Does time average induce phase space propability distribution?

Question

Lets say we have a trajectory (positions and momenta) $(x(t), p(t))$ that is the solution of the equation of motion for a system with Hamiltonian $H(x,p)$. For some function $A(x,p)$, the time average is defined as $$\langle A \rangle_{\text{time}} = \lim_{T\to\infty} \frac{1}{T} \int_{0}^{T} \mathrm{d}t A(x(t),p(t)).$$ This assigns a probability $P(M)$ to a set $M \subset \{(x(t),p(t))|t \in [0,\infty)\}$ (via the characteristic function of $M$) that increases with increasing time that the trajectory $(x(t),p(t))$ spends in $M$. I wonder if this induces a probability distribution on the phase space $\{(x,p) | H(x,p) = E\}$ if the trajectory $\{(x(t),p(t))|t \in [0,\infty)\}$ is a dense subset of the phase space. Specifically, does this induce the microcanonical probability density function? I feel it should, given that we know the velocity $v$ the system has at every point of phase space: $v =|\nabla H(x,p)|$.

A common expression for the microcanonical ensemble average is (see, e.g., Schwabl "Statistical Mechanics"): $$\langle A \rangle_{\text{ensemble}} \propto \int_{S}\mathrm{d}S \frac{A(x,p)}{|\nabla H(x,p)|}$$ where $S = \{(x,p)|H(x,p) = E\}$, so every point in the phase space is weighted by the inverse of the velocity of the system at that point, which makes sense. However, this expression is not derived from the time average in the book by Schwabl, and I can't find much more in other books or lecture notes. Any help is appreciated.

Edit: I rephrase my question. Is there a textbook or something like that (lecture notes etc.) that argues in the following way: We want to have $$ \langle A \rangle_{time} = \langle A \rangle_{ensemble} $$ because the left represents what we do in experiments. For this to hold we need to have: $$\langle A \rangle_{\text{ensemble}} \propto \int_{S}\mathrm{d}S \frac{A(x,p)}{|\nabla H(x,p)|}$$ and then continues to give examples for simple systems where this holds and were it doesnt, and maybe even derives under which requirements it holds for complex systems (or is this just impossible). This just seems to me as the more direct way to introduce the microcanical density, especially when considering the justification for the $\delta$ function distribution, e.g., in Schwabl: "it is intuitively plausible that in equilibrium no particular region should play a special role".

Answer 1

The approach you propose is indeed the most standard route to thermalization. We say an observable thermalizes precisely when its infinite time average approaches what we call, it's thermal average, i.e. the right hand side of your equation.

This was the point of view advocated first by Boltzmann. Indeed if the system is ergodic one has the equality you write for almost all initial conditions and all observables $A\in L^1$ for example.

Note however that, although this is the most accepted scenario, it is not the only possible route to thermalization. The book of Kinchin and Gamov for example provides some alternatives.

In any case for a reference on what you say, Landau's statistical mechanics book provides a discussion on the probability distribution which comes out in an ergodic system. He sort of proves that it should be indeed

$$ \rho_\mathrm{time}(p,q) \propto \delta( H(p,q)-E) $$

Using your notation. Note that this is another, perhaps more familiar way of writing the microcanonical ensemble. The equations you write can be obtained from this by a change of variable.

The mathematically precise statement is a little more involved (and essentially treats possible pathologies).

Answer 2

In statistical physics we usually assume that a system in thermodynamic equilibrium during its time evolution would pass through every point of the phase space, and the time averages of the times that it spends in each point can be interpreted as a probability distribution. This is called ergodicity hypothesis:

In physics and thermodynamics, the ergodic hypothesis says that, over long periods of time, the time spent by a system in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e., that all accessible microstates are equiprobable over a long period of time.

This reasoning is grounded in Liouville's theorem (as exaplained in the Wiki article):

Liouville's theorem states that, for a Hamiltonian system, the local density of microstates following a particle path through phase space is constant as viewed by an observer moving with the ensemble (i.e., the convective time derivative is zero). Thus, if the microstates are uniformly distributed in phase space initially, they will remain so at all times. But Liouville's theorem does not imply that the ergodic hypothesis holds for all Hamiltonian systems.

Interestingly, this reasoning fails spectacularly, if we try to use it to describe evolution towards equilibrium - it predicts constant entropy! See What maximizes entropy?

Remark
As the answer by Icv pints out, we can also derive statistical mechanics using assumptions other than ergodicity, but standard stat mech texts usually do it this way.