TH · Foundations · Five questions to investigate
How can reversible evolution produce a useful irreversible description?
For which observables, initial states, and time scales does an isolated quantum system behave thermally, and what assumptions supply the time direction?
Starting knowledge: Energy eigenstates, unitary evolution, expectation values, and entropy.
A closed quantum system evolves unitarily. A gas nevertheless loses visible memory of many details of its preparation, and simple measurements approach thermal values. There is no contradiction if information becomes stored in complicated correlations that the chosen observables do not resolve.
But this answer contains conditions that must be tested. Which observables count as simple? Which initial states relax? How fast? What prevents additional conserved quantities from retaining memory? Equilibration means remaining close to a fairly steady value for most relevant times. Thermalization further means that the value agrees with a thermal ensemble specified by a few conserved quantities. The first does not imply the second.
Many interacting quantum models exhibit eigenstate thermalization: expectation values of suitable few-body observables in individual energy eigenstates vary smoothly with energy and match thermal expectations in an appropriate regime. Numerical and experimental evidence is substantial, but this is not a theorem for every Hamiltonian, observable, or energy range. [33]
Integrable systems have additional conserved quantities that can require a more elaborate ensemble. Some systems have atypical nonthermal states or extremely slow relaxation. Disorder can strongly inhibit transport; claims about asymptotic localization require careful statements about dimensionality, interactions, and limits.
Explaining local relaxation does not by itself explain the cosmological arrow of time. Reversible laws permit time-reversed special states. A low-entropy past boundary condition and a choice of macroscopic description are additional inputs. Typicality arguments must specify a measure and what they condition on; they do not conjure a preferred past from reversible equations alone.
Write an initial state in energy eigenstates, $|\psi(0)\rangle=\sum_n c_n|n\rangle$. For an observable $A$,
$E_n$ are energies and $A_{mn}=\langle m|A|n\rangle$. Oscillating off-diagonal contributions can cancel in time averages. For nondegenerate energies, the infinite-time average is
With degeneracies, coherences inside equal-energy subspaces can survive. The resulting diagonal ensemble retains all occupation weights $|c_n|^2$. To replace it by a microcanonical ensemble, which averages over a narrow energy window, one needs extra information: a suitably narrow energy distribution and sufficiently smooth diagonal matrix elements for the observables in question. [33]
Start with a pure state of a finite spin chain and evolve it with
Here $L$ is the number of spins, $X_i,Z_i$ are Pauli matrices acting at site $i$, and $J,g_x,g_z$ have energy units. Open boundaries are specified by the first sum. At $g_z=0$ this is the integrable transverse-field Ising chain. Generic nonzero longitudinal and transverse fields remove that simple integrability, though one must still resolve exact symmetries in spectral comparisons.
Take an alternating product state and track a central spin. Entanglement with the remainder can make its reduced state mixed even though the entire chain remains pure:
The local entropy need not increase monotonically; a finite chain can oscillate and recur. Its energy may also be too close to a spectral edge for a chosen thermal argument. These are results to inspect, not defects to smooth out. A plot becomes evidence of thermalization only after comparing local observables with the correct ensemble and checking time and size dependence.
Insight. Dephasing explains why many oscillating terms cease to add coherently. Eigenstate structure explains why the remaining value can be thermal. A special initial condition and coarse access explain the practical time direction. These are three separate pieces.
Calibration questions have established answers. Diagnostics ask what follows inside a specified model or comparison. Extensions are bounded proposals whose novelty and feasibility need a focused literature check. Each answer should record its assumptions and limits.
Do: Derive the diagonal-ensemble expression from the finite-time integral of each phase. Repeat with one exact energy degeneracy.
Check: Equal-energy coherences survive; distinct energies require long averaging compared with their inverse splitting.
Meaning: Identifies both the approximation and the memory left after dephasing.
Do: Evolve a small chain with the Hamiltonian above. Calculate a one-spin reduced state and its entropy alongside global purity and energy.
Check: Global purity and energy remain constant within numerical error. Report local entropy decreases and recurrences rather than removing them.
Meaning: Reconciles unitary evolution with local mixedness in a fully specified finite model.
Do: Compare $g_z=0$ with a generic nonzero value at matched energy density. Plot diagonal matrix elements of a local observable against energy, resolving symmetries.
Check: Use several sizes and the same energy window. Smoothness alone is not enough if the initial energy distribution is broad.
Meaning: Produces a scoped eigenstate-thermalization diagnostic, not a universal law.
Do: Evolve a product state forward, then reverse the evolution exactly; repeat with a small controlled Hamiltonian error.
Check: Exact reversal returns the initial state. Quantify how errors affect a local observable and global overlap differently.
Meaning: Separates fundamental reversibility, special preparation, and practical irreversibility.
Do: In one chain family, compare local magnetization, a two-point correlation, and a subsystem entropy after matched-energy quenches. A quench is a specified sudden parameter change.
Check: Define a relaxation tolerance and time window before inspecting results. Test sizes and initial states to distinguish slow dynamics from finite-size recurrences.
Meaning: A useful model-specific hierarchy of memory; novelty requires comparison with known conserved and nearly conserved modes.
Begin with TH-1 and TH-2, then make TH-3 the first numerical study. State the ensemble, observable, and time window in every conclusion. A successful thermalization plot does not settle why the universe had a low-entropy past.
Quantum measurement and classical records · Black holes and information · Turbulence and anomalous dissipation · Strange metals and electrical resistance