A coarse description groups distinct microscopic possibilities under one recorded label. That can simplify prediction, but several different entropy quantities then appear. A finite example makes it possible to calculate exactly what was omitted and whether the resulting quantity has a direction.
Twelve states and an invertible shift
Label twelve possible microscopic states from 0 to 11. At each step, move a state forward by one label, wrapping 11 back to 0. The inverse is a backward shift. Initially, states 0, 1 and 2 each have probability one third, while all other states have probability zero.
This is an ensemble of possible states, not twelve particles moving together. The three nonzero probabilities rotate around the labels without changing their values. Their fine entropy Hfine = −Σ pi ln pi remains ln 3 at every step. After twelve steps the entire distribution returns.
Suppose the observer reports only which block contains the state. Blocks have b consecutive labels, with b dividing twelve. Let PB be a block's total probability. The entropy of the reported label is Hlabel = −Σ PB ln PB. A label has fewer alternatives than the underlying state.
Reporting a label and filling in its missing detail
A second operation reconstructs an estimated microscopic distribution by spreading each block probability uniformly over its b labels. Its entropy is Hreconstructed = Hlabel + ln b. This estimate has the same measured block probabilities but need not match the original microscopic distribution.
The entropy chain rule gives Hfine = Hlabel + H(state | block). The conditional entropy lies between zero and ln b. Consequently Hlabel ≤ Hfine ≤ Hreconstructed. These inequalities compare descriptions at one step; neither says the reconstructed entropy must grow at every later step.
For b = 3, step zero places all probability in one block: the label entropy is zero and the reconstructed entropy is ln 3. At step one, the occupied labels are 1, 2 and 3, so two blocks have probabilities two thirds and one third. The label entropy becomes about 0.636514 and the reconstructed entropy about 1.735126. At step three, all probability again lies in one block and the reconstructed entropy returns to ln 3.
Inspect the lost distinctions
FINITE REVERSIBLE MODEL · NATURAL LOGARITHMS
The three occupied states lie in one observed block. Move the step slider to follow the same reversible distribution.
Compare every step of one cycle
| Step | Fine H | Label H | Reconstructed H |
|---|
A loss of detail does not establish an arrow
The reconstruction is performed only for comparison. It is never fed back into the microscopic update. Replacing the actual state distribution by its uniform reconstruction after every step would define a different dynamical model. Such repeated resetting can change the entropy history and must be declared explicitly.
The quantities above are dimensionless information entropies. Interpreting one as physical thermodynamic entropy requires a specified state space, statistical measure and physical preparation. Multiplying an arbitrary label uncertainty by Boltzmann's constant does not, by itself, supply that identification.
The example is deliberately finite and recurrent. A useful irreversible approximation needs an observation regime, a preparation and a mechanism that keeps omitted correlations unavailable on the relevant timescale. Changing resolution may also change whether the observed labels contain enough information to predict their next values.
Which distinctions survive completion?
In SMM/DTT, the corresponding question is which inner distinctions remain in an outer record. The book's discussion of physical reduction separates reduced-state entropy from a universal entropy-growth claim. Here the source update is reversible, the observation groups states, and the estimated entropy oscillates. Each contribution can be inspected separately.
To turn this into a physical model, identify the instrument that groups states and test whether the hidden distinctions later affect an accessible result. That would distinguish temporary inaccessibility from an effective irreversible reduction.
Sources and further reading
- David Tong, The Fundamentals of Statistical Mechanics, for ensemble entropy and statistical descriptions. The twelve-state construction is an original teaching example.
- Continue with compression and lost distinctions and hidden variables and effective memory.