precisely $e^{\Delta S/k_{\mathrm B}}$ , and this has been confirmed by dragging single RNA molecules and colloidal beads through optical traps, where the machine literally watches individual trajectories momentarily decrease entropy at exactly the predicted rate. The second law survives as the statement that the average obeys $\langle\Delta S\rangle\ge0$ while individual small-system realizations fluctuate both ways. The same logic underwrites the thermodynamics of information and computation. Landauer's principle — erasing a bit costs at least $k_{\mathrm B}T\ln 2$ — is Eq. (5) read backwards: erasure compresses two macrostates into one, an entropy decrease of the memory that must be paid for by dumping $k_{\mathrm B}\ln 2$ of entropy to the environment. Reversible-computing research and the design of ultra-low-power logic take this bound seriously, and it has been measured directly in single-electron and colloidal bit-erasure experiments.

The physical picture

Think of the accessible phase space as a landscape carved into macrostate-regions of wildly unequal size. The "gas in the left half" region is a village; the "gas uniform" region is a continent so much larger that the village is invisible on its map. A trajectory, sampling microstates with equal weight, spends essentially all its time on the continent — not because the continent pulls, but because that is where the overwhelming majority of microstates live. Start it in the village (a low-entropy initial condition) and it will almost surely wander out onto the continent and stay there. That one-way drift is the second law. The disparity is not merely large, it is beyond intuition. An extensive entropy difference of $\Delta S = k_{\mathrm B}\cdot N$ means a phase-space volume ratio of $e^{N}$ . For a mole, $e^{10^{23}}$ — a number whose digit count itself has 23 digits. The figure below shows the two competing regions on a log scale, the only scale on which the small one is even visible:

Figure

Two macrostates, one continent: entropy is the log of the region size

Now the payoff: what does "overwhelmingly larger" mean as a probability? The chance of finding a fluctuation that lowers entropy by an amount $\Delta S$ is $e^{-\Delta S/k_{\mathrm B}}$ . Plot that against the size of the entropy dip and the curve falls off a cliff — a dip of even a nanojoule-per-kelvin is suppressed by $e^{-10^{14}}$ :

Figure

Probability of an entropy-decreasing fluctuation falls off a cliff: exp(−ΔS/k)

The two figures together are the whole logic: the equilibrium region is exponentially the largest (figure one), and any macroscopic excursion out of it is exponentially improbable (figure two). Neither figure invokes an asymmetric force — both are pure counting. The microscopic dynamics remains reversible; the statistics is what points.

The core equations

The bridge from counting to thermodynamics is Boltzmann's relation, owned and derived in Boltzmann Entropy; here it is the tool that converts a volume ratio into an entropy change:

S = k_{\mathrm B}\ln\Omega \qquad (1)

where: $S$ — entropy of a macrostate; $\Omega$ — its multiplicity (number of accessible microstates, or dimensionless phase-space volume); $k_{\mathrm B}=1.380649\times10^{-23}\ \mathrm{J/K}$ (exact, 2019 SI). Because $S\propto N$ , $\Omega$ is exponentially large — the source of every disparity in this article. The entropy change between two macrostates is therefore the logarithm of their volume ratio:

\Delta S = S_{\text{final}} - S_{\text{initial}} = k_{\mathrm B}\ln\frac{\Omega_{\text{final}}}{\Omega_{\text{initial}}} \qquad (2)

where: $\Omega_{\text{final}},\Omega_{\text{initial}}$ — multiplicities of the final and initial macrostates. Read physically: $\Delta S>0$ means the system moved to a macrostate realized by exponentially more microstates. The engine of the second law is what happens when an internal constraint is removed. The set of accessible microstates can only grow, because every previously-allowed microstate is still allowed:

\Omega_{\text{new}} \ge \Omega_{\text{old}} \quad\Longrightarrow\quad \Delta S = k_{\mathrm B}\ln\frac{\Omega_{\text{new}}}{\Omega_{\text{old}}} \ge 0 \qquad (3)

where: equality holds iff the removed constraint was already inactive (added no new microstates). This is the second law in its sharpest form — a set inclusion, not a dynamical claim; the accessible region is a superset after relaxation. The probability of the reverse — a spontaneous entropy decrease by $\Delta S$ — is the inverse volume ratio, and it is astronomically small:

P(\text{drop by }\Delta S) = \frac{\Omega_{\text{small}}}{\Omega_{\text{large}}} = e^{-\Delta S/k_{\mathrm B}} \qquad (4)

where: $\Delta S$ — the size of the entropy decrease. For any macroscopic $\Delta S$ this is $e^{-N}$ -class: the fluctuations plotted in the second figure. The law is violated in principle but never in practice. For the concrete case of a free expansion — gas released from volume $V_i$ into $V_f$ with energy fixed — only the configuration integral changes, giving a clean result:

\Delta S = N k_{\mathrm B}\ln\frac{V_f}{V_i} \qquad (5)

where: $V_i,V_f$ — initial and final volumes; $N$ — particle number. Each particle independently gains a factor $V_f/V_i$ of accessible positions, so $\Omega$ multiplies by $(V_f/V_i)^N$ and the log makes it extensive. Finally, the recurrence and reversal objections are quantified by the two timescales that make the statistical law effectively absolute:

t_{\text{Poincaré}} \sim e^{N},\qquad P_{\text{reversal}} \sim e^{-N} \qquad (6)

where: $t_{\text{Poincaré}}$ — the mean time for a bounded system to return near its start (in units of a microscopic time); $P_{\text{reversal}}$ — the measure of entropy-decreasing trajectories. Both are treated fully in Statistical Origin of Irreversibility; here they simply certify that "overwhelmingly probable" means "never observed."

Derivation ARemoving a constraint can only raise the entropy

This is the article's owned structural result, and it is exact — not asymptotic, not statistical, but a statement about sets. When an internal constraint on an isolated system is relaxed (a partition is punctured, a catalyst is added, two subsystems are put in contact), the set of microstates the system may occupy either grows or stays the same; it can never shrink. Since entropy is the log of the size of that set, $\Delta S\ge0$ follows immediately, with equality precisely when the constraint was doing nothing. The power of the argument is that it needs no dynamics at all — only the observation that "accessible before" implies "accessible after."

Step 1. Let the constrained macrostate have accessible set $A_{\text{old}}$ of microstates, with $\Omega_{\text{old}}=|A_{\text{old}}|$ . Every microstate in $A_{\text{old}}$ satisfies the (fewer) remaining constraints after relaxation, so it lies in the new accessible set $A_{\text{new}}$ : thus $A_{\text{old}}\subseteq A_{\text{new}}$ . Why: relaxing a constraint deletes a restriction; it cannot forbid a configuration that was previously allowed. The accessible region is a superset — this is the entire physical content.

Step 2. Set inclusion gives volume inclusion: $\Omega_{\text{new}}=|A_{\text{new}}|\ge|A_{\text{old}}|=\Omega_{\text{old}}$ . Why: a superset contains at least as many microstates (equivalently, at least as much phase-space volume, which the flow preserves by Liouville's theorem).

Step 3. Apply Boltzmann's relation and Eq. (2):

\Delta S = k_{\mathrm B}\ln\frac{\Omega_{\text{new}}}{\Omega_{\text{old}}} \ge k_{\mathrm B}\ln 1 = 0

Why: the logarithm is monotone, so $\Omega_{\text{new}}\ge\Omega_{\text{old}}$ forces $\Delta S\ge0$ . The second law drops out of a set inclusion.

Step 4. Equality holds iff $A_{\text{new}}=A_{\text{old}}$ , i.e. the removed constraint added no new microstates — it was already inactive. Otherwise the new region is exponentially larger, because $A_{\text{new}}$ typically includes the equilibrium macrostate whose $\Omega$ dwarfs everything. Why: a constraint that was "slack" (e.g. a partition between two gases already at the same pressure and temperature) changes nothing; a constraint that was "active" (gas confined to half the box) unlocks an exponentially larger region on removal, and the system falls into it. *Verified: for free expansion and for two equal ideal gases brought into thermal/particle contact, $A_{\text{old}}\subseteq A_{\text{new}}$ and $\Omega_{\text{new}}\ge\Omega_{\text{old}}$ checked explicitly (with the slack-constraint equality case) in verify_aq-sm-second-law-from-statistics.py. Cf. Reif §3.1, Callen Ch. 4.*

Go deeper · Derivation B · 1 minFree expansion: the constraint argument made numerical

Derivation BFree expansion: the constraint argument made numerical

Here we turn the set-inclusion theorem into a number, deriving the entropy of free expansion directly from microstate counting. A gas of $N$ particles, energy $E$ fixed, is released from volume $V_i$ into $V_f>V_i$ through a punctured partition — no work, no heat, purely the removal of a constraint. We compute $\Omega$ before and after, take the ratio, and recover $\Delta S=Nk_{\mathrm B}\ln(V_f/V_i)$ , the canonical illustration that entropy rises even when nothing is "done" to the gas.

Step 1. Write the multiplicity of an ideal gas as a momentum factor times a configuration factor: $\Omega(E,V,N)=g(E,N)\,V^N/N!$ , where $g$ comes from the momentum-shell integral (owned by Accessible Phase Space and Energy Shells) and $V^N$ from each particle ranging over the box. Why: with no interactions the position and momentum integrals separate; only the position part changes in a free expansion at fixed $E$ .

Step 2. Since $E$ (hence $g$ ) and $N$ are unchanged, the ratio is purely configurational:

\frac{\Omega_f}{\Omega_i} = \frac{V_f^N/N!}{V_i^N/N!} = \left(\frac{V_f}{V_i}\right)^{N}

Why: the momentum shell is identical before and after (the gas neither gains nor loses energy), so every factor except $V^N$ cancels — including the $N!$ indistinguishability factor.

Step 3. Take the log:

\Delta S = k_{\mathrm B}\ln\frac{\Omega_f}{\Omega_i} = N k_{\mathrm B}\ln\frac{V_f}{V_i}

For doubling, $V_f=2V_i$ , so $\Delta S = Nk_{\mathrm B}\ln 2$ . Why: each particle independently multiplies its accessible positions by $V_f/V_i$ ; the product over $N$ particles becomes a sum of $N$ equal logs — extensive, as entropy must be.

Step 4. Consistency with the reverse-probability formula: the chance the gas spontaneously re-crowds into $V_i$ is $\Omega_i/\Omega_f=(V_i/V_f)^N=e^{-\Delta S/k_{\mathrm B}}$ . For $V_f=2V_i$ that is $2^{-N}$ . Why: this is Eq. (4) in action — the free expansion is irreversible not because return is forbidden but because it carries probability $2^{-N}$ , which for a mole is $10^{-1.8\times10^{23}}$ . *Verified: $\Delta S=Nk_{\mathrm B}\ln 2$ for $V_f=2V_i$ , and the reverse probability $2^{-N}$ , computed for $N=1,10,100,N_A$ in the verify script; matches the thermodynamic free-expansion entropy. Cf. Reif §3.5, Pathria §1.5.*

Limiting cases

Case	Result	Meaning
Slack constraint removed	$\Omega_{\text{new}}=\Omega_{\text{old}}$ , $\Delta S=0$	mixing two identical gases at the same $T,P$ : no new microstates, no entropy change
Active constraint removed	$\Delta S=k_{\mathrm B}\ln(\Omega_{\text{new}}/\Omega_{\text{old}})\gg0$	free expansion, gas contact: exponentially larger region, large $\Delta S$
Reversible (quasistatic) process	$\Delta S_{\text{universe}}=0$	the system stays in equilibrium at every step; entropy transfers but is not generated
Small $N$	fluctuations $\Delta S<0$ occur with probability $e^{-\|\Delta S\|/k_{\mathrm B}}$	the law is probabilistic; a few-particle system visibly violates it, briefly
Thermodynamic limit $N\to\infty$	$\Delta S\ge0$ effectively absolute	anti-thermodynamic behaviour has measure $e^{-N}\to0$ ; the law becomes deterministic

Common trap“Entropy increase is caused by a time-asymmetric force or friction”

There is no such force. The microscopic equations of motion are exactly time-reversal symmetric and conserve phase-space volume (Liouville). Friction and dissipation are themselves emergent — coarse-grained descriptions of energy spreading into more degrees of freedom, i.e. into larger $\Omega$ . The arrow of time in the second law comes entirely from two facts with no dynamical asymmetry: the equilibrium macrostate is exponentially the largest, and the system started in a special, low-entropy macrostate. Attributing $\Delta S\ge0$ to a fundamental irreversibility of the equations is the deepest and most common misconception in the subject — and it is answered head-on in Statistical Origin of Irreversibility.

Worked examples

1 · The entropy of a free expansion, and how forbidden the reverse is (real number)

One mole of ideal gas ( $N=N_A=6.022\times10^{23}$ ) expands freely from $V$ to $2V$ at fixed energy. Compute $\Delta S$ and the probability that the gas spontaneously returns to the original half.

Step 1. Multiplicity ratio from the configuration integral: $\Omega_f/\Omega_i=(2V/V)^N=2^{N}$ . Each particle doubles its accessible volume.

Step 2. Entropy change: $\Delta S=k_{\mathrm B}\ln 2^{N}=Nk_{\mathrm B}\ln 2=(6.022\times10^{23})(1.381\times10^{-23}\,\mathrm{J/K})(0.6931)=\boxed{5.76\ \mathrm{J/K}}$ . This is the standard molar free-expansion entropy $R\ln 2=8.314\times0.6931=5.76\ \mathrm{J/K}$ — the same number by the thermodynamic route, confirming the microstate count.

Step 3. Probability of spontaneous return: $P=\Omega_i/\Omega_f=2^{-N}=2^{-6.022\times10^{23}}$ . In base ten, $\log_{10}P=-N\log_{10}2=-6.022\times10^{23}\times0.3010=-1.81\times10^{23}$ , so $P=10^{-1.81\times10^{23}}$ .

Step 4. Put that in perspective: the age of the Universe is $\sim10^{18}$ s, and molecular collision times are $\sim10^{-13}$ s, giving at most $\sim10^{31}$ "tries." Against a probability of $10^{-10^{23}}$ , that is no tries at all. The gas never comes back — not because it cannot, but because the returning microstates are an unimaginably tiny fraction of all microstates.

2 · Slack versus active constraint: why mixing identical gases gives zero (exam level)

Two chambers of volume $V$ each hold $N$ atoms of the same ideal gas at the same $T$ and $P$ . The partition is removed. Compute $\Delta S$ . Then repeat for two different gases (say argon and neon).

Step 1. Identical gases. Before: each side is a macrostate of the same species at the same density. After: one chamber of $2V$ with $2N$ atoms at the same density. Because the atoms are indistinguishable, the accessible set does not grow: any post-removal microstate corresponds to a pre-removal one (with the correct $1/N!$ counting on each side and $1/(2N)!$ after). Formally $\Omega_{\text{new}}=\Omega_{\text{old}}$ .

Step 2. Therefore $\Delta S=k_{\mathrm B}\ln(\Omega_{\text{new}}/\Omega_{\text{old}})=k_{\mathrm B}\ln 1=\boxed{0}$ . The removed partition was a slack constraint: it forbade nothing that indistinguishability did not already collapse. No new microstates, no entropy of mixing — the second law's equality case.

Step 3. Different gases. Now argon can access the neon chamber and vice versa — genuinely new configurations. Each species free-expands from $V$ to $2V$ : $\Delta S_{\text{Ar}}=Nk_{\mathrm B}\ln 2$ and $\Delta S_{\text{Ne}}=Nk_{\mathrm B}\ln 2$ .

Step 4. Total: $\Delta S_{\text{mix}}=2Nk_{\mathrm B}\ln 2$ . The constraint was active — it kept distinguishable particles apart — so removing it unlocks new microstates and raises the entropy. Whether $\Delta S$ is zero or $2Nk_{\mathrm B}\ln 2$ turns entirely on whether the particles are the same, which is the Gibbs paradox; the full treatment is owned by Indistinguishability and the Gibbs Paradox.

Origins · 1 minRudolf Clausius and how this was discovered

Rudolf Clausius

1822–1888 · Zürich and Bonn

Clausius gave the second law its modern voice. In 1850 he stated that heat does not spontaneously pass from a colder to a hotter body, and in 1865 he coined the word entropy (from the Greek for "transformation") and wrote the two summarizing sentences of thermodynamics: "The energy of the universe is constant; the entropy of the universe tends to a maximum." His entropy was a thermodynamic state function defined by $\mathrm dS=\delta Q_{\text{rev}}/T$ , with no reference to atoms — the statistical meaning came later from Boltzmann. But it was Clausius's sharp formulation of $\Delta S\ge0$ as an inviolable directional law that made the microscopic puzzle acute: how could a strictly one-way macroscopic law emerge from time-symmetric mechanics? The reconciliation in this article is the answer to the question Clausius's own precision forced the field to confront.

History

1850

— Clausius states that heat cannot spontaneously flow from cold to hot (an early form of the second law).

1865

— Clausius introduces the term entropy and the maximum-entropy statement for the universe.

1872

— Boltzmann's H-theorem argues a gas approaches equilibrium, giving a mechanical model of irreversibility.

1876

— Loschmidt raises the reversibility objection: reversed trajectories decrease entropy.

1877

— Boltzmann answers by identifying entropy with microstate count, making the law statistical.

1896

— Zermelo raises the recurrence objection from Poincaré's theorem.

Going further

Texts. Reif, Fundamentals of Statistical and Thermal Physics, Ch. 3 · Callen, Thermodynamics and an Introduction to Thermostatistics, 2nd ed., Ch. 4 · Pathria & Beale, Statistical Mechanics, 3rd ed., §1.5, §2.3 · Kardar, Statistical Physics of Particles, Ch. 4 · Sethna, Statistical Mechanics: Entropy, Order Parameters, and Complexity, Ch. 5.

Provenance. All results are standard. A verifier should confirm: the set-inclusion inequality $\Omega_{\text{new}}\ge\Omega_{\text{old}}$ for constraint removal, with the slack-constraint equality case; free-expansion $\Delta S=Nk_{\mathrm B}\ln(V_f/V_i)$ , giving $R\ln 2=5.76\ \mathrm{J/K}$ per mole for doubling; the reverse probability $2^{-N}$ ( $=10^{-1.81\times10^{23}}$ for a mole); the reversal-measure and recurrence-time scalings $e^{-N}$ , $e^{N}$ (stated here, owned by Statistical Origin of Irreversibility); the identical-gas mixing result $\Delta S=0$ versus $2Nk_{\mathrm B}\ln 2$ for distinct gases. The only fundamental constant is $k_{\mathrm B}=1.380649\times10^{-23}\ \mathrm{J/K}$ (exact, 2019 SI); $R=N_Ak_{\mathrm B}=8.314\ \mathrm{J/(mol\,K)}$ .

Prerequisites. Boltzmann Entropy · Microstates and Macrostates · Fundamental Postulate of Equal A Priori Probabilities · Liouville's Theorem · Accessible Phase Space and Energy Shells. Leads to. Gibbs Entropy · The Ergodic Hypothesis · Ergodic Theory and Mixing · Statistical Ensembles.

If the microscopic laws are time-symmetric, isn't it just as likely for entropy to decrease? What breaks the tie?

The tie is broken by the initial condition, not the dynamics. For every entropy-increasing trajectory there is indeed a time-reversed, entropy-decreasing partner — Loschmidt's point, and it is correct. But we do not start systems in random microstates; we start them in special low-entropy macrostates (gas in a corner, hot next to cold). From such a start the overwhelming majority of trajectories head toward larger $\Omega$ , simply because those macrostates are exponentially more numerous. The reverse partners exist but require exquisitely fine-tuned initial microstates of measure $e^{-N}$ . The asymmetry is in where we begin, and ultimately in the low-entropy early universe.

Doesn't Poincaré recurrence mean entropy must eventually decrease, so the second law is false?

Recurrence is real — Zermelo's objection — but it is defanged by its timescale. A bounded conservative system does return arbitrarily close to its start, so $S$ cannot rise forever. But the recurrence time for a macroscopic system scales as $e^{N}$ (in microscopic units), i.e. $\sim e^{10^{23}}$ — a duration next to which the age of the Universe ( $\sim10^{18}$ s) is indistinguishable from zero. So on any observable timescale entropy increases monotonically; the eventual decrease is a mathematical certainty and a physical irrelevance. The full accounting of both objections is owned by Statistical Origin of Irreversibility.

Why does removing a constraint necessarily increase entropy — couldn't the system just stay put?

It could, and then $\Delta S=0$ — that is the equality case, when the constraint was slack (already inactive). But if the constraint was active, its removal unlocks an exponentially larger region of phase space, and under equal a priori probabilities the system will, with probability $1-e^{-N}$ , be found in that larger region almost immediately. "Staying put" means staying in the tiny original region, which now holds a fraction $\sim e^{-N}$ of the accessible microstates — overwhelmingly improbable. So the entropy rises not because the system is pushed but because "not rising" would require it to avoid virtually all of its available states.

Is the entropy of the whole universe really always increasing? What about a fridge or a growing crystal?

Local decreases are fine; the law is about the total entropy of an isolated system. A refrigerator lowers the entropy of its contents, but only by dumping more entropy into the room (and ultimately the environment) via the compressor's waste heat — the sum increases. A crystal ordering from a melt lowers its own configurational entropy but releases latent heat that raises the surroundings' entropy by more. In every case $\Delta S_{\text{system}}+\Delta S_{\text{surroundings}}\ge0$ . The counting argument applies to the combined, isolated whole; subsystems may buck the trend by exporting entropy.

The formula P = e^{−ΔS/k} says small entropy dips happen all the time. Is that observed?

Yes, directly. For a single Brownian particle in an optical trap, entropy-decreasing fluctuations of order $k_{\mathrm B}$ occur constantly and have been measured — this is the content of the fluctuation theorems. What is never observed is a macroscopic dip, because $e^{-\Delta S/k_{\mathrm B}}$ with $\Delta S$ of order joules-per-kelvin gives probabilities like $e^{-10^{23}}$ . So the second law is exact on average and violated momentarily at the nanoscale — precisely as Eq. (4) predicts. The threshold between "seen constantly" and "never seen" is just how many particles the entropy change involves.

How is this different from the ergodic hypothesis — don't both say the system explores phase space?

They are related but distinct. This article makes a static counting claim: the equilibrium macrostate is exponentially the largest, so it is where almost all microstates are. That alone gives $\Delta S\ge0$ as a probability statement about where the system is likely to be found. The ergodic hypothesis is a dynamical claim: that a single trajectory actually visits the accessible region uniformly over time, so that time averages equal ensemble averages. The second law needs the counting; whether a given system truly equilibrates on a reasonable timescale (approach to the largest region, and mixing) is the further question owned by The Ergodic Hypothesis and Ergodic Theory and Mixing.