Canonical Ensemble - Issues in The Traditional Models of The Derivation of The Canonical Distribution

Issues in The Traditional Models of The Derivation of The Canonical Distribution

Understanding and clear presentation of the derivation of the canonical distribution are difficult both for students and for teachers. The difficulty is caused by the complexity of the subject, but also by the circumstance that mathematical schemes allowing to receive a desirable result are confused with physical models.

These schemes consider only the eigenstates of the system to be system states. However, a great number of quantum superpositions of the system eigenstates corresponds to the same value of the energy of the system.

In one of the schemes the system S is considered to be a part of a huge system U, usually called “Universe”. The system environment W (or addition to system U) is often called “the thermostat”. The “Universe” is described by the microcanonical ensemble. It means that the "Universe" is in equilibrium, the energy of the "Universe" lies in a very small interval, only the eigenstates of the "Universe" are possible and all eigenstates are equiprobable.

Another scheme - the method of the most probable distribution - assumes that the "Universe" consists of a very great number of systems identical to the system under consideration. In both schemes the system interaction with its environment is considered extremely weak – to make it possible to talk about certain quantum state of the system S. At the same time, the transition of the system from one eigenstate to another is considered to be caused by its energy exchange with the environment. It is obvious that the canonical distribution can be used to calculate the observed quantities only if during the measurement the system has time to visit all states of the spectrum repeatedly. However, the aforementioned schemes do not correlate the values of the contact with the environment, spectral diapason of the system energy and measurement time.

As neither the Universe, nor "Universe" are in equilibrium, nor do they consist of a great number of systems identical to the system under consideration, nor only the eigenstates of systems are possible, we can say with good reason that both schemes are artificial and have no relation to physical reality. We have got used to these artificial schemes-models for the lack of others, but our habit can't make them true.

Thus, we have enough bases to conclude that till now physics has not found a satisfactory model allowing to receive the canonical distribution as a consequence of interaction of the system with its environment.

The schemes used for the derivation of canonical distribution do not call into question the absolute accuracy of quantum mechanics. However, one must remember that both classical and quantum mechanics have resulted from the observation of systems with small number of objects. If the number of objects (e.g. particles) in a system is small and calculations are possible, the mechanics show amazing accuracy. One might assume that in systems with macroscopically great number of particles quantum mechanics would also be absolutely exact. However, this assumption contradicts the irreversibility of evolution of the macrosystems, the second law of thermodynamics and the experimental data received on concrete physical objects.

The experimental data show that quantum-mechanical probability does not cover the entire probabilistic nature of the microworld and that God plays dice not exactly the way prescribed by Schrödinger. It forces us to reflect that the possibility of using the canonical distribution can be connected with internal processes in macrosystems, not described by the existing formalism of quantum mechanics.

The experimental evidence of the existence of the probabilistic processes which are not described by the standard quantum formalism allows to consider the canonical distribution as a result of the averaging on various system states with the same energy of the squared modules of the coefficients of expansion of the system state function on the eigenfunctions. But the question arises of what exactly are the system states in light of the existence of the probabilistic processes which are not considered by the standard formalism of the quantum mechanics (in particular, recording the state function as a superposition of eigenfunctions of the whole macrosystem may not be quite adequate).

Understanding and correct derivation of the canonical distribution demands that we further develop our understanding of the nature of the microworld; it also demands that we be critical towards artificial mathematical schemes.