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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:\\
\\
{\bfseries Mapping of the classical kinetic balance equations onto the Schrodinger equation }(Adriana I Pesci, Raymond E Goldstein), and {\bfseries
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:\\
\\
\({f_j}\big({x^N},{p^N}\big)\multsp =\multsp {{\int }_{\Omega }}{f_N}\big({x^N},{p^N}\big)\multsp \prod _{l=j+1}^{N}\DifferentialD x\DifferentialD
p\)
where \({f_1}\multsp \Mvariable{and}\multsp {f_2}\)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 :\\
\\
{ }\(\frac{\partial \multsp {f_1}}{\partial t}\)\(+\) \(\frac{{p_1}}{m}\cdot \)\(\MathBegin{MathArray}{l}
\frac{\partial \multsp {f_1}}{\partial t}\multsp -\multsp \frac{\partial V({x_1})}{\partial {x_1}}\cdot \multsp \frac{\partial {f_1}}{\partial {p_1}}
\\
\noalign{\vspace{1.32292ex}} \\
=\int \DifferentialD {x_2}\int \DifferentialD {p_2}\big[{f_1}\big(p_{1}^{'}\big)\multsp {f_1}\big(p_{2}^{'}\big)\multsp \multsp -\multsp {f_1}({p_1})\multsp
{f_1}({p_2})\big]
\MathEnd{MathArray}\)\\
{ }\\
Integrating over momentum allows them to use conservation of particle number and momentum to write:\\
\\
\(\MathBegin{MathArray}{l}
\int _{-\infty }^{\infty }\DifferentialD p\multsp \bigg(\frac{\partial {f_1}}{\partial t}+\frac{p}{m}\cdot \frac{\partial {f_1}}{\partial t}-\frac{\partial
V(x)}{\partial x}\cdot \frac{\partial {f_1}}{\partial p}\bigg)= \\
\noalign{\vspace{1.07292ex}}
\hspace{1.em} 0\multsp ,\\
\MathEnd{MathArray}\)\\
\hspace*{1.ex} \\
{ }\(\MathBegin{MathArray}{l}
\int _{-\infty }^{\infty }p\multsp \DifferentialD p\multsp \bigg(\frac{\partial {f_1}}{\partial t}+\frac{p}{m}\cdot \frac{\partial {f_1}}{\partial
t}-\frac{\partial V(x)}{\partial x}\cdot \frac{\partial {f_1}}{\partial p}\bigg)= \\
\noalign{\vspace{1.07292ex}}
\hspace{1.em} 0\\
\MathEnd{MathArray}\)\\
\\
Until now, the above steps are standard. Their original contribution comes in examining these balance equations in terms of the fourier representations
of \({f_{1.}}.\)They take \\
\\
\(\MathBegin{MathArray}{l}
{f_1}(x,p,t)\multsp =\multsp \\
\noalign{\vspace{0.958333ex}}
\hspace{1.em} \frac{1}{{{(2\multsp \pi \multsp \eta )}^3}}\int _{-\infty }^{\infty }\exp\bigg(-\multsp i\multsp \multsp \frac{p\cdot y}{\eta }\bigg)\overvar{f}{\RawWedge
}(x,y,t)\DifferentialD y\\
\MathEnd{MathArray}\)\\
\\
The balance equations then become\\
\\
\(\MathBegin{MathArray}{l}
\frac{1}{{{(2\multsp \pi \multsp \eta )}^3}}\multsp \\
\noalign{\vspace{1.23958ex}}
\hspace{2.em} \int \multsp \int _{-\infty }^{\infty }\DifferentialD p\DifferentialD y\multsp \\
\noalign{\vspace{1.29167ex}}
\hspace{5.em} \Mvariable{exp}\bigg(-i\multsp \frac{p\cdot y}{\eta }\bigg)\big[\frac{\partial f}{\partial t}+\frac{\eta }{i\multsp m}\frac{\partial
}{\partial x}\cdot \frac{\partial \overvar{f}{\RawWedge }}{\partial y}\big]= \\
\noalign{\vspace{1.07292ex}}
\hspace{1.em} 0\\
\MathEnd{MathArray}\)\\
\hspace*{0.5ex} \\
\(\MathBegin{MathArray}{l}
\frac{1}{{{(2\multsp \pi \multsp \eta )}^3}}\multsp \int \multsp \int _{-\infty }^{\infty }\DifferentialD p\DifferentialD y\multsp \Mvariable{exp}\bigg(-i\multsp
\frac{p\cdot y}{\eta }\bigg)\multsp \big[ \\
\noalign{\vspace{1.86458ex}}
\hspace{1.em} \frac{\partial }{\partial t}\bigg(\frac{\eta }{i}\frac{\partial \overvar{f}{\RawWedge }}{\partial y}\bigg)-\frac{{{\eta }^2}}{m}\frac{\partial
}{\partial x}\cdot \Bigg(\frac{\overvar{{{\partial }^2}\overvar{f}{\RawWedge }}{\longleftrightarrow }}{\partial y\RawWedge 2}\Bigg)+ \\
\noalign{\vspace{1.79167ex}}
\hspace{2.em} \frac{\partial }{\partial y}\Big(y\cdot \frac{\partial V}{\partial x}\overvar{f}{\RawWedge }\Big)\big]\\
\MathEnd{MathArray}\)\\
\hspace*{0.5ex} \\
These equations can be manipulated to show\\
\\
\(\underline{\Mvariable{lim}}\big[\frac{\partial \overvar{f}{\RawWedge }}{\partial t}+\frac{\eta }{i\multsp m}\frac{\partial }{\partial x}\cdot \frac{\partial
\overvar{f}{\RawWedge }}{\partial y}\big]=0\)\\
\\
\(\MathBegin{MathArray}{l}
\underline{\Mvariable{lim}}\big[\frac{\partial }{\partial t}\bigg(\frac{\eta }{i}\frac{\partial \overvar{f}{\RawWedge }}{\partial y}\bigg)-\frac{{{\eta
}^2}}{m}\frac{\partial }{\partial x}\cdot \Bigg(\frac{\overvar{{{\partial }^2}\overvar{f}{\RawWedge }}{\longleftrightarrow }}{\partial y\RawWedge
2}\Bigg)+\frac{\partial V}{\partial x}\overvar{f}{\RawWedge }+ \\
\noalign{\vspace{1.5ex}}
\hspace{3.em} \Mfunction{O}(y)\big]=0\\
\MathEnd{MathArray}\)\\
\\
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:\\
\\
\(\int \){\itshape }\(\MathBegin{MathArray}{l}
{{\Mvariable{dx}}^{''}}{{\Mvariable{lim}\multsp }_{x'\rightarrow x''}}\big[i\multsp \eta \multsp \multsp \frac{\partial \overvar{f}{\RawWedge }}{\partial
t}+\frac{{{\eta }^2}}{2\multsp m}\bigg(\frac{{{\partial }^2}\overvar{f}{\RawWedge }}{\partial {x^{'2}}}-\frac{{{\partial }^2}\overvar{f}{\RawWedge
}}{\partial {x^{''2}}}\bigg)\big]\multsp =\multsp \\
\noalign{\vspace{1.16667ex}}
\hspace{1.em} 0\\
\MathEnd{MathArray}\)\\
\\
\(\int \){\itshape }\(\MathBegin{MathArray}{l}
{{\Mvariable{dx}}^{''}}{{\Mvariable{lim}\multsp }_{x'\rightarrow x''\multsp }}\frac{1}{2}\big(\frac{\partial }{\partial {x^'}}-\frac{\partial }{\partial
{x^{''}}}\big)\multsp \times \big[ \\
\noalign{\vspace{1.36458ex}}
\hspace{2.em} \frac{\eta }{i}\multsp \multsp \frac{\partial \overvar{f}{\RawWedge }}{\partial t}+\frac{{{\eta }^2}}{2\multsp m}\bigg(\frac{{{\partial
}^2}\overvar{f}{\RawWedge }}{\partial {x^{'2}}}-\frac{{{\partial }^2}\overvar{f}{\RawWedge }}{\partial {x^{''2}}}\bigg)\multsp +\multsp \\
\noalign{\vspace{1.16667ex}}
\hspace{3.em} \big(V\big({x^'}\big)\multsp -\multsp V\big({x^{''}}\big)\big)\overvar{f}{\RawWedge }\big]\multsp =\multsp 0\\
\MathEnd{MathArray}\)\\
\\
Here, the authors note two points: one, that the Fourier transform has linearized and made separable (up to O({\bfseries 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 {\bfseries x}\(\leftrightarrow \){\bfseries 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:\\
\\
\(\frac{\partial \rho }{\partial t}+\nabla \cdot (\rho \multsp u)=0\)\\
\\
\(\MathBegin{MathArray}{l}
\frac{\partial }{\partial t}(m\multsp \rho \multsp u)+\nabla \cdot \big[\rho \multsp \Bigg(m\multsp \Mvariable{uu}\multsp -\frac{{{\eta }^2}}{4\multsp
m}\frac{\overvar{{{\partial }^2}\Mvariable{ln\rho }}{\longleftrightarrow }}{\partial x\partial x}\Bigg)\big]\multsp +\multsp \\
\noalign{\vspace{1.5ex}}
\hspace{2.em} \rho \multsp \nabla V\multsp =\multsp 0\\
\MathEnd{MathArray}\)\\
\\
where {\bfseries u }is defined as the momentum averaged velocity {\bfseries p/}m. Moreover, this transformation is irreversible, as the authors excellently
describe: "This function will give the correct probability...when integrated over {\bfseries p}... but, clearly, it is not equal to the true reduced
probability density \({f_1}\)... This discrepancy is due to the fact that \(\overvar{f}{\RawWedge }\) has been constructed using {\itshape only}
the information given by the first two kinetic equations... To claim that anti-transforming the \(\overvar{f}{\RawWedge }\) we found keeps any resemblance
to the full probability \({f_1}\multsp \)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 \(\Psi \) \(=\) (1/2) ln \(\rho
\) \(+\) \((\eta /i)\multsp S\), where \(\Delta \)S \(=\) 2\(\pi \){\itshape n}\(\eta \), with n an integer, the authors derive the familiar:\\
\\
\(<{x^n}>\multsp =\frac{\int \Psi \multsp {x^n}{{\Psi }^*}\multsp }{{{\Mvariable{\Psi \Psi }}^*}}\)\\
\\
\(<{p^n}>\multsp =\multsp \frac{\int {{\Psi }^*}\multsp {{(-i\multsp \eta \multsp \nabla )}^n}\Psi \multsp }{\int {{\Mvariable{\Psi \Psi }}^*}}\)\\
Additionally, the authors show that an equation of motion in p-conjugate space is given by:\\
\\
\(\frac{-{{\eta }^2}}{2m}{{\nabla }^2}\Psi \multsp +\multsp V(x)\Psi \multsp =\multsp i\multsp \eta \multsp \multsp \frac{\partial }{\partial t}\Psi
\multsp \)\\
\\
with the probability as a function of {\bfseries x} only given by \(\rho \)({\bfseries x},t) \(=\) \(\Psi \)({\bfseries x,}t) \({{\Psi }^*}\)({\bfseries
x,}t).\\
\\
And, finally, the authors note that as their transformation is irreversible, the Fourier transform requires \\
\\
\(\Delta \){\itshape p}\(\Delta \){\itshape x }\(=\) \(\eta \)/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 \\
\\
\(\overvar{f}{\RawWedge }\big({x^'},{x^{''}},t\big)\multsp =\multsp {h^'}\big({x^'},t\big)\multsp {h^{''}}\big({x^{''}},t\big)\multsp +\multsp {h^'}\big({x^'},t\big)\multsp
{h^{''}}\big({x^{''}},t\big)\).\\
\\
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 \(\eta \)/2 times \(<\){\bfseries \(\sigma \)}\(>\), where {\bfseries \(\sigma
\) }is the vector made from the standard Pauli matrices. This extra, intrinsic, degree of freedom, in \(\eta \), describes vorticity and plays an
identical role in the equations of motion as spin does in the Schrodinger equation.\\
\\
\\
{\bfseries Summary of auxiliary summer reading}{\bfseries \\
\\
}{\bfseries "}{\itshape 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.{\bfseries \\
}\\
Fisher information \(\imag \multsp =\multsp \int \frac{\nabla p\cdot \nabla p}{p}\multsp dr\)\\
\\
Shannon information \(\imag =-\int p\multsp \ln(p)\multsp dr\)\\
\\
where {\itshape p} \(=\) {\itshape 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".\\
\\
\\
"{\itshape 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."
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