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.