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Online at stacks.iop.org/Non/18/211

Adriana I. Pesci, Raymond E. Goldstein, Recommended P Cvitanović

Published Jan 1, 2004• 0 citations
OpenAlex

Abstract

Here we find a mapping onto the Sturm–Liouville operator of the first two balance equations derived from Boltzmann’s equation. This mapping, which is irreversible and valid only for a subclass of solutions, is achieved by applying a Fourier transform to the momentum coordinate. In light of this irreversibility, it is necessary to develop a set of consistent prescriptions to find the probability of any physical quantity in the p-conjugate space such that it will coincide with the average over the momentum of the true probabilities obtained from the original Boltzmann equation. The one drawback of this prescription is that it is impossible to predict exactly the precise values of the position x and the momentum p at the same time. This uncertainty is limited by the relationship that all conjugate variables in a Fourier transform should obey, namely xp = η/2, where η is a free parameter of the theory. The prescriptions we have found appear to coincide with the postulates of quantum mechanics, whenη is set equal to h̄. This procedure seems to provide a statistical representation of quantum mechanics. PACS numbers: 03.65.Ta, 05.20.Dd, 45.05.+x, 03.65.Sq 1.

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Online at stacks.iop.org/Non/18/227

Adriana I. Pesci, Raymond E. Goldstein, Hermann Uys +1 more

Here we find a mapping onto the Pauli equation of the first two balance equations derived from the classical Boltzmann equation. The essence of this mapping, which we previously used to obtain the particular case of the Sturm– Liouville operator known as Schrödinger’s equation, consists of applying a Fourier transform to the momentum coordinate of the distribution function. This procedure introduces a natural parameter η with units of angular momentum. The main difference between the two cases is the conditions imposed on the probability distribution function, a difference most clearly understood at the level of the hydrodynamic equations generated in the first steps of the mapping. The case leading to the Sturm–Liouville operator corresponds to an irrotational flow, while here the ansatz leading to the Pauli equation corresponds to a fluid with non-zero vorticity. In the context of fluid dynamics, the magnitude of the angular momentum associated with the vorticity is η/2. To perform the mapping we follow the standard technique common in hydrodynamic problems, namely writing the Lagrangian for the Euler equations with the corresponding constraints expressed in terms of the Clebsch variables. PACS numbers: 03.65.Ta, 05.20.Dd, 45.05.+x, 03.65.Sq 1.

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Here we find a mapping onto the Sturm–Liouville operator of the first two balance equations derived from Boltzmann’s equation. This mapping, which is irreversible and valid only for a subclass of solutions, is achieved by applying a Fourier transform to the momentum coordinate. In light of this irreversibility, it is necessary to develop a set of consistent prescriptions to find the probability of any physical quantity in the p-conjugate space such that it will coincide with the average over the momentum of the true probabilities obtained from the original Boltzmann equation. The one drawback of this prescription is that it is impossible to predict exactly the precise values of the position x and the momentum p at the same time. This uncertainty is limited by the relationship that all conjugate variables in a Fourier transform should obey, namely xp = η/2, where η is a free parameter of the theory. The prescriptions we have found appear to coincide with the postulates of quantum mechanics, whenη is set equal to h̄. This procedure seems to provide a statistical representation of quantum mechanics. PACS numbers: 03.65.Ta, 05.20.Dd, 45.05.+x, 03.65.Sq 1.
Here we find a mapping onto the Sturm–Liouville operator of the first two balance equations derived from Boltzmann's equation. This mapping, which is irreversible and valid only for a subclass of solutions, is achieved by applying a Fourier transform to the momentum coordinate. In light of this irreversibility, it is necessary to develop a set of consistent prescriptions to find the probability of any physical quantity in the p -conjugate space such that it will coincide with the average over the momentum of the true probabilities obtained from the original Boltzmann equation. The one drawback of this prescription is that it is impossible to predict exactly the precise values of the position x and the momentum p at the same time. This uncertainty is limited by the relationship that all conjugate variables in a Fourier transform should obey, namely Δ x Δ p = η/2, where η is a free parameter of the theory. The prescriptions we have found appear to coincide with the postulates of quantum mechanics, when η is set equal to ℏ. This procedure seems to provide a statistical representation of quantum mechanics.
Here we present a study of the solutions and mathematical structure of the momentum-averaged Liouville (or Collisionless Boltzmann) equation in Fourier space. We show that the averaging procedure leads to a formalism identical to that of the density matrix of quantum mechanics. This mathematical mapping leaves the averages of all quantities unaltered and provides a unique way to construct the Hermitian version of a given operator. This seems to be the only method that resolves the ambiguity of Hermitization of operators that contain products of non-commuting variables. We also present a systematic perturbation scheme to evaluate correctly the classical solutions from the quantum ones and a formal proof of the approximate correspondence between the Poisson brackets and commutators.
In previous work we have shown that the quantum potential can be derived from the classical kinetic equations both for particles with and without spin. Here, we extend these mappings to the relativistic case. The essence of the analysis consists of Fourier transforming the momentum coordinate of the distribution function. This procedure introduces a natural parameter η with units of angular momentum. In the non-relativistic case the ansatz of either separability, or separability and additivity, imposed on the probability distribution function produces mappings onto the Schrödinger equation and the Pauli equation, respectively. The former corresponds to an irrotational flow, the latter to a fluid with non-zero vorticity. In this work we show that the relativistic mappings lead to the Klein–Gordon equation in the irrotational case and to the second-order Dirac equation in the rotational case. These mappings are irreversible; an approximate inverse is the Wigner function. Taken together, these results provide a statistical interpretation of quantum mechanics.
Here we find a mapping onto the Pauli equation of the first two balance equations derived from the classical Boltzmann equation. The essence of this mapping, which we previously used to obtain the particular case of the Sturm– Liouville operator known as Schrödinger’s equation, consists of applying a Fourier transform to the momentum coordinate of the distribution function. This procedure introduces a natural parameter η with units of angular momentum. The main difference between the two cases is the conditions imposed on the probability distribution function, a difference most clearly understood at the level of the hydrodynamic equations generated in the first steps of the mapping. The case leading to the Sturm–Liouville operator corresponds to an irrotational flow, while here the ansatz leading to the Pauli equation corresponds to a fluid with non-zero vorticity. In the context of fluid dynamics, the magnitude of the angular momentum associated with the vorticity is η/2. To perform the mapping we follow the standard technique common in hydrodynamic problems, namely writing the Lagrangian for the Euler equations with the corresponding constraints expressed in terms of the Clebsch variables. PACS numbers: 03.65.Ta, 05.20.Dd, 45.05.+x, 03.65.Sq 1.