Questions on the conceptual basis of statistical mechanics

[Abu Abdallah]

Hi,
1- In the introduction of the concepts of partition function and canonical ensemble, a system is assumed to be in direct contact with a heat bath (a thermal reservoir) where energy can be transferred between them. All thermodynamical properties of the system can be deduced from the partition function derived from this model. These results are then applied to any system and not necessarily in contact with a huge heat path nor in an ensemble of many identical systems(even to isolated systems, I think) Why?

2- Moreover, the results derived from the canonical ensemble treatement is applied on systems that no energy can be transferred between it and other systems when put in direct contact ( e.g, a system of magnetic dipoles ) or the direct contact can't be easily defined (photon gas). Suppose that two systems of magnetic dipoles (or harmonic oscillators) of different temperatures were put in direct contact with each other. Theoretically their states will change to reach a state of equilibrium where the number of available microstates to the whole system is maximum, but practically, how will this happen? How energy will be transferred between them?
In Mandl's book for example, when he considered a single dipole while explaining paramagnetism he said that the heat bath may be considered the rest of the crystal, but how can the dipole exchange energy with the rest of the crystal?

3- In trying to deduce the Planck's radiation law of the blackbody using Bose statistics derived from the grandcanonical ensemble, Pathria set the chemical potential to zero. Why? He said that because the number of photons is indefinite, but isn't this the case for the grand canonical ensemble? Also, where is the thermal reservoir for this system we used to derive the grand cannoical results? Is it the walls of blackbody? I doubt !

4- Is entropy a purely quantum phenomenon? I don't think so. However, in calculating the properties of an ideal gas, Pathria used the quantum model of a particle in a box to count the allowed combinations of the quantum numbers nx,ny,nz as the number of the available microstates to the system. Hence the relations between S, E included Planck's constant(h) [ Pathria 1.4.21]. But before the appearence of quantum mechanics, how did classical physicists count the number of microstates? What if we set the limit h -> 0 in this relation? Mandl, based on classical thermodynamics deduced a relation between S, T. Cv, but he didn't mention how to calculate Cv.

5- The postulate of equal apriori probabilities, seems logical. The system will be at the macrostate (e.g, E) that has the largest number of microstates. This is based on the assumption that at the same macrostate, the system can jump freely between different allowed microstates without restriction. For the case of ideal gas, these microstates may correspond to different energy distributions between the prticles ( or different space configurations). This of course require a means of energy transfer between the particles which truly exists in the ideal gas through collisions. But what about other system? The system of harmonic oscillators for example, that Planck used in his famous 1900 paper, how can the system jump from one microstate to another? Planck started with N harmonic oscillators and he counted the ways by which a certain amount of energy can be distributed among them. How could he reach the same result if he starte by a single harmonic oscillator (i.e, N=1) ??


6- In the same paper, Planck stated that :

Entropy depends on disorder and this disorder, according to the electromagnetic theory of radiation for the monochromatic vibrations of a resonator when situated in a permanent stationary radiation field, depends on the irregularity with which it constantly changes its amplitude and phase, provided one considers time intervals large compared to the time of one vibration but small compared to the duration of a measurement. If amplitude and phase both remained absolutely constant, which means completely homogeneous vibrations, no entropy could exist and the vibrational energy would have to be completely free to be converted into work.

This is very nice. However after taking N oscillators and counting the number of ways one can distribute the total energy Un among them (the number of allowed microstates) and using his renowned postulate of quantizing the energy of a single oscillator, he was able to find the total entropy of the N oscillators and hence the enropy of a single oscillator( see equation 6 in his paper). But how did Planck explain the source of the entropy (disorder) in the single oscillator? If he was asked that quaestion after 1927, he may have said it is due to the uncertainty relation between the two conjugate variable P, Q in the harmonic oscillator, but what if he was asked in 1900 ??

Thanks for your patience !

References:
1-On the Law of Distribution of Energy in the Normal Spectrum, Max Planck http://dbhs.wvusd.k12.ca.us/webdocs/Chem-History/Planck-1901/Planck-1901.html
2-Statistical Mechanics, 2nd edition, Pathria
3-Statistical Physics, 1st edition, F.Mandl

 

[arcnets]

1 - I think we need the heat bath to keep temperature constant. This means that our system can exchange energy with the heat bath, but since input = output, the net change is zero. Same as with an isolated system.

2 - I think we don't care about the exact mechanism, but use conservation laws. Because we are only interested in the equilibrium state.

3 - I think the role of the black body is just to make an interaction possible. Since light does not interact with light directly.

4 - I think the definition of entropy is purely mathematical. To actually calculate it, (and NOT because of QM) one often has to introduce "phase space cells". The shape and size of these cells should just give an additive constant in the result, playing no physical role.

5 - See 2.

6 - Can't really help you here. But I think Planck's equation was successful because it agreed with experiment, and not because it could be deduced (or explained) from deeper principles.

 

[charng]

Dear questioner,

Thank you for your thought-provoking questions on the conceptual basis of statistical mechanics. I am always happy to engage in discussions about fundamental concepts and theories in our field.

1. The use of the partition function and canonical ensemble allows us to describe the thermodynamic properties of a system in equilibrium, without the need for detailed knowledge of the system's microscopic dynamics. This is because the partition function encapsulates all the necessary information about the energy levels and degeneracies of the system. Therefore, we can apply this approach to any system, even if it is not in contact with a heat bath or in an ensemble of identical systems. As long as the system is in equilibrium, the partition function can be used to calculate its thermodynamic properties.

2. In the case of two systems of magnetic dipoles at different temperatures, energy will be transferred between them through collisions and interactions between the dipoles. This is similar to the way energy is transferred between particles in an ideal gas. The heat bath can be thought of as the rest of the crystal or the walls of the container, which provide the means for energy transfer between the dipoles. As for the photon gas, the walls of the blackbody act as the heat bath, allowing for energy exchange between the photons and the walls.

3. The reason for setting the chemical potential to zero in the grand canonical ensemble is because the number of photons in the system is not fixed. Therefore, the chemical potential, which is a measure of the energy required to add or remove a particle from the system, is not relevant. As for the thermal reservoir, in this case, it is the walls of the blackbody, which maintain the temperature of the system.

4. Entropy is not a purely quantum phenomenon. In fact, the classical definition of entropy was developed long before the development of quantum mechanics. However, in the case of an ideal gas, using the quantum model of a particle in a box allows us to count the number of available microstates more accurately. When taking the limit of h->0, we recover the classical result for the number of microstates. As for calculating Cv, it can be done using classical thermodynamics, by considering the work done on the gas and the heat transferred to it.

5. The postulate of equal a priori probabilities is based on the assumption that all microstates are equally likely to occur. This is a simplifying assumption that allows us to