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Time evolution equations for hydrodynamic variables with arbitrary initial data
Results in Physics 6 (2016) 252–255
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Time evolution equations for hydrodynamic variables with arbitrary initial data A. Muriel Program in Mathematics, Graduate Center, City University of New York, 506 Fifth Avenue, NY 10016, United States
a r t i c l e
i n f o
Article history: Received 26 November 2015 Accepted 4 December 2015 Available online 19 December 2015 Keywords: Time evolution equations Arbitrary initial data Field velocities Pressure Kinetic energy Solutions of Navier–Stokes equation
a b s t r a c t We prescribe an alternative procedure for arriving at the time evolution equations for hydrodynamic local variables such as density, velocity fields, and kinetic energy using a time evolution equation of the single-particle distribution. It is suggested that when applied to various combinations of pairpotential between monoatoms, geometry and initial data, the prescription will have wide applications in hydrodynamics, including solutions of the Navier–Stokes equation. Ó 2016 The Author. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction There have been a series of four recent papers that showed related solutions of the Navier–Stokes equation [1,2,3,4]. The first [1] was an existence proof of solutions in 3D. The second one [2] was a simplification of Muriel [1] to 2D with field velocities wrapped around a globe, producing flows symmetric about the equator. The third one [3] uses spherical symmetry to produce an analytic model of implosion toward the possibility of controlled nuclear fusion. The first three papers assumed an initially uniform system with delta-function momentum distributions. The fourth paper [4] is valid for an initial spatially uniform system and arbitrary initial momentum distribution. This paper generalizes the development for initial arbitrary space and momentum distributions, producing the most general approach to date. We follow definitions and conventions in Muriel and Dresden [5] but review them for clarity and consistency with previous results. Let f ðr; p; tÞ be the single-particle distribution function of a many-body system as in kinetic theory. It represents the probability that a particle in location r possesses the momentum p at time t. We use the phase space variables r; p in keeping with kinetic theory. Later, we will replace the momentum divided by the particle mass m with velocity to conform to the Navier–Stokes notation. The following time evolution equation for the single-particle distribution was derived in Muriel and Dresden [5]: E-mail addresses: [email protected], [email protected]
f ðr; p; tÞ ¼ f r pt :p; 0 m Rt R 0 0Þ @ p @ þno 0 ds1 es1 Lo dr @Vðrr f ðr; r0 ; p; 0Þ @r @p m @r 2 Rt Rs R 0 0Þ @ p @ f ðr; r0 ; p; 0Þ þno 0 ds1 0 1 ds2 es2 Lo dr @Vðrr @p m @r 2 @r Rt Rs R 0R 0 0Þ @ p0 @ f ðr; r0 ; p; p0 ; 0Þ no 0 ds1 0 1 ds2 es2 Lo dp dr @Vðrr @p m @r 2 @r Rs R 0 R 00 @ @Vðrr0 Þ @ @Vðrr00 Þ n2 R t þ 2o 0 ds1 0 1 ds2 es2 Lo dr dr @p f 3 ðr; r 0 ; r 00 ; p; 0Þ @p @r @r Rs R 0 @ @Vðrr0 Þ @ @Vðrr0 Þ n2 R t þ 2o 0 ds1 0 1 ds2 es2 Lo dr @p f 2 ðr; r 0 ; p; 0Þ @r @r @p 1 n X @ þ O @p n n¼3
ð1Þ where f 2 ðr; r0 ; p; 0Þ is the mixed probability that two particles are in r; r0 and the first particle has momentum p at time t ¼ 0. f 2 ðr; r0 ; p; p0 ; 0Þ is the mixed probability that two particles at r; r 0 each possess momentum p; p0 at time t ¼ 0. f 3 ðr; r 0 ; r00 ; p; 0Þ is the mixed probability that particles are located at r; r 0 ; r 00 where the first particle has momentum p at time t ¼ 0. These mixed probabilities come from the original formulation of the many-body problem from the Liouville equation. The mixed probability distributions, representing particle correlations at t ¼ 0 will be simplified to uncorrelated functions represented by simple products. We use the operator @ Lo ¼ mp @r . no is the average particle density. Vðr r00 Þ is the pairpotential of two particles located at r; r 00 . The existence of this pair-potential distinguishes this approach from the usual continuum model.
http://dx.doi.org/10.1016/j.rinp.2015.12.004 2211-3797/Ó 2016 The Author. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
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A. Muriel / Results in Physics 6 (2016) 252–255
We will calculate averages of ð1; p; p2 Þ, so that using integration by parts, the contribution of the last term is zero, effectively truncating the series to a finite number of terms due to the vanishing of the momentum distribution at the boundary of the momentum space. Using factored initial distributions, that is, f 2 ðr; r0 Þ ¼ f 1 ðrÞf 1 ðr 0 Þ, etc., we rewrite Eq. (1) as f ðr; p; tÞ ¼ f r pt ; 0 uðp; 0Þ m Rt R 0 þno 0 ds1 es1 Lo dr Vðr r0 Þ @r@ 0 f ðr0 ; 0Þ @p@ pmi @r@ f ðr; 0Þuðp; 0Þ j i j Rt Rs R 0 @Vðrr0 Þ þ no 0 ds1 0 1 ds2 es2 Lo dr Vðr r 0 Þ @r@ 0 f ðr0 ; 0Þ @p@ pmi @r@ f ðr; 0Þuðp; 0Þ @r j i j Rt Rs R 0 no 0 ds1 0 1 ds2 es2 Lo dr Vðr r0 Þ @r@ 0 f ðr0 ; 0Þ j R 0 0p @r@ f ðr; 0Þ @p@ uðp; 0Þ dp mi uðp0 ; 0Þ i j R 0 Rs 0 R 00 00 n2 R t þ 2o 0 ds1 0 1 ds2 es2 Lo f ðrÞ dr Vðr r0 Þ @f@rðr0 Þ dr V ðr r00 Þ @f@rðr00 Þ @p@ @p@ uðpÞ j i j i R 0 Rs n2 R t þ 2o 0 ds1 0 1 ds2 es2 Lo f ðrÞ dr ðVðr r0 ÞÞ2 @r@ 0 @r@ 0 f ðr0 Þ @p@ @p@ uðpÞ j i j
i
The first and second terms differ only in the time integrals. Third term
Rs ds1 0 1 ds2 es2 Lo S3
@ R 0 p0 0 S3 ¼ dr Vðr r 0 Þ @x@ 0 f ðr0 ; 0Þ @p@ x pmx @x f ðr; 0Þuðp; 0Þ dp mx uðp0 ; 0Þ R p @ R 0
0 p0 þ dr Vðr r 0 Þ @x@ 0 f ðr0 ; 0Þ @p@ x my @y f ðr; 0Þuðp; 0Þ dp my uðp0 ; 0Þ @ R 0 p0 R 0
þ dr Vðr r 0 Þ @x@ 0 f ðr0 ; 0Þ @p@ x pmz @z f ðr; 0Þuðp; 0Þ dp mz uðp0 ; 0Þ i R 0 p0 R 0h @ þ dr Vðr r 0 Þ @y@ 0 f ðr 0 ; 0Þ @p@ y pmx @x f ðr; 0Þuðp; 0Þ dp mx uðp0 ; 0Þ i R R 0h p @ 0 p0 f ðr; 0Þuðp; 0Þ dp my uðp0 ; 0Þ þ dr Vðr r 0 Þ @y@ 0 f ðr 0 ; 0Þ @p@ my @y y i R 0 p0 R 0h @ þ dr Vðr r 0 Þ @y@ 0 f ðr 0 ; 0Þ @p@ pmz @z f ðr; 0Þuðp; 0Þ dp mz uðp0 ; 0Þ y @ R 0 p0 R 0
dr Vðr r 0 Þ @z@ 0 f ðr 0 ; 0Þ @p@ pmx @x f ðr; 0Þuðp; 0Þ dp mx uðp0 ; 0Þ z R R 0
p @ 0 p0 þ dr Vðr r 0 Þ @z@ 0 f ðr 0 ; 0Þ @p@ my @y f ðr; 0Þuðp; 0Þ dp my uðp0 ; 0Þ z @ R 0 p0 R 0
þ dr Vðr r 0 Þ @z@ 0 f ðr 0 ; 0Þ @p@ pmz @z f ðr; 0Þuðp; 0Þ dp mz uðp0 ; 0Þ
no
Rt 0
R
z
ð4Þ
ð2Þ
In Eq. (2), the symbols r ¼ ðx; y; zÞ; p ¼ ðpx ; py ; pz Þ are standard. In Cartesian component form, use summation over repeated indices, 2. Reduction procedure
Fourth term
n2o 2
Z
Z
t
s1
ds1 0
ds2 es2 Lo f ðrÞS4
0
where We rewrite the individual terms of Eq. (2) in explicit Cartesian dot products, useful for evaluating future applications. In simplifying the expressions of Eq. (2), we use the following properties: 0
0 R 00 0 00 dr Vðr r 0 Þ @f@xðr0 Þ dr Vðr r 00 Þ @f@xðr00 Þ @p@ @p@ uðpÞ x x R 0 0 R 00 00 þ dr Vðr r 0 Þ @f@yðr0 Þ dr Vðr r 00 Þ @f@xðr00 Þ @p@ y @p@ x uðpÞ R 0 0 R 00 00 þ dr Vðr r 0 Þ @f@zðr0 Þ dr Vðr r 00 Þ @f@xðr00 Þ @p@ @p@ uðpÞ z x R 0 0 R 00 00 þ dr Vðr r 0 Þ @f@xðr0 Þ dr Vðr r 00 Þ @f@yðr00 Þ @p@ x @p@ y uðpÞ R 0 0 R 00 00 þ dr Vðr r 0 Þ @f@yðr0 Þ dr Vðr r 00 Þ @f@yðr00 Þ @p@ y @p@ y uðpÞ R 0 0 R 00 00 þ dr Vðr r 0 Þ @f@zðr0 Þ dr Vðr r 00 Þ @f@yðr00 Þ @p@ @p@ uðpÞ z y R 0 0 R 00 00 þ dr Vðr r 0 Þ @f@xðr0 Þ dr Vðr r 00 Þ @f@zðr00 Þ @p@ x @p@ z uðpÞ R 0 0 R 00 00 þ dr Vðr r 0 Þ @f@yðr0 Þ dr Vðr r 00 Þ @f@zðr00 Þ @p@ y @p@ z uðpÞ R 0 0 R 00 00 þ dr Vðr r 0 Þ @f@zðr Þ dr Vðr r 00 Þ @f@zðr00 Þ @p@ @p@ uðpÞ
S4 ¼
0
Þ Þ (a) @V ðrr ¼ @V ðrr , Newton’s Third Law of action and reaction; @r @r 0 (b) integration by parts over r0 ; and R 0 (c) dr @r@ 0 ½Vðr r0 Þf ðr0 ; 0Þ ¼ 0; j
a boundary condition we use for the first time. Because of (c) we have departed from the space integrals over a cube of the earlier papers. Property (c) is applicable to most geometries. We analyze each of the terms in Eq. (2): Zeroth term
py t pt pt F x x ;y ; z z ; 0 uðpx ; px ; px ; 0Þ m m m
z
Rt 0
z
Second term
Z þno
Z
t
ds1 0
0
þ
n2o 2
Rt 0
ds1
R s1
ds2 es2 Lo S1
z
0
ds2 es2 Lo f ðrÞS5
0
dr ðVðr r 0 ÞÞ2 @x@ 0 @x@ 0 f ðr 0 Þ @p@ @p@ uðpÞ x x R 0 þ dr ðVðr r0 ÞÞ2 @y@ 0 @x@ 0 f ðr0 Þ @p@ @p@ uðpÞ y x R 0 þ dr ðVðr r0 ÞÞ2 @z@ 0 @x@ 0 f ðr0 Þ @p@ @p@ uðpÞ z x R 0 þ dr ðVðr r0 ÞÞ2 @x@ 0 @y@ 0 f ðr0 Þ @p@ x @p@ y uðpÞ R 0 þ dr ðVðr r0 ÞÞ2 @y@ 0 @y@ 0 f ðr0 Þ @p@ y @p@ y uðpÞ R 0 þ dr ðVðr r0 ÞÞ2 @z@ 0 @y@ 0 f ðr0 Þ @p@ @p@ uðpÞ z y R 0 0 2 @ @ 0 @ @ þ dr ðVðr r ÞÞ @x0 @z0 f ðr Þ @px @pz uðpÞ R 0 þ dr ðVðr r0 ÞÞ2 @y@ 0 @z@ 0 f ðr0 Þ @p@ y @p@ z uðpÞ R 0 þ dr ðVðr r0 ÞÞ2 @z@ 0 @z@ 0 f ðr0 Þ @p@ @p@ uðpÞ
S5 ¼
ð3Þ
R
z
s1
ð5Þ
where we now suppress the t ¼ 0 qualification for all initial data henceforth. Fifth term
First term
ds1 es1 Lo S1 @ R 0
f ðr; 0Þuðp; 0Þ S1 ¼ dr Vðr r 0 Þ @x@ 0 f ðr 0 ; 0Þ @p@ pmx @x x R 0
p @ f ðr; 0Þuðp; 0Þ þ dr Vðr r 0 Þ @x@ 0 f ðr0 ; 0Þ @p@ my @y x @ R 0
f ðr; 0Þuðp; 0Þ þ dr Vðr r 0 Þ @x@ 0 f ðr0 ; 0Þ @p@ pmz @z x i R 0h @ þ dr Vðr r 0 Þ @y@ 0 f ðr0 ; 0Þ @p@ y pmx @x f ðr; 0Þuðp; 0Þ i R 0h p @ f ðr; 0Þuðp; 0Þ þ dr Vðr r 0 Þ @y@ 0 f ðr0 ; 0Þ @p@ my @y y i R 0h @ þ dr Vðr r 0 Þ @y@ 0 f ðr0 ; 0Þ @p@ y pmz @z f ðr; 0Þuðp; 0Þ @ R 0
f ðr; 0Þuðp; 0Þ dr Vðr r 0 Þ @z@ 0 f ðr0 ; 0Þ @p@ pmx @x z R 0
p @ þ dr Vðr r 0 Þ @z@ 0 f ðr0 ; 0Þ @p@ z my @y f ðr; 0Þuðp; 0Þ @ R 0
f ðr; 0Þuðp; 0Þ þ dr Vðr r 0 Þ @z@ 0 f ðr0 ; 0Þ @p@ pmz @z
R
ð6Þ
z
In [1–4], for an initially uniform system, only the zeroth term and the fifth term are non-zero. Here, the most general initial data activates the first, second, third and fourth term of Eq. (2)
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A. Muriel / Results in Physics 6 (2016) 252–255
Notice that this approach no longer depends on symbolic computation, to which purists object, the main feature of references [1–4]. 3. An example The main purpose of this work is to provide a prescription on how to generate new solutions of the Navier–Stokes equation with arbitrary initial space and momentum distribution. Thus, we stop short of applications, a potential major activity for the future. Nevertheless, we outline our approach by picking a simple example. We invite the reader to complete the following exercise – still quite long – but illustrative. Suppose
uðpÞ ¼ uðpx Þ
ð7Þ
f ðrÞ ¼ f ðxÞ
ð8Þ
Then Eqs. (3)–(6) simplify to
Z
S1 ¼
ð9Þ
@ @ px 0 dr Vðr r 0 Þ 0 f ðx0 ; 0Þ uðpx Þ @x @px m
ð10Þ
@ f ðx; 0Þ @x
S2 ¼
0
Z
@ @ px @ 0 f ðx; 0Þuðpx Þ dr Vðr r 0 Þ 0 f ðx0 ; 0Þ @x @px m @x Z 0 0 px dp uðp0 ; 0Þ m
Z S3 ¼
Z @f ðr 0 Þ 0 00 dr Vðr r 00 Þ dr Vðr r0 Þ 0 @x @f ðr00 Þ @ @ u ðpÞ @x00 @px @px
ð11Þ
Z S5 ¼
0
2
dr ðVðr r 0 ÞÞ
@ @ @ @ f ðx0 Þ uðpÞ @x0 @x0 @px @px
Z
@ 0 dr Vðr r0 Þ 0 f ðx0 ; 0Þ @x 0
2
dr ðVðr r0 ÞÞ
so that
Z
S1 ¼
S2 ¼
@ @ f ðx0 Þ @x0 @x0
@ @ px 0 dr Vðr r 0 Þ 0 f ðx0 ; 0Þ uðpx Þ @x @px m
ð15Þ
0
2
@ @ @ @ f ðx0 Þ uðpÞ @x0 @x0 @px @px
C Bv C B p B xC B x=m C C B C B B vy C Z B py=m C C B C B C B C B B v z C ¼ dpx dpy dpz B pz=m Cf ðr; p; tÞ C B C B 2 B Ex C B p =2m C C B C B x C B C B 2 @ Ey A @ py =2m A Ez p2z =2m
ð20Þ
As a reminder, the Navier–Stokes equations for this system are:
where nu is the viscosity. The pressure P is a tensor. Px is the force on a unit area perpendicular to the x-direction. As shown in Muriel [1–4], the right hand side of Eqs. (21)–(23) are all calculable, the gradient of the pressure may be evaluated, hence the pressure terms by integration. The self-consistent set of pressure and field velocities constitute an exact solution of the Navier–Stokes equation. Eq. (20) recovers all the results published to date in Muriel [1–3]. Following [4], we find that the solution of the Navier–Stokes equation is a subset of time-evolution equations shown above in Eq. (20). The choice of the pair-potential is important. To illustrate further the example, we invite the reader to evaluate all expressions with the following familiar choice of pairpotential 0 2 þðyy0 Þ2 þðzz0 Þ2 Þ
Vðr r 0 Þ ¼ geaððxx Þ
ð24Þ
and the initial data for the spatial distribution
Z
dr ðVðr r 0 ÞÞ
1
ð23Þ
Z @ @ px @ 0 0 f ðx; 0Þuðpx Þ dp dr Vðr r 0 Þ 0 f ðx0 ; 0Þ @x @px m @x p0 x uðp0 ; 0Þ m
Z
1
@Pz @v z ¼ nur2 v z v rv z @z @t
ð16Þ
S5 ¼
0
ð13Þ
Z @f ðr 0 Þ 0 00 S4 ¼ dr Vðr r0 Þ dr Vðr r 00 Þ @x0 @f ðr00 Þ @ @ u ðpÞ @x00 @px @px
1
ð22Þ
Z
S3 ¼
q
@Py @v y ¼ nur2 v y v rv y @y @t
ð14Þ
Z
0
ð12Þ
@ @ px @ 0 f ðx; 0Þuðpx Þ dr Vðr r 0 Þ 0 f ðx0 ; 0Þ @x @px m @x
@ f ðx; 0Þ @x
ð19Þ
ð21Þ
We can evaluate the integrals
Z
x
@Px @v x ¼ nur2 v x v rv x @x @t
Z
S4 ¼
f ðx; y; z; px ; py ; pz; tÞ ¼ f x pmx t uðpx Þ h i R 0 Rt 0 þ 0 ds1 es1 Lo dr Vðr r0 Þ @f@xðx0 Þ @p@ x pmx @f@xðxÞ uðpx Þ i Rt Rs R 0h 0 þno 0 ds1 0 1 ds2 es2 Lo dr Vðr r 0 Þ @f@xðx0 Þ @p@ pmx @f@xðxÞ uðpx Þ x i Rt Rs R 0h 0 no 0 ds1 0 1 ds2 es2 Lo dr Vðr r 0 Þ @f@xðx0 Þ @p@ x pmx @f@xðxÞ uðpx Þ R 0 p0 dp mx uðp0x Þ Rs R 0 0 n2o R t ds1 0 1 ds2 es2 Lo f ðxÞ dr Vðr r 0 Þ @f@xðx0 Þ 2 0 R 00 00 dr Vðr r 00 Þ @f@xðx00 Þ @p@ x @p@ x uðpx Þ Rs R 0 2 2 0 n2 R t uðpx Þ þ 2o 0 ds1 0 1 ds2 es2 Lo f ðxÞ dr ðVðr r 0 ÞÞ2 @ @xf ðx02 Þ @ @p 2 to give the time evolution equations as in Muriel [4]
@ @ px @ dr Vðr r Þ 0 f ðx0 ; 0Þ f ðx; 0Þuðpx Þ @x @px m @x 0
For this example, the simplified time evolution equation for the single distribution function is now
ð17Þ ð18Þ
f ðxÞ ¼ Heav isideðL xÞ þ Heav isideðL þ xÞ
ð25Þ
integrated over infinite space. The initial data for uðpÞ may be put last to complete the momentum integrations. As mentioned in Muriel [1], one can calculate the pressure two ways, by the Navier–Stokes equation, or by the three bottom elements of Eq. (20). The latter is more accurate, as explained in Muriel [1], which we repeat: the Navier–Stokes equation is a continuum approximation while Eq. (20) comes from fundamental kinetic theory of mono-atomic particles, reducing to the continuum model of fluids in the appropriate limit.
A. Muriel / Results in Physics 6 (2016) 252–255
With the suggested exercise and three other published cases [1,2,3], one can verify the following assertions: the field velocities and pressure are smooth and regular, the kinetic energies in the three axes are finite, and there is no blow up in time. On the question of blow up there is a previous work that shows that blow up exists for an averaged Navier–Stokes equation, but not for the classic equation itself [8]. We repeat the observation that for every choice of pairpotential, there is a unique solution for the hydrodynamic variables. But the general nature of the solutions is not sensitively dependent on the choice of the pair-potential, although it simplifies matters if the expressions containing the pair potential are evaluated in closed form. So far, there have been no pathologic solutions that can invite a description as turbulent [9]. We raise once again the proposal that turbulence cannot be explained from existing, and possibly even other future exact solutions of the Navier–Stokes equations. We must look elsewhere for turbulent solutions [6,7]. 4. Further remarks and future program Now for three remarks. (1) A very large part of the literature in kinetic theory attempts to find exact or approximate solutions of the Boltzmann transport equation. Mathematically, this is expressed as the search for the time evolution of the single distribution function f ðr; p; tÞ satisfying the Boltzmann equation. In the Boltzmann approach, this function is dependent on the collision integral applied to short-range interaction between molecules. By contrast, our time evolution Eq. (20) is determined by a pair-potential which is not necessarily a short range. One could say that our solution for the single particle distribution is more general than any prospective solution – if it exists, or eventually discovered – without approximations. Thus our approach is an alternative to the so-far unrealized solutions of the Boltzmann equation. (2) On the matter of solutions to the Navier–Stokes equation, we can also say that our proposed solutions are indeed exact and more general because they are valid for arbitrary initial data. The boundary conditions introduced by Eq. (2) contains space integrals dependent on geometry of the system. Specific applications are dependent on the geometry of the physical applications, of which there will be many. For the above two reasons, we venture to suggest that our time evolution equations cover both problems of the Boltzmann equation and the Navier–Stokes equation. Our time evolution equations may well be a useful alternative to these two formulations. Indeed it would be very productive if numerous applications of the old Boltzmann equation and Navier–Stokes equation are reformulated with our time evolution approach, a challenging opportunity we hope to address in future work. (3) We next tackle the sixth problem of Hilbert. Hilbert’s sixth problem on the mathematical basis of quantum mechanics and hydrodynamics consists of two parts, the axiomatization of quantum mechanics and hydrodynamics, which we quote from Hilbert [10]:
255
6. Mathematical Treatment of the Axioms of Physics. The investigations on the foundations of geometry suggest the problem: To treat in the same manner, by means of axioms, those physical sciences in which already today mathematics plays an important part; in the first rank are the theory of probabilities and mechanics. ‘‘As to the axioms of the theory of probabilities, it seems to me desirable that their logical investigation should be accompanied by a rigorous and satisfactory development of the method of mean values in mathematical physics, and in particular in the kinetic theory of gases. ... Boltzmann’s work on the principles of mechanics suggests the problem of developing mathematically the limiting processes, there merely indicated, which lead from the atomistic view to the laws of motion of continua.” The first part, quantum mechanics, is now considered solved [11]. We cannot say the same for hydrodynamics, the second part [12]. However, by bypassing the Boltzmann equation, our formalism in this paper seems to address the second part, as remarked by Mark Arend, advancing our solution for the hydrodynamic variables as a partial response to Hilbert’s sixth problem. Going beyond Hilbert, to complete the fundamental basis for hydrodynamics, we must also include explanations for transport coefficients like viscosity, diffusion coefficient and thermal conductivity, which we have mentioned already sometime ago in Muriel and Dresden [5]. Curiously, Hilbert did not specifically include turbulence as a fundamental problem in hydrodynamics. Given that we have discouraged the Navier–Stokes equation hope for the origin of turbulence, we must still find an explanation for it. In keeping with our earlier suggestions on the quantum origin of turbulence [6,7], we propose that turbulence will be addressed by returning to the first part of Hilbert’s sixth problem. We have already started this program by invoking quantum mechanics to explain the origin of turbulence [9]. This might be done by the quantum analog of our time evolution equations [13] presented in this work. When this is done to everyone’s satisfaction, we can then assert that Hilbert’s sixth problem will have been completely solved. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
[12] [13]
Muriel A. Results Phys 2011;1:2. Muriel A. Phys Lett A 2014;378:2529. A. Muriel, Results Phys., to be published. Muriel A. http://dx.doi.org/10.13140/RG.2.1.1972.2088. Muriel A, Dresden M. D 1997;101:299. Getreuer P, Albano AM, Muriel A. Phys Lett A 2007;101. Muriel A. Invited Editorial, Simulating multi-component systems to study turbulence. CFD Lett 2010;2(3). Tao T. J Math Soc 2015:ii–v. http://dx.doi.org/10.1090/jams/838. Article electronically published on June 30. Muriel A. Quantum theory of turbulence. Harvard Book Store; 2011. Hilbert D, Göttinger Nachrichten, 1900: 253 and Archiv der Mathematik und Physik, 3dser., vol. 1213; 1901. Wightman AS, Hilbert’s sixth problem: mathematical treatment of the axioms of physics. In: Browder FE, editor. Proceedings of symposia in pure mathematics XXVIII. XXVIII, Vol. 146. American Mathematical Society; 1976. ISBN 0-8218-1428-1. Gorban AN, Karlin I. Bull Am Math Soc 2014;51(2):186. Muriel A. Phys Rev Lett 1995;75:2255.
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Time evolution equations for hydrodynamic variables with arbitrary initial data A. Muriel Program in Mathematics, Graduate Center, City University of New York, 506 Fifth Avenue, NY 10016, United States
a r t i c l e
i n f o
Article history: Received 26 November 2015 Accepted 4 December 2015 Available online 19 December 2015 Keywords: Time evolution equations Arbitrary initial data Field velocities Pressure Kinetic energy Solutions of Navier–Stokes equation
a b s t r a c t We prescribe an alternative procedure for arriving at the time evolution equations for hydrodynamic local variables such as density, velocity fields, and kinetic energy using a time evolution equation of the single-particle distribution. It is suggested that when applied to various combinations of pairpotential between monoatoms, geometry and initial data, the prescription will have wide applications in hydrodynamics, including solutions of the Navier–Stokes equation. Ó 2016 The Author. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction There have been a series of four recent papers that showed related solutions of the Navier–Stokes equation [1,2,3,4]. The first [1] was an existence proof of solutions in 3D. The second one [2] was a simplification of Muriel [1] to 2D with field velocities wrapped around a globe, producing flows symmetric about the equator. The third one [3] uses spherical symmetry to produce an analytic model of implosion toward the possibility of controlled nuclear fusion. The first three papers assumed an initially uniform system with delta-function momentum distributions. The fourth paper [4] is valid for an initial spatially uniform system and arbitrary initial momentum distribution. This paper generalizes the development for initial arbitrary space and momentum distributions, producing the most general approach to date. We follow definitions and conventions in Muriel and Dresden [5] but review them for clarity and consistency with previous results. Let f ðr; p; tÞ be the single-particle distribution function of a many-body system as in kinetic theory. It represents the probability that a particle in location r possesses the momentum p at time t. We use the phase space variables r; p in keeping with kinetic theory. Later, we will replace the momentum divided by the particle mass m with velocity to conform to the Navier–Stokes notation. The following time evolution equation for the single-particle distribution was derived in Muriel and Dresden [5]: E-mail addresses: [email protected], [email protected]
f ðr; p; tÞ ¼ f r pt :p; 0 m Rt R 0 0Þ @ p @ þno 0 ds1 es1 Lo dr @Vðrr f ðr; r0 ; p; 0Þ @r @p m @r 2 Rt Rs R 0 0Þ @ p @ f ðr; r0 ; p; 0Þ þno 0 ds1 0 1 ds2 es2 Lo dr @Vðrr @p m @r 2 @r Rt Rs R 0R 0 0Þ @ p0 @ f ðr; r0 ; p; p0 ; 0Þ no 0 ds1 0 1 ds2 es2 Lo dp dr @Vðrr @p m @r 2 @r Rs R 0 R 00 @ @Vðrr0 Þ @ @Vðrr00 Þ n2 R t þ 2o 0 ds1 0 1 ds2 es2 Lo dr dr @p f 3 ðr; r 0 ; r 00 ; p; 0Þ @p @r @r Rs R 0 @ @Vðrr0 Þ @ @Vðrr0 Þ n2 R t þ 2o 0 ds1 0 1 ds2 es2 Lo dr @p f 2 ðr; r 0 ; p; 0Þ @r @r @p 1 n X @ þ O @p n n¼3
ð1Þ where f 2 ðr; r0 ; p; 0Þ is the mixed probability that two particles are in r; r0 and the first particle has momentum p at time t ¼ 0. f 2 ðr; r0 ; p; p0 ; 0Þ is the mixed probability that two particles at r; r 0 each possess momentum p; p0 at time t ¼ 0. f 3 ðr; r 0 ; r00 ; p; 0Þ is the mixed probability that particles are located at r; r 0 ; r 00 where the first particle has momentum p at time t ¼ 0. These mixed probabilities come from the original formulation of the many-body problem from the Liouville equation. The mixed probability distributions, representing particle correlations at t ¼ 0 will be simplified to uncorrelated functions represented by simple products. We use the operator @ Lo ¼ mp @r . no is the average particle density. Vðr r00 Þ is the pairpotential of two particles located at r; r 00 . The existence of this pair-potential distinguishes this approach from the usual continuum model.
http://dx.doi.org/10.1016/j.rinp.2015.12.004 2211-3797/Ó 2016 The Author. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
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We will calculate averages of ð1; p; p2 Þ, so that using integration by parts, the contribution of the last term is zero, effectively truncating the series to a finite number of terms due to the vanishing of the momentum distribution at the boundary of the momentum space. Using factored initial distributions, that is, f 2 ðr; r0 Þ ¼ f 1 ðrÞf 1 ðr 0 Þ, etc., we rewrite Eq. (1) as f ðr; p; tÞ ¼ f r pt ; 0 uðp; 0Þ m Rt R 0 þno 0 ds1 es1 Lo dr Vðr r0 Þ @r@ 0 f ðr0 ; 0Þ @p@ pmi @r@ f ðr; 0Þuðp; 0Þ j i j Rt Rs R 0 @Vðrr0 Þ þ no 0 ds1 0 1 ds2 es2 Lo dr Vðr r 0 Þ @r@ 0 f ðr0 ; 0Þ @p@ pmi @r@ f ðr; 0Þuðp; 0Þ @r j i j Rt Rs R 0 no 0 ds1 0 1 ds2 es2 Lo dr Vðr r0 Þ @r@ 0 f ðr0 ; 0Þ j R 0 0p @r@ f ðr; 0Þ @p@ uðp; 0Þ dp mi uðp0 ; 0Þ i j R 0 Rs 0 R 00 00 n2 R t þ 2o 0 ds1 0 1 ds2 es2 Lo f ðrÞ dr Vðr r0 Þ @f@rðr0 Þ dr V ðr r00 Þ @f@rðr00 Þ @p@ @p@ uðpÞ j i j i R 0 Rs n2 R t þ 2o 0 ds1 0 1 ds2 es2 Lo f ðrÞ dr ðVðr r0 ÞÞ2 @r@ 0 @r@ 0 f ðr0 Þ @p@ @p@ uðpÞ j i j
i
The first and second terms differ only in the time integrals. Third term
Rs ds1 0 1 ds2 es2 Lo S3
@ R 0 p0 0 S3 ¼ dr Vðr r 0 Þ @x@ 0 f ðr0 ; 0Þ @p@ x pmx @x f ðr; 0Þuðp; 0Þ dp mx uðp0 ; 0Þ R p @ R 0
0 p0 þ dr Vðr r 0 Þ @x@ 0 f ðr0 ; 0Þ @p@ x my @y f ðr; 0Þuðp; 0Þ dp my uðp0 ; 0Þ @ R 0 p0 R 0
þ dr Vðr r 0 Þ @x@ 0 f ðr0 ; 0Þ @p@ x pmz @z f ðr; 0Þuðp; 0Þ dp mz uðp0 ; 0Þ i R 0 p0 R 0h @ þ dr Vðr r 0 Þ @y@ 0 f ðr 0 ; 0Þ @p@ y pmx @x f ðr; 0Þuðp; 0Þ dp mx uðp0 ; 0Þ i R R 0h p @ 0 p0 f ðr; 0Þuðp; 0Þ dp my uðp0 ; 0Þ þ dr Vðr r 0 Þ @y@ 0 f ðr 0 ; 0Þ @p@ my @y y i R 0 p0 R 0h @ þ dr Vðr r 0 Þ @y@ 0 f ðr 0 ; 0Þ @p@ pmz @z f ðr; 0Þuðp; 0Þ dp mz uðp0 ; 0Þ y @ R 0 p0 R 0
dr Vðr r 0 Þ @z@ 0 f ðr 0 ; 0Þ @p@ pmx @x f ðr; 0Þuðp; 0Þ dp mx uðp0 ; 0Þ z R R 0
p @ 0 p0 þ dr Vðr r 0 Þ @z@ 0 f ðr 0 ; 0Þ @p@ my @y f ðr; 0Þuðp; 0Þ dp my uðp0 ; 0Þ z @ R 0 p0 R 0
þ dr Vðr r 0 Þ @z@ 0 f ðr 0 ; 0Þ @p@ pmz @z f ðr; 0Þuðp; 0Þ dp mz uðp0 ; 0Þ
no
Rt 0
R
z
ð4Þ
ð2Þ
In Eq. (2), the symbols r ¼ ðx; y; zÞ; p ¼ ðpx ; py ; pz Þ are standard. In Cartesian component form, use summation over repeated indices, 2. Reduction procedure
Fourth term
n2o 2
Z
Z
t
s1
ds1 0
ds2 es2 Lo f ðrÞS4
0
where We rewrite the individual terms of Eq. (2) in explicit Cartesian dot products, useful for evaluating future applications. In simplifying the expressions of Eq. (2), we use the following properties: 0
0 R 00 0 00 dr Vðr r 0 Þ @f@xðr0 Þ dr Vðr r 00 Þ @f@xðr00 Þ @p@ @p@ uðpÞ x x R 0 0 R 00 00 þ dr Vðr r 0 Þ @f@yðr0 Þ dr Vðr r 00 Þ @f@xðr00 Þ @p@ y @p@ x uðpÞ R 0 0 R 00 00 þ dr Vðr r 0 Þ @f@zðr0 Þ dr Vðr r 00 Þ @f@xðr00 Þ @p@ @p@ uðpÞ z x R 0 0 R 00 00 þ dr Vðr r 0 Þ @f@xðr0 Þ dr Vðr r 00 Þ @f@yðr00 Þ @p@ x @p@ y uðpÞ R 0 0 R 00 00 þ dr Vðr r 0 Þ @f@yðr0 Þ dr Vðr r 00 Þ @f@yðr00 Þ @p@ y @p@ y uðpÞ R 0 0 R 00 00 þ dr Vðr r 0 Þ @f@zðr0 Þ dr Vðr r 00 Þ @f@yðr00 Þ @p@ @p@ uðpÞ z y R 0 0 R 00 00 þ dr Vðr r 0 Þ @f@xðr0 Þ dr Vðr r 00 Þ @f@zðr00 Þ @p@ x @p@ z uðpÞ R 0 0 R 00 00 þ dr Vðr r 0 Þ @f@yðr0 Þ dr Vðr r 00 Þ @f@zðr00 Þ @p@ y @p@ z uðpÞ R 0 0 R 00 00 þ dr Vðr r 0 Þ @f@zðr Þ dr Vðr r 00 Þ @f@zðr00 Þ @p@ @p@ uðpÞ
S4 ¼
0
Þ Þ (a) @V ðrr ¼ @V ðrr , Newton’s Third Law of action and reaction; @r @r 0 (b) integration by parts over r0 ; and R 0 (c) dr @r@ 0 ½Vðr r0 Þf ðr0 ; 0Þ ¼ 0; j
a boundary condition we use for the first time. Because of (c) we have departed from the space integrals over a cube of the earlier papers. Property (c) is applicable to most geometries. We analyze each of the terms in Eq. (2): Zeroth term
py t pt pt F x x ;y ; z z ; 0 uðpx ; px ; px ; 0Þ m m m
z
Rt 0
z
Second term
Z þno
Z
t
ds1 0
0
þ
n2o 2
Rt 0
ds1
R s1
ds2 es2 Lo S1
z
0
ds2 es2 Lo f ðrÞS5
0
dr ðVðr r 0 ÞÞ2 @x@ 0 @x@ 0 f ðr 0 Þ @p@ @p@ uðpÞ x x R 0 þ dr ðVðr r0 ÞÞ2 @y@ 0 @x@ 0 f ðr0 Þ @p@ @p@ uðpÞ y x R 0 þ dr ðVðr r0 ÞÞ2 @z@ 0 @x@ 0 f ðr0 Þ @p@ @p@ uðpÞ z x R 0 þ dr ðVðr r0 ÞÞ2 @x@ 0 @y@ 0 f ðr0 Þ @p@ x @p@ y uðpÞ R 0 þ dr ðVðr r0 ÞÞ2 @y@ 0 @y@ 0 f ðr0 Þ @p@ y @p@ y uðpÞ R 0 þ dr ðVðr r0 ÞÞ2 @z@ 0 @y@ 0 f ðr0 Þ @p@ @p@ uðpÞ z y R 0 0 2 @ @ 0 @ @ þ dr ðVðr r ÞÞ @x0 @z0 f ðr Þ @px @pz uðpÞ R 0 þ dr ðVðr r0 ÞÞ2 @y@ 0 @z@ 0 f ðr0 Þ @p@ y @p@ z uðpÞ R 0 þ dr ðVðr r0 ÞÞ2 @z@ 0 @z@ 0 f ðr0 Þ @p@ @p@ uðpÞ
S5 ¼
ð3Þ
R
z
s1
ð5Þ
where we now suppress the t ¼ 0 qualification for all initial data henceforth. Fifth term
First term
ds1 es1 Lo S1 @ R 0
f ðr; 0Þuðp; 0Þ S1 ¼ dr Vðr r 0 Þ @x@ 0 f ðr 0 ; 0Þ @p@ pmx @x x R 0
p @ f ðr; 0Þuðp; 0Þ þ dr Vðr r 0 Þ @x@ 0 f ðr0 ; 0Þ @p@ my @y x @ R 0
f ðr; 0Þuðp; 0Þ þ dr Vðr r 0 Þ @x@ 0 f ðr0 ; 0Þ @p@ pmz @z x i R 0h @ þ dr Vðr r 0 Þ @y@ 0 f ðr0 ; 0Þ @p@ y pmx @x f ðr; 0Þuðp; 0Þ i R 0h p @ f ðr; 0Þuðp; 0Þ þ dr Vðr r 0 Þ @y@ 0 f ðr0 ; 0Þ @p@ my @y y i R 0h @ þ dr Vðr r 0 Þ @y@ 0 f ðr0 ; 0Þ @p@ y pmz @z f ðr; 0Þuðp; 0Þ @ R 0
f ðr; 0Þuðp; 0Þ dr Vðr r 0 Þ @z@ 0 f ðr0 ; 0Þ @p@ pmx @x z R 0
p @ þ dr Vðr r 0 Þ @z@ 0 f ðr0 ; 0Þ @p@ z my @y f ðr; 0Þuðp; 0Þ @ R 0
f ðr; 0Þuðp; 0Þ þ dr Vðr r 0 Þ @z@ 0 f ðr0 ; 0Þ @p@ pmz @z
R
ð6Þ
z
In [1–4], for an initially uniform system, only the zeroth term and the fifth term are non-zero. Here, the most general initial data activates the first, second, third and fourth term of Eq. (2)
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A. Muriel / Results in Physics 6 (2016) 252–255
Notice that this approach no longer depends on symbolic computation, to which purists object, the main feature of references [1–4]. 3. An example The main purpose of this work is to provide a prescription on how to generate new solutions of the Navier–Stokes equation with arbitrary initial space and momentum distribution. Thus, we stop short of applications, a potential major activity for the future. Nevertheless, we outline our approach by picking a simple example. We invite the reader to complete the following exercise – still quite long – but illustrative. Suppose
uðpÞ ¼ uðpx Þ
ð7Þ
f ðrÞ ¼ f ðxÞ
ð8Þ
Then Eqs. (3)–(6) simplify to
Z
S1 ¼
ð9Þ
@ @ px 0 dr Vðr r 0 Þ 0 f ðx0 ; 0Þ uðpx Þ @x @px m
ð10Þ
@ f ðx; 0Þ @x
S2 ¼
0
Z
@ @ px @ 0 f ðx; 0Þuðpx Þ dr Vðr r 0 Þ 0 f ðx0 ; 0Þ @x @px m @x Z 0 0 px dp uðp0 ; 0Þ m
Z S3 ¼
Z @f ðr 0 Þ 0 00 dr Vðr r 00 Þ dr Vðr r0 Þ 0 @x @f ðr00 Þ @ @ u ðpÞ @x00 @px @px
ð11Þ
Z S5 ¼
0
2
dr ðVðr r 0 ÞÞ
@ @ @ @ f ðx0 Þ uðpÞ @x0 @x0 @px @px
Z
@ 0 dr Vðr r0 Þ 0 f ðx0 ; 0Þ @x 0
2
dr ðVðr r0 ÞÞ
so that
Z
S1 ¼
S2 ¼
@ @ f ðx0 Þ @x0 @x0
@ @ px 0 dr Vðr r 0 Þ 0 f ðx0 ; 0Þ uðpx Þ @x @px m
ð15Þ
0
2
@ @ @ @ f ðx0 Þ uðpÞ @x0 @x0 @px @px
C Bv C B p B xC B x=m C C B C B B vy C Z B py=m C C B C B C B C B B v z C ¼ dpx dpy dpz B pz=m Cf ðr; p; tÞ C B C B 2 B Ex C B p =2m C C B C B x C B C B 2 @ Ey A @ py =2m A Ez p2z =2m
ð20Þ
As a reminder, the Navier–Stokes equations for this system are:
where nu is the viscosity. The pressure P is a tensor. Px is the force on a unit area perpendicular to the x-direction. As shown in Muriel [1–4], the right hand side of Eqs. (21)–(23) are all calculable, the gradient of the pressure may be evaluated, hence the pressure terms by integration. The self-consistent set of pressure and field velocities constitute an exact solution of the Navier–Stokes equation. Eq. (20) recovers all the results published to date in Muriel [1–3]. Following [4], we find that the solution of the Navier–Stokes equation is a subset of time-evolution equations shown above in Eq. (20). The choice of the pair-potential is important. To illustrate further the example, we invite the reader to evaluate all expressions with the following familiar choice of pairpotential 0 2 þðyy0 Þ2 þðzz0 Þ2 Þ
Vðr r 0 Þ ¼ geaððxx Þ
ð24Þ
and the initial data for the spatial distribution
Z
dr ðVðr r 0 ÞÞ
1
ð23Þ
Z @ @ px @ 0 0 f ðx; 0Þuðpx Þ dp dr Vðr r 0 Þ 0 f ðx0 ; 0Þ @x @px m @x p0 x uðp0 ; 0Þ m
Z
1
@Pz @v z ¼ nur2 v z v rv z @z @t
ð16Þ
S5 ¼
0
ð13Þ
Z @f ðr 0 Þ 0 00 S4 ¼ dr Vðr r0 Þ dr Vðr r 00 Þ @x0 @f ðr00 Þ @ @ u ðpÞ @x00 @px @px
1
ð22Þ
Z
S3 ¼
q
@Py @v y ¼ nur2 v y v rv y @y @t
ð14Þ
Z
0
ð12Þ
@ @ px @ 0 f ðx; 0Þuðpx Þ dr Vðr r 0 Þ 0 f ðx0 ; 0Þ @x @px m @x
@ f ðx; 0Þ @x
ð19Þ
ð21Þ
We can evaluate the integrals
Z
x
@Px @v x ¼ nur2 v x v rv x @x @t
Z
S4 ¼
f ðx; y; z; px ; py ; pz; tÞ ¼ f x pmx t uðpx Þ h i R 0 Rt 0 þ 0 ds1 es1 Lo dr Vðr r0 Þ @f@xðx0 Þ @p@ x pmx @f@xðxÞ uðpx Þ i Rt Rs R 0h 0 þno 0 ds1 0 1 ds2 es2 Lo dr Vðr r 0 Þ @f@xðx0 Þ @p@ pmx @f@xðxÞ uðpx Þ x i Rt Rs R 0h 0 no 0 ds1 0 1 ds2 es2 Lo dr Vðr r 0 Þ @f@xðx0 Þ @p@ x pmx @f@xðxÞ uðpx Þ R 0 p0 dp mx uðp0x Þ Rs R 0 0 n2o R t ds1 0 1 ds2 es2 Lo f ðxÞ dr Vðr r 0 Þ @f@xðx0 Þ 2 0 R 00 00 dr Vðr r 00 Þ @f@xðx00 Þ @p@ x @p@ x uðpx Þ Rs R 0 2 2 0 n2 R t uðpx Þ þ 2o 0 ds1 0 1 ds2 es2 Lo f ðxÞ dr ðVðr r 0 ÞÞ2 @ @xf ðx02 Þ @ @p 2 to give the time evolution equations as in Muriel [4]
@ @ px @ dr Vðr r Þ 0 f ðx0 ; 0Þ f ðx; 0Þuðpx Þ @x @px m @x 0
For this example, the simplified time evolution equation for the single distribution function is now
ð17Þ ð18Þ
f ðxÞ ¼ Heav isideðL xÞ þ Heav isideðL þ xÞ
ð25Þ
integrated over infinite space. The initial data for uðpÞ may be put last to complete the momentum integrations. As mentioned in Muriel [1], one can calculate the pressure two ways, by the Navier–Stokes equation, or by the three bottom elements of Eq. (20). The latter is more accurate, as explained in Muriel [1], which we repeat: the Navier–Stokes equation is a continuum approximation while Eq. (20) comes from fundamental kinetic theory of mono-atomic particles, reducing to the continuum model of fluids in the appropriate limit.
A. Muriel / Results in Physics 6 (2016) 252–255
With the suggested exercise and three other published cases [1,2,3], one can verify the following assertions: the field velocities and pressure are smooth and regular, the kinetic energies in the three axes are finite, and there is no blow up in time. On the question of blow up there is a previous work that shows that blow up exists for an averaged Navier–Stokes equation, but not for the classic equation itself [8]. We repeat the observation that for every choice of pairpotential, there is a unique solution for the hydrodynamic variables. But the general nature of the solutions is not sensitively dependent on the choice of the pair-potential, although it simplifies matters if the expressions containing the pair potential are evaluated in closed form. So far, there have been no pathologic solutions that can invite a description as turbulent [9]. We raise once again the proposal that turbulence cannot be explained from existing, and possibly even other future exact solutions of the Navier–Stokes equations. We must look elsewhere for turbulent solutions [6,7]. 4. Further remarks and future program Now for three remarks. (1) A very large part of the literature in kinetic theory attempts to find exact or approximate solutions of the Boltzmann transport equation. Mathematically, this is expressed as the search for the time evolution of the single distribution function f ðr; p; tÞ satisfying the Boltzmann equation. In the Boltzmann approach, this function is dependent on the collision integral applied to short-range interaction between molecules. By contrast, our time evolution Eq. (20) is determined by a pair-potential which is not necessarily a short range. One could say that our solution for the single particle distribution is more general than any prospective solution – if it exists, or eventually discovered – without approximations. Thus our approach is an alternative to the so-far unrealized solutions of the Boltzmann equation. (2) On the matter of solutions to the Navier–Stokes equation, we can also say that our proposed solutions are indeed exact and more general because they are valid for arbitrary initial data. The boundary conditions introduced by Eq. (2) contains space integrals dependent on geometry of the system. Specific applications are dependent on the geometry of the physical applications, of which there will be many. For the above two reasons, we venture to suggest that our time evolution equations cover both problems of the Boltzmann equation and the Navier–Stokes equation. Our time evolution equations may well be a useful alternative to these two formulations. Indeed it would be very productive if numerous applications of the old Boltzmann equation and Navier–Stokes equation are reformulated with our time evolution approach, a challenging opportunity we hope to address in future work. (3) We next tackle the sixth problem of Hilbert. Hilbert’s sixth problem on the mathematical basis of quantum mechanics and hydrodynamics consists of two parts, the axiomatization of quantum mechanics and hydrodynamics, which we quote from Hilbert [10]:
255
6. Mathematical Treatment of the Axioms of Physics. The investigations on the foundations of geometry suggest the problem: To treat in the same manner, by means of axioms, those physical sciences in which already today mathematics plays an important part; in the first rank are the theory of probabilities and mechanics. ‘‘As to the axioms of the theory of probabilities, it seems to me desirable that their logical investigation should be accompanied by a rigorous and satisfactory development of the method of mean values in mathematical physics, and in particular in the kinetic theory of gases. ... Boltzmann’s work on the principles of mechanics suggests the problem of developing mathematically the limiting processes, there merely indicated, which lead from the atomistic view to the laws of motion of continua.” The first part, quantum mechanics, is now considered solved [11]. We cannot say the same for hydrodynamics, the second part [12]. However, by bypassing the Boltzmann equation, our formalism in this paper seems to address the second part, as remarked by Mark Arend, advancing our solution for the hydrodynamic variables as a partial response to Hilbert’s sixth problem. Going beyond Hilbert, to complete the fundamental basis for hydrodynamics, we must also include explanations for transport coefficients like viscosity, diffusion coefficient and thermal conductivity, which we have mentioned already sometime ago in Muriel and Dresden [5]. Curiously, Hilbert did not specifically include turbulence as a fundamental problem in hydrodynamics. Given that we have discouraged the Navier–Stokes equation hope for the origin of turbulence, we must still find an explanation for it. In keeping with our earlier suggestions on the quantum origin of turbulence [6,7], we propose that turbulence will be addressed by returning to the first part of Hilbert’s sixth problem. We have already started this program by invoking quantum mechanics to explain the origin of turbulence [9]. This might be done by the quantum analog of our time evolution equations [13] presented in this work. When this is done to everyone’s satisfaction, we can then assert that Hilbert’s sixth problem will have been completely solved. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
[12] [13]
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