Hi all,
I'd like to ask two questions that are loosely linked to the first chapter of Nelson.
Firstly, what is everyone's understanding of entropy? This is one of those topics where everyone seems to have a different spin on entropy, depending on their particular scientific background.
My own interpretation is fairly similar to Nelson's. Entropy (to me) is a measurement of the randomness
(not the disorder, which is different) of a system, whilst the change in entropy during any thermodynamic
process indicates how well the quality of that energy has been preserved (i.e. whether the transaction makes
the energy less useful to an organism). The former seems to tie in with the definition of entropy from a statistical point of view (S = k ln g, where g is the multiplicity function) whilst the latter is more in the spirit of the classical definition (dS = dq/T for an infinitesimal exchange of heat at temperature T).
Secondly, Nelson prefaces his comments on thermodynamics with a discussion of Rumford's fluid theory of heat. I don't understand why he draws a correlation with fluid theories of electromagnetism and then discounts it.
For starters, we know conclusively that charge comes in quanta i.e. we can have time-averaged charges that are less than the electron charge, but at any given instant the charge at any point must be an integer multiple of e. This doesn't really matter until we get to a molecular scale though. The sticking point for me is that we know that energy generally comes in discrete packets too, so why should we not consider heat as being some form of fluid too? In particular, the cannon drilling `experiments' still make perfect sense within the framework of a fluid model if we allow `work quanta' to become `heat quanta'.
I suppose only that if we are being super-technical the `fluid theory of heat' is invalid, whilst
the `fluid theory of energy' is (to the same point as the fluid theory of charge is).
I know these are fairly vague questions/musings, but I thought Nelson's discussion of them was also fairly vague (not poorly focussed, just not very technical). Thoughts?
James
Interesting idea re: "heat quanta". How do observables become quantized in normal (i.e. pure-state, Hilbert Space) quantum mechanics? Can the same arguments be applied to heat? How would you formulate the "heat part" of the e.g. Hemholz free energy (-TdS) as an observable within this context? How would you go about solving for the eigenvalues of the "entropy operator" (sometimes called the Surprisal) or, if the surprisal is difficult to work with, how would you talk about the "temperature operator"? Can you consistently discuss the temperature as an observable in the normal way (i.e. as a projective measurement)? What initial conditions on these "heat operator" concepts might give you quantization?
ReplyDeleteA spelling note, James—when spelling a word with a final consonant and preceeding single vowel (e.g. 'focus'), the (interesting) accepted rule is that the final consonant is only doubled when the stress is on the final part of the word. For example, FO-cus would be changed to FO-cusing whereas fo-CUS would be changed to fo-CUSSing. For general interest, enjoy :)
ReplyDeleteTo add to your discussion on entropy: I recently encountered the notion of Information Entropy via Shanon's information theory. In this context entropy is considered to be a measure of unpredictability (or conversely of surprise). So for example one can calculate the entropy of a particular statistical distribution and from this get the amount of information which can be obtained from each sample of that distribution. The preview section in Chapter 1 eluded to this connection between entropy and information in Chapter 6 and I'm keen to see how the two will be married in a biophysics context (and if the marriage will bear any useful fruit!).
ReplyDeleteOn the heat front: I would disagree with the assertion that "...energy generally comes in discrete packets..." From my understanding of physics, (and I should stress it's fairly limited) quantization is only observed when there is some form of confinement - for example a wave function confined to the geometry of an atom and conservation laws. One can of course argue that at some level there is always confinement and thus, yes, everything can be thought of as being quantized; but I would argue this has no practical use. And besides, I don't see how the existence of a heat quanta gets the "fluid theory of heat" out of trouble. It falls apart because the "heat fluid" doesn't SEEM to directly obey any conservation laws. We now know of course that it does, but only at a much more fundamental level.
Interesting discussion points!
Cheers
Martin
Thanks for the notes Josh.
ReplyDeleteThis is a pretty puzzle you have set Seth! As I understand the interpretation of temperature in quantum mechanical systems is still up in the air... but here are some thoughts.
1) How are observables quantised?
Quantisation is a direct result of boundary conditions (particularly evident in the case of a wavefunction) i.e. an observable has a continuous spectrum only if no boundary conditions are imposed (except that the state should really have a square-integrable wavefunction, if one exists). The corresponding operators are usually found by the canonical quantisation procedure, where the classical Poisson bracket becomes i*h_bar times the quantum commutator.
2) Can we apply a similar argument to heat?
This is an interesting question! My initial reaction is to say that we cannot, because heat is not a property of a system: rather, it describes the way in which two systems interact. This is not the same as an interaction term included in the combined Hamiltonian (operator) since that has a well-defined (expectation) value at any given time (rather than just initial and equilibrium states). I would also hazard a guess that the system(s) state(s) would generally need to be described using the density operator formalism, although I don't know if entangled states would survive thermalisation.
3)The entropy 'operator' is well defined in quantum mechanics... it is -\rho\textup{ln}{\rho}, where \rho is the density operator (sorry for the LaTeX formatting of equations) (at least, the entropy is given by S = Tr(-\rho\textup{ln}{\rho})). I need to think a little more on how to solve for eigenvalues!
4)Temperature operator???
I have little idea of what this could look like. I am attempting to relate a `temperature operator' to the internal energy by using the classical ideal gas formula, U = 3NkT/2, but I don't know if this is even vaguely valid yet.
5)Initial conditions???
I think in this case the ìnitial conditions' or boundary conditions on the 'heat operator' would be related to the energy spectra of the two systems involved. For example, consider two simple harmonic oscillators (SHO) in thermal contact but isolated from their surroundings. If the frequency of one is twice that of the other then there is not way that I can transfer a single quantum of energy from the low to high frequency oscillator (the energy isn't compatible with the quantisation condition of the second oscillator). I wonder how superpositions would work?
These are all still a little vague in my head since I cannot fathom where the entropy comes from (I know it looks like the classical entropy, but I'm not sure how to link them just yet) or what temperature would mean in a quantum setting.
As a final note, I should clarify the original post by saying that the fluid theory of heat would be rubbish whilst a fluid theory of energy is just fine.
James
Hi all.
ReplyDeleteI agree with the information entropy Martin... this is an interesting branch of analysis that I would like to learn more about. We were briefly introduced to the concept of entropy as the amount of information that can be gained from a measurement in PHYS3041 (quantum), but not in depth. I think it was Seth (in BIPH2000) that likened entropy to a quantity describing how ignorant of the state of a system we are (microstate, i.e. microscopic configuration).
On the topic of quantisation: yes, it arises from some form of boundary condition which is normally associated with a confining potential (but not always, e.g. the projection of the angular momentum is quantised even without any external potentials). However, in the context of any interacting system we must have an interaction potential, so quantisation of energy must be important when considering energy transfers.
As in my last post, I would argue that the fluid theory of energy has no worries, but the fluid theory of heat does...
James
Hello everyone,
ReplyDeleteI think that the discussion we had on order was very interesting today. It seems that we can agree that the use of the terminology and thermodynamic theory of 'heat' is not really appropriate in a quasi-molecular context (as Dr Seth has said, what is the difference between heating a molecule and working on it).
So, where does that leave heat in larger systems? Is heat a form of energy transfer that also involves 'order' transfer? It is also interesting to consider that the work applied to a small system viewed in the context of a much larger system (e.g. work applied to a fish-tank sized amount of water in the ocean) can also be viewed as 'disorderly' movement of particles (when compared with the rest of the larger system's movement). Is this view appropriate?
What do you guys think that our definition of order is after today? I think that Martin's phrasing was quite good: order defines the amount of information you can get about a system from an observation. This does seem a bit distant from our traditional definition of 'order', however, since we usually think of order as being a quality that refers to similarity of a system's constituent parts. Perhaps this is why students typically have such a difficult time with the entropy?
PS: Seth, I've left my text book, by accident, in your office, so I will come and collect it sometime next week.
PPS: James, watch out for the PP (Post Police, do you get the pun?)! You might be detained for... double posting!!!?!?!
AHHHH! How does one make new posts (i.e. not comments?)
ReplyDeleteAnyway, if I could make a new post, it would be concerning this: how does one go about quantising 'order'? Any ideas? Is it necessary in many models (e.g. ecology, computational, circuit theory)
Sorry for the grammar issues, had to get off of the train....
ReplyDelete