Comparison

The following table contrasts the salient points of standard theories of Classical Thermodynamics and Statistical Mechanics with the new thermodynamic theory without entropy presented in Thermodynamics without Entropy.

Standard entropy theory	New thermodynamics without entropy
The universe evolves to maximum disorder.	Every system evolves to a balanced state.
Avoids speculating on the composition or internal dynamics of a system. Only exchange with the outside can be known.	A system is an arbitrary delineation of particles interacting by the fundamental laws of physics.
System is treated as a "black box".	All particles exist in independent modes of motion governed by interaction and system geometry. Quantum mechanics governs transition rates between modes. Particle transition rate between specific modes is the product of mode population and rate.
Particles identified only by substance.	Each stable form of particle (each phase of each substance) identified as a distinct species.
Assume all elemental particles have zero formation energy.	All particles have absolute formation energy in order to address all processes.
Mean properties determined by statistical average over possible configurations.	Mean properties determined by averaging over the period of a measurement.
Assume minus 1^st Law: All isolated systems evolve to equilibrium. Variously justified by irreversibility, irrecoverability, time-asymmetry and entropy growth arguments.	Apparent irreversibility is due to asymmetry in the number of modes a system can transition into versus modes representing its immediate past. Intrinsic mode decorrelating events in fundamental time-symmetric mechanical interactions cause isolated systems to approach a steady state.
Macroscopic energy and phase formation energy excluded from internal energy. The latter is represented by internal entropy and latent heat.	Macroscopic and formation energy included in total system energy naturally accounts for friction (dissipation) and latent heat.
Assume 2^nd Law: Heat never flows from cooler to hotter regions by itself.	Energy diffuses in all forms. Heat transport equation implies heat may flow from cooler to hotter regions "by itself" only in the absence of other strong gradients in material properties.
Assume 2^nd Law: Isolated systems evolve to maximize universal entropy.	All systems evolve toward stable material phase configurations indicated by local energy and particle transport equations.
Equilibrium defined as stationary phase-space distribution corresponding to maximal number of accessible states.	Equilibrium defined as zero mean flow in any form.
No quasi-equilibrium condition.	Equilibrium distribution is practically valid when external exchange is small relative to internal mode transition rates.
Detailed balance implies zero net transition rate between any two phase space elements.	Detailed balance implies all mode populations are steady.
A system obeys different equilibrium ensemble distributions depending on outside constraints.	All time-averaged particle mode populations obey the same form of distribution function in equilibrium.
Prohibit "impossible" (non-spontaneous) processes by postulate.	Transport equations determine direction of spontaneous system evolution.
Heat conduction is dissipative and inefficient. Maximum efficiency corresponds to quasi-static "reversible" process limit.	Heat conduction is not inefficient. Maximum process efficiency occurs when all energy besides work is absorbed at the maximum density and emitted at the minimum density during a process.
Define four interchangeable potential functions: Internal energy U, enthalpy H, and free energies A, G. Internal energy excludes macroscopic energy but includes applied body force potential energy.	Total system energy U is the fundamental measure for all analysis. Enthalpy H is a practically convenient ancillary quantity.
Only the difference in potential between two states has physical meaning.	System energy is the total absolute energy contained within the system, consistent with all mechanical analysis.
Parameters: Entropy S, Particle number N, Volume V. Properties: Temperature T, Pressure p, Chemical potential μ. Parameter and properties are interchangeable.	Parameters: T, N, V. Properties: Heat capacity C, p, Particle energy ε. Parameter and properties are not interchangeable.
Equilibrium processes with constant particle number:
1^st Law: Change in U equals heat input from outside reservoirs plus work done by applied pressure: dU = dQ^res − p dV.	System energy conservation accounts for all energy exchange: dU = dQ^res + dQ^loss + dE^{(net macro)} − p dV. Friction converts macroscopic into microscopic energy.
System entropy change: dS^sys ≡ C_V dT/T + Disgregation = dU/T + p dV/T	S^sys ≡ U/T + k_B lnZ is one of many state functions but has no predictive power and is not relevant in any evaluation.
Cyclic process: ∮dS^sys ≥ ∮dQ^res/T^sys by assumption of the 2^nd Law. Process is irreversible if not zero.	∮dS^sys = 0 for all cyclic processes terminating in equilibrium because S^sys is a state function.
3^rd Law: S^sys = 0 at absolute zero temperature.	No axioms beyond fundamental physics.
0^th Law: Two systems in equilibrium at the same temperature stay in equilibrium at the same temperature when in contact.	No axioms beyond fundamental physics.
Total entropy S^total = S^sys + S^outside. ∮dS^total ≥ 0 assuming that the universe is an adiabatically isolated system. Cyclic process is irreversible if greater than zero.	∮dS^total = ∮dQ^res/T^res reflects only the reservoir equivalence value, which can be negative, zero or positive. Non-zero result indicates the combined system is initially out of equilibrium.
dS^sys ≥ dQ^res/T^sys by assuming there exists a reversible process between any two states. Assume forces balance in quasi-static limit.	This classical statement is not valid because any system with internal friction cannot progress by a "reversible” process no matter how slow. Quasi-static does not imply reversible.
Internal entropy always increases: dS^int ≡ dS^sys − dQ^res/T^sys ≥ 0 by assumption of the 2^nd Law.	dS^sys − dQ^res/T^sys may decrease because the 2^nd Law is not true for complex systems.
Partition function: Z must be extensive and dln[Z]/dN intensive by definition of the potential functions.	Z and dln[Z]/dN are not physical variables and so do not need to be extensive or intensive.
Gibbs Paradox. Partition function must be divided by N! in classical regime.	No paradox. No ambiguity in any regime.
A material phase is stable if the second derivative of free energy with respect to particle number is positive.	Net heat, particle and momentum flow opposes any mean deviation from equilibrium in a stable material phase.
Equilibrium processes with variable particle number:
Boltzmann entropy: S^sys ≡ k_B ln[Ω] with state probability 1/Ω. Gibbs entropy: S^sys ≡ - k_B ∑_i P_i ln[P_i] given ensemble probability distribution P_i.	The number of particle configurations is sensitive to energy, momentum and particle flows. A steady state determines equilibrium population distribution.
It is unclear if all entropy definitions are consistent.	S^sys defined only by the partition function, as stated above.
System is a member of an ensemble, which is indeterminate unless the entire ensemble is present, and evolves by assumption of molecular chaos.	The actual system evolves by known mechanical laws recognizing inherent fluctuations with each mode transition event. Steady-state defines a unique equilibrium distribution, which is also valid when all flows are sufficiently small.
Ambiguous how probability should be interpreted.	Only the mode transition probability (rate) is relevant. The likelihood of an equilibrium state is defined through the steady-state mean mode population.
Ergodicity, reversibility and recurrence objections remain unresolved.	Energy, momentum and particles always diffuse toward a steady state because mode populations are uncorrelated.
Additional stochastic perturbation required to cause evolution toward equilibrium.	Quantum and external fluctuations are sufficient to de-correlate mode populations in the thermodynamic limit.
Must assume one of many distinct equilibrium ensemble distributions depending on system interaction.	In equilibrium, the mode population of all constituents follow one distribution.
Exchange of energy and particles requires temperature and chemical potential distribution parameters.	Distribution has only temperature parameter because particles transform and do not exchange sub-particles.
Chemical reactions and phase transitions are treated differently. Phase transition is indicated by change in system entropy with constant particle number. Enthalpy of phase transition = T dS^sys − p dV.	All particle transformations (reactions, transitions and flux) are governed by kinetic conditions. Heat of transformation is the net particle energy: ∑_j ε_j dN_j, with sum over transforming species.
Fundamental Equation: dU ≡ T dS^sys − pdV + ∑_j μ_j dN_j, with the sum over substances.	Internal change between equilibrium states: dU = C_V dT + (∂U/∂V) dV + ∑_j ε_j dN_j, with sum over species. Energy conservation includes formation energy ε_0j of exchanged species.
Gibbs-Duhem Equation is an independent condition: S^sys dT - V dp + ∑_j N_j dμ_j = 0	All properties, including Gibbs-Duhem and Maxwell relations, derive from the partition function.
Process constraints: Fundamental Equation, the five Laws, Gibbs-Duhem Eq., applied force Eq. of State, and stoichiometric.	Process constraints: Conservation of energy, kinetic equilibrium conditions, applied force Eq. of State, and stoichiometric.
Chemical equilibrium defined as zero free energy change with empirical "activity coefficient” correction.	All reaction rates are balanced in equilibrium.
"Latent heat" invented to account for transition of a single substance.	"Latent energy" is the net difference in particle energy of any transformation.
Quantum-Classical threshold: μ = 0.	Quantum-Classical threshold of a given species: η_j + ε_0j/k_BT = 0.
1^st order phase transitions identified by jump in 1^st derivative of any Eq. of State. Pure 1^st order phase transitions identified by jump in system entropy.	All 1^st order phase transitions are due to particle transformation between different species.
2^nd order phase transitions identified by jump in 2^nd derivative of any Eq. of State. 2^nd order phase transitions to superfluid and through terminal critical points are due to particle interaction.	All 2^nd order phase transitions are due to discrete mode spectrum, not particle transformations.
Ideal boson model shows no λ profile as observed in liquid helium superfluid transition. Excitation by particle interaction (e.g. Rotons) are presumed to cause λ profile in 2-fluid model.	λ profile in boson system properties appears below the quantum-classical threshold as particles condense in the ground mode, regardless of particle interaction. Coherent states with superfluid/superconducting properties have lower energy and are preferred at very low temperature.
Near Equilibrium:
Assume Gibbs entropy is an independent dynamical variable well defined in and out of equilibrium.	S^sys is not defined out of equilibrium.
Fundamental Equation: dU ≡ (∂U/∂S^sys) dS^sys - (∂U/∂V) dV + ∑_j (∂U/∂N_j) dN_j	Change in system properties is approximated by extrapolation from equilibrium.
Temperature is a property of a system in and out of equilibrium: T ≡ ∂U/∂S^sys.	Temperature is defined only in equilibrium. Non-equilibrium population distribution is described by more than one parameter, replacing temperature.
Boltzmann and Gibbs postulates relate state probability to system entropy, but neither quantity is known out of equilibrium.	System evolves according to transport equations representing aggregate mode transition rates.
Assume phenomenological transport relations.	Transport equations derive from quantum theory of continuous media.
Assume internal entropy density rate of change is never negative.	Transport equations govern local evolution.
Reciprocal relations follow from assuming variation in mean properties obeys detailed balance of equilibrium fluctuations.	Reciprocal relations follow from commutativity of second derivatives of the partition function, independent of fluctuations.
Knowledge of the probability of the combined system and environment state is required to analyze a process involving disruption and diffusive relaxation.	Process dynamics approximated by system equilibrium mode population and transition rates, given energy and particle exchange rates with the outside.
Information entropy is added to satisfy the Second Law during biological function and action of intelligent agents. Life is improbable.	Large molecules may exist in diffusive balance. Life evolves in suitable environments. Mode spectrum and transition rates account for any agent action.
Far from equilibrium:
No consistent formulation of entropy. No general extremal conditions found to support the Fundamental Equation.	Process dynamics are always determined by current system mode population and transition rates.