Postulates of Statistical Mechanics

Once again we remark that in statistical mechanics we build on the description of matter provided by quantum mechanics in terms of the eigenstates, or the Hamiltonian. The theory of ensembles enables us to express measurable quantitites of macroscopic systems in terms of the underlying quantum mechanical principles inspite of having incomplete information about the preparation of the system and the interactions with its surroundings. If we look carefully we can see that the ensemble-average technique depends on two postulates and then builds an enormous structure on them to do the characteristic work of the theory.
 

   

The experimental result of a measurement of an observable in a macroscopic system is the ensemble average of such observable.

    

Any macroscopic system at equilibirum is described by the maximum entropy ensemble, subject to contraints that define the macroscopic system.
 
 

The first postulate is needed to equate the ensemble average to a more elementary description of what is begin observed. To analyze this aspect, consider a variable that depends on the quantum state (or eitherwise classically, on the locations and velocities of the particles that consititute the system). With a sufficiently delicate measuring device one could measure a fluctuating observable $O(t)$, but the measured value of the observable O is usually taken to be a time average

$$\overline{O} = \lim_{\tau \rightarrow \infty} \frac{1}{\tau} \int_0^{\tau} O(t) dt,$$ (111)

due to the slow response of the measuring system when compared to the rapid changes in quantum states, or locations and velocities of the particles that make up the system.

According to the dynamical picture described above, the time dependence of $O(t)$ is due to the fact that the system changes its microscopic state in time. In order to compute the time average, introduced by Eq. (111), it is necessary to know the sequence of quantum states through which the system passes in time and this is determined by the initial conditions as well as by the interactions between the system and its surroundings. Due to the large number of degrees of freedom in a macroscopic systems, it is therefore impossible to know the sequence of quantum states since neither the initial conditions nor the detailed description of the interactions are ready available. The first postulate thus introduces an alternative way of computing the average over microscopic states. The alternative approach thus involves constructing a large number of replicas of the system of interest, compute O for each replica and average the results

$$\overline{O} = \sum_{j} p_j O(j),$$ (112)

where $p_j$ is the probability of state $j$. Note that by implementing this alternative approach there is no need to follow the sequence of quantum states as described above as long as there is an alternative way to find how many of the replica systems of the ensemble are in each quantum state $j$.

The second postulate is needed to connect the attributes of a quantum state to its probability (i.e., to the fraction of the replica systems in an ensemble that will be found in that state). We found that the maximum entropy postulate established the connection between $p_j$ and the attributes of $j$ such as its energy $E_j$ and the number of particles $N_j$. For a canonical ensemble $p_j$ is determined by the Boltzmann distribution, introduced by Eq. (44); for a microcanonical ensemble $p_j$ is independent of $j$ and is determined by inverse of the total number of states (see Eq. (45)); and for the grand canonical ensemble $p_j$ is determined by the generalized Boltzmann distribution, introduced by Eq. (80). Therefore, the second postulate established through Eqs. (44), (45) and (80) that all quantum states with the same energy and the same number of particles are equally probable. Going back to the dynamical picture, one can analyze the implications of the second postulate with regards to how much time the system lives in each microscopic state during the measurement of $O$.

In the dynamical picture the second postulate establishes that the system is observed for the same fraction time in all microscopic states with the same energy and the same number of particles. Therefore, the ensemble average, introduced by Eq. (112), is equal to the average over time, introduced by Eq. (111). When formulated in such terms, the second postulate is known as the ergodic hypothesis of statistical mechanics and a system that satisfies such hypothesis is called ergodic.