# Noise and fluctuations in nanoscale gas flow

J. Dastoor  D. M. Willerton  W. Reisner  G. Gervais  Department of Physics,
McGill University, Montreal, Quebec H3A 2T8, Canada

###### Abstract

We theoretically calculate the fundamental noise that is present in gaseous
(dilute fluid) flow in channels in the classical and degenerate quantum
regime, where the Fermi-Dirac and Bose-Einstein distribution must be
considered. Results for both regimes are analogous to their electrical
counterparts. The quantum noise is calculated for a two terminal system and is
a complicated function of the thermal and shot noise with the thermal noise
dominating when $2k_{B}T\rho\gg m\Delta P$ and vice versa. The cumulant
generating function for mass flow, which generates all the higher order
statistics related to our mass flow distribution, is also derived and is used
to find an expression for the third cumulant of flow across a fluidic channel.

Keywords: nanopores, noise, fluctuations, mass flow

##  I Introduction

Nanoscale fluid transport in dilute (gaseous) regimes is of broad fundamental
and engineering importance, relevant in diverse scenarios ranging from
understanding fluid flow in quantum and classical regimes to industrial
applications involving gas processing [1]. From a fundamental point of view,
improvements in nanofabrication have enabled the production of nanochannels
and nanopores with well-defined nanometric dimensions that can be used to
verify classical theories for gas flow in free-molecular transport (Knudsen)
regimes [2, 3, 4]; these measurements have been extended cryogenically to
explore transport of quantum fluid phases of 4He [5] where it is expected that
a one-dimensional many-body quantum state would form [6]. From an engineering
point of view, nanochannels and nanoporous materials, due to their high
surface to volume ratio and pore sizes below the molecular mean free path
and/or approaching molecular dimensions [7], have excellent absorptive
properties and can exhibit size-based molecular sieving [7, 8, 9], useful for
applications in gas separation [9] and catalysis [10].

Most experimental and theoretical efforts devoted to characterizing nanoscale
gas transport have focused on modeling the gas mass flow-rate $Q$, e.g. [11,
12]. In analogy to the case of electrical transport, this is given by
$Q=G\Delta P$, where $Q$ is the mass flow, $G$ is the flow conductance, and
$\Delta P$ is the pressure difference across the channel or pore [3]. However,
the gas mass flow-rate is not the only quantity of interest that can be
extracted via monitoring a given mass flow channel. Just as is the case for
electrical current, statistical fluctuations in the mass flow will exist (mass
flow noise). These fluctuations are also of fundamental interest, for example
providing new information about a system’s fluidic properties in both
classical and quantum regimes – e.g., its transmission properties or flow
limitations. The mass flow noise will also have practical implications,
limiting applications where mass-flow rate is used as a sensor by creating a
noise floor that observable signals need to exceed. In addition, there might
be scenarios where the degree of mass flow noise present could itself
constitute the signal of interest. Finally, gas-flow fluctuations might affect
the performance of gas based separation or catalysis devices, for example
statistical fluctuations in the stream of a low concentration catalysis or
inhibitor species might lead to large fluctuations in output.

Considerable effort has been devoted in the past to improve our understanding
of electrical noise; much of this insight can be adopted to the closely
analogous case of mass flow noise. White current fluctuations arise due to
thermal energy (Johnson-Nyquist noise) and the discrete nature of electrical
charge (shot noise) [13]. These two noise sources are classically distinct,
yet become interlinked in the quantum regime. The development of quantum shot
noise theory has seen new applications in distinguishing particles from waves
and future proposals for new entanglement detectors [14], and has led to the
spectacular experimental validation of the effective quasiparticle charges of
electrons confined to two dimensions in the fractional quantum Hall regime
[15, 16]. Somewhat surprisingly these sources of noise that set fundamental
limits in terms of signals in the case of dilute mass flow has been neglected,
and so here we propose a theoretical calculation for the thermal noise of an
ensemble of particles, forming a dilute gas, by adapting well-defined
techniques developed for electrical noise. We also calculate the noise
associated with a directed flow (shot noise) in the classical regime and then
adapt our discussion to include quantum effects by combining thermal and shot
noise. This quantum noise is sometimes referred to as the quantum shot noise
and will be derived by adapting Martin and Landauer’s wave-packet approach for
electrical noise to mass flow noise [13]. This approach is chosen because it
allows us to develop a quantum noise expression based on a similar process to
our derivation of classical noise while straightforwardly incorporating the
relevant quantum mechanical considerations. All our expressions are derived
based on the assumption that the noise is distributed equally over all
frequencies (i.e. that it is white) and conforms with a Gaussian distribution,
which is expected based on analogy with the electrical noises.

We produce general results from the fundamental flow equation, $Q=G\Delta P$,
where $Q$ is the mass flow, $G$ is the flow conductance, and $\Delta P$ is the
pressure difference across the channel. In doing so, we neglect any
consideration of turbulence since our concern is in small dilute fluidic
systems with very low Reynolds numbers, $Re\ll 2000$. It should however be
noted that under certain conditions even small mesoscopic systems may be
subject to turbulent flow, e.g. [17, 18, 19], and in such cases a new approach
may be needed. We have also confined our analysis to idealized fluidic
channels in which electromagnetic effects are negligible. We note that in many
nanofluidic systems, charges on mass carriers and surface effects can heavily
influence the mass flow and a new approach may be needed to address these
cases as well [20, 21]. The noise, $\delta Q$, will be calculated as a mean
squared fluctuation, $\langle\delta Q^{2}\rangle$, where $\langle\dots\rangle$
refers to an average with respect to time. Recent work found that $G$ is
quantized in units of $2m^{2}/h$, where $m$ is the mass of a fluid particle
and $h$ is Planck’s constant [22, 23]. The resemblance with the quantum of
electrical conductance ($2e^{2}/h$ where $e$ is the electrical charge)
highlights the close analogy between mass flow and charge flow.

Other sources of noise in mesoscopic fluidic systems, such as the irregular
motion of impurities, also become important to consider when building
sensitive devices. While these sources are not explored here, the cumulant
generating function of the quantum white noise is calculated to allow for easy
combination with other independent noise sources in future scenarios. In
addition, we also derive an expression for the third cumulant of quantum white
noise. With growing interest in the theory of full counting statistics, these
will both be valuable tools for future work. Finally, we verify our results
using the fluctuation dissipation theorem, which describes the relation
between random fluctuations of a system at equilibrium to a small
perturbation. The theorem states that fluctuations occurring in equilibrium,
i.e. in the absence of a net mass flow, are proportional to the channel
conductance [24]. This provides a baseline test for our results.

##  II Classical Noise

Noise in the classical regime is separated into thermal noise and shot noise
contributions. In dilute fluidic systems, the shot noise originates from the
discrete nature of an average mass flow signal and is thus formally only
present out of equilibrium. The thermal noise, however, is expected to exist
even at thermodynamic equilibrium as it is due to the innate, random motion of
particles that is implied by the kinetic theory of gases and the Maxwell-
Boltzmann distribution. We may conceptualize the ensuing random passage of
particles across the pore as comprising an instantaneous flow rate that
averages to zero over long times, i.e. $\braket{Q}=0$, and is thus permitted
to occur even in the absence of a net pressure differential, $\Delta P$.

We begin by considering the thermal fluctuations in a system at equilibrium by
noting that when both terminals are in equilibrium with each other, there is
no net energy transfer within the system and we can use the analogy of a
standing wave in an open pipe. We know from equipartition that each standing
wave mode has two degrees of freedom and therefore each mode has an average
energy of $k_{B}T$, where $k_{B}$ is Boltzmann’s constant and $T$ is
temperature.

Figure 1: a) A cartoon of the flow through a cylindrical fluid channel. Flow
is from the left side to the right side (source to the drain). b) Cartoon
example of mass flow fluctuations about an average flow signal,
$\langle{Q}\rangle$, occurring in a dilute nanochannel.

The total average energy of our system is found as $\Delta jk_{B}T$, where
$\Delta j$ is the range of modes that our system can occupy. The occupied
range of modes is characterised by the time taken for a particle to pass
through the channel and by the frequency range, $\Delta\nu$, that the standing
waves can occupy. An equation for the average power can now be written using
our previous results and the frequency of oscillation of our standing wave.
Lastly, we equate this power expression with the instantaneous thermodynamic
power of the fluid and derive an expression for the thermal noise in a fluid
channel:

$\langle\delta Q^{2}\rangle=4k_{B}TG\rho\Delta\nu,$ (1)

where $\rho$ is the mass density of the fluid. The fluctuation dissipation
theorem is satisfied since the noise is proportional to the conductance. Note
that equation 1 shows that thermal noise exists even when $\langle Q\rangle=0$
(at equilibrium). The Johnson-Nyquist expression for the electrical current
thermal noise, $4k_{B}TR^{-1}\Delta\nu$ (where $R$ is the electrical
resistance), is analogous to equation 1 [25].

In the case that there exists a nonzero pressure differential across the pore,
there will be a net flow of mass from one reservoir to the other and equation
1 no longer suffices to describe the fluctuations. To formulate a noise
expression that accommodates an arbitrary pressure differential across the
pore, it is necessary to model the mass flow as consisting in a stream of
discrete particles, much like an electrical current. We further note that, in
the free-molecular flow regime, interparticle collisions are negligible and we
can model the flow of particles as a Poisson process. We may then loosely
follow van der Ziel’s derivation for electrical shot noise, adapting the
approach where necessary, to establish a comprehensive expression for the
mass-flow fluctuations [26].

Suppose that on either side of a nanopore we hold reservoirs at fixed
pressures, $P_{A}$ and $P_{B}$, such that

$P_{A}=\Delta P+P_{0};\;P_{B}=P_{0},$ (2)

where we have identified reservoir $A$ as the region of higher pressure and
have used the pressure of $B$ to mark a baseline, $P_{0}$. Particles are
transmitted across the pore in both directions, each with some associated
instantaneous rate of occurrence, $r(t)$, which fluctuates in time. Defining
$N$ to be the net number of particles passing through the pore in the
direction of $A$ to $B$, we have

$N=\int_{0}^{\tau}[r_{A\rightarrow B}(t)-r_{B\rightarrow A}(t)]dt,$ (3)

for some time interval, $\tau$. Note that a negative $N$ signifies a net
passage of particles from $B$ to $A$. We may also define the fluctuation in
$N$ as the instantaneous deviation from its average,

$\delta N=N-\braket{N},$ (4)

with

$\braket{N}=\braket{r_{A\rightarrow B}}\tau-\braket{r_{B\rightarrow A}}\tau.$
(5)

The averages in the above expression may be interpreted as either ensemble or
time averages, since these are equivalent for an Ergodic process. We now
define an additional random variable, $\delta R_{\tau}$, corresponding to
fluctuations in the net rate of particle transmissions as

$\delta R_{\tau}=\frac{\delta N}{\tau}.$ (6)

Noting that the variance of $N$ is defined as $Var(N)=\braket{\delta N^{2}}$,
we may write that

$\displaystyle\braket{\delta R_{\tau}^{2}}$
$\displaystyle=\frac{Var(N)}{\tau^{2}}=\frac{Var(N_{A\rightarrow
B}-N_{B\rightarrow A})}{\tau^{2}}$ (8)
$\displaystyle=\frac{Var(N_{A\rightarrow B})+Var(N_{B\rightarrow
A})}{\tau^{2}},$

where the final step follows from the fact that the variance for a difference
on two Poisson variables is just the (positive) sum of the variances
associated with each individual process. Additionally, because the variance of
a Poisson process is equal to its mean, we have

$\braket{\delta R_{\tau}^{2}}=\frac{\braket{r_{A\rightarrow
B}}+\braket{r_{B\rightarrow A}}}{\tau}.$ (9)

Applying the Wiener-Khintchine theorem allows us to extract the zero-frequency
component of the noise spectral density as

$S_{R}(0)=\lim_{\tau\rightarrow\infty}2\tau\braket{\delta
R_{\tau}^{2}}=2\braket{r_{A\rightarrow B}}+2\braket{r_{B\rightarrow A}},$ (10)

which, for white noise, suffices to describe the entire spectrum. We now make
the conversion to units of mass-flow fluctuations by multiplying the spectral
density by the mass of the fluid particles squared:

$S_{Q}(0)=2m\braket{Q_{A\rightarrow B}}+2m\braket{Q_{B\rightarrow A}},$ (11)

where we have distributed one factor of $m$ into each average to pass from
particle flow rates to mass flow rates. We may alternatively write the above
expression as

$S_{Q}(0)=2mGP_{A}+2mGP_{B}=2mG\Delta P+4mGP_{0},$ (12)

which is legitimate because the transmission events are statistically
independent and we may consider $P_{A}$ and $P_{B}$ separately, each
constituting an effective pressure differential across the pore. The first
term on the right contains the factor $G\Delta P$, which we know to be the
average net mass flow, $\braket{Q}$. If we make this substitution and also
replace $P_{0}$ with $nk_{B}T$ via the ideal gas law, then we may multiply
through by an arbitrary frequency bandwidth, $\Delta v$, to arrive at the
result

$\braket{\delta Q^{2}}=2m\braket{Q}\Delta v+4Gk_{B}T\rho_{0}\Delta v,$ (13)

with $\rho_{0}$ being defined as a baseline mass density that exists across
both reservoirs. When cast in this form, the above expression lends itself to
a straightforward analogy with electrical circuits. Firstly, we note that when
$\braket{Q}=0$, or equivalently when the two reservoirs exist in thermodynamic
equilibrium, we recover equation 1, which was previously likened to the
Johnson-Nyquist thermal noise in electrical resistors. We also note that the
first term on the right may be identified with Schottky’s result for
electrical shot noise, $\braket{\delta I^{2}}=2e\braket{I}\Delta v$ [27].

If we interpret the first term in equation 12 as the mass-flow shot noise and
the second term as the mass-flow thermal noise, then we can construct the
unitless ratio

$\frac{\braket{\delta Q^{2}_{therm}}}{\braket{\delta
Q^{2}_{shot}}}=\frac{2P_{0}}{\Delta P},$ (14)

to identify the conditions under which each noise source is expected to
dominate. From Figure 2 it is clear that when $\Delta P$ is sufficiently small
compared with $P_{0}$, the thermal fluctuations are dominant. This is expected
because the net flow, which gives rise to the shot noise, will be small
compared with the opposing flows resulting from $P_{0}$ and thus contribute
much less to the overall fluctuations. When $\Delta P$ is large compared with
$P_{0}$, the reverse is true and the shot noise is expected to dominate.

Figure 2:  The ratio of thermal noise to shot noise plotted against the
pressure differential, $\Delta P$, across a cylindrical nanopore. Various
baseline pressures, $P_{0}$, have been distinguished by line colour. The
displayed range of pressures have been somewhat arbitrarily chosen based on
typical conditions for free-molecular 4He gas flow in nanopores of $\leq$ 100
nm diameters, such as those used in [3]. We also reiterate that, strictly
speaking, this result is only valid in the free-molecular flow regime.

##  III Quantum Noise

In this section we consider a system in which quantum effects are taken into
account and the particles may no longer be treated independently. In this
case, the thermal noise and shot noise are interlinked, and there is an
inherent probability associated with transmission through the channel. We
theoretically calculate the noise for a two-terminal cylindrical fluid channel
although this approach can easily be generalised to a multi-channel system.
The shot noise can easily be adapted at zero temperature to account for the
transmission probability, $D$, by recognising that the Poissonian noise
distribution becomes binomial. We also assume $D$ to be energy independent for
the rest of this paper. In this case, the fluctuations are given by:

$\langle\delta Q^{2}\rangle=2m\langle Q\rangle(1-D)\Delta\nu,$ (15)

where $Q$ is now defined as the outgoing flow from the channel and hence,
implicitly absorbs a transmission probability factor. Note that the ingoing
and outgoing flow now differ by a factor $D$.

When $T>0$, we must consider thermal fluctuations in the incoming and outgoing
flow. Assuming thermal equilibrium, the occupation of states at the source and
drain are governed by the Fermi-Dirac distribution for fermions, and the Bose-
Einstein distribution for bosons. Both will be referred to as $f$ in their
respective contexts, with chemical potentials $\mu_{L}$ and $\mu_{R}$,
referring to the left and right side of the channel, where
$(\mu_{L}-\mu_{R})=m\Delta P/(\rho)$. Transport from the left to right is
defined as positive in quantized units of $G$ where [22]:

$G=\frac{Q}{(\mu_{L}-\mu_{R})\rho/m}=\frac{2m^{2}}{h\rho}D.$ (16)

We now adapt Landauer’s wave packet approach for electric circuits to mass
flow to find an expression for the noise. In this approach, transmission and
reflection are characterised by wave packets that are emitted from the source
and the drain at a constant rate and each contain one quantum mechanical state
[13]. Note that due to the Pauli exclusion principle, only two fermions can
occupy this state (opposite spins) whereas there is no restriction for bosons.
Packets are assumed to be emitted in phase and simultaneously from the source
and drain, such that a transmitted wave from the source is able to map onto
the same state as a reflected wave from the drain and vice versa. This method
allows us to consider particles moving against the pressure gradient (which is
more probable at low pressures). Using counting statistics we find:

$\displaystyle\langle\delta
Q^{2}\rangle=\frac{4m^{2}}{h}\Delta\nu\int_{0}^{\infty}dE$
$\displaystyle\\{D[f_{L}(1\mp f_{L})+f_{R}(1\mp f_{R})]$ (17)
$\displaystyle\pm D(1-D)(f_{L}-f_{R})^{2}\\},$

where $f_{L}$ and $f_{R}$ denote the distribution of particles at the source
and drain respectively. Note that the upper sign is for fermions and the
bottom is for bosons. The integral serves to include wave packets of all
energies. For bosons, the integral diverges, as is also seen for bosons in
electric circuits [24]. Hence, for the remainder of this section we will focus
on fermions. For fermions, we have the exact result:

$\displaystyle\langle\delta Q^{2}\rangle=$ $\displaystyle
4k_{B}TG\rho\Delta\nu D$ (18) $\displaystyle\;\;+2m\langle
Q\rangle(1-D)\Delta\nu\coth{\left(\frac{m\Delta P}{2kT\rho}\right)}.$

The first term is our classical thermal noise with a transmission factor,
whereas the second term is our classical shot noise with a complicated cutoff
factor. When $k_{B}T\rho\gg m\Delta P$, the hyperbolic cotangent is
approximated by the inverse of its argument and we recover our classical
thermal noise expression given by equation 1. Hence, the quantum noise
satisfies the fluctuation dissipation theorem as the equilibrium noise is
proportional to the conductance. When $k_{B}T\rho\ll m\Delta P$, the
hyperbolic cotangent is approximately one and the second term dominates, so we
recover our zero-temperature shot noise expression given by equation 15. This
is further shown in Figure 3.

Figure 3: The root mean square (RMS) mass flow fluctuations for fermions in a
two terminal system plotted against the log of temperature using arbitrary
parameters. The classical thermal noise and zero-temperature shot noise have
been plotted as dashed lines for comparison. Note that the intercepts with the
$Q_{RMS}$ axis occur at zero and the square root of equation 15.

##  IV Cumulants

The cumulants provide an alternative description of a random variable to the
probability distribution, where the first cumulant is the mean, the second is
the variance, the third describes the asymmetry or skewness of the
distribution, etc. For statistically independent variables, the cumulants are
additive, making them extremely useful when seeking to sum over multiple
independent sources of noise [28]. For this reason, we now provide a
reformulation of our results for the quantum noise in terms of a cumulant
generating function.

The $n$-th cumulant, $\kappa_{n}$, can be found by evaluating the $n$-th
derivative of the cumulant generating function (CGF), $K(t)$, at $t=0$ [28].
For fermions, the CGF takes the form:

$\displaystyle K(t)=$
$\displaystyle\frac{4}{h\Delta\nu}\int_{0}^{\infty}dE\ln\\{\left(e^{m\Delta\nu
t}-1\right)f_{L}D(1-f_{R})$ (19)
$\displaystyle\;\;\;\;\;\;\;\;+\left(e^{-m\Delta\nu
t}-1\right)f_{R}D(1-f_{L})+1\\}.$

The first two factors represent transport from the left to right or right to
left of the channel, respectively. Note that if we attempt to derive a similar
expression for bosons, all our integrals diverge and are hence, meaningless.
Focusing on fermions, the exponential mass factor is explained by recognising
that the $n$-th cumulant can be thought of as a pairing of $n$ independent
variables, where each variable contributes a mass factor. The frequency factor
results from the Fourier transform. Under the zero temperature limit, equation
19 becomes:

$\lim_{T\to 0}K(t)=\frac{4m}{\rho h\Delta\nu}\Delta
P\ln\left\\{1-D+De^{m\Delta\nu t}\right\\}.$ (20)

This expression is, as expected, reminiscent of the well-defined CGF of the
binomial distribution, $n\ln\\{1-p+pe^{t}\\}$, where $n$ is the number of
trials and $p$ is the probability [28]. We are also able to use equation 20 to
recover our zero-temperature shot noise, given by equation 15, from
$K^{\prime\prime}(0)$. On the other hand, if we calculate
$K^{\prime\prime}(0)$ when $T>0$, we recover the fermion variance given by
equation 18.

Utilizing our fermion CGF, we can further improve our understanding of the
distribution of mass flow by calculating the skewness (or third cumulant).

$\displaystyle\kappa_{3}=\frac{1}{2}mGk_{B}T(\Delta\nu)^{2}(1-D)\mathrm{csch}^{2}\left(\frac{m\Delta
P}{2k_{B}T\rho}\right)$
$\displaystyle\times\left\\{6D\rho\sinh{\left(\frac{m\Delta
P}{k_{B}T\rho}\right)}+(2D-1)\frac{m\Delta P}{k_{B}T}\cosh{\left(\frac{m\Delta
P}{k_{B}T\rho}\right)}\right.$ $\displaystyle\hskip
216.81pt\left.-(1+4D)\frac{m\Delta P}{k_{B}T}\right\\}.$ (21)

The expression above is complex, and no longer similar to our usual thermal
and shot noise. Under the limit $k_{B}T\rho\gg m\Delta P$, our expression for
$\kappa_{3}$ becomes $4(G/\Delta P)(\rho k_{B}T)^{2}(2-D)(1-D)(\Delta\nu)^{2}$
and in the opposite limit it becomes $m^{2}\langle
Q\rangle(1-D)(2D-1)(\Delta\nu)^{2}$. Under the classical limit, approximately
when $D=1$, we see that $\kappa_{3}=0$. Hence, departure from the classical
regime can be detected as $\kappa_{3}\neq 0$. This further shows the non-
Gaussian like nature of mass flow noise in the quantum regime, even under the
assumption of zero frequency.

##  V Discussion and Outlook

Our results demonstrate that flow in nanofluidic channels is subject to innate
mass-flow fluctuations, which vary depending on the conditions of the system
and may be a relevant consideration in the study and applications of
nanofluidics. In the previous sections we have derived classical and quantum
expressions for the white noise prevalent in mass flow in two-terminal fluid
channels. We have also determined the CGF for fermions and provided
expressions for $\kappa_{3}$. Notably, even though we have only assumed white
noise, our expressions should still be valid at low frequencies where
$k_{B}T\gg h\nu$. However, at high frequencies where $k_{B}T\approx h\nu$, one
would expect quantum effects and must replace the classical expression for
average energy $k_{B}T$ with its quantum version:
$h\nu/[\exp{(h\nu/k_{B}T)}-1]$. Similar substitutions have been made for the
electrical case of a quantum point contact, and have been found to agree with
experiments [29].

Generalizations to multi-channel systems is possible by summing over the
different transmission probabilities corresponding to each channel. The number
of channels in a typical pipe is $A/{\lambda_{F}}^{2}$, where $A$ is the
cross-section of the pipe and $\lambda_{F}$ is the Fermi wavelength. Hence,
with larger channels this will become more important. Furthermore, there are a
number of different sources of noise and types of noise that have yet to be
studied for mass flow. For instance, there may be extrinsic sources of noise
that are sensitive to boundary layer effects and system imperfections. These
can be studied with a specific system in mind. Additionally, noise inversely
proportional to frequency, traditionally called $1/f$ noise, is common in
electric circuitry and can also be studied here. Turbulent flow might also
impact our expressions for noise and should be studied in more detail.

Note that our calculations assume that the particles colliding with device
surfaces undergo specular rather than diffuse reflections. This assumption
might not hold for measurements performed in long nanochannels fabricated via
classic nanomachining approaches as the device surfaces in these cases are not
atomically smooth [30]. However, there is extensive interest in gas transport
in materials possessing atomically smooth surfaces [7], such as channels
formed from carbon nanotubes [31], graphene [8], MoS2 and h-BN [32]. These
materials lead to enhanced gas transport while not loosing their separation
selectivity [7].

It is also interesting to consider real world applications where both ends of
the channel might not be at thermal equilibrium. A temperature difference can
be incorporated into our method by adapting $f_{L}$ and $f_{R}$. However,
adding a more complex gradient is an interesting problem that requires more
theoretical consideration. In this case, cumulants could be used to sum up
many distributions and account for varying temperature differences.

##  VI Conclusion

We have theoretically calculated the classical and quantum noise, and quantum
CGF, for mass flow in a dilute fluid channel. The result is found to be
analogous with previous calculations for electrical noise and obeys the
fluctuation dissipation theorem. We have also used the CGF to determine the
third cumulant which, even at zero frequency, is non-zero. This shows the non-
Gaussian nature of mass-flow noise.

Although our results are mathematically similar to the electric case, the
adaptation of methods for electrical noise to mass flow is important. The
classical, free-molecular flow noise can be thought of as the sum of the full
shot noises associated with two opposing mass flow currents, which each
originate from the underlying Poisson statistics of the discrete mass
carriers. The quantum noise may be similarly understood, with the
acknowledgement that in this case the statistics of the particles must conform
with the corresponding quantum mechanical prescription, thus leading to
interparticle interactions which were not accounted for in the classical case.
The theoretical prescription presented here can be used to calculate the
theoretically minimum noise expected in dilute fluidic channels, or help
justify experimentally observed signal fluctuations.

We believe that these insights are important to understand smaller and more
complex fluid systems and thus may be of use in areas of nanofluidics where
noise considerations are of crucial importance, such as in certain sensing
technologies [33]. We hope that this work may serve as a stepping stone for
more elaborate noise sources to be considered in the future. For instance,
future work may include an extension of our classical expressions into other
gas flow regimes, such as into the transition and continuum regimes, since
strictly speaking we have confined our classical analysis to free-molecular
flow. The noise theory of dilute fluid channels should also be extended to
high frequencies, considering other sources of noise, and adapting the results
to fit more realistic systems with temperature gradients and numerous
channels.

## Acknowledgments

This work has been financially supported by NSERC (Canada), the New Frontier
in Research Fund (Canada), FRQNT (Québec) and the McGill Tomlinson fund.

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