GENERALIZING THE MAXWELL HYPOTHESIS AS A
CONCEPT OF OPTIMAL ENTROPY
University of Hagen / Germany
CONCEPT OF OPTIMAL ENTROPY
University of Hagen / Germany
Let us take the combined work of Briton James Clark Maxwell and Austrian Ludwig Boltzmann from the second part of 19th century as starting point. In order to study the phenomena of Statistical Mechanics Boltzmann has created a model of gas molecules representing them as N hard discs -- comparable billiard balls – within a container being imposed on the Newtonian dynamics.
The great perception of Maxwell, i.e. the so-called Maxwell Hypothesis, states that the equilibrium distribution of momenta is a normal one; its variance being determined – up to a multiplicative constant -- by the temperature in [K], we speak here as of the Maxwell-Boltzmann distribution.
Although generally accepted a direct examination of the Maxwell Hypothesis by laboratory physics is not possible; a way out is offered by computer experimentation.
By a computer experiment we can show that the equilibrium distribution of momenta is indeed a normal one. The momentum distribution is estimated in the statistical word sense based on the Boltzmann model of moving gas molecules from a 2-dimensional space implemented on the computer.
A remark concerning the estimation technique:
The momenta realized by the computer experiment in the momentum space IU = IR2 are projected onto the linear subspace Lß of IU with polar angle ß varying between 0 and 360 degree in order to estimate the density-graph of the momentum distribution by a non-parametric procedure and parallel to this the variance of the actual distribution is estimated parametrically, i.e. within the class of centered normal distributions, which yields a second estimate of the momentum distribution. If these two estimates coincide we know the type of the distribution on Lß but also the actual parameter, 0° <= ß <= 360°, which determines by a Corollary of a Theorem of Cramer and Wold, cf. Billingsley (1986) Theorem 29.4, also the momentum distribution on IU.
The estimated variances of the projected normal distribution are used – according to the Maxwell Hypothesis – to calculate the 'temperature' of the virtual system of moving molecules implemented on the computer. By the rotational symmetry of the Maxwell-Boltzmann distribution in even higher dimensions than 1 and the relation between temperature and variance it is thereby clear, that temperature is a scalar quantity.
The Maxwell-Boltzmann distribution reveals itself – by mathematical considerations – as the one having maximal entropy under the condition of conservation of energy – we speak here of the entropic momentum distribution.
If an equilibrium distribution coincides with the entropic distribution, then we always have -- as a necessary condition – that temperature is a scalar quantity. In the case of such a coincidence we say that the generalized Maxwell Hypothesis or the Entropy Principle holds true.
As an important question we have: Is the (generalized) Maxwell Hypothesis resp. the Entropy Principle strictly confined to the standard Newtonian dynamics already treated by Maxwell and Boltzmann or does this fact open a window to a more general insight?
To this end we examine based on computer experiments representing moving gas molecules being imposed on various other dynamics than the standard Newtonian one, whether the estimated momentum equilibrium distribution coincide with the entropic one or in other words we examine the validity of generalized Maxwell Hypothesis for various types of dynamics being different from the standard Newtonian one.
What happens, if we substitute – in the sense of non-real physics – the mass matrix m I (m mass of a molecule, I identity matrix) typical for the standard Newtonian dynamics by a general positive definite matrix M with the consequence that the entropic momentum distribution is still a normal distribution but by contrast to the standard Newtonian dynamics now with elliptical contours. In other words: As the rotational symmetry is lost, as we have it for the standard Newtonian dynamics, the question arises, whether temperature is still a scalar quantity. The latter is a necessary condition for the validity of generalized Maxwell Hypothesis.
But even so -- by a computer experiment an affirmative answer can be given for the validity of the generalized Maxwell Hypothesis.
The same affirmative answer can be given, when in a next experiment the causality concept as we have it in classical physics is given up. Till now an energy splitting of two molecules is caused by an impact of them. The link of the energy splitting and the collision of the molecules is broken now. Any two molecules can split their energies and exchange their momentum according to the laws of momentum and energy conservation. The partners of an energy splitting as well as the time points and the places of it are determined randomly.
A quite new situation we have treating a system of moving molecules being imposed on the relativistic dynamics due to Albert Einstein. We have not only a totally new dynamics but also the entropic momentum distribution is of a total another statistical type.
Following Werner Heisenberg we consider finally a discrete momentum space; i.e. the momentum space is – according to quantum mechanics – a lattice. This means energy and also the momentum components cannot assume any value of IR, the continuum of real numbers. These values are restricted now to a sub-lattice of IR.
Also in the following experiments causality is given up. Notice in this context, that that causality plays no role in quantum mechanics.
Denote by e the unit vector of the (classical) momentum exchange direction, connecting the both molecules i and j.
Conservation of momentum leads in classical physics to the following ansatz, relating the momenta u*i, u*j and u i , u j of the molecules i and j shortly after and shortly before momentum exchange, respectively:
u*i = u i + s e
u*j = u j -- s e
u*j = u j -- s e
where the scalar s is determined by the condition of energy conservation.
Implementing now a micro-model of moving gas molecules for the case of a discrete momentum space IU being a lattice a problem arises, because the unit vector e may fail to be an element of IU (being a sub--lattice of IR).
One may try to overcome the sketched problem determining a dynamics for which the unit vector e is substituted by a unit vector e* being an element of IU such that the angle between e and e* becomes minimal.
If the generalized Maxwell Hypothesis (Entropy Principle) should be fulfilled for the described dynamics, then temperature should be a scalar quantity!
But the computer experiment shows an another result. There are at least two temperatures, a 'horizontal' and a 'vertical' one, depending on the fact on which subspace the momentum data are projected. In other words we have found a dynamics for which the generalized Maxwell Hypothesis, i.e. the Entropy Principle does not hold true.
Is it possible to determine a slightly different dynamics with the same momentum space and the same Hamiltonian (not explicitely introduced here), such that the generalized Maxwell Hypothesis, i.e. the Entropy Principle, is fulfilled?
To this end consider the set of all nodes of the momentum space of a pair of molecules for which the sums of energies and momenta remain constant; i.e. for which the laws of energy and momentum conservation are fulfilled; we speak of the set of 'possible' nodes.
This set is finite and not empty, so the dynamics for the discrete momentum space is defined in such a way, that for any energy splitting one of these 'possible' nodes is realized randomly according to the uniform distribution determining the next momentum configuration.
With this additional rule to determine a dynamics for a discrete momentum space the computer experiments shows that the validity of the generalized Maxwell Hypothesis, i.e. the Entropy Principle, is fulfilled.
The experiment is meaningful insofar as the presented micro-model by the chosen Hamiltonian (not made explicit here) supports the theoretically postulated probabilities of the excited energy levels of the harmonic oscillator from quantum mechanics.
Physicists have always postulated, i.e. they have always – successfully believed – in the Entropy Principle, but a respective micro-model confirming this postulate for the harmonic oscillator did -- to our knowledge – not exist.
Implementing now a micro-model of moving gas molecules for the case of a discrete momentum space IU being a lattice a problem arises, because the unit vector e may fail to be an element of IU (being a sub--lattice of IR).
One may try to overcome the sketched problem determining a dynamics for which the unit vector e is substituted by a unit vector e* being an element of IU such that the angle between e and e* becomes minimal.
If the generalized Maxwell Hypothesis (Entropy Principle) should be fulfilled for the described dynamics, then temperature should be a scalar quantity!
But the computer experiment shows an another result. There are at least two temperatures, a 'horizontal' and a 'vertical' one, depending on the fact on which subspace the momentum data are projected. In other words we have found a dynamics for which the generalized Maxwell Hypothesis, i.e. the Entropy Principle does not hold true.
Is it possible to determine a slightly different dynamics with the same momentum space and the same Hamiltonian (not explicitely introduced here), such that the generalized Maxwell Hypothesis, i.e. the Entropy Principle, is fulfilled?
To this end consider the set of all nodes of the momentum space of a pair of molecules for which the sums of energies and momenta remain constant; i.e. for which the laws of energy and momentum conservation are fulfilled; we speak of the set of 'possible' nodes.
This set is finite and not empty, so the dynamics for the discrete momentum space is defined in such a way, that for any energy splitting one of these 'possible' nodes is realized randomly according to the uniform distribution determining the next momentum configuration.
With this additional rule to determine a dynamics for a discrete momentum space the computer experiments shows that the validity of the generalized Maxwell Hypothesis, i.e. the Entropy Principle, is fulfilled.
The experiment is meaningful insofar as the presented micro-model by the chosen Hamiltonian (not made explicit here) supports the theoretically postulated probabilities of the excited energy levels of the harmonic oscillator from quantum mechanics.
Physicists have always postulated, i.e. they have always – successfully believed – in the Entropy Principle, but a respective micro-model confirming this postulate for the harmonic oscillator did -- to our knowledge – not exist.
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