PHY.8:4.2 - Construct admissible alternatives and their weighting grounds

Choose the microscopic variables and their physical constraints. Use the theory and preparation to determine possible states, conserved quantities, accessible transitions and coupling to the surroundings. Preserve collective restrictions: fixing total energy or particle number can couple otherwise separate constituents.

Then say what supports the weights. A controlled preparation can supply frequencies; an admitted dynamical law can transport an initial distribution; an equilibrium argument can supply a statistical ensemble under its physical assumptions. An ensemble is a statistical description of possible preparations or states, not an additional physical population that must exist.

For an equilibrium construction, identify what the surroundings hold fixed and what can be exchanged. If a weakly coupled subsystem exchanges energy with a large equilibrated reservoir, a canonical distribution may be appropriate. If the total energy is fixed, begin with that restriction instead. Retain state multiplicities: several distinct states with one energy contribute separately. Check equivalence of proposed ensembles for the observable and regime being used before substituting one for another.

A maximum-entropy inference selects a distribution relative to specified alternatives, a reference measure and constraints. It can provide a useful conditional prediction. Its inferential grounds remain distinct from an argument that this preparation physically equilibrates to that distribution.

For a quantum account, use the prepared state and the relevant observable or measurement operators. A density operator can represent a subsystem correlated with its environment. Its decomposition into weighted pure states need not identify a unique physical preparation. Compute the probabilities of the chosen measurement using the theory; do not replace the state by presumed simultaneous values for incompatible measurements.

If the grounds leave several weightings possible, carry their different consequences far enough to see whether the unresolved choice matters. A bound or a common consequence can already answer the work question.