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\layout Title


\size large 
\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 
<mpf@cise.ufl.edu>
\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