1. Lecture 2 The microcanonical ensemble.
Quantum states and the phase space.
Some paradoxes in statistical physics.
Ergodic hypothesis.
Quasi-ergodic systems.
Some model systems in statistical physics

2. The microcanonical ensemble
When a system consisting of a great number of particles (more generally a system having a great number of degrees of freedom) is isolated for a long time from its environment, it will finally reach a thermal equilibrium state.
In this case ensemble is defined by the number of molecules N, the volume V and the energy E.
The energy of the system is constant, so that it is presumed to be fixed at the value E with certain allowance .
It means that we specify a range of energy values, say from (E - ) to (E+).
With a specified macrostate, a choice still remains for the systems of the ensemble to be in any one of a large number of possible microstates.

3. In the phase space, correspondingly, the representative points of the ensemble have a choice to lie anywhere within a "hypershell" defined by the condition
(2.1)
The volume of the phase space enclosed within the shell is given by
(2.2)
where the primed integration extends only over that part of the phase space which conforms to the condition (2.1). It is clear that  will be a function of the parameters N,V,E and .

4. Now the microcanonical ensemble is a collection of systems for which the density function  is, at all times, given
(2.3)
If (q,p)=const, it means that we are dealing with a swarm of representative points uniformly distributed over the relevant region of the phase space (outside the relevant region  is identically zero).

5. Physically it corresponds to an ensemble of systems, which at all times are uniformly distributed over all possible microstates. In this case the ensemble average can be written in the following way
(2.4)
where  denotes the total «volume» of the accessible region of the phase space.
Clearly, in this case, any member of the ensemble is equally likely to be in any one of the various possible microstates, inasmuch as any representative point in the swarm is equally likely to be in the neighborhood of any phase point in the allowed region of the phase space.

6. <f> the ensemble average of f
This statement is usually referred to as the «postulate of equal and a priory probabilities» for the various possible microstates (or for the various volume elements in the allowed region of the phase space); the corresponding ensemble is referred to as the microcanonical ensemble.
Let us try to clarify the physical meaning of the ensemble average <f>, as given by (2.4). Since the ensemble under study is a stationary one, the ensemble average of any physical quantity f must be independent of time; accordingly, taking a time average thereof will not produce any new result.
Thus
<f> the ensemble average of f
= the time average of (the ensemble average of f).
Now the, the processes of time averaging and ensemble averaging are completely independent processes, so the order in which they are performed may be reversed without causing any change in the value of <f>. Thus

7. <f>= the ensemble average of (the time average of f).
Now, the time average of any physical quantity whatsoever, taken over a reasonably long interval of time, must be the same for every member of the ensemble, for after all, we are dealing with only the mental copies of a given system. Therefore, taking an ensemble average thereof should be inconsequential and we may write
<f>= the long-time average of f.
where the latter may be taken over any member of the ensemble. We further observe that the long time average of a physical quantity is all one obtains by making measurements of that quantity on a given system; therefore, it should be identified with the value one expects to obtain through experiment. Thus we finally have

8. (2.5)
This brings us to the most important result: the ensemble average of any physical quantity f is identical with the value one expects to obtain on making an appropriate measurement on the given system
The next that we have to establish is the connection between the mechanics and microcanonical ensemble and the thermodynamics of the member systems.
To do this we observe that there exists a direct correspondence between the various microstates of the given system and the various locations in the phase space.
The volume  (of the allowed region of the phase space) is, therefore, a direct measure of the multiplicity  of the microstates obtaining in the system.

9. To establish a numerical correspondence between  and , we must discover a fundamental volume 0, which could be regarded as "equivalent to one microstate". Once this is done, we can right away conclude that, asymptotically
(2.6)
The thermodynamics of the system would then go through the relationship
(2.7)
The basic problem then consists in determining 0. From the dimensional considerations (see eqn.2.2), 0 must be in the nature of an “angular momentum raised to the power 3N”.

10. To determine it exactly, we consider in the sequel certain simplified systems, both from the point of view of the phase space and from the point of view of the distribution of quantum states. We find that
(2.8)
where is the Planck constant.
In quantum theory the construction of the microcanonical ensemble is based on the same postulate of equal a priori probabilities for the various accessible states. Accordingly, the density matrix mn (which, in the energy representation, must be a diagonal matrix) will be of the form
(2.9)
with

11. (2.10)
where g(E) is the degree of energy level degeneration (or degree of levels degeneration), which energies are in the interval of microcanonical ensemble gives the statistical interpretation of Equilibrium State.
Some paradoxes of statistical physics.
Accordingly to the zero low of thermodynamics the relaxation process of the system to equilibrium is irreversible. If the Liouville equation will be a simple (real) motion equation of the macroscopic system, this point has to taken into account.

12. However, the Liouville equation is generally equivalent to the ordinary motion equations and transferred to them in the case of «pure» microstates and that is why it reversible in time. Moreover, the limited motion has the periodical cycles (Poincare cycles), that is bringing us to the contradiction with the equilibrium. Such contradictions are formulated as paradoxes.
Paradox of reversibility (Poincare, Zermilo) The mechanical conservative system in finite motion is passing the state as much as close to the initial state (as much as far from equilibrium)
Paradox of convertibility (Loshmidt) as a result of time reversibility (or particle velocities) the system returned to the initial (not necessary equilibrium) state.

13. Poincare Recurrence Theorem (1890 - 1897)
If you play bridge long enough you will eventually be dealt any grand-slam hand, not once but several times. A similar thing is true for mechanical systems governed by Newton's laws, as the French mathematician Henri Poincare ( ) showed with his recurrence theorem in 1890: if the system has a fixed total energy that restricts its dynamics to bounded subsets of its phase space, the system will eventually return as closely as you like to any given initial set of molecular positions and velocities. If the entropy is determined by these variables, then it must also return to its original value, so if it increases during one period of time it must decrease during another.

14. This apparent contradiction between the behavior of a deterministic mechanical system of particles and the Second Law of Thermodynamics became known as the "Recurrence Paradox." It was used by the German mathematician Ernst Zermelo in 1896 to attack the mechanistic worldview. He argued that the Second Law is an absolute truth, so any theory that leads to predictions inconsistent with it must be false. This refutation would apply not only to the kinetic theory of gases but to any theory based on the assumption that matter is composed of particles moving in accordance with the laws of mechanics.

15. Boltzmann had previously denied the possibility of such recurrences and might have continued to deny their certainty by rejecting the determinism postulated in the Poincare-Zermelo argument. Instead, he admitted quite frankly that recurrences are completely consistent with the statistical viewpoint, as the card-game analogy suggests; they are fluctuations, which are almost certain to occur if you wait long enough. So determinism leads to the same qualitative consequence that would be expected from a random sequence of states! In either case the recurrence time is so inconceivably long that our failure to observe it cannot constitute an objection to the theory.

16. Loschmidt's paradox states that if there is a motion of a system that leads to a steady decrease of H (increase of entropy) with time, then there is certainly another allowed state of motion of the system, found by time reversal, in which H must increase.
This puts the time reversal symmetry of (almost) all known low-level fundamental physical processes at odds with the second law of thermodynamics which describes the behaviour of macroscopic systems. Both of these are well-accepted principles in physics, with sound observational and theoretical support, yet they seem to be in conflict; hence the paradox.

17. May be that are not paradoxes and is a real situation
May be that are not paradoxes and is a real situation. One of the indirect supports of this assertion is the example with «spin echo». It is known that the changing of the direction of magnet particle motion under the influence of magnetic fields leads in some time to the non-Equilibrium State where the particles are gathered.
In any case we have to take into consideration the Boltzmann’s answer: «Long time one have wait...» and «try to turn them back...». It means here that the prolongation of Poincare cycle of the big systems much more then the age of macrocosm, and it is almost impossible to reverse the velocities of the molecules without changing there positions.
All that statements means that in order to base the statements of statistical mechanics one have to develop the way of obtaining the irreversible equations from the Liouville equation. It can be reached by some simplification of macroscopic system description.

18. Ergodic Hypothesis
The measured physical quantities are the time average values in one system
(2.11)
At comparatively long times T (longer then relaxation times) <f> is equilibrium value of the physical quantity f . In this case one can write:
(2.12)
where 0 is the equilibrium distribution function. This statement for the closed systems with microcanonical distribution 0 embodies the so-called ergodic hypothesis, which was first introduced by Boltzmann (1871).

19. According to this hypothesis, the trajectory of a representative point passes, in the course of time, through each and every point of the relevant region of the phase space.
A little reflection, however, shows that the statement as such cannot be strictly true; we better replace it by the so-called quasi-ergodic hypothesis, according to which the trajectory of a representative point traverses, in the course of time, any neighborhood of any point of the relevant region. An ergodic system behaves accordingly to the ergodic hypothesis.

20. Some model systems of statistical mechanics
Most of the macroscopic systems properties are connected with the great number of degrees of dynamical freedom only.
In order to study of these properties it will be very useful to use the simplest model systems, that allowed there detail consideration on the base of classical or quantum mechanics.
If we will consider the system where the interaction between the particles is so small that we can neglect it in the calculation of energy spectrum, this system can be defined as an ideal model system.

21. Possible energetic levels are defined by the energy spectrum of independent particles. The model systems that are usually used in statistical physics are the following:
The system of the non interactive particles in the box with the ideal reflected walls (ideal gases)
The system of non interactive spins in the external magnetic field (ideal paramagnetic)
The systems of non-interactive oscillators with specified oscillation frequency (Einstein model of oscillation in solids).

22. Ideal spin system (classical consideration)
Let us consider an ideal spin 1/2 system of N independent particles, each bearing a magnetic moment  which may directed either parallel or antiparallel to an external magnetic field B. The energy of each particle is E=B, according to the orientation of the magnetic moment.
Fig.2.1
The system of N spins equal to 1/2.
Every arrow shows the direction of magnetic moment.

23. Let us calculate the probability distribution of the total magnetic moment M of the system in the absence of the magnetic field.
We know of course that the average value of M in this conditions is zero, but we are interested in the probability distribution w(M).
In zero magnetic fields the projection of each moment is equally likely to be .
We are interested in the number of arrangements, which result in ½ (N+n) moments being positive and ½ (N-n) moments being negative.
The problem for B=0 is essentially identical with the problem of the random walk, in one dimension, the steps taken as of equal length.

24. We observed first that the (normalized) probability of a given specific sequence of particles is
as for each individual moment there is a probability 1/2 that it will take the orientation required by the assignment, and there are N particles required to have their spins ordered in the specific sequence.
We mean by a specified sequence that, for example, particle A should be up, particle B up, particle C down... However, there are a number of different ways in which we can satisfy the weaker requirement that any ½ (N+n) of the particles one way and the remainder ½ (N-n) point the other way.

25. We note that N particles can be ordered among themselves in N. ways
We note that N particles can be ordered among themselves in N! ways. There are N ways of selecting the first particle to be drawn, N-1 ways of selecting the second, and so on, to give N! for the number of ways of ordering the particles.
Many of these N! ways do not give independent distinguishable arrangements into groups of ½ (N+n) and ½ (N-n) particles.
Interchanges of the ½ (N+n) particles purely among themselves lead to nothing new, and there are [ ½ (N+n)]! such interchanges.
Similarly the [ ½ (N-n)]! Interchanges of the (N-n) particles do not give new arrangements. Thus the total number W(n) of independent arrangements or sequences giving a net moment M=n is
(2.13)

26. and the probability w(M) of a net moment M=n is obtained on multiplying (2.13) by the probability (1/2)N of specific sequence giving
(2.14)
For large values of the factorials we may use Stirling’s approximation
(2.15)
(2.16)

27. Thus (2.14) gives, on taking the ln of both sides and using (2.16)
(2.17)
For n<<N we use the series expansion
(2.18)
and so, after some transformations and collecting terms we have
(2.19)

28. and so
(2.20)
We see from this result that the magnetization has a Guassian distribution about the value zero. Thus the average value of the magnetization in the absence of a filed is zero. The probability distribution has its maximum at this point, so the most probable value coincides with the average value.

29. Ideal spin system (the quantum consideration)
The spectrum of one spin can be determined by the Schredinger equation H1= with H1=-B=SB, where  is the magnet moment of the particle,  is the giromagnetic relation and B the permanent external magnetic field directed along the axes z.
The spectrum consist from two nondegradated energy levels 1/2 (in units  B), that are related to the stationary states (orbits in the case of individual particles) oriented along and the field and the opposite direction.
Figure 2.2 Spectrum of spin ½

30. The spectrum of all the system consist of N+1 equidistant levels, that are state symmetrically around zero (figure 2.3).
The minimal energy -N/2 () is corresponds to the state with orientation of all spins against the field. The inversion of any spin increase the energy on one unit. The level m corresponds to the state where (N/2)+m spins are already inverse and the rate of degradation of that level is equal
Figure 2.3 Spectrum of the system consisting of N spin S=1/2

31. (2.23)
When m <<N this relation is transferred to the Gauss distribution. It can be obtained taking ln g or from direct using of Stirling formula.
(2.24)
The width of the distribution m/N is decreasing with increasing of N . It decreases according to 1/N.