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.
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.
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.
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.
A variational solution of the Liouville equation is used to obtain collective coordinates for simple classical fluids. Comparison is made with macroscopic analogs determined from relevant hydrodynamic equations. Approximate eigenfunctions of the Liouville operator, when constructed from linear combinations of the spatial Fourier transforms (fluctuations) of the mass, momentum, and energy-density operators for the fluid, are found appropriate to low-frequency disturbances. When time derivatives of fluctuations are included among trial functions, the variational procedure provides fluid-state counterparts of usual solidstate phonons.
In (1), the Schrodinger equation is derived using a hydrodynamical continuity of density approach. Thus, spatial density represents x probability. There, however, is also a flow velocity field in the problem v(x) and it is assumed this is irrotational. From this it is stated in (1) that v(x) is proportional to 1/m d/dx S(x) ((1)) where S(x) is some function and m is mass. In this note, we argue that a particle moving in one dimension with constant p (momentum) is automatically irrotational, but it should satisfy ((1)) from (1). In other words, a constant p particle hence constant velocity is associated with a function of x, S(x) even though its density in space is not. If one considers S(x)=ln(B(x)), then ((1)) is equivalent to: p = d/dx ln(B(x)). If one tries to associate B(x) with a probability, then ln(B(x)) is information = px. B(x), however, is a different kind of x probability from density(x) which should be constant for a free moving particle. In fact, using -id/dx instead of d/dx and exp(ipx) for B(x), one finds exp(ipx) as a second x probability. Its modulus yields the constant value of density(x). Thus the irrotational flow assumption leads to p= dS/dx. Incidentally, this form is equivalent to Fermat’s least time principle for light with p=p (x component). We argue that applying p=dS/dx associates an x probability distribution even to a constant p, so the assumption p=dS/dx is essentially a quantum mechanical statement, even if one does not derive the Schrodinger equation. The time independent Schrodinger equation may be obtained by using exp(ipx) as a relative probability and using energy conservation.
Quantum theory and relativity exhibit several formal analogies with fluid mechanics. This paper extends upon known analogies by showing that under specific assumptions, an Euler-Korteweg vortex model can be cast into equations that are mathematically equivalent to the Schrödinger and Klein-Gordon equations. By assuming that the angular momentum of an irrotational vortex in an inviscid, barotropic, isothermal fluid with sound speed c is equal in magnitude to the reduced Planck constant, and incorporating Korteweg capillary stress, a complex wave equation describing the momentum and continuity equations of an Euler-Korteweg vortex is obtained. When uniform convection is introduced, the weak field approximation of this wave equation is formally equivalent to Schrödinger's equation. The model is shown to yield analogues to de Broglie wavelength, the Einstein-Planck relation, the Born rule and the uncertainty principle. Accounting for the retarded propagation of the wave field of a vortex in convection requires the Lorentz transformation and yields a wave equation mathematically equivalent to the Klein-Gordon equation, with Schrödinger's equation appearing as the low-Mach-number limit.