Understanding the Uniform Probability Distribution in Statistical Ensembles

[stevendaryl]

To make physical sense of an expectation value of an observable, you have to say what that expectation value means for an observation. And what is that? It isn't that a measurement of quantity $A$ will always produce value $\langle A \rangle$. It isn't that it will always produce something in the range $\langle A \rangle \pm std(A)$, where $std(A)$ means the standard deviation. It seems to me that to connect expectation values with observations, you have to get into probabilities. So expectation values have all the same conceptual problems that probabilities do.

 

[A. Neumaier]

I don't want to make a fool of you, but I think that you are claiming insights that you don't actually have. You aren't doing anything different than "shut up and calculate".

I am just claiming that the meaning of an abstract concept is determined by its use, not by its historical origin. I know very well how expectations are used in statistical mechanics, and nowhere does one make the slightest use of probabilities. These probabilities are as fictitious as the ensembles Gibbs introduced to justify the expectation calculus (because in his time abstract algebra was still far in the future). Whereas one makes frequent use of the meaning discussed in the post referred to in post #164, which stands for itself without any reference to probabilities.

 

[stevendaryl]

But the measurement is done on the single system only. The others are just fictitious copies (as Gibbs told us) without any influence on the measured system. Your expectation would be an average over fictitious measurements, which makes no sense.

I don't know why it doesn't make sense to you, but everybody has his own limitations.

 

[A. Neumaier]

It seems to me that to connect expectation values with observations, you have to get into probabilities.

Only into uncertainty.

But to connect classical observables with observation you also have to get into uncertainty. Measuring the side and the diagonal of a square posed the basic conflict already 25 centuries ago.

It is illegitimate to equate uncertainty with probability, as you constantly do. Uncertainty had a meaning many centuries before probabilities were even conceived as a concept. And today it still has a different, far more encompassing meaning, as the link to wikipedia shows.

 

[stevendaryl]

I am just claiming that the meaning of an abstract concept is determined by its use, not by its historical origin. I know very well how expectations are used in statistical mechanics, and nowhere does one make the slightest use of probabilities.

Okay, what does it mean, in practice, to say that a thermodynamic quantity $A$ has expectation $\langle A \rangle$?

 

[A. Neumaier]

I don't know why it doesn't make sense to you, but everybody has his own limitations.

Because fictitious systems cannot be measured! The measurement result on a single system must be a property of the single system, and cannot depend on properties of imagined copies.

 

[stevendaryl]

Because fictitious systems cannot be measured! The measurement result on a single system must be a property of the single system, and cannot depend on properties of imagined copies.

You're getting confused. The measurement result is not an expectation. I'm talking about the relationship between the measurement result and the theoretically computed expectation value.

 

[A. Neumaier]

Okay, what does it mean, in practice, to say that a thermodynamic quantity $A$ has expectation $\langle A \rangle$?

I gave the link stating the precise meaning repeatedly in this discussion, last in post #164.

In a slightly fuzzy (but still fully correct) version, one can say one can measure (in principle) $\langle A \rangle$ with a negligible uncertainty if the system is large enough. There is no uncertainty at all in the thermodynamic limit that is usually invoked when deriving thermodynamics from statistical mechanics.

 

[stevendaryl]

I gave the link stating the precise meaning repeatedly in this discussion, last in post #164.

The question isn't how to CALCULATE expectation value, the question is, what is the physical significance of saying that the expectation value of $A$ is $\langle A \rangle$? A physical theory has two parts: one is mathematical, which tells you how to compute various quantities, and the second is observational, which is how those quantities relate to our observations. I'm asking about the second.

 

[A. Neumaier]

You're getting confused. The measurement result is not an expectation. I'm talking about the relationship between the measurement result and the theoretically computed expectation value.

What was it exactly that you claimed? Did you mean to say no more than that the theoretically computed value is the average over an ensemble of similar systems? This does not give any relation to a measurement result, it only relates a theoretical value to other theoretical values. Moreover, the theoretical value depends on which ensemble you use to define which systems are similar. Each time I get a different result. Which one is the one related to the measurement?

Hence your proposed relationship amounts to nothing. (One has the same problem with classical probability: The probability to get lung cancer depends a lot on whether you choose the ensemble of all people or the ensemble of all heavy smokers. Which one is the correct theoretical probability? And how do you check it on a particular person who didn't get lung cancer?)

 

[stevendaryl]

What was it exactly that you claimed? Did you mean to say no more than that the theoretically computed value is the average over an ensemble of similar systems? This does not give any relation to a measurement result, it only relates a theoretical value to other theoretical values.

That's what your definition of "expectation value" does.

 

[A. Neumaier]

The question isn't how to CALCULATE expectation value, the question is, what is the physical significance of saying that the expectation value of $A$ is $\langle A \rangle$? A physical theory has two parts: one is mathematical, which tells you how to compute various quantities, and the second is observational, which is how those quantities relate to our observations. I'm asking about the second.

I was answering the second. The observation gives approximately the expectation, with an uncertainty given by the standard deviation. No probabilities are involved in either asserting or checking this. Why do we have all the error bars in scientific reports on measurements?

 

[A. Neumaier]

That's what your definition of "expectation value" does.

I was asking two questions. Your comment is answering neither.

 

[stevendaryl]

I was answering the second. The observation gives approximately the expectation

But it doesn't. You're not going to get the expectation.

with an uncertainty given by the standard deviation.

Then what does "You will get $\langle A \rangle$ with uncertainty $\delta A$" mean? What does it mean that the uncertainty is $\delta A$?

It doesn't mean that you will get a value between $\langle A \rangle - \delta A$ and $\langle A \rangle + \delta A$. So you haven't actually connected the theoretical result with observations.

 

[A. Neumaier]

You're not going to get the expectation.

I didn't claim I would. If you are measuring the diagonal of a square of side 1 you are also not getting $\sqrt{2}$.

Then what does "You will get $\langle A \rangle$ with uncertainty $\delta A$" mean? What does it mean that the uncertainty is $\delta A$?

It means that with high quality measurement equipment, the difference is bounded by a small multiple (typically less than 3, but 5 in case you want to have very high confidence) of the uncertainty. If this is not the case you expect to have an error in either the prediction procedure, or the experimental setting, or the numerical evaluation of the measurement protocol. (Or you try to publish your result as a failure of the laws of quantum mechanics. But it is unlikely your paper will be accepted unless others can reproduce your result.)

 

[stevendaryl]

I didn't claim I would. If you are measuring the diagonal of a square of side 1 you are also not getting $\sqrt{2}$.

It means that the difference is bounded by a small multiple (typically less than 3, in case you want to have very high conficence) 5 of the uncertainty.

But that's not actually true. The fact that the expectation value of $A$ is $\langle A \rangle$ and that the standard deviation is $\delta A$ doesn't actually imply that my measurement will be between $\langle A \rangle - \delta A$ and $\langle A \rangle + \delta A$. So what does it imply?

 

[A. Neumaier]

But that's not actually true. The fact that the expectation value of $A$ is $\langle A \rangle$ and that the standard deviation is $\delta A$ doesn't actually imply that my measurement will be between $\langle A \rangle - \delta A$ and $\langle A \rangle + \delta A$. So what does it imply?

Again I did not claim that. Why do you object to things I didn't say?

 

[stevendaryl]

Again I did not claim that. Why do you object to things I didn't say?

What you said was "The observation gives approximately the expectation, with an uncertainty given by the standard deviation". But that has two additional undefined terms in it: "approximately" and "uncertainty". How do you make sense of those two words, in a non-circular way?

Your claim that expectation is less problematic than probability is just false.

 

[A. Neumaier]

What you said was "The observation gives approximately the expectation, with an uncertainty given by the standard deviation". But that has two additional undefined terms in it: "approximately" and "uncertainty". How do you make sense of those two words, in a non-circular way?

By assuming that my readers understand English.

It is impossible to give definitions in which every word used is defined as well. You can't define anything at all in this way. I place the residual uncertainty in my definition in the location where they actual are when people are doing experiments.

 

[stevendaryl]

By assuming that my readers understand English.

But the usual interpretations of "uncertainty" and "approximately" are subjective. So your move from "probabilities" to "expectations" doesn't actually accomplish anything, as far as making the subject less problematic.

 

[A. Neumaier]

But the usual interpretations of "uncertainty" and "approximately" are subjective.

Not more than language in general. In spite of this subjectivity, people have a good (though also subjective) sense of what objectivity means.

The purpose of objectivity is to enable a group of cooperative people called scientists to arrive at a reliable, objective, and hence predictive consensus. Not to make everything look unambiguous and logically 100.0000000000000...% correct to nitpickers like you.

 

[stevendaryl]

Not more than language in general. In spite of this subjectivity, people have a good (though also subjective) sense of what objectivity means.

It seems to me that you are just hiding problems under the rug. You don't like probability, because it's so subjective, so you replace it by expectation, which is subjective in the exact same sense.

 

[stevendaryl]

I really have to get out of this thread...

 

[A. Neumaier]

But the usual interpretations of "uncertainty" and "approximately" are subjective.

There are standardization efforts to reduce even this amount of subjectiveness. See, e.g., the "Guide to the Expression of Uncertainty in Measurement" (GUM) published by ISO, the National Institute for Standards and Technology (NIST) Technical Note 1297, "Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results", and the Eurachem/Citac publication "Quantifying Uncertainty in Analytical Measurement".

But as you can see from these documents, every attempt to define something accurately results only in much more voluminous explanations using even more undefined words.

Language, and hence science is therefore intrinsically circular. But this benign form of circularity doesn't matter.

The standard practice is to state your assumptions in as clear terms as possible (using standard language without defining it) and start from there. Expectation (using ''approximate'' and ''small multiple'' as self-explained words in terms of which uncertainty is definable precisely) is a far better starting point than betting - which in science is completely hypothetical.

 

[A. Neumaier]

You don't like probability, because it's so subjective, so you replace it by expectation, which is subjective in the exact same sense.

It is not the same sense. Every child can interpret ''a small multiple, typically 3 or 5'', which is used in my explication of approximate, uncertain, and expectation, while probability is a fairly confusing concept even for adults, as the story with the 3 doors demonstrates.

 

[A. Neumaier]

It seems to me that you are just hiding problems under the rug.

Only under the rug of common language (for simple phrases like ''a small multiple") where everyone hides stuff since it is already full of philosophical problems. But common language is necessary to do any kind of science.

Whereas subjective probability (the rug where you are hiding the problems under) is another can of worms involving additional, much more severe philosophical problems.

 

[A. Neumaier]

the usual interpretations of "uncertainty" and "approximately" are subjective.

whereas I interpreted both very carefully in terms of the only undefined notion of ''a small multiple, typically 3 or 5'', which I can assume to be understood by everybody.

 

[Jilang]

I am going with Dirac. Probabilities need not be positive.
http://arxiv.org/pdf/1008.1287.pdf

 

[A. Neumaier]

Probabilities need not be positive.

This statement is correct with probability 1 - since probabilities can be zero it is a triviality.

 

[Demystifier]

The Hamiltonian H is invariant in time, hence does not fluctuate at all.

In QM, fluctuation is not a random change with time. Fluctuation is just another name for uncertainty. So the Hamiltonian fluctuates in any state which is not an eigenstate of $H$.

 

[vanhees71]

Apply in the sense that statistical mechanics applies to a single glass of water. One uses ensemble expectation values for the single quantum system [and, according to Gibbs, nonphysical, imagined repetitions to justify the ensemble language for the single use case] to assign a temperature and other things that can be measured.

Single, nonrepeated measurements of temperature, pressure and volume can be used to check the predictions of quantum mechanics in equilibrium. These measurements have nothing to do with any of the mock measurements of identically prepared systems discussed in the traditional interpretations of quantum mechanics.

The single glass of water is described by thermodynamic quantities like temperature and pressure. If you measure its temperature you have to put a thermometer for a sufficiently long time into the water. Then the thermometer equilibrates with the water, and you can read off a temperature, which is a time-averaged kinetic energy per particle. Also when looking at macroscopic quantities of a single system these macroscopic quantities are averaged (in this case over time) microscopic quantities. The thermometer as a measurement apparatus doesn't resolve the thermal (and quantum) fluctuations of energy, and thus averages out these fluctuations delivering a macroscopic quantity we define as temperature.

 

[A. Neumaier]

In QM, fluctuation is not a random change with time. Fluctuation is just another name for uncertainty. So the Hamiltonian fluctuates in any state which is not an eigenstate of $H$.

True, but this doesn't conform with stevendaryl's use of the term, which I was using in the discussion with him.

 

[vanhees71]

If not in a stationary state, there are also quantum fluctuations of quantities in time, right?

 

[A. Neumaier]

The single glass of water is described by thermodynamic quantities like temperature and pressure. If you measure its temperature you have to put a thermometer for a sufficiently long time into the water. Then the thermometer equilibrates with the water, and you can read off a temperature, which is a time-averaged kinetic energy per particle. Also when looking at macroscopic quantities of a single system these macroscopic quantities are averaged (in this case over time) microscopic quantities. The thermometer as a measurement apparatus doesn't resolve the thermal (and quantum) fluctuations of energy, and thus averages out these fluctuations delivering a macroscopic quantity we define as temperature.

Yes; you make my point: Compare your description with what the quantum mechanics 1 postulates claim a measurement of a quantum system is.

Single, nonrepeated measurements of temperature, pressure and volume can be used to check the predictions of quantum mechanics in equilibrium. These measurements have nothing to do with any of the mock measurements of identically prepared systems discussed in the traditional interpretations of quantum mechanics.

 

[A. Neumaier]

If not in a stationary state, there are also quantum fluctuations of quantities in time, right?

$H$ is time invariant, hence doesn't fluctuate in time, no matter which state is considered. It is also translation invariant, hence doesn't fluctuate in space. Thus fluctuations have neither a dynamic nor a spatial meaning - the term is used in the same figurative way as vacuum fluctuations in a vacuum whose particle number is zero at all times and everywhere.