Mini Review Volume 7 Issue 1

Res. Ord. Professor, Bio-Med. Engineering, Visiting Scholar at CUA, Catholic University of America, USA

**Correspondence:** Harold H Szu, PhD (The Rockefeller U), Fellows (AIAA, INNS, AIMBE, IEEE, OSA, SPIE) Academician (RAS) (RNL. S. V. Malrosov, 135, 1999), Res Ord Prof., CUA, Wash D.C, USA

Received: June 10, 2023 | Published: June 21, 2023

**Citation: **Harold S. Reentry vehicle friction heat estimated by means of Boltzmann integral-differential equation of fluctuation-dissipation theory. *MOJ App Bio Biomech*. 2023;7(1):78-81. DOI: 10.15406/mojabb.2023.07.00178

We begin with Atmosphere Einstein Brownian motions toward Knudsen Gas Boltzmann Kinetic formulism of the fluctuation-dissipation theory and applied the formulism to space travelers reentry vehicles. Early experimental results are estimated on the worst situation upper bound when a reentry vehicle is frictionally passing through a dense atmosphere. In reality, the vehicle has spent more time in a dilute atmosphere space. Accordingly, the fluctuation and dissipation theorem, derived from the Boltzmann kinetic equation, predicted a much less friction heat than the final reentry stage.

We wish to estimate for Astronauts returning to the Earth with limited payload heat shields. In other words, we wonder whether the reentry vehicles bottom with an already **4-meter thick heat insulation** layers can sustain the increasing burning off by frictional heating under different returning speed. The heat shield’s about **13-foot-diameter** composite structure — located at the bottom, blunt end of the Dragon capsule — is detachable and interchangeable between the reusable spacecraft in ** SpaceX’s Dragon fleet** (e.g. Dragon 1, flew 23 cargo missions to the Int’l Space Station (ISS) between 2010 and 2020 before being retired. SpaceX's CEO,

**Figure 1** ** Dragon capsule** — is detachable and interchangeable between the reusable spacecraft in

If it stays below the corridor, it may burn up. ${F}_{drag}=\frac{1}{2}\rho {V}^{2}{C}_{D}A$
, in terms of air density $\rho $
velocity $V$
, drag coefficient ${C}_{D}.$
high Mach number collision cylinder with negligible sound speed ${v}_{o};$
$\left(V+{v}_{o}\right)\cdot A\cong V\cdot A$
. e.g. ** H. Julian Allen** NASA, Ames Research Center Moffett Field.

The **shortfall**, if any, is that^{1} it costs ** $10,000** to put a

**In this paper**, we wish to provide a kinetic estimation with a safe margin from the statistical mechanics viewpoint to take into account the material thermal accommodation coefficient $0<\gamma <1$
changes and the atmosphere molecular distribution function varies $f\left(\overrightarrow{x},\overrightarrow{v},t\right).$
*from an *extreme **rarefied Knudsen gas** at an ** infinite mean free path** (

$y={x}_{o}+{v}_{o}(t-{t}_{mfp})+{x}^{2};t{t}_{mfp}$

**Figure 2** The Smoke comes off in linear free flight until multiple collisions with air molecules to reach parabolic curvature.

$\frac{m}{2}\langle {u}^{2}\rangle =\frac{1}{2}{k}_{B}{T}_{w}\left(t\right);{u}_{rms}=\sqrt{\raisebox{1ex}{${k}_{B}{T}_{w}\left(t\right)$}\!\left/ \!\raisebox{-1ex}{$m$}\right.};$

${t}_{mfp}\cong {x}_{mfp}/{u}_{rms}$

The ** flower pollens** motions are named after the botanist

The diffusion equation has been derived by **Adolf Fick** in *1855*.from the continuity equation, which states that a change in density is. ** $\frac{\partial \rho \left(t\overrightarrow{,x}\right)}{\partial t}$
; $\overrightarrow{J}\equiv -{D}_{o}\overrightarrow{\nabla}\rho \left(t.\overrightarrow{x}\right);\frac{\partial \rho \left(t\overrightarrow{,x}\right)}{\partial t}={D}_{o}\overrightarrow{\nabla}\cdot \overrightarrow{\nabla}\rho \left(t\overrightarrow{,x}\right)$
**due to inflow and outflow of material where the vector flux of the diffusing material $\overrightarrow{J}$
and the diffusion constant ${D}_{o}$
is the proportionality.

**Theorem 1: Kinetic derivation of diffusion equation**

Given kinetic distribution function $f\left(\overrightarrow{x},\overrightarrow{v},t\right)$ of space-velocity & time, we average away the velocity to derive the spatial density function $\rho \left(\overrightarrow{x},t\right)$

$\rho \left(\overrightarrow{x},t\right)\equiv <f\left(t,\overrightarrow{x},\overrightarrow{v}\right){>}_{\overrightarrow{V}}$
, which will satisfy ** a **diffusion equation $\frac{\partial \rho}{\partial t}={D}_{o}{\overrightarrow{\nabla}}^{2}\text{\rho};$

In 1855, physiologist **Adolf Fick** first reported flux $\overrightarrow{j}$
is the **diffusion flux**, of which the dimension is the amount of substance per unit area per unit time. $\overrightarrow{j}$
measures the amount of substance that will flow through a unit area during a unit time interval. *D* is the **diffusion coefficient** or **diffusivity**. Its dimension is area per unit time.

$\overrightarrow{j}=-D\overrightarrow{\nabla \text{\rho}};\frac{\partial \rho}{\partial t}+\overrightarrow{\nabla}\cdot \overrightarrow{j}=0;\frac{\partial \rho}{\partial t}=D{\overrightarrow{\nabla}}^{2}\text{\rho}$

Kinetic theory would begin that a regional time changes equals to its spatial change. The temporal density function $\rho \left(\overrightarrow{x},t\right)$ change in finite time $\tau $ must equal to its spatial changes:

**Proof:**

$\rho \left(\overrightarrow{x},t\right)+\frac{\partial \rho}{\partial t}){{\displaystyle \int}}^{\text{}}\phi \left(\overrightarrow{\Delta}\right)d\overrightarrow{\Delta}=\frac{\partial \rho}{\partial \overrightarrow{x}}{{\displaystyle \int}}^{\text{}}\phi \left(\overrightarrow{\Delta}\right)\overrightarrow{\Delta}d\overrightarrow{\Delta}+\text{}\frac{1}{2}\frac{\partial}{\partial \overrightarrow{x}}\cdot \frac{\partial \rho}{\partial \overrightarrow{x}}\left({{\displaystyle \int}}^{\text{}}\phi \left(\overrightarrow{\Delta}\right)\left|\overrightarrow{\Delta}\cdot \overrightarrow{\Delta}\right|d\overrightarrow{\Delta}\right);$ $\frac{\partial \rho}{\partial t}={D}_{o}{\overrightarrow{\nabla}}^{2}\text{\rho}$

where diffusion constant ${D}_{o}\equiv \frac{1}{2}\langle {\left|\overrightarrow{\Delta}\right|}^{2}\rangle $ with jump $\overrightarrow{\Delta}$ probability density ${{\displaystyle \int}}^{\text{}}\phi \left(\overrightarrow{\Delta}\right)d\overrightarrow{\Delta}=1.$

**Theorem 2:** **Solving the diffusion equation** $\frac{\partial \rho}{\partial t}={D}_{o}{\overrightarrow{\nabla}}^{2}\text{\rho}$

Assuming the reentry space craft $A\left(\overrightarrow{x},t=0\right)$
in the vast space may be represented by Dirac delta generalized function $\delta \left(\overrightarrow{x}\right)$
If $A\left(\overrightarrow{x},t=0\right)=\delta \left(\overrightarrow{x}\right),then{a}_{\overrightarrow{k}}\left(0\right)=1$
in terms of its Fourier transforms. We will adopt Fourier transform to replace the ** Laplacian** operator ${\overrightarrow{\nabla}}^{2}\leftrightarrow {\overrightarrow{k}}^{2}$

$A\left(\overrightarrow{x},t\right)\equiv {{\displaystyle \int}}^{\text{}}d\overrightarrow{k}{a}_{\overrightarrow{k}}\left(t\right){e}^{i\overrightarrow{k}\overrightarrow{x}};{a}_{\overrightarrow{k}}\left(t\right)\equiv \frac{1}{{(2\pi )}^{3}}{{\displaystyle \int}}^{\text{}}d\overrightarrow{x}{e}^{-i\overrightarrow{k}\overrightarrow{x}}A\left(\overrightarrow{x},t\right);\text{}$

the diffusion equation is the Fourier space

$\frac{1}{{(2\pi )}^{3}}{{\displaystyle \int}}^{\text{}}d\overrightarrow{k}{e}^{i\overrightarrow{k\cdot}\overrightarrow{x}}[\frac{d{a}_{\overrightarrow{k}}}{dt}+{D}_{o}{k}^{2}{a}_{\overrightarrow{k}}]=0$

Since the first order time integral gives an exponential function, then $\frac{d{a}_{\overrightarrow{k}}}{dt}+{D}_{o}{k}^{2}{a}_{\overrightarrow{k}}=0$ ; yields the final answer: ${a}_{\overrightarrow{k}}=$ ${a}_{\overrightarrow{k}}\left(0\right)\text{exp}\left(-{D}_{o}{k}^{2}t\right)$ = $\text{exp}\left(-{D}_{o}{k}^{2}t\right)$ in the Fourier space, equivalently in the spatial domain:

$A\left(\overrightarrow{x},t\right)={{\displaystyle \int}}^{\text{}}\frac{d\overrightarrow{k}}{2\pi}\text{exp}\left(i\overrightarrow{k}\overrightarrow{x}\right)\text{exp}\left(-{D}_{o}{k}^{2}t\right)$
; **Q.E.D.**

Since the rocket thrust must be sufficient to take care the mass weight of rocket, we can avoid involve the computation with mass, geometry, etc. consideration. We will concentrate on the kinetic collision boundary condition. Thus, for geometry simplicity of all kind of reentry vehicles, we develop the molecular fluctuation dissipation theorem, e.g. French Physics ** Maurice Couette s**uggested simplified object geometry, i.e. one plate moves with respect to the other plate with gases between two plates. Now we wish to verify the boundary condition based on molecular thermal motion as the origin of the fluctuations. Its local property shall have nothing to do with the geometry object of immersed object and its boundary interaction condition.

**Figure 3** Plane couette flow, $u\equiv U{(\frac{2{K}_{B}T}{m})}^{\frac{1}{2}}$
; distance in terms of the mean free path $\lambda ={({n}_{o}\sigma )}^{-1}$
interns of number density ${n}_{o}$
, and molecular collision cross section $\sigma $
.

We can prove the **fluctuation and dissipation theorem** without further ado; A **solid body of mass** ** M** under Brownian motion is described by

$M\frac{du}{dt}=-\beta u+\tilde{F}\left(t\right);\langle F\left(t\right)\rangle =-\beta u;\tilde{F}\left(t\right)\equiv {F}_{T}\left(t\right)-\langle F\left(t\right)\rangle ;\langle \tilde{F}\left(t\right)\stackrel{\u02c7}{F}\left({t}^{\prime}\right)\rangle =2{D}_{o}\delta \left(t-t"\right)$ ;

$<u\left(t\right){>}^{{u}_{o}}={u}_{o}\mathrm{exp}\left(-\frac{\beta}{M}t\right)+W[1-\text{exp}\left(-\frac{\beta}{M}t\right);\mathrm{lim}\text{t}\to \infty ;\text{}u\left(t\right){}^{{u}_{o}}\to W$

$<(\text{u-W)}\left({u}^{\prime}-W\right)=\langle \left(u-W\right)\rangle \langle \left({u}^{\prime}-W\right)\rangle +\frac{D}{\beta M}\text{{}\mathrm{exp}(-\frac{\text{\beta}}{\text{M}}|t-{t}^{\prime}|\text{)}-\mathrm{exp}(-\frac{\text{\beta}}{\text{M}}|t+{t}^{\prime}|)\};$

Setting the equal partition law per degree of freedom:

$\underset{t\to \infty}{\text{lim}}\frac{M}{2}{(u-W)}^{2}=\frac{{D}_{o}}{2\beta}=\frac{1}{2}{k}_{B}T$ ,

we have derived the **fluctuation –dissipation theorem. ${D}_{o}={k}_{B}T\beta $
;**

$\tilde{F}\left(t\right)\stackrel{\u02c7}{F}\left({t}^{\prime}\right)>=2{k}_{B}T\beta \delta \left(t-t"\right);$

The devil of fluctuation-dissipation truth is in the detail of molecular-boundary collisions and inter-collisions.

**Fluctuating Kinetic Theory of discrete molecular phase space**$f\left(\overrightarrow{x},\overrightarrow{v},t\right)$ formulation augmented theIntegral-differential equation with fluctuation Fox sources. We shall simplify the collision kernel by*Ludwig Boltzmann*relaxation equation{2} (Am. Math Soc. Providence RI, 1963, Ch. IV), and keep the fluctuation*L. Bhatnagar, E.P. Gross, M. Krook (BGK)**Fox sources $\tilde{S}\left(t,\overrightarrow{v}\right)$**(The Rockefeller Univ. 1970 Thesis of Ronald Fox)*

$\frac{\partial f}{\partial t}+{\overrightarrow{v}}_{\alpha}\frac{\partial f}{\partial {x}_{\alpha}}={\left(\frac{\partial f}{\partial t}\right)}_{Col.}\cong -\frac{f-{f}^{\left(o\right)}}{{\theta}_{o}}+\tilde{S}\left(t,\overrightarrow{v}\right);$

$\tilde{S}\left(t,\overrightarrow{v}\right)\tilde{S}\left(t\text{'},{\overrightarrow{v}}^{\text{'}}\right)=\delta \left(t-{t}^{\prime}\right)\delta (\overrightarrow{v}-{\overrightarrow{v}}^{\text{'}}\}$

Where in the LHS, we have omitted the acceleration force term ${\overrightarrow{a}}_{\alpha}\frac{\partial f}{\partial {\overrightarrow{v}}_{\alpha}}$
while in the RHS we have simplified the ** Boltzmann collision** term with

${f}^{\left(o\right)}=n{(\frac{m}{2\pi {K}_{B}T})}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{e}^{-\frac{m}{2{K}_{B}T}{(v-u)}^{2}}$

First of all we must introduce a fluctuation sources at solid boundary condition. Our goal is to demonstrate for an arbitrary Maxwell accommodation coefficient for absorbed and reemitted condition with the probability $0\gamma \le 1$ that a Maxwellian distribution by the plate temperature dynamic condition

${f}^{\left(o\right)}\left(\pm \frac{\alpha}{2};c\right)={n}_{o}{(\frac{m}{2{K}_{B}T})}^{\frac{3}{2}}\text{exp}\left[-{c}_{x}^{2}-{c}_{y}^{2}-\left({c}_{z}\pm \frac{U}{2}{)}^{2}\right]\cong {f}_{o}\cdot \left[1\pm {c}_{z}U\right)+O\left({U}^{2}\right)\right]$

Before fluctuation and dynamic case, we first analyze equilibrium case that a stationary flow and the plate at the same temperature and a constant density of gas:

$\frac{\partial h}{\partial \tau}=0;\left(1,h\right)=\left({c}^{2}-\frac{3}{2},h\right)=0;g\left(x\right)=\frac{2}{U}\left({c}_{z},h\right);g\left(x\right)=-g\left(-x\right)$

Linearized BGK equation for Cuette flow becomes

${c}_{x}\frac{\partial h}{\partial x}+h={c}_{z}Ug\left(x\right)$

This must add boundary condition ${h}^{+}\left(x,{c}_{x}\right)$ and ${h}^{-}\left(x,-{c}_{x}\right)$

We can write the Maxwell boundary conditions $at\alpha =\frac{d}{\lambda}$ is the inverse Knudsen number, and $\gamma $ is Maxwell thermal accommodation coefficients.

${h}^{+}\left(-\frac{\alpha}{2},{c}_{x}\right)=-\gamma {c}_{z}U+\left(1-\gamma \right){h}^{-}\left(-\frac{\alpha}{2},-{c}_{x}\right)$

${h}^{-}\left(+\frac{\alpha}{2},-c{\text{'}}_{x}\right)=+\gamma {c}_{z}U+\left(1-\gamma \right){h}^{+}\left(+\frac{\alpha}{2},{c}_{x}^{\text{'}}\right)$

Solution of Fluctuation and Dissipation Relationship with BGK Boltzmann integral-differential equation with Fox sources, under an arbitrary Maxwell thermal accommodation $0<\gamma \le 1$ has been first given by Harold Szu’s dissertation in 1971 under Prof. George E. Uhlenbeck at the Rockefeller Univ. (Unpublished), the results can be compared with experiment data done by NASA and SpaceX Shuttle flights.

${f}^{\left(o\right)}\left(\pm \frac{\alpha}{2},c\right)={n}_{o}{(\frac{m}{2{K}_{B}T})}^{\frac{3}{2}}\text{exp}[-{c}_{x}^{2}-{c}_{y}^{2}-\left({c}_{z}\pm \frac{U}{2}{)}^{2}\right]={f}_{o}\left(1\pm {c}_{z}U\right)+O\left({U}^{2}\right)$

The complete Fluctuation-Dissipation solution can be found in Ph D dissertation of Harold Szu, Rockefeller University, 1971 (unpublished).

Further lab work

We shall estimate the Shuttle payload mass its average heat capacity of soil compared to the common sense, water 4,18 Joules of heat (1 kilocalorie) for the temperature of one kilogram of water to increase 1°C The specific heat of possible shuttle materials are listed in the following table specific heat defined as follows

[https://en.wikipedia.org/wiki/Specific_heat_capacity]

${\mathcal{C}}_{o}\equiv \frac{\mathcal{C}}{M}=\frac{1}{M}\frac{dQ}{dT}$

Specific Heat ${\mathcal{C}}_{o}$ capacity at a const. pressure versus const. volume (${\beta}_{T}$ isothermal compressibility, $\alpha $ coeff. Of thermal expansion; $\u2102\text{op}-{\text{C}}_{\text{oV}}=\frac{{\text{\alpha}}^{2}\text{T}}{{\text{\rho \beta}}_{\text{T}}}\text{}$

Silicon won’t oxidize in the outer surface, as it is coated with **silicon carbide** designed by about **13-foot-diameter** composite structure and **4 meter thick** provided a burn-off mass generating the heat thermal energy should be equivalent, its general proof in the Appendix A.

The ** conclusions and future works** are given as follows: Some suggestions are put forward to measure how we can reduce realistically due to the

None.

None.

The author declares no conflicts of interest.

- THE AERODYNAMIC HEATING OF ATMOSPHERE ENTRY VEHICLES - A REVIEW By H. Julian Allen NASA, Ames Research Center Moffett Field, California Paper for Symposium on Fundamental Phenomena in Hypersonic Flow, Cornell Aeronautical Laboratory, Buffalo, New York. June 25-26, 1964.
- A study detailing the habitability of a nearby exo-planet appears to have caught the attention of
**SpaceX**CEO**Elon Musk**, who has wished to transform humanity into a mult-planetary species with plans to colonise**Mars**within the next few decades. (Credit: Anthony Cuthbertson, Yahoo News Tue, June 13, 2023) - Bhatnagar PL, Gross EP, Krook M. A model for collision processes in gases.
*Phys Rev.*1954;94 (1954):511. Bhatnagar-Gross-Krook model. Encyclopedia of Mathematics. - The Boltzmann equation, Gyu Eun Lee, https://www.math.ucla.edu/~gyueun.lee/writing/gso_boltzmann.pdf.
- Harold Hwaling Szu. The dissertation. The Rockefeller Univ. 1971:1–100.

**Appendix A ****the Mass Energy Equivalence Relationship of Albert Einstein**

Einstein's mass-energy equivalence becomes important in Space travel, because the *payload mass* is equivalent to the *energy consumption.* Conversely, the *heat energy consumption* is equivalent to the* mass energy usage*.

**$F=\frac{d\left(mv\right)}{dt}=m\frac{dv}{dt}+v\frac{dm}{dt}$
**

$\Delta E=\underset{0vacuum}{\overset{f}{{\displaystyle \int}}}F\cdot ds=\underset{0}{\overset{f}{{\displaystyle \int}}}\left(m\frac{dv}{dt}+v\frac{dm}{dt}\right)\cdot ds=\underset{0}{\overset{f}{{\displaystyle \int}}}mv\cdot dv+\underset{0}{\overset{f}{{\displaystyle \int}}}v\frac{dm}{dt}\cdot ds=\frac{1}{2}\underset{0}{\overset{f}{{\displaystyle \int}}}md\left({v}^{2}\right)+\underset{0}{\overset{f}{{\displaystyle \int}}}{v}^{2}dm$

Lorentz transform

$m=\frac{{m}_{0}}{\sqrt{1-\frac{{v}^{2}}{{c}^{2}}}}$
** ${v}^{2}={c}^{2}\left(1-\frac{{m}_{o}^{2}}{{m}^{2}}\right)$
$\frac{d\left({v}^{2}\right)}{dm}=\frac{2{m}_{o}^{2}{C}^{2}}{{m}^{3}}$
**

**$\Delta E=\frac{1}{2}\underset{0}{\overset{f}{{\displaystyle \int}}}md\left({v}^{2}\right)+\underset{0}{\overset{f}{{\displaystyle \int}}}{v}^{2}dm=\underset{0}{\overset{f}{{\displaystyle \int}}}m\frac{{m}_{o}^{2}{C}^{2}}{{m}^{3}}dm+\underset{0}{\overset{f}{{\displaystyle \int}}}{c}^{2}\left(1-\frac{{m}_{o}^{2}}{{m}^{2}}\right)\text{dm}=\underset{0}{\overset{\text{f}}{{\displaystyle \int}}}{c}^{2}\text{dm}$
**

$\because \Delta E={C}^{2}\Delta m$
; $\therefore E=m{C}^{2}$
**Q.E.D.**

**Verification of Lorentz transform in terms of slow speed limit as follows:**

$E=\frac{{m}_{0}{C}^{2}}{\sqrt{1-\frac{{v}^{2}}{{c}^{2}}}}\cong {m}_{0}{C}^{2}\left\{1+\frac{1}{2}\frac{{v}^{2}}{{c}^{2}}+\frac{3}{8}\frac{{v}^{4}}{{c}^{4}}+\dots \right\}={m}_{0}{C}^{2}+\frac{1}{2}{m}_{0}{v}^{2}+\frac{3}{8}{m}_{0}\frac{{v}^{4}}{{c}^{2}}$

where the first term is zero velocity mass energy equivalence, and the second term is Newtonian kinetic energy, etc.

**Appendix B Fluctuation-Dissipation Theory Estimated by BGK collision kernel.**

There is more scientific eluciation in the 1971 Harold Szu PhD thesis of 100 pages (cf. Reference 5). For example, besides the bulk entropy source fluctuation that Ronny Fox derived and discussed in his 1970 Rockefeller Univ. PhD thesis, there are the boundary wall fluctuations that Harold Szu derived and discussed in a dilute atmosphere medium where *Fox"s*** molecular collisions** become rare, but the

** $\frac{\partial f}{\partial t}+{\overrightarrow{v}}_{\alpha}\frac{\partial f}{\partial {x}_{\alpha}}={\left(\frac{\partial f}{\partial t}\right)}_{Col.}\cong -\frac{f-{f}^{\left(o\right)}}{{\theta}_{o}}+\tilde{S}\left(t,\overrightarrow{v}\right);$
**

** $\tilde{S}\left(t,\overrightarrow{v}\right)\tilde{S}\left(t\text{'},{\overrightarrow{v}}^{\text{'}}\right)=\delta \left(t-{t}^{\prime}\right)$
**

For the plane Couette flow, the stochastic BGK equations can be simplified to:

** $\frac{\partial h}{\partial \tau}+{c}_{x}\frac{\partial h}{\partial x}=-h+2{c}_{z}\left({c}_{z},h\right)+\tilde{B}\left(\overrightarrow{x},\overrightarrow{c},\tau \right)$
;** $\langle \tilde{B}\rangle =0$

$<\tilde{B}\left(\overrightarrow{x},\overrightarrow{c},\tau \right)\tilde{B}\left({\overrightarrow{x}}^{\text{'}},{\overrightarrow{c}}^{\text{'}},{\tau}^{\prime}\right)\ge \frac{2}{{n}_{o}{\lambda}^{3}}[{\pi}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\text{exp}(\frac{1}{2}\left(\left|\overrightarrow{c}{|}^{2}+\right|{\overrightarrow{c}}^{\text{'}}{|}^{2}\right)$ $\delta \left(\overrightarrow{c}-\overrightarrow{c}\text{'}\right)-2\overrightarrow{{c}_{z}}{\overrightarrow{{c}_{z}}}^{\text{'}}]$

$\delta \left(\overrightarrow{x}-\overrightarrow{{x}^{\prime}}\right)\delta \left(\tau -{\tau}^{\prime}\right)$ ];

Putting $h=\langle h\rangle +\left[h-\langle h\rangle \right]\equiv \langle h\rangle +\tilde{h}$ and assuming $\langle h\rangle $ independent of $\tau $ , one gets back for $\langle h\rangle :$

${c}_{x}\frac{\partial \langle h\rangle}{\partial x}=-\langle h\rangle +2{c}_{z}\left({c}_{z},\langle h\rangle \right)$ ; while for $\tilde{h}\left(x,\overrightarrow{c},\tau \right)$ , one obtains

** $\frac{\partial \tilde{h}}{\partial \tau}+{c}_{x}\frac{\partial \tilde{h}}{\partial x}=-\tilde{h}+2{c}_{z}\left({c}_{z},\tilde{h}\right)+\tilde{B}$
**

**Note that **

**(1) Since the correlation of fluctuation source $\tilde{B}$
is related to the collision kernel $K\left(\overrightarrow{c},\overrightarrow{{c}^{\prime}}\right)$
if the intermolecular collision can be neglected in the Knudsen limit, then the effect of $\tilde{B}$
will disappear. **

**(2) The drag depends on the accommodation coefficients $\gamma $
, while the Fox bulk fluctuations are independent of $\gamma $
. So that these fluctuations alone, the fluctuation-dissipation theorem cannot follow.**

**(3) There is a second source of fluctuation, which is due to the wall called wall fluctuations [Harold Szu 1971]. More details computation will not be presented but found in the dissertation The Rockefeller Univ., 1971, pp.1, 100. **

©2023 Harold. This is an open access article distributed under the terms of the, which permits unrestricted use, distribution, and build upon your work non-commercially.