# Tag Info

41

The ergodic hypothesis is not part of the foundations of statistical mechanics. In fact, it only becomes relevant when you want to use statistical mechanics to make statements about time averages. Without the ergodic hypothesis statistical mechanics makes statements about ensembles, not about one particular system. To understand this answer you have to ...

27

As for references to other approaches to the foundations of Statistical Physics, you can have a look at the classical paper by Jaynes; see also, e.g., this paper (in particular section 2.3) where he discusses the irrelevance of ergodic-type hypotheses as a foundation of equilibrium statistical mechanics. Of course, Jaynes' approach also suffers from a number ...

18

Why would you ever try to motivate a physical theory without appealing to experimental results??? The motivation of quantum mechanics is that it explains experimental results. It is obvious that you would choose a simpler, more intuitive picture than quantum mechanics if you weren't interested in predicting anything. If you are willing to permit some ...

9

No macroscopic quantum system is described by a pure state. For example, notions like temperature or pressure, which apply to macroscopic systems do not even exist for systems described by a pure state. The description of macroscopic objects (discussed in statistical mechanics) is always in terms of a density matrix (or the essentially equivalent notion of a ...

7

I am late to this party here, but I can maybe advertize something pretty close to a derivation of quantum mechanics from pairing classical mechanics with its natural mathematical context, namely with Lie theory. I haven't had a chance yet to try the following on first-year students, but I am pretty confident that with just a tad more pedagogical guidance ...

6

You should use history of physics to ask them questions where classical physics fail. For example, you can tell them result of Rutherford's experiment and ask: If an electron is orbiting around nucleus, it means a charge is in acceleration. So, electrons should release electromagnetic energy. If that's the case, electrons would loose its energy to collapse ...

6

If I'm only allowed to use one single word to give an oversimplified intuitive reason for the discreteness in quantum mechanics, I would choose the word 'compactness'. Examples: The finite number of states in a compact region of phase space. See e.g. this Phys.SE post. The discrete spectrum for Lie algebra generators of a compact Lie group, e.g. angular ...

6

There are several forms of discreteness in quantum theory. The simplest one is the discreteness of eigenvalues and the associated countable eigenstates. Those arise similarly to the discrete standing waves on a guitar string. The boundary conditions only allow certain standing waves that nicely fit into the enforced region in space. Even though the string is ...

5

It seems that perhaps you are missing a crucial piece of the puzzle. The stochastic approach you describe is equivalent to performing a unitary which is chosen stochastically (essentially applying a superoperator) and then measuring in some fixed basis. In principle, however, you can go further, by stochastically choosing whether or not to measure at all, ...

5

Let me try to convince you that the density operator is a mathematical convenience and not a fundamental aspect of quantum mechanics by describing a very general setup for states and observables in both classical and quantum mechanics. This may not directly answer your question, but hopefully it will settle whatever motivated this question. Briefly, ...

5

I searched for "mixing" and didn't find it in other answers. But this is the key. Ergodicity is largely irrelevant, but mixing is the property that makes equilibrium statistical physics tick for many-particle systems. See, e.g., Sklar's Physics and Chance or Jaynes' papers on statistical physics. The chaotic hypothesis of Gallavotti and Cohen basically ...

5

Though there are many good answers here, I believe I can still contribute something which answers a small part of your question. There is one reason to look for a theory beyond classical physics which is purely theoretical and this is the UV catastrophe. According to the classical theory of light, an ideal black body at thermal equilibrium will emit ...

5

If I would be designing an introduction to quantum physics course for physics undergrads, I would seriously consider starting from the observed Bell-GHZ violations. Something along the lines of David Mermin's approach. If there is one thing that makes clear that no form of classical physics can provide the deepest law of nature, this is it. (This does make ...

5

There's two (ultimately related) answers. For the first answer, just forget about $\hbar$ (but say $c=1$), we are doing a classical relativistic field theory. The first is that you can consider the field profile around a static, spherically symmetric source of mass $M$ (you need to add a coupling to the action of the form $g \phi J$, where $J$ is an ...

5

Mathematical proof is to physics roughly what syllogism (or some other fundamental inference rule) is to logic. Namely, it begins from assumptions modelling our conception of some physical reality and shows what must be so if the assumptions hold, but it cannot say anything about the underlying assumptions themselves. A simple example was given by dmckee in ...

5

Well, the problem is still open. Although maybe the axioms were taken as self-evident for mathematics, Hilbert did not really want mathematically self-evident axioms to be the basics for physical axioms. Since Gauß and the hyperbolic space, it is well known that you can get equally valid models from different assumptions that could all be seen as ...

4

All the key parts of quantum mechanics may be found in classical physics. 1) In statistical mechanics the system is also described by a distribution function. No definite coordinates, no definite momenta. 2) Hamilton made his formalism for classical mechanics. His ideas were pretty much in line with ideas which were put into modern quantum mechanics long ...

4

Given a $C^\ast$-algebra $A$, its "Bohr topos" (see there for a survey) is the presheaf topos on its commutative subalgebras. The idea here is that if we think of $A$ as the algebra of quantum operators of a quantum mechanical system (for instance all the bounded operators on the Hilbert space of states of a system), then the commutative subalgebras ...

4

Hilbert's Sixth problem is not the same as finding the theory of everything and then making the maths rigorous. This is a very common misconception, and has led to people thinking that making renormalisation in QFT rigorous was the main thing to do. But in fact Hilbert stated explicitly that it would be just as important to axiomatise false physical ...

3

As an initial aside, there is nothing uniquely ‘quantum’ about non commuting operators or formulating mechanics in a Hilbert space as demonstrated by Koopman–von Neumann mechanics, and there is nothing uniquely ‘classical’ about a phase space coordinate representation of mechanics as shown by Groenewold and Moyal’s formulation of Quantum theory. There does ...

3

My impression from the literature is that physicists are still divided on this question. The Quantum Information Theory camp says the latter, but the Quantum Optics people say the former. A related, but distinct, issue is whether one regards the concept of «open system» as a mere mathematical convenience, or as a fundamental concept. This issue has ...

2

I do not agree with Marek's statement that ''in many practial applications of statistical mechanics, the ergodic hypothesis is very important, but it is not fundamental to statistical mechanics, only to its application to certain sorts of experiments.'' The ergodic hypothesis is nowhere needed. See Part II of my book Classical and Quantum Mechanics via Lie ...

2

You may be interested in these lectures: Entanglement and the Foundations of Statistical Mechanics The smallest possible thermal machines and the foundations of thermodynamics held by Sandu Popescu at the Perimeter Institute, as well as in this paper Entanglement and the foundations of statistical mechanics. There is argued that: "the main postulate ...

2

Before answering, I would like to say that the difference between macroscopic and microscopic is not made in terms of ensembles of systems; in fact, quantum mechanics has an ensemble interpretation. About your questions, my answers are the following: Yes. General relativity is a pre-quantum theory, which means that does not account for the discrete ...

2

Was Einstein explicit about this in deriving it? Or did he simply start by assuming that matter can be modelled as a continuously subdivisible fluid and take it from there? Judge by yourself with this excerpt from the Princeton lectures (1921), published in english as "The Principle of Relativity". When departing from Poisson's equation in his heuristic ...

2

General relativity is a classical theory, so it makes sense at all levels, though that's different from being correct at all levels (it shouldn't be). The energy-momentum tensor doesn't intrinsically have anything to do with statistical mechanics or fluids at all. Its size just reflects that gravity is a spin-2 field. For a particle with charge $q$ in its ...

2

First of all one should define what illusion means. 1) a) obsolete : the action of deceiving b) (1) : the state or fact of being intellectually deceived or misled : misapprehension (2) : an instance of such deception 2) a) (1) : a misleading image presented to the vision (2) : something that deceives or misleads ...

1

(Classical) field theory lives naturally in more general spaces than Hilbert (and even more general than Banach). The space of smooth sections of a fiber bundle is a Fréchet manifold (if the basespace is compact). If this fiber bundle thing is new to you, you can consider your fields just as smooth functions $\psi: \text{spacetime} \rightarrow R$. In field ...

1

Thomas's Calculus has an instructive Newtonian Mechanics exercise which everyone ought to ponder: the gravitational field strength inside the Earth is proportional to the distance from the centre, and so is zero at the centre. And, of course, there is the rigorous proof that if the matter is uniformly distributed in a sphere, then outside the sphere it ...

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