2.4.9 · D5Thermodynamics & Statistical Mechanics (Advanced)
Question bank — Boltzmann's entropy S = k_B ln(Ω)
Reminder of the notation you'll need (all earned in the parent note):
- (Greek "omega") = the number of microstates matching a macrostate — a count of hidden possibilities.
- = the natural logarithm, the function that turns products into sums.
- = Boltzmann's constant, the units-carrier.
- = entropy.
True or false — justify
A microstate lists every particle's exact state; a macrostate is just the bulk numbers .
True — a microstate is a full snapshot of all positions/momenta (or all spins); a macrostate is the coarse description many microstates share.
Doubling a system's size doubles its entropy.
True (for extensive systems) — is extensive, so joining two identical copies gives ; this additivity is exactly what forced the in the first place.
Doubling a system's size doubles .
False — multiplies to (each half's microstates combine independently), and , which is what makes double.
If then .
True — , so a system locked into a single microstate (e.g. a perfect crystal at ) has zero entropy; this is Boltzmann's route to the Third Law of Thermodynamics.
Entropy is measured in the same units as energy.
False — is a pure number, so carries the units of , namely J/K, not joules; that per-kelvin is what makes dimensionally work.
Two systems in thermal equilibrium have equal entropy.
False — they have equal (equal temperature), not equal ; a large body and a small body at the same have very different entropies.
Adding heat to any system always raises its entropy.
False — for a capped-energy system (like spins in a field) can fall as rises, so drops and goes negative; see Negative Temperature & Spin Systems.
The Boltzmann formula and the Gibbs formula are unrelated.
False — when all accessible microstates are equally likely (), Gibbs' formula collapses exactly to ; Boltzmann is the equal-probability special case of Gibbs Entropy $S=-k_B\sum p_i\ln p_i$.
Spot the error
"The combined entropy of two independent systems is ."
Wrong — the counts multiply () but the entropies add: . The whole point of the logarithm is to convert that product into a sum.
" is always a whole number because you're counting."
In discrete systems yes, but in continuous classical phase space is a phase-space volume divided by , which is generally not an integer; the divisor keeps dimensionless and cures Gibbs Paradox & Indistinguishability.
"This particular arrangement of molecules is a high-entropy configuration."
A single microstate has no entropy — is a property of the whole macrostate (the distribution over all compatible microstates), not of one snapshot.
"For a free gas expanding into double the volume, ."
The factor is missing — every one of the molecules gains the two-way position choice, so and .
"Since and , entropy can never be negative."
In this idealized counting yes, but if you drop the and factors, inside the log can be dimensionful or less than 1, giving unphysical negative or ill-defined — that error is exactly what produces Gibbs' paradox.
"Temperature is defined as ."
It's the inverse that appears: . Temperature measures how fast microstates multiply with energy, so a slower growth means a higher ; see Temperature as $\partial S/\partial E$.
"Melting increases entropy because heat is added."
The deeper reason is that liquid molecules access vastly more position/orientation microstates than the rigid crystal (); the heat is the mechanism, the rise in is the cause.
Why questions
Why a logarithm and not, say, a square root or the count itself?
Only satisfies , the unique continuous function turning multiplied counts into added entropies — additivity demands it.
Why does the constant have to be specifically and not just any number?
The number sets the scale, but 's J/K units convert the pure count into thermodynamic entropy and make come out in kelvin when we use .
Why is the most probable macrostate the one with the highest entropy?
More microstates point to it (larger ), so nature — wandering blindly among equally likely microstates — lands there far more often; high is high probability is high .
Why does entropy connect to our "ignorance" rather than to disorder in the object?
counts how many microscopic arrangements we can't distinguish given only ; more hidden possibilities = more missing information, linking directly to Shannon Information Entropy.
Why divide the classical phase-space volume by ?
Because identical particles are truly indistinguishable — swapping two of them gives no new physical state, so counting them as distinct overcounts and breaks extensivity (the Gibbs paradox).
Why does the assumption "all accessible microstates are equally likely" matter?
It's the founding postulate of the Microcanonical Ensemble; only under equal likelihood does hold, and it's what lets Gibbs' formula reduce to Boltzmann's.
Edge cases
What is (and ) if a system can be in exactly one microstate?
, so — the degenerate zero-entropy case, realized by a perfect crystal at absolute zero.
Can ever be zero, and what would be?
No — would mean the macrostate is impossible (no microstate realizes it), so it simply doesn't occur; is not a physical entropy, it flags a forbidden macrostate.
What happens to at the maximum energy of a capped spin system?
Just below the max, adding energy reduces , so and — hotter-than-infinite; at the very peak is largest and (see Negative Temperature & Spin Systems).
If you split one gas into two identical halves and recombine them, does change?
With the indistinguishability factor, no net change (mixing identical gases is a non-event); without it you'd wrongly compute an entropy of mixing — the essence of Gibbs Paradox & Indistinguishability.
For a truly reversible, isolated process at fixed , how does behave?
It stays constant (the accessible microstate count doesn't change), so — the equality boundary of the Entropy and the Second Law of Thermodynamics where entropy neither rises nor falls.
In the limit , why do fluctuations away from the max-entropy macrostate vanish?
Because of the most probable macrostate exponentially dominates all others, so the probability of any noticeably different macrostate is — astronomically tiny for macroscopic .
Recall Two-second self-check before you move on
If you can answer these instantly, you've internalized the traps. Counts of joined systems: multiply or add? ::: Multiply (). Entropies of joined systems: multiply or add? ::: Add (). Whose property is entropy — a microstate or a macrostate? ::: The macrostate (the whole distribution). when ? ::: Exactly . Does more energy always mean more entropy? ::: No — not for capped-energy systems.