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I'm studying Landau and Lifshitz's Statistical Physics, Part 1, 3rd edition and am looking for clarification on the following statement, which appears on page 11 in the section on The Significance of Energy:

As we know from mechanics, there exist only seven independent additive integrals of motion: the energy, the three components of the momentum vector and the three components of the angular momentum vector. We shall denote these quantities for the ath subsystem (as functions of the coordinates and momenta of the particles in it) by $E_a$(p,q), $\mathbf{P}_a(p,q)$, $\mathbf{M}_a(p,q)$ respectively.

Landau/Lifshitz go on to say that since $\log{\rho_a}$ (that is, the distribution function for the $a$th subsystem) is an additive integral of the motion, it must be expressible as a linear combination:

$$\log{\rho_a} = \alpha_a + \beta E_a(p,q) + \mathbf{\gamma \cdot P}_a(p,q) + \mathbf{\delta \cdot M}_a(p,q)$$ "with constant coefficients $\alpha_a, \beta, \mathbf{\gamma}, \mathbf{\delta}$ of which $\beta, \mathbf{\gamma}, \mathbf{\delta}$ must be the same for all subsystems in a given closed system."

I do not understand which assumptions were made to justify the statement that these coefficients must be the same for all subsystems.

My best guess is that it follows from the assumption that the subsystems are weakly interacting in a sufficiently small time interval such that the distribution function of two subsystems together, $\rho_{12}$ is equal to the product of the distribution function of each subsystem ($\rho_1$, $\rho_2$) separately: $$\rho_{12} = \rho_1 \rho_2.$$

(Landau/Lifshitz call this "statistical independence" for sufficiently small times in "quasi-closed" subsystems.)

They also state that the closed system as a whole is in "statistical equilibrium," for which the time scales in question are long compared to the relaxation time. I don't know if this assumption is used to justify the claim about the coefficients or if it is in conflict with the time scale for which quasi-closed systems are statistically independent.

It is not clear to me how the result follows from the weakly interacting or statistical equilibrium assumptions, but the former is certainly required otherwise $\log{\rho_a}$ wouldn't be an additive integral and hence the linear combination wouldn't necessarily hold.

Any help would be greatly appreciated. Also for clarity, I am willing to take for granted Landau/Lifshitz's claim that there exist only seven independent additive integrals of motion.

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For the $a$ subsystem:

$$ \log \rho_a = \alpha_a + \beta_a E_a + \mathbf{\gamma}_a \cdot \mathbf{P}_a + \delta_a \cdot \mathbf{M}_a . $$

Writing down this equation assumes that Liouville's theorem is applicable, which itself assumes that the subsystem $a$ is closed (or quasi-closed for relatively short time intervals).

For another subsystem labelled $b$:

$$ \log \rho_b = \alpha_b + \beta_b E_b + \mathbf{\gamma}_b \cdot \mathbf{P}_b + \delta_b \cdot \mathbf{M}_b . $$

For the combined subsystem $ab$:

$$ \log \rho_{ab} = \alpha_{ab} + \beta_{ab} (E_{a} + E_{b}) + \mathbf{\gamma}_{ab} \cdot (\mathbf{P}_{a} + \mathbf{P}_{b}) + \delta_{ab} \cdot ( \mathbf{M}_{a} + \mathbf{M}_{b}) . $$

Substituting the three expressions above into the equation $\log \rho_{ab} = \log\rho_a + \log\rho_b$ (which follows from the "statistical independence" assumption) and equating the coefficients of the independent variables $E_a, E_b, \mathbf{P}_a, \mathbf{P}_b, \mathbf{M}_a, \mathbf{M}_b$ gives:

$$ \alpha_{ab} = \alpha_a + \alpha_b ,$$ $$ \beta_{ab} = \beta_a = \beta_b , $$ $$ \gamma_{ab} = \gamma_a = \gamma_b , $$ $$ \delta_{ab} = \delta_a = \delta_b . $$

So $\beta$, $\gamma$ and $\delta$ don't depend on the subsystem label.

Any closed macroscopic system will be in statistical equilibrium if it is observed for a long enough time (longer than the relaxation time), but I think that statistical equilibrium is an instantaneous quality of a system. Landau and Lifshitz' definition is:

If a closed macroscopic system is in a state such that in any macroscopic subsystem the macroscopic physical quantities are to a high degree of accuracy equal to their mean values, the state is said to be in a state of statistical equilibrium.

So I think that a system can be both in statistical equilibrium and quasi-closed.

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