Hot answers tagged foundations
38
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 ...
25
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 ...
10
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 ...
7
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 ...
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
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
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 ...
4
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 ...
4
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 ...
4
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,
...
4
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 ...
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
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 ...
2
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 ...
1
I've been thinking about this question on and off since it was posted, and I have some thoughts that I hope will shed some light on this stuff. I think it helps to begin with the following:
An Analogy from Mechanics.
Consider the following expression that you'll often encounter in classical mechanics:
$$
L(x, \dot x) = \frac{1}{2}m\dot x^2
$$
and let's ...
1
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 ...
1
Hilbert spaces are occur everywhere where the Lagrangian\Hamiltonian is quadratic in derivatives. If the Lagrangian is non-quadratic then the Hilbert spaces are no longer so convenient. In particular in analysis of Navier–Stokes equations the Banach spaces(not Hilbert spaces) are active used.
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
There are classical systems without trajectories with the particles 'going through' all possible classical paths. Check for instance Poincaré resonances and the limits of trajectory dynamics.
The concept of trajectory is an approximation both in quantum and classical mechanics (check above ref.); we recover trajectories when the states are localized $\sigma ...
1
This is an interesting as still open question.
@Quiaochu Yan The point is that we are using a mathematical model to understand how the world is working. We are using vectors and matrices to make prevision of some feature of the nature. You cannot answer to the question "is the mathematical model that we are using describing completely the nature or not?" ...
1
Your question is ambiguous. If you are asking if the density operator $\hat{\rho}$ formalism is more fundamental than the state vector $|\Psi\rangle$ formalism, the response is yes, because the density operator formalism applies to open quantum systems for the which no state vector exists. Moreover, it has been shown in last decades that there exists a kind ...
1
If you want you can go back to Planck's derivation of the black body energy spectrum, otherwise known as Planck's law, as well as Einstein's use of Planck's work in his explanation of the Photo Electric Effect (which garnered him the Nobel prize) in order to first understand some of the experimental motivation. However, to understand the roots of quantum ...
1
I have recently published an important paper, Some special cases of Khintchine's conjectures in statistical mechanics: approximate ergodicity of the auto-correlation function of an assembly of linearly coupled oscillators. REVISTA INVESTIGACIÓN OPERACIONAL VOL. 33, NO. 3, 99-113, 2012
http://rev-inv-ope.univ-paris1.fr/files/33212/33212-01.pdf
which advances ...
1
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 ...
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