Certainty and Classical Physics

[reilly]

There is a presumption that classical physics describes a world of certainty. For example, moving classical objects are, supposedly, described by well defined trajectories, and so on. And, in fact, this take seems to work wonderfully well in practice in the prescribed theaters of physics.

But... In physics labs, at any level, we learn that we must take errors of measurement into account. One measurement is virtually never sufficient to pin something down. In the ususal drill, errors are assumed to be Gaussian, and standard statistics is usually sufficient to set the standard error.

So, given the reality of experimental errors, what can be said about the certainty of classical physics?

Regards,
Reilly Atkinson

 

[russ_watters]

Classical physics is know to not conform with things like quantum uncertainty, but errors in measurement are strictly a technology thing, so the fact that there are errors in experiments says nothing about classical physics.

 

[BioBen]

So, given the reality of experimental errors, what can be said about the certainty of classical physics?

If you really really want to head in that direction, then i think the more correct way to say it is that it may lead to chaotic systems (chaos theory, because we can never make "perfect" experiments => you don't know exact the intial conditions), but in no way it is linked with QM indetermination principle.

I would say that the first one is technological wheras the other one is "intrinsic" to your system/ to how Nature works (or seems to work).

 

[thtadthtshldntbe]

I think Einstein used the term determinism, or at least his camp did to desribe Relativity and classical mechanics, whereas they used the term indeterminism to describe quantuam mechanics, chromodynamics, or whatever the quantum catch phrase is now.

In a deterministic universe, one could be certain that the result of a given calculation for something meant something precise or was based on something precise, whereas in an indeterministic universe the result at best could represent some probability or the calculation was in and of itself based on some probability.

 

[Galileo]

In the description of classical mechanics, we associate a definite point in space at each time with particle. The idea is that the particle really IS there, exactly. That we always have some error in measurement when we want to know where that particle is doesn't relate to that. Any error in measurement can in priniciple be made as small as we like (>0). It's just a matter of how clever we are.

 

[Theoretician]

I remember reading a point made by Feynman that, in a classical world, even the most minute of errors in measurement of one particle in a system would result in indeterminism eventually because of error propagation.

 

[reilly]

Note that I said that classical physics has a domain in which it is considered the theory de jour. Quantum theory has nothing to do with my question. In fact, the errors of measurement are important in QM, but that's another story. And, it makes absolutely no difference whether errors of measurement are "technical" or not.

Theoretician has done me a great favor; Feynman's point is the one I was leading to. And, indeed, the possibility of chaotic motion makes the issue all the more problematic.


A more practical version of my question is: how much do you trust radar?

Regards,
Reilly Atkinson

 

[SGT]

A more practical version of my question is: how much do you trust radar?

Regards,
Reilly Atkinson

Radar measurements contain errors, but a radar takes several measurements by second, so filters can be used to reduce the error. If you are measuring a stationary object, the mean is an optimal estimation of the position. If you are measuring a moving object, you must use a filter that uses the dynamic equations of the movement, like a Kalman Filter, to estimate not only the position but velocity and acceleration.

 

[Juan R.]

There is a presumption that classical physics describes a world of certainty. For example, moving classical objects are, supposedly, described by well defined trajectories, and so on. And, in fact, this take seems to work wonderfully well in practice in the prescribed theaters of physics.

But... In physics labs, at any level, we learn that we must take errors of measurement into account. One measurement is virtually never sufficient to pin something down. In the ususal drill, errors are assumed to be Gaussian, and standard statistics is usually sufficient to set the standard error.

So, given the reality of experimental errors, what can be said about the certainty of classical physics?

Regards,
Reilly Atkinson

You really obtain in an experiment for an observable A is

A = <A> + deltaA

and classical physics (EXCEPT thermodynamics) traditionally focuses ONLY on average values.

For example Newton second law usually wrote as

ma = F

would be written as

m<a> = <F>

The most general equation is Langevin one

ma = <F> + F_random

Nobody can prove that F_random was the result of a deterministic underlying force (between others requirements one would use a laboratory instrumental with INFINITE precision, which is obviously imposible)

What can be said is that determinism is a philosophical option: Ones argue (newer prove) that F_random is the reflect of some ASSUMED underlying deterministic force f, others argue that the world is inherently non-determinist (e.g. Juan R., Prigogine, etc.) and that f does not exist.

From a scientific point of view our universe is stochastic (newer determinist) and in real laboratory measurements we ALWAYS obtain

ma = <F> + F_random

and this is the reason that we repeat experiments for obtaining average values and write down equations for those average values.

m<a> = <F>

The equations of classical mechanics, EM, etc are valid only for the averages.

Also usual (elementary) equations of others disciplines are valid for averages. For example for the chemical reaction A --> B

the kinetic equation

d[A]/dt = - k [A]

that appears in any elementary textbook of chemistry is only valid for <[A]>.

d<[A]>/dt = - k <[A]>

The most general expresion is

d[A]/dt = - k [A] + c

where c is a chemical random force. This is the reason that when one does an experimental measurement of REAL [A] versus time one do NOT obtain the exponential predicted by the equation valid for the average. Then one extract <[A]> from the real data for A via statistical analysis and compute via theoretical methods the value of k for that reaction. In macroscopic chemical kinetics one is generally interested only in k; in single molecular dynamics, however, one is really interested in c. In fact, c plays a crucial role in the chemistry of biological systems, for example in biological chanels in membranes.

P.S: If i am not wrong we have found a possible demonstration that f does not exist using a relativistic quantum formulation that includes also quantum gravity, in the Center for CANONICAL |SCIENCE). If this research is correct, relativistic quantum gravity restrictions to spacetime 'foam' add an inherent indeterminism to universe.

Remember that the dynamical structure of QM -far from usual beliefs- is DETERMINIST. Indeterminism in QM arises only in the quantum measurement process WHICH is not explained by the Schrödinger equation. That quantum gravity may play a special role in the quantum measurement process is also maintained by a number of authors, e.g. Penrose. In Penrose theory, the indeterminism of quantum mechanics arises due to structure of spacetime in quantum gravity.

 

[reilly]

SGT -- Indeed, having actually worked on a radar and communications system (for the FAA), I chose radar quite purposefully. And, actually, it's worse than you suggest. Ground reflection (& scattering) and ground absorption can't always be neglected, particularly for ground based radars (See Sommerfeld's Partial Differential Equations for discussions of these types of issues.) And, there are various sources of noise -- especially in the receiver circuits -- in the actual E&M wave propagation -- effectively random indices of refraction, from moisture, turbulence, and whatever. It's quite complicated. For a radar and communication system, once an airplane is detected, one wants to know where it will be on the next beam sweep so communications an be accomplished, so forecasting target positions and dynamics is also an issue. technically it's a big stochastic mess. (One of the first attempts to deal with such forecasting came from Norbert Weiner during WWII in his work on anti-aircraft tracking and targeting. Also, to see how complicated some of these propagation issues are, without stochastic influences, take a look at Dean Duffy's book, Transform Methods for Solving Partial Differential Equations.) But, given years of experience, we've become quite good at dealing with these issues, Kalman Filters and worse (non-linear), and the empirical record says we can trust radar quite highly in may circumstances, and that Maxwell's Eq.s, work nicely as the basis for describing and understanding radar.

Juan R -- Just a couple of points, as I'm running out of time. If QM is deterministic, via the Schrödinger eq., then necessarily Newtonian and Maxwellian E&M dynamics are as well -- the equations + boundary conditions completely specify solutions for all time. This is pretty well known. Physics has it's own form of determinism. But, under many circumstances. nature does appear to have a stochastic aspect. The usual approach is to assume that any stochastic errors, even if not small, can be averaged out.

But as control engineers will tell you, that usual approach does not always work. And, your version of the Langevin Eq. is not the most general. In fact, usually, you must combine both measurement error propagation and system dynamics to give a general description of a system's dynamic evolution. This gets into Kalman Filters (Optimal Control and Estimation, RF Stengel, Dover), the estimation scheme of Larson & Peschon (Optimization Theory with Applications DA Pierre, Dover). As far as the Langevin Eq. is concerned, I'll grant you the <F> term rather than F although I'm dubious, but as Chandrasekhar points out in his classic Rev. Mod. Phys (19430 article, Stochastic Problems in Physics and Astronomy that justification of a Langevin eq. requires very drastic assumptions. The article is reprinted in Selected Papers on Noise and Stochastic Processes, (Nelson Wax, ed, Dover) along with Uhlenbeck's two fundamental papers on Brownian Motion (with both Langevin and Fokker-Planck approaches. Worse yet, Doob's paper on Stochastic Differential eq., in which he shows that the speed described by the Langevin Eq. is actually nondifferentiable. Doob gets around this by various tricks, including going to equations for the distribution of speeds.

We learn classical physics as deterministic (except for obvious subjects as thermo, statistical mechanics, ..). We then quite independently devise theories of measurement, which we apply to measure basic classical physics. (Unless we want to look at airplanes traveling through a turbulent atmosphere, or track cascades of cosmic rays through the atmosphere...) When all else fails, Monte carlo simulations are available.
(Would like to know more about your work with gravity,...)
Regards,
Reilly Atkinson

 

[Juan R.]

Juan R -- Just a couple of points, as I'm running out of time. If QM is deterministic, via the Schrödinger eq., then necessarily Newtonian and Maxwellian E&M dynamics are as well -- the equations + boundary conditions completely specify solutions for all time. This is pretty well known. Physics has it's own form of determinism. But, under many circumstances. nature does appear to have a stochastic aspect. The usual approach is to assume that any stochastic errors, even if not small, can be averaged out.

But as control engineers will tell you, that usual approach does not always work. And, your version of the Langevin Eq. is not the most general. In fact, usually, you must combine both measurement error propagation and system dynamics to give a general description of a system's dynamic evolution. This gets into Kalman Filters (Optimal Control and Estimation, RF Stengel, Dover), the estimation scheme of Larson & Peschon (Optimization Theory with Applications DA Pierre, Dover). As far as the Langevin Eq. is concerned, I'll grant you the <F> term rather than F although I'm dubious, but as Chandrasekhar points out in his classic Rev. Mod. Phys (19430 article, Stochastic Problems in Physics and Astronomy that justification of a Langevin eq. requires very drastic assumptions. The article is reprinted in Selected Papers on Noise and Stochastic Processes, (Nelson Wax, ed, Dover) along with Uhlenbeck's two fundamental papers on Brownian Motion (with both Langevin and Fokker-Planck approaches. Worse yet, Doob's paper on Stochastic Differential eq., in which he shows that the speed described by the Langevin Eq. is actually nondifferentiable. Doob gets around this by various tricks, including going to equations for the distribution of speeds.

We learn classical physics as deterministic (except for obvious subjects as thermo, statistical mechanics, ..). We then quite independently devise theories of measurement, which we apply to measure basic classical physics. (Unless we want to look at airplanes traveling through a turbulent atmosphere, or track cascades of cosmic rays through the atmosphere...) When all else fails, Monte carlo simulations are available.
(Would like to know more about your work with gravity,...)
Regards,
Reilly Atkinson

Well you would agree that there is nothing undeterministic in the Schrödinger equation. Indeterminism is introduced via collapse postulates where the Schrödinguer equation does not aply: this is the basis of the so-called dualistic structure of current QM.

It is true that Newtonian and Maxwellian E&M dynamics are as well deterministic. I think that i already said that. But in the same form as i explained that ma= F is only VALID for DETERMINISTIC averages (the most general form is ma =/= <F>) the same can be said of Schrödinger equation.

The most general form, which is well-known in modern research (specially in studies of collapse via quantum gravity effects) is the so called Ito-Schrödinger equation (which is also a special case of our equation. The Ito-Schrödinger equation is UNDETERMINISTIC.

Physics has it's own form of determinism. But, under many circumstances. nature does appear to have a stochastic aspect.

As i already explained it is the inverse. Determinism arised ONLY in early physics due that physics began in astronomy and here random components are zero. Chemistry newer claimed that nature was deterministic. Interestingly, centuries after now physics claim that universe is NOT via QM.

Determinism is not exact, it is an approximation when random forces vanish.

The usual approach is to assume that any stochastic errors, even if not small, can be averaged out.

Yes, by the own definition of average used! But the equation for the average describes an ensemble of systems (in the theory of physical ensembles) not the own system.

And, your version of the Langevin Eq. is not the most general.

I do not understand can you write it?

I do not agree that justification of a Langevin eq. requires very drastic assumptions. Simply that spacetime in the small scales is not differentiable. Curiously, the same conclusion is derived from several recent quantum gravity approaches (e.g. non commutative geometry).

We learn classical physics as deterministic

Well also textbooks in special and general relativity 'explain with detail' why aether does not exist in Nature, but that is very different that people did the theory (and several modern authors also. I am undecided) think (for example string theorists attack loop quantum gravity arguing that contain a new kind of aether in the sipn structure).

In a letter to Lorentz dated 17 June 1916, Einstein wrote

I agree with you that the general relativity theory admits of an ether hypothesis as does the special relativity theory.

 

[SGT]

SGT -- Indeed, having actually worked on a radar and communications system (for the FAA), I chose radar quite purposefully. And, actually, it's worse than you suggest. Ground reflection (& scattering) and ground absorption can't always be neglected, particularly for ground based radars (See Sommerfeld's Partial Differential Equations for discussions of these types of issues.) And, there are various sources of noise -- especially in the receiver circuits -- in the actual E&M wave propagation -- effectively random indices of refraction, from moisture, turbulence, and whatever. It's quite complicated. For a radar and communication system, once an airplane is detected, one wants to know where it will be on the next beam sweep so communications an be accomplished, so forecasting target positions and dynamics is also an issue. technically it's a big stochastic mess. (One of the first attempts to deal with such forecasting came from Norbert Weiner during WWII in his work on anti-aircraft tracking and targeting. Also, to see how complicated some of these propagation issues are, without stochastic influences, take a look at Dean Duffy's book, Transform Methods for Solving Partial Differential Equations.) But, given years of experience, we've become quite good at dealing with these issues, Kalman Filters and worse (non-linear), and the empirical record says we can trust radar quite highly in may circumstances, and that Maxwell's Eq.s, work nicely as the basis for describing and understanding radar.


Regards,
Reilly Atkinson

I have also worked in radar information processing. I am quite aware of the sources of process noise you mentioned. For naval radars it is still worse. The antenna moves with the ship and the sea surface contains waves that are continuously moving, instead of the fixed obstacles in ground.
That is why radar processing filters model all those incertitudes as process noise. If the radar is used to track an aircraft, there are also incertitudes about the trajectory, due to wind disturbances and the manoeuvers commanded by the pilot. All of those incertitudes are also lumped in the process noise.
Measurement noise comes from the finite length of the radar pulse, the aperture of the antenna lobe and glint caused by the nonhomogeneous shape of the target and its movements around the center of gravity.