A system of 100 distinguishable coins starts with all 100 showing heads. After being shaken randomly, it overwhelmingly reaches a disordered state. The best statistical mechanics explanation for this is:
APhysical forces preferentially push the system toward disorder
BThe coins are driven toward equilibrium by entropy maximization acting as a physical force
CThere are astronomically more microstates corresponding to mixed outcomes than to all-heads, so random access to microstates overwhelmingly produces disordered-looking macrostates
DEnergy is minimized in disordered states, making them thermodynamically favorable
Disorder wins by pure combinatorics, not by any preference or force. For 100 coins, there is exactly 1 microstate with all heads, but C(100,50) ≈ 10²⁹ microstates near the 50/50 split. The equal a priori probability postulate says all microstates are equally likely. So 'disordered' macrostates are overwhelmingly more probable simply because they correspond to vastly more microstates. Entropy doesn't 'push' the system anywhere — the system is just equally likely to be in any microstate, and almost all microstates look disordered.
Question 2 Multiple Choice
In the microcanonical ensemble, temperature is defined by 1/T = (∂S/∂E)_{V,N}. What does this definition mean physically?
ATemperature is the average kinetic energy of particles in the system divided by Boltzmann's constant
BTemperature is the inverse of the rate at which the system's entropy increases as energy is added — a high-T system gains little additional entropy per unit of added energy
CTemperature measures the number of accessible microstates at a given energy
DTemperature in the microcanonical ensemble is defined independently of entropy and only relates to particle velocities
The definition 1/T = (∂S/∂E) says temperature is derived from entropy, not the other way around. When you add a unit of energy to a cold system (low T, high ∂S/∂E), entropy increases a lot — the system has many new accessible states. When you add the same energy to a hot system (high T, low ∂S/∂E), entropy barely changes — the system was already exploring a vast number of states. This definition also explains heat flow: when two systems with different T are brought into contact, energy flows from high-T to low-T because doing so increases total entropy.
Question 3 True / False
The equal a priori probability postulate — that most accessible microstates are equally probable for an isolated equilibrium system — is derived from Newton's laws of mechanics.
TTrue
FFalse
Answer: False
The equal a priori probability postulate is a foundational assumption, not a derived result. It is justified by its extraordinary predictive success — the entire edifice of equilibrium statistical mechanics is built on it — but it cannot be rigorously derived from classical or quantum mechanics alone. Attempts to derive it from ergodic theory (that systems explore all accessible states over time) provide partial justification but not a complete proof. The postulate's status as a postulate rather than a theorem is important to recognize.
Question 4 True / False
The microcanonical ensemble is conceptually foundational but computationally impractical for most systems, because the constraint that energy is exactly E makes the multiplicity Ω very difficult to calculate.
TTrue
FFalse
Answer: True
Calculating Ω(E) requires counting the exact number of microstates with energy precisely equal to E — a combinatorially hard problem for most realistic systems. The canonical ensemble avoids this by allowing energy to fluctuate around a fixed average (controlled by temperature), which introduces the Boltzmann factor e^(−βE) and makes the mathematics far more tractable. In the thermodynamic limit, the two ensembles give identical predictions for average quantities, so physicists almost always work with the canonical ensemble in practice while understanding the microcanonical ensemble as the conceptual foundation.
Question 5 Short Answer
Why does S = k_B ln Ω, rather than S = k_B · Ω, correctly capture thermodynamic behavior? What property of physical systems does the logarithm capture?
Think about your answer, then reveal below.
Model answer: Entropy must be additive: the entropy of two independent systems combined equals the sum of their individual entropies. When two independent systems are combined, their total number of microstates is the product of their individual multiplicities (Ω_total = Ω₁ × Ω₂), because each state of one system can be combined with each state of the other. The logarithm converts this multiplicative combination into an additive one: ln(Ω₁ × Ω₂) = ln Ω₁ + ln Ω₂. This ensures entropy is extensive — proportional to system size — as thermodynamics requires.
The logarithm also converts astronomical numbers into manageable ones. For a mole of gas, Ω might be on the order of 10^(10^23), but ln Ω is a tractable number proportional to N. The Boltzmann constant k_B then gives the correct dimensional units (J/K) to match macroscopic thermodynamics. Both the additivity requirement and the need for extensive quantities point uniquely to the logarithm as the correct function relating multiplicity to entropy.