Two glass samples are made from the same material — one cooled rapidly, one cooled slowly. Their densities and optical properties differ measurably. What does this observation demonstrate about ergodicity?
ARapid cooling produces a crystalline microstructure while slow cooling produces an amorphous one, explaining the different properties
BThe system is history-dependent: different cooling paths trap the system in different metastable free-energy valleys, demonstrating ergodicity breaking — the final state cannot be predicted from temperature and pressure alone
CThe glass transition temperature differs between the two samples, indicating different chemical compositions after cooling
DEntropy production is greater during rapid cooling, which permanently elevates the free energy of the rapidly cooled sample
History-dependence is the observable signature of ergodicity breaking. In an ergodic system, the equilibrium state is determined entirely by macroscopic state variables (temperature, pressure, composition). In an ergodicity-broken system, the system is trapped in one of many metastable free-energy valleys, and which valley it occupies depends on its history — including cooling rate. Both glass samples have the same composition and are held at the same temperature, but they occupy different basins of the landscape. This path-dependence is impossible in a truly ergodic equilibrium system.
Question 2 Multiple Choice
A ferromagnet below its Curie temperature is trapped in one magnetization direction and never spontaneously reverses. How does this ergodicity breaking differ from that of a glass?
AFerromagnets break ergodicity kinetically (slow dynamics), while glasses break it through a sharp thermodynamic phase transition with a well-defined order parameter
BFerromagnets break ergodicity through spontaneous symmetry breaking at a sharp phase transition with a well-defined order parameter; glasses break ergodicity kinetically, without a sharp transition or obvious order parameter, and their properties are history-dependent
CBoth systems break ergodicity identically — high free-energy barriers separate equivalent ground states in both cases, so the distinction is merely quantitative
DOnly glasses truly break ergodicity; ferromagnets remain ergodic because thermal fluctuations can eventually reverse the magnetization
The distinction is thermodynamic versus kinetic. In a ferromagnet, symmetry breaking is a genuine equilibrium phase transition: below T_c, the free energy has two equivalent minima separated by a barrier that diverges in the thermodynamic limit. This is sharp, reversible at T_c, and described by an order parameter (magnetization). Glass ergodicity breaking is kinetic: there is no sharp transition, no obvious order parameter, and the system falls out of equilibrium because structural relaxation times exceed experimental timescales. Glass properties are path-dependent in a way that reflects this kinetic trapping.
Question 3 True / False
In an ergodicity-broken system, time averages measured over experimental timescales differ from ensemble averages because the system cannot explore all of phase space within those timescales.
TTrue
FFalse
Answer: True
This is the operational definition of ergodicity breaking. Ergodicity, in statistical mechanics, equates time averages with ensemble averages — the justification for using equilibrium ensembles to predict measurable properties. When a system is trapped in a metastable free-energy valley, it samples only that valley's portion of phase space over any realistic observation time. Its time-averaged properties reflect the local valley, not the full ensemble of all accessible microstates — which is why different samples (different valleys) exhibit different properties.
Question 4 True / False
Ergodicity breaking typically requires a sharp thermodynamic phase transition — systems that remain in a disordered, amorphous state as they are cooled can seldom break ergodicity.
TTrue
FFalse
Answer: False
Glasses are the paradigm counterexample. A glass-forming liquid cooled below T_g becomes kinetically arrested in an amorphous, disordered state — no crystallization, no sharp phase transition, no obvious order parameter. The ergodicity breaking is kinetic: structural relaxation times grow faster than experimentally accessible timescales as temperature drops, eventually trapping the system. This is distinct from symmetry-breaking ergodicity breaking at equilibrium phase transitions. Glass demonstrates that ergodicity breaking can arise from dynamics alone, without any thermodynamic transition.
Question 5 Short Answer
What is the key observable signature that distinguishes an ergodicity-broken system from a truly equilibrated one, and why does it arise from the free-energy landscape picture?
Think about your answer, then reveal below.
Model answer: The key signature is history-dependence: two samples prepared with the same macroscopic conditions (temperature, pressure, composition) but via different routes arrive at states with measurably different properties. In a truly ergodic equilibrium system, macroscopic state variables alone determine the equilibrium state — history is irrelevant. In an ergodicity-broken system, the preparation path determines which metastable free-energy valley the system occupies, and different valleys have different structural and physical properties.
The free-energy landscape picture explains why: if the landscape has many valleys separated by high barriers, different paths through configuration space lead to different valleys. Once trapped, the system cannot hop between valleys on experimental timescales, so the valley it occupies — determined by its history — determines its measured properties. A truly ergodic system has barriers low enough that thermal fluctuations explore all valleys, making the eventual state path-independent regardless of how the system arrived there.