Register to reply 
Is Classical Statistical Mechanics dead? 
Share this thread: 
#1
Nov406, 03:25 PM

P: n/a

With the (oftquoted) tremendous success of Quantum Mechanics (QM)
predicting all types of phenomena, is there still any application for the subject of Classical Statistical Mechanics (CSM)? The two subjects do not represent limits of each other in any rigorous sense, so CSM cannot be justified as a convenient approximation to QM in the same way that Newtonian Mechanics can for Special Relativity Theory. So are there any natural phenomena that find a prediction/explanation only in CSM? I say 'natural' because obviously we could invent one, by building a macroscopic object out of a large ensemble of ballbearings, say, if we wanted to, but this would surely be a bit selfserving. Vonny N. 


#2
Nov406, 03:25 PM

P: n/a

Vonny N. wrote:
> [snip] is there still any application for > the subject of Classical Statistical Mechanics (CSM)? The two subjects > do not represent limits of each other in any rigorous sense, so CSM > cannot be justified as a convenient approximation to QM in the same way > that Newtonian Mechanics can for Special Relativity Theory. Classical statistical mechanics is a perfectly reasonable limit of quantum statistical mechanics, though perhaps not at the level of rigor you're considering. For example, the MaxwellBoltzmann distribution function is a reasonable approximation to both the FermiDirac and BoseEinstein distributions, in the appropriate limits (kT much larger than the intrinsic level spacings of the system). > So are there any natural phenomena that find a prediction/explanation > only in CSM? I say 'natural' because obviously we could invent one, by > building a macroscopic object out of a large ensemble of ballbearings, > say, if we wanted to, but this would surely be a bit selfserving. Are there any natural phenomena that find a prediction/explanation *only* in Newtonian mechanics? The kinetic theory of gases is pretty well described by classical statistical mechanics. I should also point out that there is active research still going on in what you might consider analogous to your ensemble of cannon balls: granular media (like sand piles). Amazingly, there are still many unanswered questions about basic things like mixing and force distributions in ensembles of classical grains, even though this seems like the sort of thing that should have been solved 150 years ago by a dead French mathematician whose name probably starts with "L". 


#3
Nov406, 03:25 PM

P: n/a

In article <1138560039.263237.85710@g14g2000cwa.googlegroups.com>,
Vonny N. <vonnyn@hotmail.com> wrote: >With the (oftquoted) tremendous success of Quantum Mechanics (QM) >predicting all types of phenomena, is there still any application for >the subject of Classical Statistical Mechanics (CSM)? The two subjects >do not represent limits of each other in any rigorous sense, so CSM >cannot be justified as a convenient approximation to QM in the same way >that Newtonian Mechanics can for Special Relativity Theory. > >So are there any natural phenomena that find a prediction/explanation >only in CSM? I say 'natural' because obviously we could invent one, by >building a macroscopic object out of a large ensemble of ballbearings, >say, if we wanted to, but this would surely be a bit selfserving. Maybe I'm missing your point, but isn't this precisely what the kinetic theory of gases is? We model a gas as a collection of many ballbearings undergoing elastic collisions with each other, and pretend the particles are classical rather than quantum. Kinetic theory's still useful, even though everyone knows that the exact theory is quantum rather than classical. Ted  [Email me at name@domain.edu, as opposed to name@machine.domain.edu.] 


#4
Nov406, 03:25 PM

P: n/a

Is Classical Statistical Mechanics dead?
Vonny N. wrote:
> With the (oftquoted) tremendous success of Quantum Mechanics (QM) > predicting all types of phenomena, is there still any application for > the subject of Classical Statistical Mechanics (CSM)? The two subjects > do not represent limits of each other in any rigorous sense Neither classical nor quantum statistical mechanics is applicable to real phenomena 'in any rigorous sense'; hence this argument does not prove classical statistical mechanics dead. Indeed, most of chemical engineering is done on the basis of classical statistical mechanics, quantum methods being only used selectively where absolutely essential. Arnold Neumaier 


#5
Nov406, 03:25 PM

P: n/a

In article <1138560039.263237.85710@g14g2000cwa.googlegroups.com>,
"Vonny N." <vonnyn@hotmail.com> wrote: > With the (oftquoted) tremendous success of Quantum Mechanics (QM) > predicting all types of phenomena, is there still any application for > the subject of Classical Statistical Mechanics (CSM)? The two subjects > do not represent limits of each other in any rigorous sense, so CSM > cannot be justified as a convenient approximation to QM in the same way > that Newtonian Mechanics can for Special Relativity Theory. > > So are there any natural phenomena that find a prediction/explanation > only in CSM? I say 'natural' because obviously we could invent one, by > building a macroscopic object out of a large ensemble of ballbearings, > say, if we wanted to, but this would surely be a bit selfserving. > > Vonny N. > This is not exactly my field, but I know there has been a lot of effort to link classical statistical mechanics to dynamics, especially (conservative) chaotic dynamics. This work develops relationships between a system's lyapunov exponents and stat mech quantities, among other things. Check out the name Dorfman (at University of Maryland, USA) for more info. I'm pretty sure there are others workin on this, too.  Lou Pecora (my views are my own) REMOVE THIS to email me. 


#6
Nov406, 03:25 PM

P: n/a

"Vonny N." <vonnyn@hotmail.com> schrieb
> With the (oftquoted) tremendous success of Quantum Mechanics (QM) > predicting all types of phenomena, is there still any application for > the subject of Classical Statistical Mechanics (CSM)? The two subjects > do not represent limits of each other in any rigorous sense, so CSM > cannot be justified as a convenient approximation to QM in the same way > that Newtonian Mechanics can for Special Relativity Theory. It can. If there are some loopholes in the justification (I don't want to argue about their existence) there are similar loopholes also in the justification of this limit from classical Hamiltonian physics. Some people argue that the justification of the limit is even easier with QM as the starting point. Last not least, we already start with probability distributions. > So are there any natural phenomena that find a prediction/explanation > only in CSM? Everything in its domain of application too complicate to compute it in full quantum beauty. Ilja 


#7
Nov406, 03:25 PM

P: n/a

> Maybe I'm missing your point, but isn't this precisely what the
> kinetic theory of gases is? We model a gas as a collection of many > ballbearings undergoing elastic collisions with each other, and > pretend the particles are classical rather than quantum. Kinetic > theory's still useful, even though everyone knows that the exact > theory is quantum rather than classical. My question, rather than my point, is: What exactly do people mean when they say, as you do, things like "Kinetic theory's still useful"? We know for sure that the predictions of the classical theory are incorrect. We know for sure that the predictions of the quantum theory fair much much better. We also now know that the quantum theory does not support a picture of reality that is anything at all like the 'elastic ballbearing' models  or even a picture of microscopic reality at all for that matter. Given these assertions (which, of course, I open for debate), I am wondering where classical statistical mechanics still gets used productively. Of course, if a system 'really is' composed of an ensemble of macroscopic bodies, one would expect the classical theory to work, and I would be interested in practical, rather than pedagogical, examples of such systems. But in modeling gases, and matter in general, as ensembles, the classical picture is so completely wrong that I just can't imagine it being used by real physicists and engineers to solve real problems. So again, if this reasoning is incorrect, I would be very interested in (counter?)examples. Vonny 


#8
Nov406, 03:25 PM

P: n/a

Vonny N. wrote:
> With the (oftquoted) tremendous success of Quantum Mechanics (QM) > predicting all types of phenomena, is there still any application for > the subject of Classical Statistical Mechanics (CSM)? The two subjects > do not represent limits of each other in any rigorous sense, so CSM > cannot be justified as a convenient approximation to QM in the same way > that Newtonian Mechanics can for Special Relativity Theory. > > So are there any natural phenomena that find a prediction/explanation > only in CSM? I say 'natural' because obviously we could invent one, by > building a macroscopic object out of a large ensemble of ballbearings, > say, if we wanted to, but this would surely be a bit selfserving. Classical statistical mechanics is not dead, it is very much alive. However, it has merged with field theory. That may be a reason why you wouldn't hear much about it. The Euclidean version of the path integral formulation of quantum mechanics (specifically quantum field theory, the QM of fields) is equivalent to a classical statistical mechanical system with one extra space dimension. So 3 (space) + 1 (time) dimensions in field theory goes into 4 (space) dimensions in statistical mechanics. There is no time considered in statistical mechanics since we assume everything is in equilibrium. The identification between the two theories puts in correspondance the Wick rotated phase factor with the Boltzman factor, while the Wick rotated classical action corresponds to the Hamiltonian. With the same identification, hbar corresponds to inverse temperature. In short, classical statistical mechanics is as alive today as is field theory. Igor 


#9
Nov406, 03:25 PM

P: n/a

Douglas Natelson wrote: > Classical statistical mechanics is a perfectly reasonable limit > of quantum statistical mechanics, though perhaps not at the level > of rigor you're considering. > > For example, the MaxwellBoltzmann distribution function is a > reasonable approximation to both the FermiDirac and BoseEinstein > distributions, in the appropriate limits (kT much larger than the > intrinsic level spacings of the system). Here is a problem I have with your answer. If these distributions have already been established using quantum theory, then there is no need to rederive them using classical theory. The fact that the latter might be simpler, for example, is irrelevant since we already have the theorem, so the work is done. Nor can we claim that the classical derivation gives some sort of physical insight into what is going on because we already know that classical statistical mechanics is founded on a microscopic picture of reality which is completely and utterly incorrect and untrustworthy. If we take a result from classical statistical mechanics, whose quantum counterpart has not yet been established, would anybody trust it without doing the quantum calculations first? I would imagine not. In this case, we once again have a result which has been established quantum mechanically, and I can't see how we benefit from an untrustworthy rederivation using flawed classical methods. > > > So are there any natural phenomena that find a prediction/explanation > > only in CSM? I say 'natural' because obviously we could invent one, by > > building a macroscopic object out of a large ensemble of ballbearings, > > say, if we wanted to, but this would surely be a bit selfserving. > > Are there any natural phenomena that find a prediction/explanation > *only* in Newtonian mechanics? I take your point :) In retrospect I shouldn't have inserted 'only' since that really wasn't my point. > The kinetic theory of gases is pretty well described by classical > statistical mechanics. This is just a tautology. The kinetic theory of gases is a theory established within a classical framework. My question relates precisely to the empirical usefulness of such theories. I should also point out that there is > active research still going on in what you might consider > analogous to your ensemble of cannon balls: granular media > (like sand piles). Amazingly, there are still many unanswered > questions about basic things like mixing and force distributions > in ensembles of classical grains, even though this seems like the > sort of thing that should have been solved 150 years ago by > a dead French mathematician whose name probably starts with "L". I would certainly be interested in finding out a bit about this kind of research. One would expect classical statistical mechanics to work in dealing with ensembles of 'real' macroscopic objects. On the other hand, I remain unconvinced that this theory is useful in dealing with matter, such as gases, assumed to be made of such ensembles. Vonny N. 


#10
Nov406, 03:25 PM

P: n/a

Vonny N. wrote:
> My question, rather than my point, is: What exactly do people mean when > they say, as you do, things like "Kinetic theory's still useful"? We > know for sure that the predictions of the classical theory are > incorrect. We know for sure that the predictions of the quantum theory > fair much much better. We also now know that the quantum theory does > not support a picture of reality that is anything at all like the > 'elastic ballbearing' models  or even a picture of microscopic > reality at all for that matter. Given these assertions (which, of > course, I open for debate), I am wondering where classical statistical > mechanics still gets used productively. I could make analogous statements about Newtonian gravity and general relativity: everyone knows that GR is more accurate at calculating things like deflection of starlight by the Sun's gravity, and the precession of the perihelion of Mercury. In some philosophy of science sense that you're using, this makes Newtonian gravity "incorrect". It is nonetheless extremely useful for calulating, say, the period of Jupiter's orbit around the sun, to a high (though not arbitrary) accuracy. > But in modeling gases, and matter in general, as > ensembles, the classical picture is so completely wrong that I just > can't imagine it being used by real physicists and engineers to solve > real problems. Why do you say this? Kinetic theory lets one derive the "ideal gas law". In the regime where kinetic theory's approximations are reasonable, the ideal gas law works very very well. Real physicists and engineers use it all the time to do very useful things. Just because a theory doesn't describe every aspect of a system flawlessly under every circumstance doesn't render the theory absolutely "incorrect" and somehow useless. It seems like you're confusing a "theory" with some platonic ideal of understanding.... 


#11
Nov406, 03:25 PM

P: n/a

Vonny N. wrote:
> Here is a problem I have with your answer. If these distributions have > already been established using quantum theory, then there is no need > to rederive them using classical theory. The fact that the latter > might be simpler, for example, is irrelevant since we already have the > theorem, so the work is done. I doubt that classical statistics is simpler than quantum statistics. There is so much trouble in classical statistics which is avoided in quantum statistics (and then taking the classical limit when appropriate to make practical calculations simpler), e.g., Gibbs's paradoxon (about the mixing entropy of, e.g., identical parts of an ideal gas). I don't know whether there is a simple "classical solution" for this paradox, while it is simply not there in quantum statistics. There in manybody theory we know from quite basic principles that there are only bosons and fermions (if the space dimension ist >=3), and this solves the riddle from the very beginning. Another example is the derivation of Boltzmann's transport equation which is quite straightforwardly given in the SchwingerKeldysh timecontour formalism (see vol. 10 of Landau and Lifgarbages's textbook on theoretical physics). Why should we bother ourselves (and students!) with classical statistics, when we have quantum statistics? We can take the classical limit anyway, if it is applicable for a given system in question and this helps to solve the problem.  Hendrik van Hees Texas A&M University Phone: +1 979/8451411 Cyclotron Institute, MS3366 Fax: +1 979/8451899 College Station, TX 778433366 http://theory.gsi.de/~vanhees/ mailto:hees@comp.tamu.edu 


#12
Nov406, 03:25 PM

P: n/a

Douglas Natelson wrote:
> I could make analogous statements about Newtonian gravity > and general relativity: everyone knows that GR is more accurate > at calculating things like deflection of starlight by the > Sun's gravity, and the precession of the perihelion of Mercury. > In some philosophy of science sense that you're using, this makes > Newtonian gravity "incorrect". It is nonetheless extremely > useful for calulating, say, the period of Jupiter's orbit around > the sun, to a high (though not arbitrary) accuracy. I disagree, because in the case of gravity the relationship between the Newtonian and General Relativistic accounts is well understood. That is, one can derive the Newtonian version from the General Relativistic version and see where and why the former will provide perfectly adequate, and much simpler, approximations in many cases. My argument is that the same cannot be said in statistical mechanics because there is no connection between the classical and quantum versions, with respect to which each can be seen as a limiting case of the other. Of course, we are all presented with heuristic arguments in text books that such a limit exists, but as physicists and philosophers working on the foundations of quantum theory are well aware, there are serious holes in these reductionistic claims. The radical, and poorly understood, distinction between the quantum and classical theories surely makes it dubious to solve a problem in statistical mechanics using the classical theory, arguing that it will give an approximation to the quantum version. Who says so? There are endless examples of statistical predictions which the classical theory gets hopelessly wrong, and there is no prescribable set of conditions one can use to decide that classical statistical mechanics will be reasonably accurate. In Newtonian mechanics this is not the case. We know what conditions on velocity and mass are necessary for this theory to give good answers. The limiting process between the quantum and classical statistical theories, on the other hand, is at best a mystery, and at worst, nonexistent. > Just because a theory doesn't describe every aspect of a > system flawlessly under every circumstance doesn't render > the theory absolutely "incorrect" and somehow useless. > It seems like you're confusing a "theory" with some platonic > ideal of understanding.... I have no such platonic view, and am perfectly comfortable in the knowledge that no theory is 'correct' or 'incorrect' in any objective sense. Hopefully the above explanation clarifies my real issue here. Vonny N. 


#13
Nov406, 03:25 PM

P: n/a

Vonny N. wrote:
> > Maybe I'm missing your point, but isn't this precisely what the > > kinetic theory of gases is? We model a gas as a collection of many > > ballbearings undergoing elastic collisions with each other, and > > pretend the particles are classical rather than quantum. Kinetic > > theory's still useful, even though everyone knows that the exact > > theory is quantum rather than classical. > > My question, rather than my point, is: What exactly do people mean when > they say, as you do, things like "Kinetic theory's still useful"? We > know for sure that the predictions of the classical theory are > incorrect. We know for sure that the predictions of the quantum theory > fair much much better. We also now know that the quantum theory does > not support a picture of reality that is anything at all like the > 'elastic ballbearing' models  or even a picture of microscopic > reality at all for that matter. Given these assertions (which, of > course, I open for debate), I am wondering where classical statistical > mechanics still gets used productively. Rethinking of space and time in Special Relativity changes the basis of classical mechanics. Nothing is more fundamental to mechanics than its spatial framework. It is natural to talk about succession here. The situation is different in statistical mechanics. Of course, one can say: >>>For example, the MaxwellBoltzmann distribution function is a >>>reasonable approximation to both the FermiDirac and BoseEinstein >>>distributions, in the appropriate limits (kT much larger than the >>>Intrinsic level spacings of the system). But it is not about fundamentals of classical or quantum statistics. Probability distributions in classical statistical mechanics can be obtained from simple assumptions of underlying kinetic theory ("a large ensemble of ballbearings"). The same kind of basis is not present in quantum statistical theory (I would be happy to hear that it is wrong). If someone could provide "quantum kinetic theory" and get statistical distributions from it, we could talk about succession in the same sense as in SR. A new understanding of kinetic theory could be significant in attempt to look beyond QM. Alex 


#14
Nov406, 03:25 PM

P: n/a

Hendrik van Hees wrote:
> Why should we bother ourselves (and students!) with classical > statistics, when we have quantum statistics? We can take the classical > limit anyway, if it is applicable for a given system in question and > this helps to solve the problem. This reminds me of a discussion several of us had back in grad school. We were all TAing different sections of introductory mechanics (for premeds, for engineers, and for wouldbe physicists). One of the TAs was advocating starting off with Lagrangians, since teaching people Newton's laws and forces and such was silly and hid the beauty of things like Noether's theorem. While his point was well taken, the rest of us all made the case that that would be a pedagogically unwise approach, particularly for undergrads who had never seen variational calculus before and who would undoubtedly ask, "But *why* is the action extremized?" Of course, our colleague really would've preferred starting off with FeynmanHibbs path integrals and explaining taking the classical limit, but that, too, was not considered useful for teaching freshmen. 


#15
Nov406, 03:25 PM

P: n/a

> If we take a result from classical statistical mechanics, whose quantum
> counterpart has not yet been established, would anybody trust it > without doing the quantum calculations first? I would imagine not. One example that springs to mind is liquid state theory (see eg Hansen and McDonald, "Liquid state theory"). This subject is highly nontrivial (eg how to compute things near the critical point? A partial answer is given by the socalled "Hierarchical reference theory" of Parola, Reatto et al). Understanding the structure of the liquid state is also nontrivial, because it's not close to a nice, wellunderstood system (unlike solids, which are close to a periodic lattice, or gases, which are close to ideal gases). Most of the results in this area are obtained using purely classical statistical mechanics, and yes, specialists in the field believe them. There are other examples as well, that have already been mentioned; for example, granular matter is tentatively being explored. Spin glasses, and, in general, disordered systems, are much harder to understand than they look at first glance. And so on. It's just that subjects such as these are not well known, because they don't sound as exciting as (say) particle physics. 


#16
Nov406, 03:25 PM

P: n/a

Vonny N. wrote:
> I disagree, because in the case of gravity the relationship between the > Newtonian and General Relativistic accounts is well understood. That > is, one can derive the Newtonian version from the General Relativistic > version and see where and why the former will provide perfectly > adequate, and much simpler, approximations in many cases. My argument > is that the same cannot be said in statistical mechanics because there > is no connection between the classical and quantum versions, with > respect to which each can be seen as a limiting case of the other. Of > course, we are all presented with heuristic arguments in text books > that such a limit exists, but as physicists and philosophers working on > the foundations of quantum theory are well aware, there are serious > holes in these reductionistic claims. > > The radical, and poorly understood, distinction between the quantum and > classical theories surely makes it dubious to solve a problem in > statistical mechanics using the classical theory, arguing that it will > give an approximation to the quantum version. Who says so? There are > endless examples of statistical predictions which the classical theory > gets hopelessly wrong, and there is no prescribable set of conditions > one can use to decide that classical statistical mechanics will be > reasonably accurate. In Newtonian mechanics this is not the case. We > know what conditions on velocity and mass are necessary for this theory > to give good answers. The limiting process between the quantum and > classical statistical theories, on the other hand, is at best a > mystery, and at worst, nonexistent. Rethinking space and time in Special Relativity changes the basis of classical mechanics. Nothing is more fundamental in mechanics than its spatial framework. It is natural to talk about succession, when fundamentals are "improved". The situation is different in statistical mechanics. Of course, one can say: >>For example, the MaxwellBoltzmann distribution function is a >>reasonable approximation to both the FermiDirac and BoseEinstein >>distributions, in the appropriate limits (kT much larger than the >>Intrinsic level spacings of the system). But it is not about fundamentals of classical or quantum statistics. Probability distributions in classical statistical mechanics can be obtained from simple assumptions of underlying deterministic kinetic theory ("a large ensemble of ballbearings"). The same kind of basis is not present in quantum statistical theory. However, technically it is possible to change the assumptions of kinetic theory. As a result, related statistical outcome (distributions) will vary. It would be nice, if one could provide "quantum kinetic theory" and get statistical distributions from it ... then we could talk about succession in the same sense as in SR and Newtonian Mechanics, for example. At least such exercise would help to understand better the origins of quantum statistics. Alex 


#17
Nov406, 03:25 PM

P: n/a

Vonny N. wrote:
> With the (oftquoted) tremendous success of Quantum Mechanics (QM) > predicting all types of phenomena, is there still any application for > the subject of Classical Statistical Mechanics (CSM)? The two subjects > do not represent limits of each other in any rigorous sense, so CSM > cannot be justified as a convenient approximation to QM in the same way > that Newtonian Mechanics can for Special Relativity Theory. > > So are there any natural phenomena that find a prediction/explanation > only in CSM? I say 'natural' because obviously we could invent one, by > building a macroscopic object out of a large ensemble of ballbearings, > say, if we wanted to, but this would surely be a bit selfserving. > > Vonny N. Certainly not  whenever the MaxwellBoltzmann statistics apply (distinguishable particles) you're in the classical realm. drl 


#18
Nov406, 03:25 PM

P: n/a

Douglas Natelson wrote:
> This reminds me of a discussion several of us had back in > grad school. We were all TAing different sections of introductory > mechanics (for premeds, for engineers, and for wouldbe physicists). > One of the TAs was advocating starting off with Lagrangians, > since teaching people Newton's laws and forces and such was > silly and hid the beauty of things like Noether's theorem. > While his point was well taken, the rest of us all made the case > that that would be a pedagogically unwise approach, particularly > for undergrads who had never seen variational calculus before and > who would undoubtedly ask, "But *why* is the action extremized?" > Of course, our colleague really would've preferred starting off > with FeynmanHibbs path integrals and explaining taking the > classical limit, but that, too, was not considered useful for > teaching freshmen. It was not my intention to propagate a "Bourbakistyle" curriculum for physics undergrads, i.e., starting with a set of axioms which fall from heaven ;). I think, one has to learn Newtonian mechanics (although I was a little startled about noncalculus courses here in the US; that really makes things more complicate than having time derivatives at hand or introducing them in an intuitive way as Newton did when he invented calculus) in order to get an idea, how physics thinking works and how it's different from everyday experience, and so on. It's most important to get a good intuition for physical problem solving strategies and theory building than just to learn a closed subject which looks as dead as Bourbaki's books on mathematics (although they are great books, and it's very nice to see axiomatised clearly formulated theories, but not for learning mathematics the first time; the same holds true for physics). You also do not have the chance to understand quantum theory with a solid knowledge of analytical mechanics (Lagrange, Hamilton, HamiltonJacobi, and of course as the highlight Noether's theorem), but to understand this you need good old Newton's physical intuitive way of the foundations of mechanics. But all this you can learn without even mentioning statistical mechanics. In the case of statistical mechanics, I think, the foundations become really much simpler in the quantum version than classical statistical mechanics can be presented, because to my feeling you always have to "repair" trouble with the classical statistics by refering to quantum theory anyway (Gibbs's paradox, to mention only one of the many paradoxes), while quantum statistics, based on some information theory is a quite clear and intuitive concept (see, e.g., A. Katz book on statistical mechanics, where he treats both, classical and quantum statistical mechanics based on Shannon's information theoretical interpretation of entropy).  Hendrik van Hees Cyclotron Institute Phone: +1 979/8451411 Texas A&M University Fax: +1 979/8451899 Cyclotron Institute, MS3366 http://theory.gsi.de/~vanhees/ College Station, TX 778433366 


Register to reply 
Related Discussions  
Statistical mechanics 6.1  Advanced Physics Homework  8  
Classical statistical mechanics: dimensions of partition function  Classical Physics  2  
Classical mechanics  Advanced Physics Homework  1  
Is Classical Statistical Mechanics dead?  General Physics  18  
QM and Classical Mechanics  General Physics  11 