What is the difference between classical  and ?  To me, they are greatly different but are different approaches for explaining same thing. But I do prefer Statistical mechanics. The former deals in a macroscopic sense while the later investigates in a microscopic approach. It is evident in the definition of Second law given by them.  So, are there differences between the two? If so,what are they?
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What is the difference between  and ?
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I'm having trouble understanding the motivation between wanting to transition between the two descriptions.  I figure that switching from  is useful if we want to recover a particle's trajectory maybe? Is there any other reason to do so? and then what are we measuring by switching from Lagrangian to Eulerian?
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What is the precise difference between the Mean, Average and Expectation ? Is there any mathematical formaulation to show the difference ? I see that often they are used interchangeably.
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I am a physicist. I always heard physicists used the terminology "", "Hermitian", "", and "essentially self-adjoint" operators interchangeably.  Actually what is the difference between all those operators? Presumably understandable by a physicist.
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Are both expressions correct? If yes, do they have different meanings? It seems to me that I came across both of them in books, but I'm not sure.
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What is the difference between implicit, explicit, and total time dependence, e.g. $\frac{\partial \rho}{\partial t}$ and $\frac{d \rho} {dt}$?  I know one is a partial derivative and the other is a total derivative. But physically I cannot distinguish between them. I have a clue that my doubt really might be understanding the difference between implicit, explicit and  total time dependence.
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Measurement of a quantum observable (in an appropriate, old-fashioned sense) necessarily involves coupling to a system with a macroscopically large number of degrees of freedom. Entanglement with this "apparatus" takes care of the decoherence. It is often said (I can provide references upon request) that the remaining problem is the one of "selection", and this is the point where one invariably invokes something philosophically radical, like many-worlds interpretation.  In the above (pretty standard) context, I am trying to make sense of the following observation. Looking at the measuring system from a statistical mechanics point of view, it seems that triggering a particular macroscopic outcome requires spontaneous symmetry breaking via a (thermodynamically) irreversible transition of the "apparatus" from a metastable to a higher entropy final state. My attitude is that "statistical mechanics point of view" is not far from "decoherent large quantum system".  So, the question is:  Is it fair to say that statistical irreversibility ("the second law") and quantum measurement irreversibility (the "wave-function collapse") are necessarily linked? Can this link be made more concrete (e.g., traced in details in a particular model)? Can you give references to approaches to the measurement problem that explore this connection?
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Note: I am aware of  differences between Bayesian and frequentist statistics. For example &quot;what is the probability that the coin on the table is heads&quot; doesn't make sense in frequentist statistics, since it has either already landed heads or tails -- there is nothing probabilistic about it. So the question has no answer in frequentist terms. But such a difference is specifically not the kind of difference I'm asking about. Rather, I would like to know how their predictions for well-formed questions actually differ in the real world, excluding any theoretical/philosophical differences such as the example I mentioned above. So in other words: What's an example of a question, answerable in both frequentist and Bayesian statistics, whose answer is different between the two? (e.g. Perhaps one of them answers &quot;1/2&quot; to a particular question, and the other answers &quot;2/3&quot;.) Are there any such differences?  If so, what are some examples?  If not, then when does it actually ever make a difference whether I use Bayesian or frequentist statistics when solving a particular problem? Why would I avoid one in favor of the other?
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I have come across the term 'functional'. How is a 'functional' different from a 'function'? The exact term I came across was 'statistical functional.'  In terms of the background, can you please focus on the first few lines followed by the four examples in  ?  The definition is seeming to be quite generalized given the examples. Is there a crux of the definition that is not too general nor too restricted?
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In Sean Carroll's book "The Big Picture," he states ():      Classical mechanics, the system of equations studied by Newton and   Laplace, isn't perfectly deterministic. There are examples of cases where a unique outcome cannot be predicted from the current state of the system. This doesn't bother most people, since cases like this are extremely rare—they are essentially infinitely unlikely among the set of all possible things a system could be doing. They are artificial and fun to think about, but not of great import to what happens in the messy world around us.   What are some of these situations he is referring to?   My first guess was something like the conceptual equivalent of a ball sitting exactly atop a hill and asking which direction it will roll. But that doesn't seem right—this would be a case where the "true" solution is that the ball wouldn't roll at all, while in reality there will always be tiny disturbances that cause the ball to roll in some direction. I don't think this is what Carroll is referring to. He seems to suggest that the "true, exact mathematical outcome" is not uniquely determined for certain classical systems.  I've heard that uniqueness theorems for solutions to differential equations don't generally apply to non-linear equations, and wonder if this may have something to do with it. Elucidation and examples would be much appreciated.
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Following up on this question, , which addresses the issue of explaining covariance to a lay person, brought up a similar question in my mind.  How would one explain to a statistics neophyte the difference between covariance and correlation? It seems that both refer to the change in one variable linked back to another variable.  Similar to the referred-to question, a lack of formulae would be preferable.
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My book uses the argument that the multiplicities of a few macrostates in a macroscopic object take up an extraordinarily large share of all possible microstates, such that even over the entire lifetime of the universe, if each microstate had an equal chance of being accessed, fluctuations away from equilibrium would never occur.  My question to this statistical proof is this: In the real world, is there really an infinitesimal but nonzero chance that macroscopic systems could access some of the more unlikely macrostates (e.g. heat transfer from a cold object to a hot object)?
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How do you effectively study physics? How does one read a physics book instead or just staring at it for hours?   (Apologies in advance if the question is ill-posed or too subjective in its current form to meet the requirements of the FAQ; I'd certainly appreciate any suggestions for its modification if need be.)
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Let me start by explaining my particle physics background is very patchy, so this question may not be as coherent as  I would like it to be.   In general terms, what is the difference between a resonance found during a particle experiment and a particle?   From reading Wikipedia it seems to be mostly based on timescales. If a heavy particle decays quickly, how can we distinguish it from a resonance?  I would not be posting this question except for the fact that the Wikipedia page says "This page has some issues".        and I would be interested in knowing more about the subject.
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What is the difference between "algebra" and "field"? In term of definition in Abstract algebra. (In probability theory, sigma-algebra is a synonym of sigma-field, does this imply  algebra is the same as field?)
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What are the difference between:    the action $\int_{t_{1}}^{t_{2}}(L+H) dt$ that we use in the principle of least action, and  the action integral $\int_{t_{1}}^{t_{2}}L dt$ that we use in Hamilton's variational principle?   References:   H. Goldstein, Classical Mechanics, Section 8.6. H. Goldstein, Classical Mechanics, Chapter 2.
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I have tried to explore the information but still not very clear on the exact difference between ,  and . Can anyone help me out to clear my concept?