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\layout Title
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\noun on
UF Reversible Computing Project Memo #M14
\size default
\noun default
\newline
The Adiabatic Principle
\newline
\size large
A generalized derivation
\layout Author
Michael Frank
\newline
UF CISE Dept.
\newline
\family typewriter
\size normal
\layout Date
Started 8/13/01, draft of 8/15/01.
\layout Standard
In this research memo, we consider the origins of what we consider to be
the general principle of adiabatic systems, namely that the total energy
dissipation
\begin_inset Formula \( E_{diss} \)
\end_inset
of an adiabatic process asymptotically scales down in proportion to the
speed at which that process takes place (that is, in inverse proportion
to the total time over which the process is carried out,
\begin_inset Formula \( E_{diss}\asymp 1/t_{tot} \)
\end_inset
).
We wish to do this in a very generalized way that can be applied to any
sort of adiabatic physical mechanism, whether it be electronic, mechanical,
quantum-mechanical,
\emph on
etc.
\emph default
This will help us to justify the sort of technology-independent models of
adiabatic/reversible computing which much of our work centers on.
\layout Standard
We guess that probably some similar derivation has been published long ago
somewhere in the phyics literature, but we deemed it easier to reinvent
this simple result than to locate its first appearance.
However, eventually, we should find the appropriate credit.
The idea for the form of this general analysis was inspired by a reading
of the analysis of the special case of electrical resistance in the textbook
\emph on
University Physics
\emph default
, 6th ed., by Sears, Zemansky, and Young (
\latex latex
{
\backslash
S}
\latex default
28-9, pp.
556-557).
\layout Standard
As a general model, let us consider the physical system in question as being
characterized at any moment by a generalized coordinate
\begin_inset Formula \( x \)
\end_inset
giving the system's position in configuration space.
For any system composed of a fixed number of particles,
\begin_inset Formula \( x \)
\end_inset
might simply be the vector of all particles'
\begin_inset Formula \( x,y,z \)
\end_inset
coordinates in 3-D space.
In an electronic system, the
\begin_inset Quotes eld
\end_inset
position
\begin_inset Quotes erd
\end_inset
could characterize the voltage states of different circuit nodes.
For quantum processes involving particle creation and destruction,
\begin_inset Formula \( x \)
\end_inset
might characterize the occupancy numbers of various particle states.
But the general conception is that
\begin_inset Formula \( x \)
\end_inset
characterizes the instantaneous state of the system, in whatever representation
framework is important.
\layout Standard
Actually, in classical mechanics, spatial position alone does not completely
describe a system; momentum coodinates are also necessary.
Quantum mechanics loosens this restriction, however, since momentum can
be represented as an emergent phenomenon, arising from the wavelength of
particle wave-packets in a Schr
\latex latex
{
\backslash
"o}
\latex default
dinger wavefunction that ranges over the system's
\emph on
position
\emph default
configuration space alone.
In any event, explicit momentum coordinates could be included if needed,
but we do not worry about doing so in the present analysis.
\layout Standard
Now, consider a process in which a system migrates along some (possibly
complexly-shaped) desired path
\begin_inset Formula \( P \)
\end_inset
through its configuration space.
Note that
\begin_inset Formula \( P \)
\end_inset
is the
\emph on
desired
\emph default
path; the system's real path may merely closely approximate
\begin_inset Formula \( P, \)
\end_inset
due to small unwanted interactions that disturb its trajectory.
Let the total length of
\begin_inset Formula \( P \)
\end_inset
(the path integral of segments
\begin_inset Formula \( ds \)
\end_inset
) be
\begin_inset Formula \( \ell \)
\end_inset
.
Suppose the system makes progress along the desired path at some roughly
constant velocity magnitude
\begin_inset Formula \( v \)
\end_inset
(despite any shifts in its direction of motion).
Call this its
\emph on
path velocity
\emph default
.
Then the total time for the trip is
\begin_inset Formula \( t_{tot}=\ell /v. \)
\end_inset
Given an
\emph on
effective mass
\emph default
\begin_inset Formula \( m \)
\end_inset
of the system, its approximate kinetic energy along path
\begin_inset Formula \( P \)
\end_inset
at any moment during the trip is
\begin_inset Formula \( E_{k}=\frac{1}{2}mv^{2} \)
\end_inset
(ignoring relativistic corrections).
\layout Standard
Suppose, now, that the system is subject to very frequent small
\begin_inset Quotes eld
\end_inset
frictional
\begin_inset Quotes erd
\end_inset
interactions with its environment, and that the nature of these interactions
is that each of them saps some fraction
\begin_inset Formula \( f \)
\end_inset
of the system's kinetic energy along the path
\begin_inset Formula \( P \)
\end_inset
(converting this energy to heat).
For example, in a flow of current through a resistor, the drifting electrons,
in their rapid thermal motions, frequently scatter off of atoms of the
material.
Each time this happens, some fraction of the electron's drift kinetic energy
(its extra energy in the direction of current flow, which itself is a small
fraction of the total drift kinetic energy of the entire current) is thermalize
d.
\layout Standard
Similarly, for an object falling in a viscous fluid, each collision between
the object and an atom of the fluid is an elastic collision which saps
a tiny fraction of the object's kinetic energy.
The lost kinetic energy is assumed, in our context, to then be replished
by an external force,
\emph on
i.e.
\emph default
, some potential energy bias favoring the system's forward motion along
the desired path.
Let
\begin_inset Formula \( r_{int}=1/t_{f} \)
\end_inset
be the average rate at which these interactions occur, where where
\begin_inset Formula \( t_{f} \)
\end_inset
is the
\emph on
mean free time
\emph default
of the system between interactions.
\layout Standard
A crucial assumption at this point is that
\begin_inset Formula \( r_{int} \)
\end_inset
is
\emph on
independent
\emph default
of the system's path velocity, at least for small
\begin_inset Formula \( v. \)
\end_inset
That is, the interactions occur at some base rate, regardless of whether
the system is moving or not, and this rate not increased significantly
so long as
\begin_inset Formula \( v \)
\end_inset
is reasonably small.
(For sufficiently high
\begin_inset Formula \( v, \)
\end_inset
the rate of interactions may vary, but that is OK since we are only interested
here in the behavior as
\begin_inset Formula \( v\rightarrow 0. \)
\end_inset
)
\layout Standard
This independence assumption is true in scenarios such as the following:
(a) Electrons in a resistor current, where electron thermal velocity is
much greater than drift velocity; electrons frequently encounter atoms
even when drift velocity goes to zero.
(b) Object falling slowly through a viscous fluid; object encounters atoms
frequently due to thermal motions in fluid, even when object is falling
very slowly.
\layout Standard
Given
\begin_inset Formula \( r_{int}, \)
\end_inset
the expected number of these interactions that take place during the entire
process is
\begin_inset Formula \( n_{int}=r_{int}\cdot t_{tot}. \)
\end_inset
Each time the interaction happens, an amount
\begin_inset Formula \( E_{int}=f\cdot E_{k} \)
\end_inset
of the system's kinetic energy is lost (but, we assume, replenished by
some source so that velocity remains roughly constant).
Therefore, the total loss over the entire process is
\begin_inset Formula \( E_{diss}=n_{int}\cdot E_{int}. \)
\end_inset
Expanding the definitions, we see that
\begin_inset Formula \begin{eqnarray*}
E_{diss} & = & n_{int}\cdot E_{int}\\
& = & r_{int}\, t_{tot}\cdot f\, E_{k}\\
& = & t_{f}^{-1}t_{tot}\cdot f\, \frac{1}{2}m\, \left( \frac{\ell }{t_{tot}}\right) ^{2}\\
& = & \frac{1}{2}\cdot \frac{f\, m\, \ell ^{2}}{t_{f}\, t_{tot}}.
\end{eqnarray*}
\end_inset
Already, we can see that the energy dissipated is inversely proportional
to the time
\begin_inset Formula \( t_{tot} \)
\end_inset
for the whole process, the key adiabatic principle which we wished to derive.
Note that if the number of dissipative events was independent of time,
the energy dissipated would scale down with kinetic energy, as
\begin_inset Formula \( t_{tot}^{-2} \)
\end_inset
, but the increasing number of dissipative events with time raises this
to order
\begin_inset Formula \( t_{tot}^{-1}. \)
\end_inset
Note also that the total dissipation is inversely proportional to the mean
free time
\begin_inset Formula \( t_{f} \)
\end_inset
as well, so if we can better isolate the system from interactions (
\emph on
e.g.
\emph default
mechanical motion in vacuum, or electrical current in superconductors),
the dissipation will decrease, as expected.
\layout Standard
This attains the basic goal of this memo, but let us now explore some other
convenient ways to express this dissipation relation, so as to better understan
d it.
\layout Standard
Given
\begin_inset Formula \( f \)
\end_inset
and
\begin_inset Formula \( r_{int} \)
\end_inset
, we can now characterize an
\emph on
energy decay coefficient
\emph default
\begin_inset Formula \( d_{E}=f\cdot r_{int} \)
\end_inset
which characterizes how rapidly a system's kinetic energy tends to decay
away over time.
That is, if
\begin_inset Formula \( e_{diss}(t) \)
\end_inset
is the total energy lost to friction so far at time
\begin_inset Formula \( t \)
\end_inset
, its time derivative
\begin_inset Formula \( \dot{e}_{diss}=d_{E}E_{k} \)
\end_inset
(where the decay is modeled as continuous due to the large number of small
cumulative energy losses).
Note that the units of
\begin_inset Formula \( d_{E} \)
\end_inset
are
\begin_inset Formula \( time^{-1} \)
\end_inset
, and so its inverse
\begin_inset Formula \( t_{d}=d_{E}^{-1} \)
\end_inset
is a
\emph on
decay time constant
\emph default
.
If the kinetic energy lost to friction is not replenished, an initial kinetic
energy of
\begin_inset Formula \( E_{k0}=E_{k}(t=t_{0}) \)
\end_inset
would decay exponentially for
\begin_inset Formula \( t\geq t_{0} \)
\end_inset
according to
\begin_inset Formula \( E_{k}(t)=E_{k0}\cdot e^{-t/t_{d}} \)
\end_inset
, the result of solving the differential equation describing this decay
process.
\layout Standard
Anyway, expressed in terms of
\begin_inset Formula \( d_{E}, \)
\end_inset
we have
\begin_inset Formula \( E_{diss}=\frac{1}{2}d_{E}\, m\, x^{2}/t. \)
\end_inset
\layout Standard
The product of
\begin_inset Formula \( d_{E} \)
\end_inset
and the effective system mass
\begin_inset Formula \( m \)
\end_inset
we will call the
\emph on
effective viscosity
\emph default
\begin_inset Formula \( \eta =m\cdot d_{E} \)
\end_inset
because it has the same physical units (
\begin_inset Formula \( mass/time \)
\end_inset
) as ordinary fluid viscosity, and indeed plays the same role in this more
general setting.
The frictional effect of energy loss can be characterized by a force aligned
opposite the path direction, with magnitude scaling in proportion to viscosity
and velocity, just as frictional forces scale when dealing with fluid viscosity.
With our definitions, the exact relation is
\begin_inset Formula \( F_{fric}=\frac{1}{2}\eta v \)
\end_inset
, .
In terms of
\begin_inset Formula \( F_{fric} \)
\end_inset
as we have so defined it, the equation for
\begin_inset Formula \( E_{diss} \)
\end_inset
then simplifies to:
\begin_inset Formula \[
E_{diss}=F_{fric}\cdot \ell ,\]
\end_inset
\emph on
i.e.
\emph default
, total energy lost equals the frictional force times the displacement distance.
\layout Standard
Note that in order for
\begin_inset Formula \( E_{diss} \)
\end_inset
to truly represent the total energy dissipation of the process, a key assumptio
n was that only
\emph on
kinetic
\emph default
energy (but not potential energy or rest mass-energy) was lost as the system
moved along the desired trajectory.
But more generally, a system may experience some constant minimum rate
of loss of its
\emph on
total
\emph default
energy, not just kinetic energy, away from the desired trajectory, due
to effects such as thermal disturbance of the trajectory, wavefunction
spreading, or quantum tunneling (all of which can be seen to be different
variations on one general phenomenon).
\layout Standard
A later memo will discuss the implications of this other type of loss on
the limits of the adiabatic principle.
\the_end