## Problema 1

## The Principle of Equal Probability

If a system is completely isolated, the system will stay forever in a definite state if it is initially in that state. But, as was already pointed out, it is useless to speak of a completely isolated system. We have some uncertainty in the energy of the system because of uncontrollable interaction between the system and

the external world.

Nevertheless, we can consider a system which is nearly isolated, and assume the validity of Liouville's theorem during some interval of time. We shall further admit that the time average of a mechanical quantity of a system under a macroscopic equilibrium state is equal to the ensemble average (ergodic

hypothesis). This ensemble must be time-independent or stationary. It is a consequence of Liouville's theorem that, if the ensemble is stationary, its density is a function of the energy of the system. Such an ensemble was first clearly mentioned by W. Gibbs, and thus it is called Gibbs' ensemble. It satisfies the

requirement that the statistical ensemble should be compatible with mechanics. The requirement is fundamental to statistical mechanics. Thus, energy plays an important special role in statistical mechanics, and

it is usually assumed that there is no invariant other than energy conservation. In mechanics, there are total momentum and total angular momentum as conserved quantities. However, for a system confined in a box, we have no momentum conservation, and if some asymmetry of the shape of the box is introduced,

the total angular momentum will be no longer conserved.

From the classical Liouville's theorem, we conclude that the weight is proportional to the volume of the portion of phase space for a stationary statistical ensemble. The correspondence with quantum mechanics leads to the assertion that every quantum state of the same energy E has the same weight w(E). This

is the fundamental principle, which is called the principle of equal probability or the assumption of equal a priori probability. In short, every quantum state is considered on equal footing. In other words, the a priori probability for a system to be in a particular energy level is the same for all levels.

That the time-average is the same as the ensemble-average and the principle of equal a priori probability, are two basic principles of statistical mechanics. After adopting these principles, we have only to construct the general probabilistic theory.

The ergodic problem aims at deriving the above principles from mechanics. This problem has been attacked rather from a mathematical point of view. In Chap. 5, we shall discuss the ergodic problem in some detail from a rather

physical viewpoint.

Fuente: Kubo: Statistical Physics I

(pg3) Before stating the postulates, we must introduce the concept of an **ensemble of systems**. An ensemble is simply a (mental) collection of a very large number of systems, each constructed to be a replica on a thermodynamic

(macroscopic) level of the actual thermodynamic system whose properties we are investigating. For example, suppose the system of interest has a volume V, contains N molecules of a single component, and is

immersed in a large heat bath at. temperature T. The assigned values of N, V, and T are sufficient to determine the thermodynamic state of the system. In this case, the ensemble would consist of R systems, all of which

are constructed to duplicate the thermodynamic state (N, V, T) and environment, (closed system immersed in a heat bath) of the original system. Although all systems in the ensemble are identical from a thermo-

dynamic point of view, they are not all identical on the molecular level. In fact, in general, there is an extremely large number of quantum (or classical) states consistent with a given thermodynamic state. This is to

be expected, of course, since three numbers, say the values of N, V and T are quite inadequate to specify the detailed molecular (or "microscopic") state of a system containing something in the order of 10^20 molecules.

Incidentally, when the term "quantum state" is used here, it will be understood that we refer specifically to energy states (i.e., energy eigenstates, or stationary states).

We now state our first postulate: the (long) time average of a mechanical variable M in the thermodynamic system of interest is equal to the ensemble average of M, in the limit as $R \rightarrow \infty$ provided that the systems of the ensemble replicate the thermodynamic state and env, ironment of the actual system of interest.

Three most important thermodynamic environments:

-an isolated system (N,V and E given)

-a closed isothermal system (N,V, T given)

-an open isothermal system ($\mu$, V, T given.)

The respresentative ensembles in the above three cases are:

-microcanonical

-canonical

-grand canonical

Second postulate

In an ensemble $(R \to \infty)$ representative of an isolated thermodynamic system, the systems of the ensemble are distributed uniformly, that is, with equal probability of frequency, over the possible quantum states consistent with the specified values of N, V and E. In other words, each quantum state is represented by the same number of systems in the ensemble; or, if selected at random from the ensemble, the probability that it will be found a particular quantum state is the same for all possible quantum states = *principle of equal a priori probabilities*

(pg5) Combined with the first state is: The single isolated system of actual interest (which serves as the prototype for the systems of the ensemble) spends equal amount of time over a long period of time, in each of the avalaible quantum states = *ergodic hypothesis*

Fuente: Hill: An introduction to statistical thermodynamics

(1)