# How to prove that assuming equal a priori probability implies thermodynamic equilibrium

I am assuming (please correct me, if I am wrong) that thermodynamic equilibrium refers to a state where there is minimal fluctuations in the macroscopic variables and that macroscopic variables are a function of averages of position and momentum of all the particles in a particular microstate. Can we show that if we assume equal a priori probability, then the averages of position and momentum for different microstates converge to delta function and therefore the system can be assumed to be in equilibrium?

The question is based on a misconception about what is meant by thermodynamic equilibrium (while the reasoning is logical, statistical physics texts do provide a definition of equilibrium on which their derivations are based). E.g., see Thermodynamic equilibrium:

Thermodynamic equilibrium is an axiomatic concept of thermodynamics. It is an internal state of a single thermodynamic system, or a relation between several thermodynamic systems connected by more or less permeable or impermeable walls. In thermodynamic equilibrium, there are no net macroscopic flows of matter nor of energy within a system or between systems. In a system that is in its own state of internal thermodynamic equilibrium, no macroscopic change occurs.

Systems in mutual thermodynamic equilibrium are simultaneously in mutual thermal, mechanical, chemical, and radiative equilibria. Systems can be in one kind of mutual equilibrium, while not in others. In thermodynamic equilibrium, all kinds of equilibrium hold at once and indefinitely, until disturbed by a thermodynamic operation. In a macroscopic equilibrium, perfectly or almost perfectly balanced microscopic exchanges occur; this is the physical explanation of the notion of macroscopic equilibrium.

A thermodynamic system in a state of internal thermodynamic equilibrium has a spatially uniform temperature. Its intensive properties, other than temperature, may be driven to spatial inhomogeneity by an unchanging long-range force field imposed on it by its surroundings.

In systems that are at a state of non-equilibrium there are, by contrast, net flows of matter or energy. If such changes can be triggered to occur in a system in which they are not already occurring, the system is said to be in a meta-stable equilibrium.

• My question regarding the relationship between equal apriori probability and equilibrium stems from this definition " An isolated system that satisfies the postulate of equal a priori probabilities is said to be in equilibrium" web.stanford.edu/~peastman/statmech/…
– veke
Commented Jan 23, 2023 at 15:10
• @veke stat mech texts usually provide calculation of the fluctuations of thermodynamic variables, with the relative error behaving as $\sqrt{\langle (\delta A)^2\rangle}/\langle A\rangle \sim\frac{1}{\sqrt{N}}$, with the number of particles being of the order of Avogadro number ($N\sim10^{24}$). This is what you wanted to know? Commented Jan 23, 2023 at 15:33
• Is the expression you are referring to, a consequence of equal apriori probability or thermodynamic limit?
– veke
Commented Jan 23, 2023 at 17:18
• @veke it is derived assuming equal a priori pribability - like all the results in equilibrium stat mech. It becomes exact in thermodynamic limit $N\rightarrow\infty$. Commented Jan 23, 2023 at 17:27
• can you share the derivation of the expression?
– veke
Commented Jan 23, 2023 at 17:28

It is a consequence of Liouville's theorem, for classical systems, or the equation of motion for the density matrix, for quantum systems.

The correct statement of the equal a priori probability is that all the microstates corresponding to the fixed value of the energy have the same probability.

In the classical case, this implies that the probability density has a Poisson parenthesis with the Hamiltonian equal to zero. In the quantum case, a zero commutator. The consequence, in both cases, is a time-independent probability density or density matrix. Therefore the statistical averages are time-independent.