A3. some physics at last
The full meaning of the thermodynamic laws involve the added physical assumptions e.g. that:
energy is conserved in transfer between systems, which impacts the inferential 0th & 1st laws,
external-world and/or observer uncertainty about (more generally: correlation with) an isolated system will not decrease (likely increase) with the passage of time, which impacts the inferential 2nd law,
most systems are finite, which impacts the inferential 3rd law by ensuring that uncertainty-slope dS/dE ≡ 1/kT is also finite even though negative values are allowed e.g. for inverted population-states & spin-systems, and
accessible-state multiplicity W ≡ eS/k is proportional: to VN (ideal gas law), to EνN/2 (equipartition), and/or to ζN/N! (mass-action) where ζ is the multiplicity of states accessible to a given molecule.
An illustrative model for specific probabilities may be obtained if we further assume that only integer-multiples of some base velocity vo ≈ h/(2mL) are possible in a gas of volume V = L3. From this, kinetic-theory velocity-distributions and more follow as well.
For instance the number of accessible states W ≡ eS/k of a monatomic gas whose average energy ⟨E⟩ = (3/2)kT is specified by contact with a thermal reservoir at absolute temperature kT might look like:
,
based on an estimation of state-multiplicities for fixed energy E with a bit of help from quantum "particle-in-box" insights (see sections below) and some laws of large numbers. Note that the first factor (after ≅) in this expression has units of meter3N while the second has units of 1/meter3N so that the number of accessible states is indeed dimensionless.
Using this a mole of Argon atoms at room temperature and about 1 atmosphere of pressure would then have W ≈ e1.1×1025 accessible states, whose log gives an accessible-state uncertainty (entropy) of about 1.1×1025 [nats] ≈ 153 [J/K]. Historical units are clearly more reasonable in size for these applications, even though they were developed for this use before the connection to natural information-units (e.g. S/kB in nats) was well understood.
The entropy expression more generally is in fact the canonical Sakur-Tetrode relation:
,
with a partition-function that looks like:
.
The ideal-gas equation of state with fluctuation standard-deviations becomes:
.
and the equipartition-relation with fluctuation standard-deviations becomes:
.
More generally: Given the Physical Axiom A that energy is conserved on exchange between systems, the role of 1/kT as an uncertainty-slope makes it a quantity that: (i) will equilibrate on stochastic-sharing between systems i.e. Law Zero, and (ii) remains finite for finite systems so that absolute-zero T is unreachable from either the negative or positive direction i.e. Law Three. The First Law relation from statistical inference now puts not just energy changes, but also energy-flows between systems, into the disordered (δQin) and ordered (TδSirr - pδV) categories.
For the special case of an isolated system, the Physical Axiom B that one's information about an isolated-system's state (which may include any sub-system/observer assemblage) can only increase with time suggests that in that case δSirr/δt ≥ 0 i.e. Law Two. Note that informatic flows in physical systems also include the buildup of subsystem-correlations between systems that are not isolated. These KL-divergence increases play a big role in information-theory application of the above equations, even though they are less important in classical thermodynamic applications where entropy can be considered an extensive quantity.
A physical approximation, which might be considered Physical Axiom C for classical thermodynamic systems, is that entropy is an extensive variable i.e. that the uncertainty about a system (e.g. SAB) equals the sum of uncertainties about the state of that system's sub-systems (e.g. SA+SB). This is true only if the mutual-information about correlations between subsystems (e.g. IAB ≡ SA+SB-SAB) can be neglected, which of course is an excellent approximation in sub-systems with Avogadro's number of particles about whose inter-relationship we only know a few facts. This is a requirement of the Gibbs-Duhem relation which e.g. was crucial in putting together the ideal gas ensemble table below.
Multiple ensembles
In this table, we began with a given partition function in terms of the micro-canonical ensemble's control-variables i.e. Z[U,V,N] or equivalently the Sakur-Tetrode expression for ideal-gas accessible-state multiplicity. Partition functions for the other ensembles were obtained by substituting known values for the control parameters into the micro-canonical expression for state-multiplicity W = eS/k. Quantities moving down in a given ensemble were generally functions of that ensemble's Z.
Note that in the micro-canonical ensemble, changes to U, V and N predictably cause decorrelation incrementsunless δU+PδV-μδN=0. In the other ensembles, the rule is not so hard and fast because some changes (like reversible expansion with ðWout = -ðWin = PδV) can avoid decorrelation while others (like free-expansion with ðWout=0) do not.