Equilibrium and Functional
Periodicity: Fundamental
Long-Term Stability Conditions
for Design of Nature and
Engineered Systems
1
Nam P. Suh
Department of Mechanical Engineering
Massachusetts Institute of Technology
npsuh@mit.edu
Abstract
All matter – subatomic particles to human beings -- that exist in nature owe their existence to their
long-term stability. The complexity theory presented by Suh in recent years shows that for long-term
stability, both engineered systems and natural systems must be either at a stable equilibrium state or
must have a functional periodicity. In engineering and physics, the former, i.e., equilibrium, is well
known, but the concept of functional periodicity has been introduced only recently. There are many
different kinds of Functional Periodicity that govern natural and engineered systems. The
performance of engineered systems has been improved by introducing a functional periodicity by
design. Nature has evolved based on the stability provided by equilibrium states or by having
functional periodicity. Classical physics such as Newtonian mechanics and thermodynamics are
based on the assumption that equilibrium states exist. The basic postulates of modern physics such as
quantum mechanics and superstring theory are consistent with the proposed existence of functional
periodicity in natural systems. The particle/wave duality of matters that forms the basis of quantum
mechanics can be explained in terms of the stability criterion presented in this paper rather than in
terms of the p robability argument presented in the past.
Keywords
design, stability, nature, engineered systems, complexity, functional periodicity,
equilibrium
1
This paper is similar to the paper “On Functional Periodicity as the Basis for Long-term Stability of
Engineered and Natural Systems and Its Relationship to Physical Laws”, which has recently appeared
in Research in Engineering Design, Vol. 15, 2004
2
1. Introduction
Designs must be evaluated in terms of the functions they perform rather than in terms of
specific physical or biological arrangements and characteristics. Functions or functional
requirements (FRs) are satisfied by physical or biological things that are created by design
or by evolution. The characteristics of the physical things can be characterized using design
parameters (DPs). The purpose of engineering design is to satisfy the FRs by choosing a
correct set of DPs. Starting from the postulate that there must exist fundamental axioms that
characterize a good design, Axiomatic Design theory was advanced [Suh 1978],[Suh 1990],
which has been used to design many engineered systems, including large systems for
aerospace applications, software, hardware, and materials. Following Axiomatic Design
theory, a complexity theory was presented, which showed that there are four types of
complexity and that the introduction of Functional Periodicity can transform a system with
time-dependent combinatorial complexity, which eventually becomes chaotic, to a stable
system that has a time -dependent periodic complexity [Suh 1999],[Suh 2001]. This work
has led to the view that for long-term stability, Functional Periodicity must be present in
both natural or engineered systems [Suh 2004a],[Suh 2004b].
Nature and well-designed engineered systems share basic common characteristics. The
two basic characteristics of engineered systems and natural matter are: they must be stable
for a long-time and serve specific functions. Functions are “what the system is designed to
achieve.” These functions are satisfied by physical or biological entities. For example,
biological cells that are made up of various constituents achieve a set of functions.
Furthermore, the fact that these biological systems have survived billions of years of
evolution proves that they are stable. Similarly, an engineered system that performs a
desired set of functions must be stable throughout its lifetime. Unstable matters, if they ever
existed, would have disappeared after a transitory existence.
A complexity theory was advanced to characterize the nature of complexity by
examining how well the functions – not the physical entities -- of engineered systems and
natural systems are satisfied [Suh 1999],[Suh 2001],[Suh 2004a]. To achieve this goal,
complexity is defined as a measure of uncertainty in satisfying the functional requirements
(FRs) within the specified (or allowable) accuracy or variation. According to the
complexity theory, the complexity of cutting a rod to 1 m +/- 1 micron is greater than the
complexity of cutting to 1 +/- 0.1 m, because the physical implements that can be used to
satisfy the FR, i.e., cut the rod to within one micron, introduce a greater uncertainty in
achieving the FR.
The complexity theory states that there are four types of complexity: time-independent
real complexity, time-independent imaginary complexity, time -dependent combinatorial
complexity, and time-dependent periodic complexity. They are defined as follows [Suh
2001],[Suh 2004]:
(1)
Time -independent real complexity exists when a system cannot
satisfy its FRs within their specified design ranges. An example of real
complexity may be illustrated using conventional internal combustion
3
(IC) engines. One of the FRs of IC engines is the elimination of
hydrocarbon emission. However, the design of typical commercial IC
engines cannot satisfy this FR within the engine’s specified range.
Therefore, there is an uncertainty associated with satisfying the FR and
hence, the IC engine is a system with a finite time -independent real
complexity. Similarly, if an FR is to determine the exact position and
velocity of an electron around the hydrogen atom at a given instant, it
would be impossible to locate the electron within the desired range.
(According to Heisenberg’s principle if the uncertainty in position is 0.1
A, the corresponding momentum uncertainty is ~10-24 kgms -1 .) Therefore,
the real complexity associated with satisfying the FR is expected to be
large.
(2)
Time -independent imaginary complexity refers to the
complexity created because of our lack of knowledge about a system,
although the system itself is not complex. For example, a combination
lock is easy to open once we know the sequence of numbers forming the
combination, but in the absence of this information, the task of opening
the lock would appear to be complex. This uncertainty, which is not real
but associated with the lack of knowledge, is defined as imaginary
complexity.
(3)
Time -dependent combinatorial complexity exists when it
becomes increasingly more difficult to satisfy FRs with time or the
progression of a given set of events. For example, suppose that we have a
snowstorm around the Detroit area so that airplanes cannot land and take
off during the day. Then the airplanes for Boston and other cities cannot
take off from Detroit. As time goes on, the flights from Boston to other
cities will be disrupted since there will not be enough airplanes to
dispatch them according to the original schedule. Therefore, the airlines
will not be able to satisfy their FRs of sending airplanes on schedule. The
situation is going to get worse as time passes and the snowstorm
continues. This is an example of time-dependent combinatorial
complexity.
(4)
Time -dependent periodic complexity is the type of complexity
that exists within a period when a set of FRs repeats itself. For example,
in the case of the airline-scheduling problem, the airplanes may resume a
regular schedule the next morning at 6 a.m. because airlines can
reinitialize the system by moving planes during the night (when not many
planes fly) to resume their original schedule. Since the airline schedule is
periodic each day, all of the uncertainties introduced during the course of
a day terminate at the end of a 24-hour cycle, and hence this
combinatorial complexity does not extend into the following day. That is,
each day, the schedule starts all over again, i.e., it is periodic and thus
uncertainties created during the prior period are irrelevant. However,
4
during a given period there are uncertainties.
This type of
time-dependent complexity is defined as time-dependent periodic
complexity.
2. Functional Periodicity
If a system has time-dependent combinatorial complexity, the system will become
“chaotic” or “uncontrollable” and eventually fail. For a system to survive for a long time, it
must have a time -dependent periodic complexity. The complexity of a system with
time-dependent combinatorial complexity can be reduced if it can be transformed into a
system with a periodic complexity. This can be done when the system has a Functional
Periodicity, i.e., a set of FRs of the system that repeat, which can be re-initialized at the
beginning of each period. The functional period is determined by a repeating set of
functions. Reinitialization involves knowing the state of the FRs that repeat periodically at
the beginning of each period, which may or may not be the same at the beginning of each
period. Knowing the initial state, the system can be made stable for the subsequent period.
There are many different kinds of Functional Periodicity: temporal, geometric,
biological, electrical, material, information processing, chemical, thermal, and
manufacturing. Biological cells provide an example of a natural system with
time-dependent periodic complexity. Biological functional periodicity is associated with
the cell cycle, i.e., a cell divides into two cells cyclically when all the functions that the cell
must satisfy have been completed within a functional period. An example of chemical
functional periodicity is the periodic table of the chemical elements. Wire rope represents
an example of geometric functional periodicity and the undulated surface for low friction at
the interface between two sliding surfaces is another example [Suh 1986]. The Carnot cycle
of an engine is an example of thermal functional periodicity. Language and music have
different forms of functional periodicity, which enable the processing of information. Table
1 provides a list of examples:
Table 1. Examples of Functional Periodicity in Engineered Systems and Nature [Suh 2004a],[Suh
2004b]
Examples of Functional
Periodicity
Examples in Nature
Examples in Engineered
Systems
Temporal periodicity
Planetary system, solar
calendar
Geometric periodicity
Crystalline solids, surfaces
of certain leaves
Biological periodicity
Cell cycle, life -death cycle,
Airline/train schedules,
resonant mechanical
systems, computers,
pendulum
Undulated surface for low
friction, “woven” electric
connectors, optical
diffraction gratings
Fermentation processes such
5
Manufacturing/ processing
periodicity
Chemical periodicity
Thermal periodicity
Information processing
periodicity
Electrical periodicity
Circadian periodicity
Material periodicity
plants, grains,
Biological systems
Periodic table of chemical
elements/atoms
Temperature of Earth,
Language
Thunder storm
Living beings,
Wavy nature of matter,
atomic structure,
crystallinity
as wine making
Scheduling a clustered
manufacturing system,
Polymers
Heat cycles (e.g., Carnot
cycle),
Re-initialization of software
systems, music
LCD, alternating current,
Light sensitive sensors
Fabric, wire drawing,
microcellular plastics
The list of functional periodicity presented in Table 1 is not an exhaustive list.
What is evident from the table is that for long-term stability and survival, a system – both
natural and engineered -- must be at an equilibrium state or have a Functional Periodicity.
The long term of an engineered system is its life cycle and for a natural matter, it is the life
cycle of living beings or the elements, which change depending on how well the stability
can be maintained.
3. Postulate on Long-Term Stability of Natural Systems or Engineered
Systems
Based on the complexity theory, the following postulate was proposed [Suh 2004a],[Suh
2004b].
For long-term stability of natural and engineered systems, the system
must be either in an equilibrium state with its surrounding or have a
functional periodicity. Systems without stability because of the lack of
functional periodicity have a transitory existence or are in a chaotic state
and eventually disappear or mutate into another system or matter which
is either in equilibrium or it is periodic.
3.1. Equilibrium State for Long-Term Stability
If an engineered system or a natural matter is at an equilibrium state with its surroundings, it
is going to be stable until its equilibrium state is disturbed. If it is at a meta-stable
equilibrium state, it will go into an equilibrium state or unstable equilibrium state or another
meta-stable state, if the system is disturbed by the application of external energy that is
larger than energy barriers surrounding the local minimum. Classical physics such as
6
Newtonian mechanics and thermodynamics have been developed based on the postulate
that equilibrium states exist. In Newtonian mechanics, forces are assumed to be in
equilibrium: the first and third Newtonian laws are based on static stable equilibrium and
the second law is based on a dynamic stable equilibrium relationship. In thermodynamics,
the equilibrium condition forms the basis for both the first and second law of
thermodynamics [Hatsopoulos 1981]. Nature continues to seek the lowest energy state so as
to be in stable equilibrium.
3.2. Functional Periodicity for Long-Term Stability
Some natural and engineered systems, or natural matter, are not at an equilibrium state at all
times. However, they are stable if they possess a functional periodicity. That is, an absolute
static equilibrium is not a requirement for long-term stability and survival if and only if the
system can renew itself through functional periodicity. Modern physics such as quantum
mechanics assumes a sinusoidal probability function of finding an electron within a volume
about the given position at a given time. Such an assumption is equivalent to assuming the
existence of a functional periodicity in natural matter such as atoms, electrons, and
subatomic particles.
a. Quantum States
Physicists generally accept that quantum mechanics provides the best description of matter
and energy at tiny scale. Numerous experiments have confirmed the basic quantum
mechanical postulate which states that basic particles such as electrons have the
characteristics of both a particle and a wave. Moreover, according to the accepted view of
quantum mechanics, the position and velocity of the particle are not deterministic; they
must rather be treated in terms of probability of the particle occupying a region in position
and momentum spaces. Furthermore, Heisenberg’s uncertainty principle states that the
accuracy of determining position and velocity of a particle simultaneously is bounded, so
that if one sought infinite accuracy in one variable, say the position, then the other variable
(velocity) would become indeterminable. Schroedinger’s equation was constructed by
seeking a wave function that represents the complex amplitude of displacement as a
function of time and space, and thereby satisfies the conservation of energy and the
quantum nature of energy and matter. The square of the wave function represents the
probability of finding the particle in a given region. Although the predictions made by
quantum mechanics are the basis for modern electronic devices and modern physics, for
many people, the particle/wave duality has been a difficult concept to understand [Giancoli
2000],[Merzbacher 1998]. The generally accepted “Copenhagen” interpretation applies the
duality principle in a statistical sense, i.e. when large numbers of particles are involved (or a
large number of observations is made on a particle). Still unsolved paradoxes such as the
famous “Schroedinger cat” dilemma illustrate the difficulty of wave/particle duality in the
limit of a single particle.
If the wave function is treated in quantum mechanics as representing the functional
periodicity rather than the probability of finding an electron in a given volume about a
position at a given time, the mathematical form of the Schroedinger equation should still
7
hold. However, the physical interpretation of, for example, the motion of an electron would
be quite different. In terms of functional periodicity, electrons are simply pursuing a
functional periodic motion to be stable 2 . The functional period is likely to depend on the
size of the nucleus and the number of electrons present.
Based on the postulate presented in this paper, the particle/wave duality of electrons, for
example, can be rationalized. Electrons will be stable if they are at equilibrium states. When
electrons are at an equilibrium state, they may be identified as a particle. Even when
electrons are not at an equilibrium state such as when they are moving around a nucleus of
an atom, they can be stable if they have a functional periodicity. When electrons seek a
stable state by having a functional periodicity, they will appear to have the characteristics of
a wave with a functional periodicity. This view is consistent with the quantum mechanical
view, which assumes that the electrons in a crystal exist in the form of waves, obeying the
Schroedinger equation. When a stable electron interacts with other particles, it may begin as
a particle and then acquire a functional periodicity, which may be the basis for Young’s
double slit experiment with electrons.
b. String Theory
Superstring theory is the latest theory that is believed to unite all fundamental laws of nature.
String theory can describe the behavior of elementary particles by representing them as
vibrating strings. The length of the string is varied to represent various particles and the
strings can interact. Superstring theory can derive Einstein’s equation by demanding that
the string move self-consistently in space-time [Kaku 1994]. An interesting aspect of the
superstring theory is that it involves functional periodicity, indicating that the concept of
functional periodicity may be fundamental for stability of any existing matter, including
materials and living beings. “This particular use of strings” may be one of many different
ways of representing a functional periodicity of matter.
3.3. Equilibrium State versus Functional Periodicity
Based on complexity theory, it was argued that for long-term stability, natural matter or
engineered systems must be at an equilibrium state or have a functional periodicity. In some
systems with functional periodicity, there may not be a sharp demarcation between an
equilibrium state and functional periodicity because of limitations to knowledge or
technology.
A system that is stable as a result of functional periodicity could appear to an observer as
operating at an equilibrium state if the period were short relative to the observation period.
Therefore, the laws of Newtonian mechanics and thermodynamics may be applicable even
when the underlying system, such as that described by superstring theory, possesses a
functional periodicity with a functional period much shorter than the time over which the
system changes from one equilibrium state to another equilibrium state.
2
Functional periodic motion does not necessarily imply that electrons are spinning around a
nucleus.
8
4. Conclusions
1. For the long-term stability of a system, it must be at a stable equilibrium state or must
have a functional periodicity.
2. The laws of nature are consistent with the long-term stability criterion.
3. Natural systems satisfy the long-term stability criterion, since they have evolved and
survived.
4. Through the application of the stability criterion, superior engineered systems has been
designated and created.
Acknowledgement
The author is grateful to his colleagues at MIT, Professors Gang Chen, Jung-Hoon Chun,
Bora Mikic and George Barbastathis, for their comments and suggestions.
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