On the Monotonous Evolutions: Model and Applications
ILYAS HAOUAM
Laboratoire de Physique Mathématique et de Physique Subatomique (LPMPS),
University of Frères Mentouri,
Constantine 25000,
ALGERIA
Abstract: This article discusses and explains some phenomena that monotonously develop and presents a
mathematical model that controls and describes these monotonous evolutions. Furthermore, this model is linked
to some applications in several different fields of physics. Knowing that this model consists of a set of
differential equations and their solutions with some mathematical properties.
Key-Words: Monotony; Linear differential equation; Evolution; Exponential increase; Radioactive decay; RC
circuit; Progress of chemical reaction X
Received: June 16, 2022. Revised: February 12, 2023. Accepted: March 3, 2023. Published: March 15, 2023.
1 Introduction
There are many phenomena that develop in one
direction and this is what we call the monotony of
evolution. The evolution of these phenomena is
described and governed by differential equations
and their solutions give us an accurate description of
these developments. The most known developments
are exponential. Here in this work, we try to shed
light on the most important phenomena that are
increasing exponentially in one direction and those
that are decreasing exponentially in one direction.
Such phenomena are common. We also give some
of the mathematical properties of these phenomena.
A differential equation is an equation that relates
one or more unknown variables (functions) and their
derivatives, [1], [2]. In applications, the variables
generally represent physical quantities, the
derivatives represent their rates of change and the
differential equation defines the evolution.
Differential equations can be classified according to
their order, which is determined by the highest
derivative that appears in the equation. Historically,
the concept of the differential equation first came
into existence with the invention of calculus in the
late 17th century by the mathematicians Isaac
Newton and Gottfried Wilhelm Leibniz
independently of each other, [3], [4], [5]. Thereafter,
the development of calculus and its uses have
continued to the present day. Differential equations
play a prominent role in many disciplines including
physics, engineering, biology, and even economics
and sociology. The beauty in the study of
differential equations is how solvable explicit
formulas are; as well, many properties of its
solutions may be determined without calculating
them intricately. Differential equations play an
important role in modelling virtually every
biological process or physical technical, from
celestial motion to an interaction between particles.
In general, several techniques and approaches have
been developed to solve differential equations,
including undetermined coefficients and separation
of variables methods, and numerical approaches
such as Runge-Kutta and Euler's methods, [2].
However, in particular, differential equations such
as those used to solve physical problems and
describe complicated phenomena may not
necessarily be directly solvable. Instead, solutions
can be approximated using numerical methods.
Among the most important physicists who were
credited with the application of differential
equations in physics, were the following scientists:
Jean le Rond d'Alembert, Leonhard Euler, Daniel
Bernoulli, Joseph-Louis Lagrange, and Joseph
Fourier. Now, as practical examples in physics
about differential equations since their discovery,
we mention: (i) the problem of a vibrating string
such as that of a musical instrument, [6]. (ii)
D’Alembert as well, discovered the one-dimensional
wave equation in 1746, and later within ten years
the 3D wave equation was discovered by [7]. (iii)
The Euler–Lagrange equations, [8], [9], which are a
system of second-order ordinary differential
equations, were developed in 1750 by Euler and
Lagrange in connection with their studies of the
tautochrone problem, [10]. This later led to the
development of Lagrange's method and applied it to
mechanics, which yielded the formulation of
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Lagrangian mechanics. (iv) The work on the heat
flow in the analytic theory of heat by Fourier in
1822, [11], which was about heat equation for
conductive diffusion of heat. This last partial
differential equation has become an essential
equation of mathematical physics that is taught to
students today.
There are several monotonous developments in
nature, and we will highlight some of these
phenomena later. Monotonous development refers
to a process or behavior that consistently increases
or decreases without any significant deviation or
reversal. The monotony here is merely an
exponential increase or decrease in one direction. It
is not an increase, then a decrease, or a combination
of the two. The mathematical model that controls
and describes these developments will be presented
and linked to some applications in several different
fields, including nuclear transformations, electricity,
chemical reactions in aqueous solutions, and
Newtonian mechanics.
In fact, there are many other examples,
including: population growth, which is exponential
growth, and light intensity in optics where the
intensity of light decreases exponentially as it
travels through a medium (Beer-Lambert law). Also,
the decay of the magnetic field because when a
magnetic field is removed or disrupted, its strength
decreases over time following an exponential decay
curve. Such as in MRI machines, where the decay of
magnetic fields is caused by the relaxation
processes, which is why MRI machines must be
calibrated regularly. Heat transfer can be also an
example of monotonous evolution in certain cases.
Monotonic evolution refers to a continuous change
in a system over time that either always increases or
always decreases, without any fluctuations or
reversals. Heat transfer can exhibit monotonous
evolution if the transfer of heat energy is always in
the same direction and does not fluctuate. For
example, when a hot object is placed in contact with
a cooler one, heat energy will transfer from the
hotter object to the cooler one until both objects
reach thermal equilibrium. This transfer of heat
energy is always in the same direction, from hot to
cold, and does not fluctuate or reverse. As a result,
the change in temperature over time between the hot
object and the cool one will follow a monotonous
trend, with the temperature of the hot object
decreasing and the temperature of the cool one
increasing until they reach equilibrium.
This paper is outlined as follows. In Section 2,
the monotonous evolution model is presented. In
Section 3, some applications and illustrations are
proposed and presented. In sub-section 3.1, we
study radioactive decay. Then in sub-sections 3.2 &
3.3, as an illustration of the proposed model in both
electricity and mechanics, we present a resistance-
capacitor circuit and the motion of a real fall of a
solid object in the air, respectively. Sub-section 4 is
devoted to the application in chemistry, in which the
proposed model is associated with the way
quantities of the substance are formed or reacted
during chemical reactions in an aqueous solution.
We present our conclusion in Section 4.
2 Monotonous Evolution Model
The pattern of monotonous developments we
present in this work is of two types: one is an
exponential increase, and the other one is an
exponential decrease, both in one direction. We start
with the following differential equation
(1)
where is the physical variable (function in time);
is the maximum value of and is the time
constant, which will be interpreted as the image of
the crossing of the tangent of the function curve
at the zero point with the asymptote . It
has the dimension of time and its unit is the second
(s). Note that the resolution of equation (1) yields
Figure 1 illustrates the given function in equation
(2) as follows:
Fig. 1: Plot of for and .
We say that equations (1) and (2) constitute a
system of evolution of the exponential increase in
one direction. From the graph, we note that there are
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two phases during the evolution of the proposed
system, one of them is transitional, in which the
system develops exponentially, while in the second
phase, the system is constant at a specific value.
Now for the completely decreasing evolutions in
one direction, we have the following differential
equation
thus the resolution of equation (3) gives
Figure 2 illustrates the given function in equation
(4) as follows:
Fig. 2: Plot of for and .
We may add certain characteristics such as:
1. The constant: it is the reciprocal of (in),
sometimes referred to as simply increasing or
decreasing rate. Thus, we have the expression
2. The half-life : it is the time taken for the
increasing or decreasing of a given amount of
to half of its initial value , which means it is
the time that correspond to . So from
equation (2) or equations (4), we have
It is easy to deduce that for , we
have
.
Last but not least, it can be concluded that: (i)
is defined as twenty percent of the time of
reaching the permanent system. It can be used to
indicate how rapidly an exponential function
increase or decreases. Physically, in a decreasing
system, represents the elapsed time required for
the system response to decay to zero, it is
of .
In an increasing system, is the time for the
system's step response to reach
of . (ii)
The systems of monotonous exponential increase (or
even decrease) that we study consist of a transitional
regime and a permanent one. The beginning of the
permanent regime is always at .
3 Applications
3.1 The Radioactive Decay
As known about nuclear transformations, there are
two types of transformations: Stimulated (non-
spontaneous) nuclear transformations, namely
nuclear fission, and thermonuclear fusion; and
spontaneous nuclear transformations, namely
radioactivity. In this part of the article, we will focus
on spontaneous nuclear reactions known as
radioactivity.
Radioactivity, [12], [13], [14], is the
phenomenon of the spontaneous disintegration of
unstable (i.e., radioactive) atomic nuclei, so in the
process of decay, one or more types of energetic
ionizing radiation (particles or electromagnetic
radiation) are emitted. However, in the random
process of radioactive decay, a nucleus loses energy
by emitting radiation, where the nature of the
produced radiation depends first on if the unstable
nucleus is heavy or not, in the case of heavy nuclei
the produced radiation Is usually in the form of
particles (Helium nuclei
). If the emitted
radiation is or particles, namely electrons
and positrons
, respectively. This is done
according to the number of neutrons and protons
present inside the unstable nucleus, where if the
number of protons is greater, the radioactive
transformation is according to thedecay pattern
i.e.,
, while if the number of neutrons
is greater, the radioactive transformation pattern is
according to i.e.,
. There are also
gamma rays
(high-energy photons), which
automatically accompanies the previous emissions,
and it occurs at the energy level, rather than at the
particle level of the nucleus. Knowing that the
nucleu’' energy reduces, making it more stable. In
all decay processes mass, charge, and lepton number
are conserved. Once -decay occurs the radioactive
nucleus changes into a different more stable one,
with two fewer protons and two fewer neutrons, and
particle is emitted. On the other hand, when or
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-decay occurs, the number of nucleons in the
nucleus remains the same, so the mass number does
not change, but as a neutron is converted into a
proton the atomic number increases by one, and as a
proton is converted into a neutron the atomic
number decreases by one.
Radioactive decay is a random process, which
means that it is impossible to predict when a
particular radioactive nucleus will decay. It is also
spontaneous; you cannot cause or influence the
decay. However, with large numbers of nuclei, it is
possible to predict statistically the behavior of the
entire group through radioactive decay. So le’'s
consider the case of a nuclide that decays into
another one through some radioactivity .
e.g., -decay, which is the emission of positrons
or electrons or -decay or -decay. Statistical
study of the behavior of several unstable nuclei
together yields the suggested model. The decay of
an unstable nucleus is completely random in time so
according to quantum theory, it is not possible to
predict when a particular nucleus will decay,
regardless of how long the nucleus has existed.
Thus, for this reason, it appears in the equation later
on the so-called probability of disintegration per
second. Therefore, given a sample of a particular
radioisotope, the number of decay events
expected to occur in a small interval of time and
is proportional to the number of radioactive present
nuclei , which means and .
Thus, and this yields
(negative because the number of nuclei decreases
over time), which means that the rate of change of
radioactive nuclei is proportional to the number of
remaining radioactive nuclei . Therefore, we have
the differential equation above is similar to equation
(3) presented in the monotonous evolution model,
and as well its solutions are
which is the radioactive decay law, where is
the quantity of remaining radioactive (undecayed)
nuclei at time , is the initial quantity (at time
), and the constant is called the decay
constant or disintegration constant, or even
transformation constant. So is the activity
of radioactive nuclei, which is the number of decays
per second. Its unit is the Becquerel () in SI,
where , also the Curie
() and Rutherford () in a non-SI, where
and , [15]. Since the
activity is proportional to the number of radioactive
atoms, it decreases exponentially with time as well
The decay rate of a radioactive substance is
characterized by the following time-independent
parameters:
1. The half-life of a particular species of
nuclei is the time that it would take for the
number of nuclei in a given sample to decay
to halve. The larger the half-life of a nuclei, the
less likely it is to decay in a given time.
2. Mean lifetime , which is the average lifetime
of a radioactive particle before decay.
3. Decay constant , which is the reciprocal of the
mean lifetime (in), sometimes referred to as
simply decay rate.
4. The equation that combines the aforementioned
properties is
Knowing that depends only on of the
nuclei. It is always the same; the amount of time for
the number of nuclei to decrease from 40 million to
20 million is the same amount of time as it takes the
number to decrease from 4.8 to 2.4. The initial
number does not have an effect.
Figure 3 illustrating the decay of remaining
radioactive nuclei is as follows:
Fig. 3: The graph shows the exponential decay of a
radioactive element. It shows three different
exponential decays, each with a different decay
constant i.e., 25, 5, 1, 1/5, and 1/25.
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The remaining radioactive nuclei undergo
exponential decay, where larger decay constants
make the quantity vanish much more rapidly.
Now, if we have , where is the
transformed nuclei (more stable), thus using
equations (8), we have
We also note that for any quantity related to the
number of remaining radioactive nuclei , the same
previous laws of a differential equation and its
solutions are applied to it. For example, we derive
the law of radiative decrease in terms of remaining
radioactive mass . We have
, where , , are the amount of substance,
Avogadro number, and the atomic molar mass,
respectively. So by using equation (8), we obtain
where is the initial mass of radioactive nuclei.
3.2 Resistor–Capacitor Circuit (RC Circuit)
The RC circuit is an electric circuit composed of
resistor R and capacitor C. The simplest RC circuit
consists of a resistor and a charged capacitor
connected in a single loop.
A capacitor is one of several kinds of devices
used in electric circuits as in computers, radios, and
other such equipment. The capacitors provide
temporary storage of energy in circuits and the
property of a capacitor that characterizes its ability
to store energy is called its capacitance. When
energy is stored in a capacitor, an electric field
exists within the capacitor.
3.2.1 Case of Capacitor Charging Process
(Presence of a Voltage Source):
We realize the electrical circuit consists of the
components: a voltage source E (generator), a
resistor, and a capacitor, connected to one another in
a single loop. The diagram of RC circuit in the
presence of a voltage source is presented in Figure
4.
Fig. 4: Diagram of RC circuit in the presence of a
voltage source.
Once the circuit is closed, the capacitor starts to
charge its stored energy through E. The system we
study will be described by a linear differential
equation and its solution is the voltage across the
capacitor , which is time-dependent, and it can be
found by using Kirchhoff's current law (sum of
voltage law) as follows
By using Ohm's law
and
we have
which is a linear differential equation similar to that
of equation (1). Knowing that , are resistor
and capacitor voltages. is the time constant (in
seconds), is the electric resistance (in ohms) and
is the electric capacitance (in farads). Noting that
each variable has a relationship with the capacitor
voltage, the studied system can be described by a
differential equation according to it such as the
charge on the poles of the capacitor
By solving the differential equation (16)
describing the studied circuit, the voltage across the
capacitor is
where .
3.2.2 Case of Capacitor Discharging Process
(Without a Voltage Source):
We use the same circuit as before, but without an
electrical voltage source E. The diagram of RC
circuit in the absence of a voltage source is
presented in Figure 5.
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Fig. 5: Diagram of RC circuit in the absence of a
voltage source.
Once the circuit is closed, the capacitor starts to
discharge its stored energy through the resistor. The
voltage across the capacitor as well is time-
dependent. Both the linear differential equation that
describes the system above and its solution (the
voltage across the capacitor ) can be found by
using the sum of voltages law as follows
Now using equations (14) and (15), we then obtain
which is a linear differential equation similar to that
of equation (3). Each variable has a relationship
with the capacitor voltage can be used to study
the system. Such as the following differential
equation according to the charge on the poles of the
capacitor
As well, the equation from equation (19) using
the voltage across the resistor is
which is a Fredholm integral equation, and once we
derive it we get
The same regarding the equation from the
current, so by deriving equation (21), we have
Now the solution to equation (18) is
In this part, we can conclude that is the time
required to charge the capacitor, through the
resistor, from an initial charge voltage of zero to
approximately 63.2% of E or to discharge the
capacitor through the same resistor to approximately
36.8% of E.
3.3 Motion of a Real Fall of a Solid Object
in the Air
An example of modelling a real problem using
differential equations is the determination of the
velocity of a solid object (s) falling through the
air, considering only gravity and air resistance. The
ball's acceleration towards the ground is the
acceleration due to gravity minus the deceleration
due to air resistance. Gravity is considered constant,
and air resistance may be modelled as proportional
to the ball's velocity. This means that the ball's
acceleration, which is a derivative of its velocity,
depends on the velocity, which depends on time.
Finding the velocity as a function of time involves
solving a differential equation and verifying its
validity. Now, we consider the system in question to
be a solid object (s) falling from a certain height
toward the ground (one-dimensional motion). Figure
6 presentes th scheme of the various forces to which
the body is subject.
Fig. 6: Scheme of the various forces to which the
body is subject.
The reference level for gravitational potential
energy is the surface of the Earth. The external
forces acting on the object (s) during its fall are the
force from gravity and
denotes the buoyant force applied onto the
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submerged object (according to Archimedes'
principle) and the force of friction
,
where for low object velocities and for high
ones .
M is the mass of the body; g is Earth's
gravitational acceleration;
is a
volumetric mass of air (fluid) and is the volume of
the solid object immersed in the fluid (equal to that
of the displaced fluid i.e., ). is the friction
constant and is the solid object velocity. By
applying Newton's second law (the basic principle
of motion), we have
where is the acceleration of the object (s). By
projecting the equation (26) on the axis of motion
and after simplifying, we find
with
is the volumetric mass of the solid
object (s).
In the case of low velocities, we have
which is a 1st-order differential equation similar to
equation (1), its solution is like equation (2), which
is
Here is the terminal velocity and
. In the
permanent regime, we have
, thus we obtain
The terminal velocity of the object increases with
the increase in the volumetric mass of the solid
object, and the following table (Table 1) gives some
examples:
Table 1. Some solid objects and their terminal
velocities .
Solid Object
Terminal Velocity
(m/s )
Paratrooper in free vertical
fall
8,5
Paratrooper with an
opened parachute
6,5
Table tennis ball
7
Golf ball
30
A steel ball with a radius
of 2 cm
80
Stone of radius 1 cm
30
A drop of water
10
In the case of high velocities, we have
where the terminal velocity is
Now, Figure 7 presents a graph from equation
(24) as follows:
Fig. 7: Plot of ) where and .
3.4 Formed and Reacted Substance
Amounts during Chemical Reactions
As another example regarding the phenomena that
develop monotonously, let's consider the evolution
of the formed or reacted (hidden) substance amounts
and the progress of reaction (as the extent of
reaction) in the chemical reactions in aqueous
solutions. By considering the simple chemical
equation as , where the chemical
reactants on the left and those of the chemical
products on the right. The amount of substance is
of the hidden reactant A and is of the formed
product C and the progress of chemical reaction ,
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which expresses the number of times the reaction
occurs; it is at the macroscopic level. So, in a
chemical reaction in an aqueous solution, the
evolution of the amount of the formed substance
will be exponentially increasing while the amount of
the reacted substance will be exponentially
decreasing. As a result, we present the plot of
) where in Figure 8.
Fig. 8: Plot of ; ) where .
In addition, the equation of the progress of
reaction in time will be
where is the maximum number of times the
reaction occurs, it also corresponds to the smallest
value of the progress for which the final quantity of
at least one of the reactants is zero. We also have the
half-life , which is intended to compare two
chemical reactions in terms of speed or to examine
whether the reaction is slow or fast. So, of a
chemical reaction is the time required for the
reaction to progress half its final progression, i.e.,
for , we have
. Once the reaction
is complete, then .
We draw attention to that any amount related to
the amount of substance, such as concentration,
could be used in the study.
Finally, yet importantly, we note that with each
application study, the picture becomes clearer. As
well, the permanent regime (the speed constancy) is
reached after a period of .
4 Conclusion
Many phenomena develop monotonously where the
evolution of these phenomena is described by
differential equations and their solutions give
accurate descriptions of these developments. The
most known developments are those that develop
exponentially. In this work, we have put a
mathematical model used to describe these
phenomena and shed light on the most common
important phenomena that are increasing
exponentially in one direction and those that are
decreasing exponentially in one direction. Such as
the radioactive decay, charging, and discharging of
a resistor–capacitor circuit and the motion of a real
fall of a solid object in the air, as well, as the
progress of quantities of substance when formed or
reacted during chemical reactions in an aqueous
solution. Besides, we explained some of the
mathematical properties of these phenomena.
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Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
This research received no external funding.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The author contributed in the present research, at all
stages from the formulation of the problem to the
final findings and solution.
Conflict of Interest
The author has no conflict of interest to declare that
is relevant to the content of this article.
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