Special Problem Reading Summary
Summer 2004, Dr. Predrag Cvitanovic
by Odell Austin Collins


This summer's reading focused on two categories: reviewing two preprints to nonlinearity on relating Eulerian fluid flows and mappings to the Schrodinger equation, and independent readings on various information metrics and their relation to physical phenomena.

First, we review the two non-linearity pre-prints:

Mapping of the classical kinetic balance equations onto the Schrodinger equation (Adriana I Pesci, Raymond E Goldstein), and Mapping of the classical kinetic balance equations onto the Pauli equation (Adriana I Pesci, Raymond E Goldstein, and Hermann Uys)

In this paper, the authors start with Liouville's equation and using what appears to be a standard procedure generate the "BBKGY hierarchy" of N coupled reduced probability distributions:

(Subscript[f, j])((x^N),(p^N)) = (∫_Ω)(Subscript[f, N])((x^N),(p^N)) (∏_(l = j + 1)^N)dxdp

where (Subscript[f, 1]) and (Subscript[f, 2])determine the kinetic and potential energy of some collection of particles.

These equations can be decoupled by varying choices of assumptions. In particular, one assumption credited as the "Bogliobuv ansatz" which corresponds to no rotational motion about the centroid of motion can decouple the above reduced probability functions to produce Boltzmann's equation for a single particle :
 
  ∂ ((Subscript[f, 1])/∂)t+ ((Subscript[p, 1])/m)·∂ ((Subscript[f, 1])/∂)t - ∂((V((Subscript[x, 1])))/∂)(Subscript[x, 1])· ((∂(Subscript[f, 1]))/(∂(Subscript[p, 1])))
=∫d(Subscript[x, 2])∫d(Subscript[p, 2])[(Subscript[f, 1])((Subscript[p, 1]^('))) (Subscript[f, 1])((Subscript[p, 2]^(')))  - (Subscript[f, 1])((Subscript[p, 1])) (Subscript[f, 1])((Subscript[p, 2]))]
  
Integrating over momentum allows them to use conservation of particle number and momentum to write:

  (∫_(-∞)^∞)dp (∂((Subscript[f, 1])/∂)t+(p/m)·∂((Subscript[f, 1])/∂)t-∂((V(x))/∂)x·((∂(Subscript[f, 1]))/(∂p)))=0 ,
  
  (∫_(-∞)^∞)p dp (∂((Subscript[f, 1])/∂)t+(p/m)·∂((Subscript[f, 1])/∂)t-∂((V(x))/∂)x·((∂(Subscript[f, 1]))/(∂p)))=0

Until now, the above steps are standard. Their original contribution comes in examining these balance equations in terms of the fourier representations of (Subscript[f, 1.]) .They take 

(Subscript[f, 1])(x,p,t) = (1/((((2 π η))^3)))(∫_(-∞)^∞)exp(- i  ((p·y)/η))\(f\&^\)(x,y,t)dy

The balance equations then become

 (1/((((2 π η))^3))) ∫ (∫_(-∞)^∞)dpdy exp(-i ((p·y)/η))[∂(f/∂)t+(η/(i m))(∂/∂)x·((\(∂f\&^\))/(∂y))]=0
 
 (1/((((2 π η))^3))) ∫ (∫_(-∞)^∞)dpdy exp(-i ((p·y)/η)) [(∂/∂)t((η/i)((\(∂f\&^\))/(∂y)))-((η^2)/m)(∂/(∂x))·(((\(\(∂\^2 f\&^\)\&⟷\))/(∂y^2)))+(∂/(∂y))(y·((∂V)/(∂x))\(f\&^\))]
 
These equations can be manipulated to show

(lim_(y -> 0))[\(∂f\&^\/∂t\)+(η/(i m))(∂/(∂x))·((\(∂f\&^\))/(∂y))]=0

(lim_(y -> 0))[(∂/∂)t((η/i)((\(∂f\&^\))/(∂y)))-((η^2)/m)(∂/(∂x))·(((\(\(∂\^2 f\&^\)\&⟷\))/(∂y^2)))+((∂V)/(∂x))\(f\&^\)+O(y)]=0

The authors highlight the fact that these limits take the same functional form that Frohlich found in his derivation of quantum hydrodynamics. Using an algebraic change of variables and much manipulation, the authors produce:

∫ \(dx\^''\)(Subscript[lim , x'->x''])[\(i\ η\ \ ∂f\&^\/∂t\)+((η^2)/(2 m))(((\(∂\^2 f\&^\))/(∂(x^(' 2))))-((\(∂\^2 f\&^\))/(∂(x^('' 2)))))] = 0

∫ \(dx\^''\)(Subscript[lim , x'->x'' ])(1/2)((∂/(∂x'))-(∂/(∂x''))) ×[\(η\/i\ \ ∂f\&^\/∂t\)+((η^2)/(2 m))(((\(∂\^2 f\&^\))/(∂(x^(' 2))))-((\(∂\^2 f\&^\))/(∂(x^('' 2))))) + (V(x') - V(x''))\(f\&^\)] = 0

Here, the authors note two points: one, that the Fourier transform has linearized and made separable (up to O(y)) the non-linear part of the earlier differential equation, and this non-linear term will later become the convective term. The second is that these equations are invariant under the exchange x↔p.

Furthermore, the authors state that there are only separable solutions for a small subclass of solutions, and require that initial and boundary conditions be separable as well. However, they claim this subclass is the same (and only) subclass earlier demonstrated to map the Sturm-Liouville operator to the Schrodinger equation.

These separable solutions can eventually be manipulated to show:

∂(ρ/∂)t+∇·(ρ u)=0

(∂/∂)t(m ρ u)+∇·[ρ (m uu -((η^2)/(4 m))(((Overscript[((∂^2) 
                                    lnρ), ⟷]))/(∂x∂x)))] + ρ ∇V = 0

where u is defined as the momentum averaged velocity p/m. Moreover, this transformation is irreversible, as the authors excellently describe: "This function will give the correct probability...when integrated over p... but, clearly, it is not equal to the true reduced probability density (Subscript[f, 1])... This discrepancy is due to the fact that \(TraditionalForm\`f\&^\) has been constructed using only the information given by the first two kinetic equations... To claim that anti-transforming the \(TraditionalForm\`f\&^\) we found keeps any resemblance to the full probability (Subscript[f, 1]) would be similar to claiming that the polynomial built with the first two coefficients of a Taylor expansion of a function would be equivalent to said function everywhere."

Thus, a method of evaluating averages in the p-conjugate space must be obtained. Using the Hopf-Cole transformation ln Ψ = (1/2) ln ρ + (η/i) S, where ΔS = 2πnη, with n an integer, the authors derive the familiar:

<(x^n)> =\(\(∫\(\(Ψ\)\(\ \)\(x\^n\) \(Ψ\^*\)\(\ \)\)\)\/\(ΨΨ\^*\)\)

<(p^n)> = ((∫\(Ψ\^*\) (((-i η ∇))^n)Ψ )/(\(∫\(ΨΨ\^*\)\)))
Additionally, the authors show that an equation of motion in p-conjugate space is given by:

((-(η^2))/(2  m)) (∇^2) Ψ + V(x)Ψ = i η  (∂/(∂t))Ψ 

with the probability as a function of x only given by ρ(x,t) = Ψ(x,t) \(TraditionalForm\`\(Ψ\^*\)\)(x,t).

And, finally, the authors note that as their transformation is irreversible, the Fourier transform requires 

ΔpΔx = η/2

Thus, the familiar postulates of quantum mechanics are obtained from purely kinetic considerations.

In their second paper, the authors extend this method to include fluids with rotational movement by changing from an assumption of full separability (which does not introduce enough degrees of freedom to consider vortical flows), to an assumption of the form 

\(TraditionalForm\`\(f\&^\)(\(x\^'\), \(x\^''\), 
          t)\  = \ \(\(h\^'\)(\(x\^'\), t)\)\ \(\(h\^''\)(\(x\^''\), 
              t)\)\  + \ \(\(h\^'\)(\(x\^'\), t)\)\ \(\(h\^''\)(\(x\^''\), 
              t)\)\).

This provides eight equations, which is sufficient to encompass vortical solutions. In this form, an extra term is introduced into the earlier equations of motion derived for a fully separable term that is proportional to η/2 times <σ>, where σ is the vector made from the standard Pauli matrices. This extra, intrinsic, degree of freedom, in η, describes vorticity and plays an identical role in the equations of motion as spin does in the Schrodinger equation.


Summary of auxiliary summer reading

"Limitation on entropy increase imposed by Fisher information," B. Nikolov, B. Roy Frieden (Physics Review E, Volume 49, Number 6)

This article discussed, in part, the different basis for Fisher and Shannon information metrics.  While both information metrics have the intuitive inverse dependence on variance, the Fisher information contains what the authors call a "measure-estimation" criteria related to the information on a distribution provided by a single measurement,  while Shannon information uses the distinguishability of signals in a channel to measure entropy and information.

The authors make a persuasive argument that Fisher information is more appropriate for deriving physical phenomena as measurement error is a more physical description of a system than is signal distinguishability.

Fisher information I = ∫((∇p·∇p)/p) dr

Shannon information I=-∫p ln(p) dr

where p = p(r|t).

"Fisher and Jaynesian statistics compared in the description of classical fluids", R E Nettleton (Journal of Physics A, 2002, 295-304)

This paper examines the use of Fisher entropy as a thermodynamic measure and compares it to the "Jaynes-Shannon" approach, which it references does not describe. It demonstrates that the two methods are equivalent at equilibrium, for  non-equilibrium solutions the Fisher entropy gives results not phenomenologically consistent with known non-equilibrium behavior. They also show that Fisher entropy solutions are not consistent with "Onsager-Casimir reciprocity".


"Concept of entropy for nonequilibrium states of closed many-body systems", J. L. del Rio-Correa (Physical Review A, Volume 43, Number 12).

This paper also analyzes non-equilibrium states using Jaynesian statisics, developing a variational method along the way. Furthermore, it relates the behavior of the system to the information available when the system is initially prepared, not an arbitrary condition at some later time t. This point is illustrated well by Jaynes' own rather emphatic remark, "If a macro-phenomenon is found to be reproducible then it follows that all microscopic details that were not under the experimenter's control must be irrelevant for understanding and predicting it."