The answer lies in the ensemble theory. You can derive the canonical and grand canonical ensembles from the microcanonical one, see e.g. the book by L. E. Reichl. The key step is to assume ergodicity; however, it should be noted that this cannot be rigorously proven in general.

Yes, ensembles give different additional conditions so that distributions will not be uniform. And Boltzmann distribution could be derived by assuming maximal entropy assumption. But I still don't see where ergodicity hypothesis functions?

I think the simplest answer is that in systems with Boltzmann distribution you always look on the energy of each state - if you sum these up you will end up with the total energy of the system, which is simply the Hamiltonian.

The Hamiltion equations then just describe the change of the total energy with your canonical coordinates.

The ergodic theorem comes in when you like to deal with systems that have an invariant measure. This is very important for thermodynamics since the defined measure properties (to be seen here: http://en.wikipedia.org/wiki/Measure_(mathematics) ) in statistics asure that there is no energy of a part of your entire system that is >= 0 and that the superposition of energies is satisfied.

The invariance of your ergodic theorem states that this holds for every! thermodynamical system regardless of the changes you apply to it.

To add to Saman, the (fundamental) microcanonical ensemble can be derived from deterministic Hamiltonian dynamics assuming (quasi-)ergodicity, and from this thermodynamics follows.

Thank you all. I got the answer. This is my understanding.

Consider microcanonical ensemble which is the basic postulation of statistical mechaniss, it states when system is in equilibrium, all accessible microstates are equiprobable. Thus the density function is constant everywhere. Then Liouville's theorem suggests that the measure is invariant along the Hamiltonian flow. That results time average makes sense. By ergodicity hypothesis, the time average coincides with space average.

What will happen if we don't have ergodicity hypothesis?

I cordially join to this interesting discussion. First, I think that ergodicity does not exactly implies that "all the accesible microstates are equiprobable", but implies more than that: "all the microstates compatible with a given energy are indeed accesible (and of course, equiprobable)". This subtetly is, in my opinion, important. When ergodicity is broken, there may be some microstates such that, even if they are compatible with a given fixed energy, they are unaccesible dynamically. In this case the equivalence between time and ensemble averages only works within particular ergodic components of the phase space. Think for example in a ferromagnet at zero temperatute with no applied magnetic fields. There are microstates with the same energy either with positive or negative magnetization. Dynamically, depending on initial conditions, only one of these two separate basins of atraction is reachable. Equal a priori probabilities for all the states with a given energy is not compatible with this fact, since you will count on the same footing states which are forbidden. In this simple case, you could correctly recover a nice ensemble theory if you asign some prior probabilities (in constradistinction with equal a priori probabilities). These priors just introduce the dynamical information that the system is in a given ergodic component of phase space, inforcing zero probabilistic weight to those states with the same energy but with opposite magnetization...

Usually we are interested on measuring the macroscopic properties, such as temperature, pressure, etc., which are related to the microscopic information of the system, such as particles positions and velocities. On the other hand, microscopic properties, positions and velocities, are dynamic, i.e. they fluctuate, and their dynamics are described by the Hamiltonian equations (for all particles, not just one particle!), which are derived from the Hamiltonian Function (assuming here that the system under constraints is Hamiltonian system) in accordance with the Hamiltonian principle, which is subject of classical mechanics.

However, the number of particle in the object under investigation is quite large, and solving these equations is quite impossible for the existing computer power! On the other hand, we have to have the microscopic information about our system at any time, that is if we have missing information about one or some particles (e.g. positions or velocities), then we will not be able to find out about the macroscopic properties related to them.

At this point, the statistical mechanics plays an important role, by introducing the concept of the ensemble and the probability distribution of this ensemble under certain constraints, as we know microcanonical ensemble (N, V, E constraints), canonical ensemble (N,V,T) and others, which are all equivalent under the thermodynamical limit condition (N --> infinity). Each member of this ensemble is a possible state of the system under the constraints and represents a possible configuration of system (i.e., microscopic information), we also say they are copies of the same system, or representation of the microscopic information (position, velocities) at a certain time of the dynamical evolution as we could have obtained by solving the equations of motion. This equivalence is represented by the so called ergodic hypothesis: the time averages are equal to ensemble averages; this mean that if we wait long enough obtaining a trajectory of the microscopic information by solving the equations of motion (for systems that we can do this), and along this trajectory we perform a time average to obtain the related macroscopic property, then this time average will be equal to ensemble average, which is the average of the macroscopic property along all configurations of the ensemble weighted by Boltzmann-Gibbs probability of obtaining that particular configuration.

this is an interesting connection between the Hamiltonian dynamics and the theory of ensembles, and has to be well understood

The connection is that the phase space probability distribution function f(x,t) of a given ensemble satisfies Liouville's equation which is ∂f/∂t + {f,H}=0. Here H is Hamiltonian of the system and the curly brackets are Poisson brackets. This is a fundamental result on which Statistical Mechanics is built up.