Probability of ideal gas macrostate¶
Suppose there is a vessel of volume \(V\) with \(N\) identical ideal gas particles whose movement is described by classical mechanics. Let us divide the volume of the vessel into \(m\) small enough cells with volumes \(V_1\) up to \(V_m\). Let us for a moment also attach a number to each gas particle to be able to tell them apart.
The microstate of the gas is the complete description of
the particle count of each cell of the vessel
the particles of which numbers appear in each cell.
A particle moving within the same cell does not change the microstate of the gas, but a particle moving from one cell to another does.
The macrostate of the gas only requires the description of the particle count in each cell of the vessel. If we assume the particles to be identical, it doesn’t make a difference, from a macroscopic standpoint, particles of which number appear in which cells. Hence the probability of the macrostate is larger than that of the microstate by the factor of the number of permutations of the particles among the cells, which is exactly the statistical weight of the macrostate.
Conditions:
There are no external fields acting on the system.
All particles are identical.
The gas is ideal.
Probabilities must sum up to \(1\).
Links:
Formula 80.8 on p. 298 of “General Course of Physics” (Obschiy kurs fiziki), vol. 1 by Sivukhin D.V. (1979).
- macrostate_probability¶
Probability of the macrostate.
- Symbol:
P_macro- Latex:
\(P_\text{macro}\)
- statistical_weight¶
Statistical weight of the macrostate.
- Symbol:
G
- particle_in_cell_probability¶
Probability of finding at least one particle in cell \(i\).
- Symbol:
p_i- Latex:
\(p_i\)
- particle_count_in_cell¶
Number of particles in cell \(i\).
- Symbol:
N_i- Latex:
\(N_i\)
- law¶
P_macro = G * Product(p_i^N_i, i)- Latex:
- \[P_\text{macro} = G \prod_i p_i^{N_i}\]