Mechanics Phenomenological Econophysics For The Description Of Microeconomical Systems Of Stocks

International Journal of Business Marketing and Management (IJBMM)
Volume 5 Issue 10 October 2020, P.P. 01-51
ISSN: 2456-4559
www.ijbmm.com
International Journal of Business Marketing and Management (IJBMM) Page 1
Mechanics Phenomenological Econophysics For The Description
Of Microeconomical Systems Of Stocks
Mihai Petrov
Republic of Bulgaria, town Burgas, University "Asen Zlatarov", Department of Real Sciences, section of
physics and mathematics.
Abstract: Econophysics as an integrated platform of physics together with other economic sciences has a
broad perspective of phenomenological physics description of the processes of economic activities. This paper
suggests methods of phenomenological physics of mechanical kinematics and the model of gravitational
acceleration for the description of the activity of microeconomical systems of stocks. A criterion of continuous
instant stability of microeconomic systems is established by the description of the phase trajectory which is a
necessary condition that this shape of the trajectory to be unchangeable with time. The conception of the
econophysical acceleration is described which is related to the sold inventory. Bigger is the sold inventory then
the smaller is the acceleration. The following formulation of the interconnection between the acceleration and
the sold inventory is suggested: The continuous decreasing of the acceleration with time is the indicator of the
continuous increasing of the sold inventory. The validation of the acceleration concept is performed by the real
example of the sold inventory. The result of the average acceleration coincides with value of the rating
coefficients of the stocks and respectively with the values of thermodynamical temperatures.
key words: econophysics, distribution of Pareto, phase trajectory, econophysical acceleration, sold inventory.
I. Introduction to econophysics. Prerequisites of the continuous development of
econophysics
Technical and scientific progress involves an integrational development of various scientific fields in
order to solve new major goals and proxies in the field of medicine, economy, pharmaceutical industry, high
modern technologies, social processes and the Human being in the new life conditions taking into account the
evolution of climatic and ecological conditions. Also new philosophical conceptions about Life imply a
widespread application of knowledge from different fields of science and eventual their application into a new
integrative scientific fields such as: biophysics, bioinformatics, econophysics, bioeconophysics etc.
Econophysics is an interdisciplinary research field, applying theories and methods originally developed
by physicists in order to solve problems in economics, usually those including uncertainty or stochastic
processes and nonlinear dynamics. Some of its application to the study of financial markets has also been
termed statistical finance referring to its roots in statistical physics. [1]
Econophysics was started on 1990s by several physicists working in the subfield of statistical mechanics.
Unsatisfied with the traditional explanations and approaches of economists – which usually prioritized
simplified approaches for the theoretical models to matching financial data sets, and then to explain more
general economic phenomena.
The worldwide scientist Harry Eugene Stanley has developed the contributions to statistical physics and is one
of the pioneers of interdisciplinary science and is one of founding fathers of econophysics. Stanley has
developed the term of econophysics for the description of the large number of papers written by physicists in the
problems of markets and presented in a conference on statistical physics in Kolkata in 1995 and first appeared
in its proceedings publication in Physica A 1996.[1][2] The inaugural meeting on econophysics was organized
in 1998 in Budapest by János Kertész and Imre Kondor.
The multidisciplinary field of econophysics uses theory of probabilities and mathematical methods developed in
statistical physics to study statistical properties of complex economic systems consisting of a large number of
complex units or population (firms, families, households, etc.) made of simple units or humans. [3]
Consequently, Rosario Mantegna and Eugene H. Stanley have proposed the first definition of econophysics as a
multidisciplinary field, or “the activities of physicists who are working on economics problems to test a
variety of new conceptual approaches deriving from the physical sciences”. “Economics is a pure subject in

Mechanics Phenomenological Econophysics For The Description Of Microeconomical Systems ..
statistical mechanics,” said Stanley in 2000: “It’s not the case that one needs to master the field of economics to
study this.” Econophysics is a sociological definition, based on physicists who are working on economics
problems. [4] Another, more relevant and synthetic definition considers that econophysics is an
“interdisciplinary research field applying methods of statistical physics to problems in economics and finance”.
[5]
A main peculiarity related to econophysics is its distinctiveness from the mainstream economics, although both
sciences share the same subject of research. It seems quite strange, since physics has long been a source of
inspiration for economists. Unquestionably, in the second half of the 19th century, physics significantly
accelerated the development of economics by providing a necessary methodological framework. [6]
A lot of scientists working on the subjects of econophysics define various points of view regarding the
econophysics. For example the physicist A. Leonidov noted that "The study of economics as a quantitative
science is one of the urgent, exciting and complex problems of cognition. The depth and diversity of the
problems that arise makes the subject of study extraordinarily attractive for specialists in various fields of
knowledge, from psychologists to mathematicians. Of course, representatives of one of the most developed and
successful quantitative disciplines, physics, could not stand aside. [7]
The term “econophysics” [8] was introduced also by analogy with similar terms, such as astrophysics,
geophysics, and biophysics, which describe applications of physics to different fields. Particularly important is
the parallel with biophysics, which studies living creatures, which still obey the laws of physics. It should be
emphasized that econophysics does not literally apply the laws of physics, such as Newton’s laws or quantum
mechanics, to humans, but rather uses mathematical methods developed in statistical physics to study statistical
properties of complex economic systems consisting of a large number of humans. So, it may be considered as a
branch of applied theory of probabilities. However, statistical physics is distinctly different from mathematical
statistics in its focus, methods, and results. Originating from physics as a quantitative science, econophysics
emphasizes quantitative analysis of large amounts of economic and financial data, which became increasingly
available with the massive introduction of computers and the Internet. Econophysics distances itself from the
verbose, narrative, and ideological style of political economy and is closer to econometrics in its focus. Studying
mathematical models of a large number of interacting economic agents, econophysics has much common
ground with the agent-based modeling and simulation. Correspondingly, it distances itself from the
representative-agent approach of traditional economics, which, by definition, ignores statistical and
heterogeneous aspects of the economy. Two major directions in econophysics are applications to finance and
economics, statistical distributions of money, wealth, and turnover among interacting economic agents.
Econophysics that is a new branch of the study of economy includes not only proper sense of econophysics as
usual but also physical economics [9] that explains the economical processes by the application of physical
phenomena and has a large priority to choose the adequate physical model for the quantitative description of the
processes of pharmaceutical marketing. [10]
Physics (from Ancient Greek: υυσική (ἐπιστήμη), translit. physikḗ (epistḗmē), lit. 'knowledge of nature',
from υύσις phýsis "nature") is the natural science that studies matter, its motion, and behavior through space and
time, and that studies the related entities of energy and force. So, physics studies the general laws of nature and
explains phenomena with appropriate patterns using mathematical methods. The traditional question of physics
is: why does this phenomenon happen? And the answer is given according to the appropriate model. The
question arises logically, but why physics? What physics, which is a very widespread science with modern new
compartments, is not enough of its own domain? Surely, the development of physics has reached such limits that
it is now becoming interdisciplinary. The human being always at different historical stages is accustomed to
observing phenomena in nature and studying them in detail, to explain why these phenomena occur and the
cause of their defense. So, namely physics is called science that deeper insight studies the essence of all things
in the Nature. Logically, we can ask ourselves in the following way, since physics explains the essence of all
things, then it really does explain everything like: historical evolution of society and eventually statistical
repetition of some historical processes, economic phenomena, periodical physico-statistical variations of some
social and economical processes, market processes described by analogical physical laws, etc.
Physics aims to observe the given phenomenon, and as a result of observation, the quantitative mathematical
apparatus is performed, the final result of which is the quantitative law that contains the numerical parameters
describing the given phenomenon.
It is worth noting that many now-famous economists were originally educated in physics and engineering. The
well known Italian scientist Vilfredo Pareto that is considered as a parent of modern science of econophysics
earned a degree in mathematical sciences and a doctorate in engineering at the ends of 19th century. Working as
a civil engineer, he collected statistics demonstrating that distributions of turnover and wealth in a society follow
a power law [11].
The word economy from Greek translation means order and discipline inside the house. Keeping this sense, then
this order and discipline can be created somehow by the application of the principle of Pareto, especially if we

are referring to the stock markets. Nowadays, this principle is largely applied not only for economical systems,
but for social, healthcare and organizational activity.
Nowadays the Pareto’s principle also has become a popular area of focus in the world of business and
management and the statement of this principle is: 80 percent of effects always come from 20 percent of the
causes. Pareto first observed this ratio when he realized that 80 percent of land and wealth in Italy was owned by
20 percent of the population. He then went on to observe the same phenomenon in his garden: 80 percent of
peas came from 20 percent of pea pods. [11] Since he published these findings, the magical ratio of 80:20 (or
the “80-20 rule”) has been found to be scattered throughout society and nature. The 80 percent of any
company’s profits come from 20 percent of their best products, 80 percent of traffic comes from 20 percent of
roads, 80 percent of food production comes from 20 percent of the best crops. The ratio is everywhere and
frequently even tipped to a 90-10 or 95-5 division. However the 80-20 phenomenon is the distribution most
often cited as a universal baseline and especially the application to the practice of hospital medicine [12]:
 80 percent of the clinical and problematic issues on any given day will arise from 20 percent of the patients.
 80 percent of telephone calls and pages will always come from 20 percent of nurses.
 80 percent of valuable medical information that is received will come from only 20 percent of what are
communicating.
Healthcare has its own Pareto principle: 80% of healthcare costs are attributed to 20% of the populace: the
chronically ill.
The Pareto principle last time is applied largely and is combined with ABC analysis for supply management
purpose. [13] Therefore the effective supply management ensures uninterrupted availability of quality approved,
safe and effective products. The econophysical studies that include the principle of Pareto were reflected in [10]
which shows that each stock article of pharmaceutical products is characterized by so-named econophysical
temperature and this term of econophysical temperature is the capacity of the generating power of turnover
(revenues) during one day of one stock article and respectively for each rating marketing groups A, B, C, X, Z
of the stocks these values of temperatures are KA=21; KB=13; KC=8; KX=5; KZ=3 that coincide with the
numbers of Fibonacci which stay on the basement of so-named “Golden ratio” of Nature’s structures and
economical structures [10], [14]. The Fibonacci sequence are applicable for various kinds of the stocks.
The econophysical studies presented in [10] apply the physical model of the “ideal gas” of the pharmaceutical
stocks and this model is related to the marketing state of hyper competition. The sold and reserve inventory of
stocks is described by the equation of marketing state [10]:
KNNP arttotp  (1)
here pP  is the average price of one pharmaceutical product; totN - total amount of products of the
inventory; artN - total amount of the names of articles; K - the value of econophysical temperature and for the
full ensemble of stocks this value is 65,5K which is calculated on the base of KA=21; KB=13; KC=8;
KX=5; KZ=3 by the consideration of the peculiarities of ABC analysis and this value 65,5K is a worldwide
constant that is independent on national currencies [10].
Similar expression like (1) is described in the paper [15]. The difference is that the econophysical temperature is
the volatility in [15]. The greater the volatility, the greater the opportunity to sell the stocks at high prices. [16]
Otherwise, the higher the econophysical temperature K described by expression (1) , the greater the opportunity
to sell the stocks at high prices.
It is clear that in order to be a good specialist in the field of econophysics, is necessary the fundamental initial
studies in the fields of physics, statistics and economics. Only then can one understand the processes that are
described this scientific integrative complex system.
Generalizing the introductory information, then the definition of econophysics could be given as follow:
Econophysics is a multidisciplinary philosophical scientific integrative system that studies the general laws
of the evolution of economical and social processes by the application of physics - mathematical and
statistical methods of philosophical, social and economical sciences.
Econophysics like physics could also contain the similar chapters like mechanics, thermodynamics and
statistics, electricity, optics, quantum mechanics, etc., exactly as phenomenological conception of econophysics
that is described in [15]
According to the point of view described in the paper [15] the equilibrium and crises in economies are explained
well by phenomenological conception of econophysics.
Logically, the first chapter could be mechanics. Historically, classical mechanics emerged first and is originated
with Isaac Newton's laws of motion in the paper [17] "Philosophie Naturalis Principia Mathematica".
Classical mechanics describes the general laws of the motion of macroscopic material bodies.

Microeconomics is the study of individuals, households and firms, behavior in decision making and allocation
of resources. It generally studies the markets of goods and services and deals with individual and economic
issues [18]. It is focused on the study deals with what choices people make, what factors influence their choices
and how their decisions affect the goods markets by affecting the price, the supply and demand [19].
The behavior of the activity of the markets of stocks is studied last time not only by classical theories of
economics but also by modern integrative means of interdisciplinary branches of sciences like statistical
mathematics and theory of probabilities, and modern statistical integrative science that is called econophysics.
[20], [21], [22].
II. Application of the distribution of Pareto to the ABC analysis for the description of
supply-demand marketing processes. The equation of the state of microeconomical
systems of stocks
According to the conception that is developed in the paper [15] the motor force of the prosperity and
good activity of the microeconomical systems is namely the Human being. He makes plans and orders of the
activity and this order depends both on the customers and the dealers or the sellers. The main aim of the
microeconomical research is to find such reasonable equilibrium between the supplied and demanded quantities.
In this topic the application of the principle of Pareto combined with ABC analysis will give the possibility to
obtain the quantitative analytical expression that contains the information about the prices of one product,
quantity of articles and quantity of packing products of each respective rating marketing groups A, B, C, X, Z.
ABC analysis [23]-marketing tool that improves the efficiency of the activity of the markets. This analysis is
performed in order to analyze the sales and priorities in the management of marketing activity. ABC analysis
that is a part of marketing starts from policies of marketing mix [24], [25], [26],[27] which is a complex of
controlled marketing varieties that the market uses in order to achieve the desire result and increasing of
turnovers by attending to consumption necessity of customers (buyers).
The VI-th Congress of Pharmacy with International Participation [13] and III-rd International Conference of
Econophysics [28] presented the information about the rating of the stocks by statistical distribution of Pareto
with ABC analysis [29], [30]. The distribution of Pareto allows to describe quantitatively these rating groups A,
B, C, X, Z of the stocks by special parameter K named rating coefficients of the stocks [13], or econophysical
temperatures [10] and have the meaning of the power of the turnovers of one stock article during one day.
In order to present the generalized information about the amount quantities of stock articles in the form of
relative position of the stock articles in the distribution of Pareto the modification was performed [13] like:
10,)1(1)(  xxxF K
(2)
where F is the cumulative turnovers , x is the relative position of the stock articles.
The respective graphic is presented on Fig. 1. The ABC analysis combined with Pareto analysis can be
represented into one diagram [13] as shown on the Fig. 2
Total shares of the stocks ABC gives approximately 80% of total turnovers and this total stock ABC includes
20% from the total stock articles of all products. The rating coefficients K of the stocks is calculated from
expression (2) for the intervals of times from unspecified random first day till several months like 72 months for
the pharmaceutical products. [13]
)1ln(
))(1ln(
x
xF
K


 (3)
Fig. 1. The modified theoretical distribution of Pareto:  к
xxF  11)(
for different numerical values of K

Fig. 2. Rating shares of the stocks on the diagram of Pareto
The found values of K of the respective rating stock groups A, B, C, X, Z depend on the interval of time as is
represented on Fig. 3
There is an important peculiarity of the dependence K=f(t) that is important to be mentioned. The starting time
can be chosen randomly and the same values of K are obtained during the same interval of time Δτ as shown
schematically on the Fig. 4. This situation might corresponds to the one of criteria of instant progressive activity
of the market.
These values of K are arranged on stationary numerical series of Fibonacci numbers (KAst=21, KBst=13, KCst=8,
KXst=5, KZst=3) for relative big intervals of time as of order of 72 months. These stationary values represent the
average turnovers of the selling per one stock article during a day and if these values of the average turnovers
are divided by the price P0j of one packing product, then it means the result of sold packing products N0j of the
respective stock article during a day [13].
The index j corresponds to the respective rating group A, B, C, X, Z , so (Z≤ j≤A). So, the quantity of sold
products N0j of respective stock article during a day is calculated as follow:
AjZ
P
K
N
j
j
j  ;
0
0 (4)
Fig.3. The dependence of rating stock coefficients K on the interval of time
Fig.4. The independence of the starting time of K=f(t)
The turnover from the selling of one stock article per day is:
AjZNPK jjj  ;00 (5)

The sum of all turnovers of the selling per one stock article T0j during the interval of time Δt is the definite
integral from zero till Δt:
AjZtNPdttKT
t
jjjoj  

;)(
0
00 (6)
Taking into consideration, that one rating group of the stocks contains quantity of stock articles jartN , then
the turnover of entire rating group Tj during the interval of time Δt is the sum of all turnovers per stock articles:
AjZtNPNTNdttKNTT
t
jjartojartjartojj jjj
 

;)(
0
00
(7)
The average value <Kj> during the interval of time Δt is calculated as:
AjZ
t
dttK
K
t
j
j 




;
)(
0
(8)
Then: AjZPNNKN jjartjart jj
 ;00 (9)
The total quantity of packing products jtotN for the full rating group is:
AjZNNN jarttot jj
 ;0 (10)
Then the expression 10 is written as:
AjZPNKN jtotjart jj
 ;0 (11)
For the big systems of quantities of stocks is better to use the average price of one packing product pjP  for
the respective rating group j, and this average price pjP  is calculated as:
jtot
jj
pj
N
PN
P
 00
(12)
The expression 5 can be generalized by the sum of the right and the left part of whole rating group j:
AjZNPK jjj   ;00 (13)
The sum   jK is repeated jartN times and, then:
 jartj KNK j
(14)
Taking into consideration the expression 12 and 14, then the expression 13 can be written as:
jj totpjjart NPKN  (15)
Taking into consideration that jarttot NNN jj 0 , then the expression 15 can be written as:
AjZNPK jpjj  ;0 (16)
More important moment is the average quantity of the packing products  jN0 per one stock article and
this value can be calculated as:
j
artj
jart
art
tot
art
jart
art
j
j N
N
NN
N
N
N
NN
N
N
N j
j
j
j
j
j
0
000
0  
(17)
So, the expression 16 can be written as: AjZNPK jpjj  ;0 (18)
Then the average price of one stocking product is:
AjZ
N
K
P
oj
j
pj 


 ; (19)

This expression 19 describes quantitatively the process of supply and demand of the stocks, which states that
if the prices of products are decreased the demanded stocks from customers are increased and vice versa [31],
(  jP is inverse proportional to ojN and respectively to  ojN ).
The process of interaction between the seller and the customer is a stochastic process and the final
result is the event of the purchasing of the demanded products. Suppose a situation of such type of existence of
substitutes products of the same type but with different price. The seller has the tendency to offer the expensive
one in order to have more revenues. So, supply process is related to the seller, dealers and producers. The
dealers and producers supply the products depending on the turnovers and salaries of the customers and as the
salaries of customers are increased they supply more expensive substitutes. Always the tendency exist that
customers are demanded more cheapest substitutes but dealers supply the more expensive substitutes. As the
result of this complex stochastic situation there is a equilibrium point where the price P*
and quantity Q*
are
stable. Such equilibrium point is obtained when the supply and demand shapes are joined in one diagram and the
point of intersection of the shapes is equilibrium point E as shown in Fig. 5.
Generally speaking, an equilibrium is defined to be the price-quantity pair where the quantity
demanded is equal to the quantity supplied. The analysis of equilibrium is a fundamental aspect
of microeconomics:
Market Equilibrium is a situation in a market when the price is such that the quantity demanded by
consumers is correctly balanced by the quantity that firms wish to supply. In this situation, the market clears.
[32]
The equation (15) is named the equation of the state of microeconomical system of stocks. It is a expression of
interdependence of the prices of one product and the quantities of articles and the quantity of products of the
respective articles at the fixed values of rating coefficient of the stocks K .
Regarding the expression (11), it can be observed that :  jojj KNP0 . Here, there is an inverse
proportionality between ojN and jP0 for the fixed stable value constKj  at the respective moment of
time. The respective graphic of the dependence of )(0 ojj NfP  is represented on the Fig. 6.
Fig. 5. Equilibrium of supply and demand
Fig. 6. The dependence of the price P0j of one packing product vs. the demanded quantity of products N0j
on one stock article

The process of the prices creation is influenced by a lot of very complicated factors such as the initial costs of
the materials, the performed work, etc., and it is expected that the planned value of the prices can be validated in
practice namely by the supply-demand process.
It is important to mention that the curve of demands have the form of hyperboles that are represented on the Fig.
6 showing the consequently decreasing of the prices of the products Poj with the increasing of the demanded
amounts Noj, and experimentally it will be expected to have namely such hyperbolic forms and is described by
such dependence
oj
j
oj
N
K
P

 . Logically, the average price of one product pjP  can be found for each
respective rating marketing group A, B, C, X, Z and is expected qualitatively that the price of one product is
highest for the A group than of Z group. Respectively, the rating coefficients of the stocks  jK are higher
for A than of Z. It is known that if the demand is instant higher then the prices of products are fixing
consequently to higher values or have the tendency of increasing in comparison with those which have small
demand. If the demand is higher then the stock reserve of the respective items will have the planning of the
increasing or they are in great quantity. So, it is expected that for the group A the stock reserve will be higher
than of Z group.
The qualitative estimation of the amounts of products of each rating group allows to represent the curve of
supply S on the Fig. 7
Fig. 7. The dependence of the price P0j of one packing product vs. the demanded quantity of products N0j
on one stock article ; S - the curve of supply
The intersection points of the curve of supply with those of demands allow to obtain the information of optimal
stock reserve. The minimal limits of the stock reserves are the values ZXCBA NNNNN ,,,, and the
respective prices of one product are ZXCBA PPPPP ,,,, that are represented on Fig. 7. Real observation of
such position of points are expected to be almost real.
In such a way the Pareto distribution combined with ABC analysis gives two very important topics: 1) equation
of the state of microeconomical systems of stocks; 2)The curve of demands-supply gives the real idea about the
numerical values of the equilibrium prices and the respective quantities of the rating groups.
III. Kinematics phenomenological econophysics of
microeconomical systems of stocks
3.1. The definition of econophysical kinematics. The notion of the speed of movement, displacement and
the vector. The instantaneous speed. The prerequisites of the possible development of the oscillator model
of the inventory
Kinematics is the chapter of mechanics dealing with the study of the coordinates of the moving bodies
and how these coordinates are variable with the time. Mechanics is the science concerned with the motion of

bodies under the action of forces, including the special case in which a body remains at rest. Force is nothing but
the ability of the customer to purchase the products by payments.
In order to give a mechanical model to the system of stocks, it is necessary to mention that the system is
described by four main parameters that are resulting from the equation of the state of microeconomical system
of stocks: 1) the total amount of the products Ntot; 2) average price of one product <P>p; 3) quantity of stock
article (varieties of the names) Nart; 4) the rating coefficient of the stock Kj (or the econophysical temperature of
the system of the stocks). The system could be into the "rest" state or into the state of the "movement". This
system is considered as the "material" body that moves with time into the space of the coordinates Ntot. Namely,
the change with time of Ntot means that the system "moves". Changing with the time of the quantities Ntot means
the changeable stock reserve (changeable inventory).
The chapter which study the movement of the body without taking into consideration the reason of the emerging
of the movement is named kinematics.
Kinematics is the part of mechanics that studies the motion of a particle (body), ignoring its causes.
A particle is a point-like mass having small size. The econophysical mass is nothing but the margin (or
the profit), or the difference between the selling price and the price of dealers. For example, an inventory of
100000$ has a mass of about 20000$. This econophysical mass is comparative smaller in comparison with the
value of the inventory.
The movement of the body could be of two types: 1) uniform motion; 2) non-uniform motion.
1)This type of the motion is defined as such motion of the body which coordinate Ntot is variable with the same
constant value ΔNtot in equal intervals of time Δt .
Regarding this type of the motion it is necessary to define the speed of motion V. Namely, if the stock reserve
that is determined by the value Ntot is changeable with the time, then is a criterion of the selling of stocks. The
quantity of stocks that are sold during one unit of time is the speed of the motion V of the system.The speed of
the motion V is the path traveled in the unit of time. The expression of the speed V is written as:
12
12 )()(
tt
tNtN
V tottot


 (20)
where )( 1tNtot and )( 2tNtot are the amounts of products of the stock reserve (inventory) respectively at the
initial moment of time t1 and the final moment of time t2. If the respective variation
)()( 12 tNtNN tottottot  is the same for the same interval of time Δt, then this motion is uniform. The
measurement unit of the speed of motion V is: (products/s; products/min; products/h; products/day;
products/month; etc.). So, the speed of motion V is constant all time. (V=const)
The Fig. 8 shows two cases when the system moves with the constant speed. The case (a) is referring to the case
when the inventory is increasing uniformly. This case (a) could be the case when the supplying with new stocks
is greater than the quantity of sold products. The case (b) is referring to the case when the inventory is
decreasing uniformly due to of stable uniform selling of products. In this situation the selling products are in
great quantity than the supplied quantity from dealers.
Fig. 8 The uniform variation of inventory: a) case of uniform increasing of inventory;
b) case of uniform selling of products
2) The non-uniform motion is such motion of the system which coordinate Ntot is variable randomly in equal
intervals of time Δt . Such type of the movement could be like the trajectory that is represented on Fig. 9

Fig. 9. Example of non-uniform movement of the system.
This type of movement could be for the cases when the products have the seasonal character, finite life-cycle of
some products, substitution with other similar products, seasonal character of the entire system.
When the variation of the amounts Ntot takes place then it is important to emphasize that the variation is not
continuous (is not a slow transition from one state with those four parameters of the system into another state),
but is discrete. The discrete transition is represented by black points on the Fig. 8 and 9. Only one difference
exist between uniform and non-uniform movement. A linear straight transition from one state to another state
takes place for the case of uniform movement, but for the case of non-uniform movement the chaotic transition
from one point to another point takes place.
The points represent an event of the selling or purchasing. Usually, if the event of the purchasing takes place
then the value Ntot is decreased and vice versa if the supply from dealers takes place then Ntot is increased. If
another event of purchasing from customers takes place, then another transition into another point takes place.
The segment between the two points is considered the "rest" state of the system. The lengths of the segments of
the rest states could be various due to of the stochastical character of the processes.
The case of uniform movement is very rare. It can only occur in relatively short time intervals. More often, non-
uniform movements could occur. In the classical mechanics of physics, the use of the notion of vector is applied.
The vector is the right oriented segment that unites the initial and the final point. The orientation of the vector by
the arrow shows the direction of the movement.
Respectively the transition from one state to another (from one point to another) is nothing but the
displacement. The displacement in this case coincides with the traveled road.
In classical mechanics the notion of the reference body is used. The reference body is the body with respect to
which the movement of the system is studied. The reference body coincides with the origin of coordinates O.
The reference body O in this mechanical description will be none other than himself own microeconomic
system. This reference body will be considered strictly as something very initially zero with zero stock and a
initial moment of time fixed at the zero value.
Referring to the recent econophysical description, then the three dimensional system of coordinates will be
applied (Ntot, No, Nart). The values of Ntot are dependent on No and Nart as Ntot= No∙Nart.
Why is necessary three coordinates? It will give more information, because the total amount Ntot is changeable
as the result of the changes of No and Nart. Sometimes, the same value of Ntot could be for the case when No is
not changeable but Nart could be changeable due to of the apparition on the market of the new product (new
name) or could be withdrawn, or could be a situation that No is changeable but Nart is fixed. The changeable
value of No could be for the cases when the amount of products for one stock article is variable due to of
seasonable character of the product. Therefore, the application of three dimensional system is more informative.
The Fig. 10 represents schematically the possible variation of the inventory on three dimensional system. The
positional vectors 1Z and 2Z shows the consequent positions of the states 1 and 2 of the system at the
respective moment of time 1t and 2t . The vector of the displacement is 12 ZZZ  . This vector of
displacement Z shows the direction of the variation of Nart, No and Nart on the Fig. 10. This exact example on
this Fig. 10 shows that all components Nart, No and Nart are increasing. In general, such situations could be when
two of them are increasing but another is decreased. For example if No is increasing and Nart is decreasing then
the result of Ntot is increased due to of the fact that the increasing of No is several more times bigger than of Nart.

Fig.10 The three - dimensional schematically representation of the variation of the inventory with time
The projection of the displacement vector XNCLAMZ  . The respective module of the vector
Z is:
           222222
12 totarto NNNXNMACLZZZZ 
(21)
The vector of the speed of movement:
t
Z
tt
Z
V Z






12
(22)
The respective decomposition of the velocity vector ZV by the components of the axes is:
XNCLAMZ VVVV  (23)
The respective speeds components by axes are written as:
t
AM
tt
AM
VAM




12
;
t
CL
VCL

 ;
t
XN
VXN

 (24)
The respective modules of the vectors of speeds of the expression (24) are written as:
t
AM
t
AM
VAM



 ;
t
CL
t
CL
VCL



 ;
t
XN
t
XN
VXN




(25)
The module of the vector ZV is written as:      222
XNCLAMZ VVVV  (26)
So, the transition from one state into another state is like a way that is travelled during the interval of time Δt.
Then, the speed ZV is considered like average speed:
t
Z
t
Z
V Z





  (27)
For two respective neighbour segments with the length ΔZ1 and ΔZ2, then the average speed:
21
21
tt
ZZ
V Z


  (28)
The respective segments are shown on the Fig. 11 with the two segments.
Fig. 11 The way of transitions with two segments.

The expression (29) could be written for the case of arbitrary quantity of segments, and then the average speed
is written as:







 n
i
i
n
i
i
Z
t
Z
V
1
1
(29)
If the intervals of the time are very short 0 it , then the speed is limited to a point of the way and the
respective speed is called instantaneous speed V (the speed at the given moment of time) and this instantaneous
speed is a function of time:




















 n
i
i
n
i
i
t
Z
t
Z
tV
i
1
1
0
lim)( (30)
The case of the short time interval is highly idealized. This is the case when each buyer is served one after the
other without any rest of the system. This is the case when the buyers wait in queue without any disobeying of
this queue. The expressions of the instantaneous speed as a function of time could be various like:
cbtattV  2
)( , or in the form of exponential functions:
bt
Z eatV 
 )( ; where a, b, c are the constant
coefficients. The traveled way also is the function of time ΔZ(t) and the expression of the instantaneous speed
could be written in this case as:
dt
dZ
tV Z  )( (31)
The values of the instantaneous speed could be variable in time also by sign. Sometimes the could be negative,
sometimes positive values. The negative value of the instantaneous speed means that at this moment the reserve
quantity of inventory is decreasing and if the instantaneous speed is positive, then the reserve inventory is
increased. The increasing takes place by supplying of new stocks from the dealers.
The curve of the way in the case of very short time of transitions is a continuous curve without any rest states
and without any fast thresholds Fig. 12.
Fig. 12. The continuous curve of the way for the case of continues serve of the customers
The infinitesimal small interval of time dt corresponds to a very small traveled way d(ΔZ) and the respective
momentary speed is calculated by the expression (31)
The full way ΔZ (the variation of the inventory during the interval of time Δt (one day, one months, etc.) is
found by the integration of the expression (31):
   CdttVtZ Z )()( (32)
where C is the constant of integration, that is find by initial condition. One of initial conditions could be like as
for the initial moment of time t0=0 the value ΔZ0=78 products, then C=78 products.
The numerical value of the traveled way ΔZ can be calculated by the definite integral if the limits of the
integration are known:  
2
1
)(
t
t
Z dttVZ (33)
If the way ΔZ that is traveled during the interval of time Δt=t2-t1 is known (Fig.13), then the average speed can
be calculated as:
12
12 )()(
tt
tZtZ
V Z


  (34)

Fig. 13 The mean of the calculation of the average speed <VΔZ> by the traveled way ΔZ
The traveled way of the system from the moment of time t1 till the moment of time t2 according to the Fig. 13 is:
  
2
1
)()( 12
t
t
Z dttVtZtZ and substituting into the expression (34), then:
1212
12
2
1
)(
)()(
tt
dttV
tt
tZtZ
V
t
t
Z
Z






 (35)
One very important moment is necessary to mention. What value must be taken into consideration Ntot or Z ?
Taking into consideration the expression (21) then:
                  222222222
1 artartooartartototarto NNNNNNNNNNZ 
Here is necessary to mention that for the big values of Nart, the numerical value    22
1 artart NN  ,
because a microeconomical systems of stocks could contain an amount of order 1000 names or bigger and
   22
1 artart NN  . Then:
              111
222222
 oartoartartarto NNNNNNNZ
(36)
For the case when the amount of products that corresponds to one article No is relative big numerical value,
then:    22
1 oo NN  and finally    totoartoart NNNNNZ  1
2
(37)
The task is the study of the value of No that makes the coincidence of the values of Z and totN , and another
task is the precision of the expression (36)
The numerical simulations of the expressions (36) and (37) for the fixed values of ΔNo and various values of
ΔNart with the consequent representation on the graphic of the Fig. 14 allows to observe any peculiarity.
Fig. 14 The numerical simulation of the values Z and totN as the function of artN for the fixed
values of No

The graphic of the Fig. 14 shows that the values Z and totN are almost the same and practically coincide for
values of 10000 artN . The coincidence of Z and totN takes place for the values of 50 N and
10000 artN .
In order to calculate the precision of the formula of ΔZ , first the average value <ΔZ> between ΔZ and ΔNtot is
calculated. Then, the deviation totNZdevZ  is calculated. After that, the relative error is calculated
%100% 


Z
devZ
 . The dependence of )(% oNf  is presented on the Fig. 15.
Fig. 15 The relative error of ΔZ as the function of ΔNo
The graphic )(% oNf  shows that for the quantities of 100 N the error has tendency to reach the zero
value. In order one method to be validated it's necessary the error do not exceed the value 20% [33].
In such a way both methods could be applied either ΔNtot or ΔZ. The method of position vector Z is more
informative and gives more general information about how all three values Ntot, No and Nart are changeable with
time. So, the values ΔZ ≈ ΔNtot and for the further description the values ΔZ are considered simply as the
amount of inventory.
Example 1. The instantaneous speed of the variation of the inventory is described by the following function
t
Z etV 
  2.0
50)( . Find the analytical expression of the inventory ΔZ as a function of time. Calculate the
inventory at the third day, if the inventory of the first day is 100 products and the unit of time is considered one
day. Represent the graphic of the function of the inventory with the path 1 day as the function of time. Calculate
the average speed of the movement from: a) third day till seventh day; b) third day till tenth day.
Solution: The momentary speed is:
dt
dZ
tV Z  )( ; The respective analytical expression of the inventory as the
function of time is calculated as the integration like:    CdttVtZ Z )()( ;
CeCeCdtetZ ttt


 

2.02.02.0
250
)2.0(
1
5050)( ;
The constant of integration C is found from the initial conditions: t =1 day; ΔZ=100;
68.30468.204100
2214.1
250
100
250
100100250;250100 2.0
2.02.0
 
e
eCCe t
;
Then: 68.304250)( 2.0
  t
etZ ;
The respective inventory of the third day is:
)(167204.1376.304
8221.1
250
6.304
250
60.30468.304250)3( 6.0
32.0
products
e
eZ  
.
The respective graphic of the function is represented on the Fig. 16:
Referring to this expression of the given solved example 68.304250)( 2.0
  t
etZ , it could be
observed that for the values of time t ;  productsZ 30568.304)(  .

Fig. 16. One example of the dependence of the inventory with time
The average speed of the movement from third day till seventh day is:
a)  4.16.0
32.072.0
4
250
4
68.30425068.304250
37
)3()7( 

 




 ee
eeZZ
V Z ;
     dayproductseeV Z /1988.182466.05488.05.62
4
250 4.16.0
 
 ;
The method of integration:
b)  26.0
32.0102.0
7
250
7
68.30425068.304250
310
)3()10( 

 




 ee
eeZZ
V Z ;
     dayproductseeV Z /1576.141353.05488.071.35
7
250 26.0
 
 ;
It can be observed that the average speed in this case is decreasing gradually with the increasing of the interval
of time from the initial moment.
The ways of the movement of the system that is characterized by variation with time of the inventory ΔZ(t)
could be very various. The next figure 17 shows a possible type of the movement of the system.
Fig. 17 The possible variety of the movement of the microeconomical system of stocks
Specifically, for this Fig. 17 is that the system has seasonal character. More selling of the stocks is for the period
at the start of the summer (minimal value of the inventory ΔZ). If the seasonality is repeating instant all time (a
lot of years), then characteristically for this system is that this system is more active during the summer. The
system has sufficient financial resource to increase its inventory that is represented by maxima on the Fig. 17.
The respective policies of the marketing mix of this system are processed and stated. The stocks are checked by
seasonality and the supplying is performed according to the respective seasonal demand.
Another interesting situation could be for the case that is represented on the Fig. 18. Suddenly, the system is
forced to be transferred from one "macro-" state with big values of inventory ΔZ into another "macro-" state that
is characterized by smaller values of inventory ΔZ. It is like a "change of the phase" of the system.
Characteristically is that when the system is transferred into another "phase" and if it lasts for a long time to stay
into this new macro-state, then it means that the system is already adapted for new conditions. Possible
transition could takes place due to of a lot of factors like: social and economical crises, demographic problem of
the given geographical place of the microeconomical system. In these new conditions the new policies of
marketing mix are elaborated.

Fig. 18. The transition of the system from one "macro-" state into another
If the values of ΔZ are permanently decreasing then the system will reach the situation when will not be able to
continue its activity and has the peculiarity of default trend (Fig. 19) until the new policies of the marketing mix
are elaborated.
Fig. 16 The default trend of the microeconomical system
The next examples allow to understand better the suggested method of inventory and how it varies with time.
Example 2. One shop decides suddenly to sell all remaining inventory of 10000 products of various articles
with the average quantity of one article ΔNo=3. How long time is necessary to sell all products from the moment
of decision if the law of the variation of the inventory is ttZ  20030000)( . What is the average speed
of the selling?. Find the instantaneous speeds for the first and third day from the moment of decision. Find the
initial quantity of the products and the initial quantity of stock articles. The unit of time is considered one day.
Solution: . First is necessary to find the moment of time when the inventory contains 10000 products:
daythttt  100;20000200;2003000010000 .
The day when all inventory is sold is find as the consideration that 0)(  tZ :
daythtt  150;200300000 .
So, the interval of time during which the remaining of inventory will be sold is 150-100=50 .
2. The instantaneous speed of the selling is found by first derivative with time:









day
products
dt
Zd
tV Z 200
)(
)( . The sign minus of the speed indicates that the inventory every time
is decreasing. It remains all time the same. So, the instantaneous speed for the first and third day from the
moment of decision is the same 






day
products
200 ;
3. The average speed of the selling is the traveled way during the interval of time 50 days.












 
day
productsZZ
V z 200
50
)150100(200
50
1002003000015020030000
50
)100()150(
The average speed can be found also by the integration:











  
day
productst
dtdttVV ZZ 200
50
)100150(200
50
|200
)200(
50
1
)(
50
1 150
100
150
100
150
100
;
4. The initially quantity of products (initially inventory) is found for the start moment of the time t=0;
)(30000)0( productsZ 
The initially quantity of stock articles is found by )(10000
3
30000)(
articles
N
oZ
N
o
art 

 .
Example 3. One shop decides suddenly to sell all remaining inventory of 10000 products of various articles
with the average quantity of one article ΔNo=4. How long time is necessary to sell all products if the law of the
variation of the inventory is
t
etZ 02.0
30000)( 
 . What is the average speed of the selling. Find the
instantaneous speeds for the first, ninth and fortieth day from the moment of decision. Find the initial quantity of
the products and the initial quantity of stock articles. The unit of time is considered one day.
Solution: 1. First is necessary to find the moment of time when the inventory contains 10000 products:
daythtteee ttt
 
50;102.0;3;
3
1
;3000010000 02.002.002.0
.
The day when all inventory is sold is find as the consideration that 1)(  tZ (formally considering one
because practically will not be sold till absolute zero inventory):
thtteee ttt
 
516
02.0
)30000ln(
);30000ln(02.0;30000;
30000
1
;300001 02.002.002.0
So, the interval of time during which the remaining of inventory will be sold is 516 -50=466
2. The instantaneous speed of the selling is found by first derivative with time:
tt
Z ee
dt
Zd
tV 02.002.0
60002.030000
)(
)( 
 

 ;
 dayproductsee
dt
Zd
V Z /2163605.0600600600
)(
)51( 02.15102.0


 
 ;
dt
Zd
V Z /18430727.0600600600
)(
)59( 18.15902.0


 
 ;
dt
Zd
V Z /991652.0600600600
)(
)90( 8.19002.0


 
 .
The instantaneous speed is decreased gradually with time by absolute value. The decreasing takes place by the
fact that the remaining reserve inventory is decreasing gradually.
3. The average speed of the selling is the traveled way during the interval of time from 50-th day till the
uncertainty day.










 
day
productsZZ
V z 22
466
100000
466
)50()516(
The average speed by the integration:










 

 

day
products
eedte
dte
V tt
t
Z 2237.64|
23302.0
300
233
300
466
600
02.050
516
50
516
50
02.002.0
516
50
02.0
4. The initial quantities of products is found by: 3000030000)0( 0
 eZ . The initial quantity of stock
articles is found by: )(7500
4
30000
articlesNart 
The Fig. 17 that is like a oscillation movement represents an special interest. The values of ΔZ are changeable
similar to Sinus or Cosinus laws with the amplitudes during summer-autumn each year. The system oscillates

periodically because the demand of the customers has the periodical character. In general the conditions of the
apparition of the movement of the systems of stocks is explained by dynamical point of view of mechanics.
Namely, the practical major cases show that a lot of activities of microeconomical systems of stocks have
seasonal periodical character if are not observable the criteria of the default trends as that represented on the Fig.
16 . This fact lead us to one of prerequisites of the emerging of the idea of the oscillator model of the system of
stocks.
Considering that the values of ΔZ oscillates by Cosinus law, then:
)cos()(   tAZtZ ech (38)
where A - amplitude of oscillation; ΔZech - the equilibrium value of the inventory; ω-cyclical frequency; t-
interval of time; υ-initial phase;
The respective oscillations of the values ΔZ with time is represented on Fig. 17.
Fig. 17. The oscillation character of the stock inventory
The equilibrium value of the inventory is such a value around which the values ΔZ are changeable within the
interval [ΔZmin res; ΔZech+A]. The value ΔZmin res is the minimal value of the stock inventory. The minimal stock
inventory ΔZmin res is such minimal reserve, when the system cannot fully satisfy buyers' needs and demands,
therefore the system is supplied by the new stocks from the dealers.
The instantaneous speed is calculated as: )sin()(   tA
dt
dZ
tV Z (39)
The cyclical frequency ω expressed by period of oscillation:

2
T (40)
The period of oscillation T for a lot of cases is one year as for the Fig. 17.
The average speed of movement during one period is:
   
T T
T
T
ZZ t
T
A
tdt
T
A
dtt
T
A
dttV
T
TV
0 0
0
0
|)cos()()sin()sin()(
1
)( 

  0
2
2
sin
2
22
sin2cos)2cos(cos)cos( 










 



T
A
T
A
T
A
T
T
A
;
The fact that the average speed within one period of time is zero means that the system returns back to its initial
state with the initially value of the inventory.
Another method of the calculation of the average speed by the method of displacement is:
   




  


cos)2cos(
2
coscos
0
)0()(
)(
A
T
AZTAZ
T
ZTZ
TV echech
Z
0
2
2
sin
2
22
sin2
2











 



A
;
The calculation of the average speed can be checked also on the intervals of time [t;t+T]:
 

 
Tt
t
Tt
t
Tt
t
ZZ t
T
A
dtt
T
A
dttV
T
TtV |)cos()sin()(
1
)( 


    tt
T
A
t
T
A
Tt
T
A
cos)2cos()cos())(cos(
;0
2
2
sin
2
222
sin2 










 

 t
T
A
The calculation of the average speed by the method of displacement:
  




 
T
tAZTtAZ
ttT
tZTtZ
TtV echech
Z
)cos()(cos)()(
)(

     0
2
2
sin
2
222
sin2cos)(cos 










 



t
T
A
tTt
T
A
;
The average speed of the movement during one half of the period is:
  
2/
0
2/
0
2/
0
|)cos(
2
)sin(
2/
)(
2/
1
)2/(
T
T
T
ZZ t
T
A
dtt
T
A
dttV
T
TV 

  










 

2
sin
2
2
sin4cos)cos(
2
cos
2
)
2
cos(
2 

T
A
T
A
T
AT
T
A
;0cos
2
cos
2
4
cos
4
2
sin
4






 








AA
T
A
T
A
;
In the case when φ=0; then ;0
2
0cos
2
)2/(  



 AA
TV Z
The sign "-" means that the inventory is decreasing during this interval of time [0;T/2].
The respective method of displacement:
  













  

cos
2
cos
2
2/
cos)2/(cos
)2/(
T
T
A
T
AZTAZ
TV echech
Z
   0cos
2
cos
4
2
sin
2
sin
4
coscos
2








 





A
T
A
T
A
T
A
;
The next example allow to understand all practical peculiarities about the suggested method of the oscillator
model of the inventory and how it behaviors with time.
Example 4. One shop has the equilibrium permanent stock of 6500 articles with the average amount of products
per articles No=4 products. It has seasonal character with the period of one year. The peak of inventory rises
36000 products. The minimum reserve within "inactive" period reaches 16000 products. The variation of the
inventory takes place by Cosinus law. Calculate: a) equilibrium inventory expressed in products; b) the
amplitude of oscillations of the inventory; c) calculate the initial phase υ if the starting moment has the
inventory of 30000 products; d) calculate the cyclical frequency if the unit of time is one month; e) the variation
of the stock inventory for the moments of time 6 months, 9 months from the start moment of time and 12 month;
f) the average speed for the interval of time 6 month and 12 month; g) the instantaneous speed at sixth month
and tenth month;
e) represent the graphics of the dependence of inventory and instantaneous speed as the function of time on the
same frame.
Solution: a) The equilibrium inventory expressed in products is ΔZech≈ΔNtot=No∙Nart=4∙6500=26000 products;
b) The picture will give the idea how the inventory is changed:

The amplitude A is the maximum deviation from equilibrium position A=ΔZmax-ΔZech=36000-26000=10000
(products)
c) The starting moment of time has the inventory of )(30000)0( productsZ  ;
The initial phase  is found from the relation:
;40002600030000)0(cos);0cos()0(  echech ZZAAZZ 
4.0
5
2
10000
4000
cos  ;
The value 0.4 has the meaning as that for the starting moment of time the inventory "is somehow planned" to
have such an inventory that consists 40% of the "future possible maximum of the inventory".
1592.1)4.0arccos(  (rad)=66.450
;
d) the cyclical frequency )(52.0
12
28.62 1
 month
T

 ;
The meaning of the cyclical frequency is the quantity of radians that corresponds to one month.
e) The variation of the stock inventory for 6 months from the start moment of time:
    cos6coscos)6cos()0()6()6(var AAZAZZZZ echech





 





 





 











 

2
32.212.3
sin20000
2
52.06
sin
2
16.1252.06
sin20000
2
6
sin
2
26
sin2

A
  )(8176999.04092.020000)56.1sin(72.2sin20000
2
12.3
sin products





 ;
It means that the inventory is decreased during 6 moths with 8176 products.
The variation of the stock inventory for 9 months from the start moment of time:





 





 





 











 

2
32.268.4
sin20000
2
52.09
sin
2
16.1252.09
sin20000
2
9
sin
2
29
sin2

A
  )(50347184.0)35038.0(20000)34.2sin(5.3sin20000
2
68.4
sin products






It means that the inventory for the moment of time 9-th month is bigger with 5034 products higher than of the
starting inventory.
The variation of the stock inventory for 12 months from the start moment of time:





 





 





 











 

2
32.224.6
sin20000
2
52.012
sin
2
16.1252.012
sin20000
2
12
sin
2
212
sin2

A
)(0
2
24.6
sin products





 ;

It means that the inventory of 12-th month coincides with the staring inventory. The starting
inventory:
)(30000400026000)1592.1cos(1000026000cos)0( productsAZZ ech  
f) The average speed for the interval of time 6 month:
     




 




  )3sin()3sin(
3
3sin
2
26
sin
6
2
6
cos6cos
6
)0()6(
)6( 
 AAAZZ
V Z
   )56.1sin()72.2sin(
3
10000
)56.1sin()16.156.1sin(
3
10000
)52.03sin(16.152.03sin
3
10000







month
products
13649999.04092.0
3
10000
The comment of this result is that during six months from the start moment the inventory is decreasing with
1364 products every month.
The average speed for the interval of time 12 month:
     




 




  )6sin()6sin(
6
6sin
2
212
sin
12
2
12
cos12cos
12
)0()12(
)12( 
 AAAZZ
V Z
   )12.3sin()28.4sin(
6
10000
)12.3sin()16.112.3sin(
6
10000
)52.06sin(16.152.06sin
6
10000







month
products
3302159.0)9079.0(
6
10000
The comment of this result is that during 12 months from the start moment the inventory is increasing with 33
products every month.
g) The instantaneous speed at sixth month, tenth month and twelve month .
The expression of instantaneous speed: )sin()(   tAtV Z ;
The instantaneous speed at the moment sixth month is:
;4716)907.0(5200)16.112.3sin(5200)16.1652.0sin(1000052.0)6( 






month
products
V Z
Exactly, at this moment of the time, this result means that the inventory is increasing its quantity by 4716
(products/month).
The instantaneous speed at the moment tenth month is:
;395076.05200)16.12.5sin(5200)16.11052.0sin(1000052.0)10( 






month
products
V Z
Exactly, at this moment of the time, this result means that the inventory is decreasing its quantity by 395
(products/month).
The instantaneous speed at the moment twelve month is:
;46738987.05200)16.124.6sin(5200)16.11252.0sin(1000052.0)12( 






month
products
V Z
Exactly, at this moment of the time, this result means that the inventory is decreasing its quantity by 4673
(products/month).
e) The graphics of the dependence of the inventory and the instantaneous speed as the function of time is
represented on Fig. 18.

Fig. 18. The numerical simulation of the dependence of the inventory ΔZ(t) and the instantaneous speed
VΔZ(t) vs. time
So, the graphic of the speed is displaced with respect to the inventory with the phase difference 900
. The point A
that is the minimal value of the inventory corresponds to the zero value of the instantaneous speed (point K).
Major practical cases namely such situation takes place when the system of stocks sells firstly products without
any payments. The decreasing of the inventory takes place (the segment MA) as the result of the selling. The
financial resources are earned and they are spent for the new stocks (the speed is increasing on the segment NK).
The inventory is still increasing on the segment AB and the respective speed is continuing its increasing on the
segment KE. The supplying with the new stocks gradually is decreasing (the segment ED) and the inventory
slowly reaches its maximal value (the point C). The processes are repeating periodically. This is an ideal model
of the oscillations and it takes place really for the every day big turnovers and continuous supplying with the
new stocks exactly with the same amounts that were sold every day. In this case the expenses for passive assets
are comparative small with respect to active assets and the movement of the system reaches the ideal case of
harmonic oscillator. The phase trajectory in this case is an ellipse in the two - dimensional coordinate system
(VΔZ; ΔZ) (Fig. 19).
Fig. 19 Phase trajectory of the microeconomical system of stocks
The system starts the movement from тхе point 1 and consequently the speed is passing through the minimal
value -Aω, then the value 0 and finally the maximal value Aω. The values of ΔZ are oscillating within the
interval [ΔZmin; ΔZmax].
The trajectory of the three-dimensional spatial phase has a spiral shape located on the lateral surface of the
cylinder with the height equal to the time interval and with the bases coinciding with the ellipses in the two-
dimensional space (VΔZ; ΔZ). (Fig. 20)

Fig. 20 Phase trajectory of the perfect microeconomical system in three dimensional phase space
If the oscillations are continuously with the same amplitudes without any attenuations then this cylinder is
infinite and perfectly with the same bases as the form of ellipses. If attenuations emerge as the result of the
decreasing of turnovers and the increasing of the passive assets then the final basement of the cylinder will have
smaller area as the initial one (S1>S2) and if the final basement is continuously decreasing all time then the
peculiarities of default of the system are observed. (Fig.21).
Fig. 21 The attenuated elliptical cylinder in the conditions of the default of the system
The qualitative description of the activity of microeconomical systems of stocks by three dimensional phase
space allow to conclude about the behavior of the system with time. Qualitatively, it could be stated that smaller
instantaneous surfaces S(t) of the ellipses suggest about smaller turnovers in comparison with bigger
instantaneous surfaces of ellipses of bigger turnovers (Fig. 22).
Fig. 22 The possible real seasonal character of microeconomical system of stocks

3.2. The notion of the acceleration of the movement of systems. The acceleration as the degree of the
turnovers and stability of the systems
The previous topic describes the notion of the speed and the trajectory. And the speed is related to the
change of the quantity of inventory expressed as the measurement unit (product/unit time). In the recent topic
the situations when the speed is not constant (is changeable with time) are studied.
In physics, acceleration is the rate of the change of the velocity with respect to time. The IS unit in physics for
the acceleration is meter per second squared (m⋅s−2
). It is expected that the econophysical measurement unit is
(product/unit time-2
).
Let's examine the trajectory that the regarded system moves with changeable speed. Let Z
V 1 and Z
V 2 are the
movement speeds of the system at the moment of time t1 and t2 (Fig.22) and the respective small interval of time
is 12 ttt  .
Fig. 22 The trajectory of the movement of the system
Imaginary the velocity vector is paralleled transferred from the point 2 into the point 1 and then according to the
triangle rule we can see what is the velocity variation ZV . The variation of the speed by the triangle rule
during this interval of time is: ZZ
VVV Z 
  12 (41)
The vector size:
t
V
a Z
Z


 
 (42)
is named the acceleration of the body at the moment of time t2. According to the definition, the acceleration is a
vector. The system moves with acceleration every time when the vector of the speed ZV changes its direction,
its value or both the value and its direction. These changes every time of the speed value and the direction of the
speed leads to this fact that the acceleration is instantaneous for the fixed moment of time and respectively the
acceleration and the module of acceleration is a function of time: )(ta Z ; )(ta Z .
The trajectory of the system represented on three dimensional system (Ntot, Nart, No) (Fig. 23) shows the vectors
of acceleration )(ta Z for several moments of time t1, t2, t3 and t4.
The vector of the acceleration )(ta Z that coincides with the direction of the variation of the vector ZV (Fig.
22) in general is not tangent to the trajectory but forms an angle as shown in the Fig. 23.
The acceleration )(ta Z can be represented for each point of the trajectory as the sum of two components:
)()()( tatata ZZZ n   (43)

Fig. 23 The trajectory of the movement of the system of stocks on the three dimensional space (Ntot, Nart,
No)
The component )(ta Z lies on the tangent to the trajectory and is called the tangential acceleration. The
meaning of the tangential acceleration is the change in the magnitude of the velocity per unit time. If the
velocity is increasing, the direction of the tangential acceleration coincides with the direction of the velocity or
with the direction of the travel (Fig. 23, p. 1, p. 4). If the speed decreases, the direction of this acceleration is
opposite to the direction of the speed. (Fig. 23, p. 2)
The component )(ta nZ is called normal acceleration. This acceleration indicates only the change of the
direction of the speed per unit time. It is always directed to the center of the curvature of the trajectory. (Fig. 23)
Only for the case of linear motion, the normal (centripetal) acceleration is zero because in this case the velocity
direction does not change (Fig. 23, p. 3). Characteristically for the p. 3 of the Fig. 23 is that the radius of the
curvature of the trajectory r is very big (r→∞) and the normal acceleration )(ta nZ tends to zero. It is important
to mention that if the inventory has the continuous tendency of the increasing as for the point 4 of the Fig. 23,
then the resultant )(ta Z is oriented up (in the direction of the increasing of Ntot). For example, the point 2 of
the Fig. 23 has the tendency of the decreasing of the value Ntot and therefore the resultant )(ta Z is oriented
down.
In order to see better how each component of N0, Nart and Ntot varies separately as the dependence of the
orientation of the acceleration resulting vector )(ta Z , it is necessary to project this acceleration vector
)(ta Z on the plane (N0; Nart) and on the axis Ntot. (Fig. 24)
The vector AB that corresponds to the resulting vector )(ta Z for the moment of time 1 has the
projection 11BA on the plane (N0; Nart). The orientation of the vector 11BA indicates on the increasing of the
quantity N0 (vector 22BA ) and the decreasing of Nart (vector 33BA ), so that the result of Ntot is the decreasing
(vector 44BA ). (Fig. 24). The another moment of time 2 is characterized by the vector CD. Its projection on
the plane (N0; Nart) indicates on the vector 11DC and its projections on the axes N0 and Nart has the vectors
22DC and 33DC . The increasing of N0 is more bigger than of the decreasing of Nart, so that the final result
gives the increasing of Ntot (vector 44DC ) than in comparison with the previous case of the moment of time 1.
In this way it is solved the problem how each of the components of the inventories Ntot, Nart, No varies according
to the projections of the acceleration resulting vector )(ta Z on the coordinate axes.
The absolute value of the acceleration is :

   22
)()()( tatata nZZZ   
(44)
The tangential acceleration
dt
tdV
ta Z
Z
)(
)( 
 
is the first derivative with time. Only for the case when the
speed is constant (the case of uniform movement), then the acceleration Za is zero. The normal acceleration
depends on the radius of the curvature of the trajectory r and is determined by
r
tV
ta Z
Zn
2
)(
)( 
  .
Fig. 24 A modality of the explanation of the components quantities variations of N0; Nart ; N tot according
to the projections of the resulting acceleration vector )(ta Z
Taking into consideration the expression (37): totoart NNNZ  , then the momentary speed is
written as:
dt
tdN
dt
tdZ
tV tot
Z
)()(
)(  (45)
So, the changeable in time of the inventory )(tNtot depends both on the amounts of products )(tNo of one
article and the quantity of articles )(tNart . The quantity of articles )(tNart also in general is dependent on
the time because the articles could have in general the seasonal character (during summer more various articles
for example, but during winter more limited to a limited quantity). In general the supply-demand processes has
the seasonal character.
In order to describe quantitatively the supply-demand processes the following system formed of two subsystems
can be examined: 1) the subsystem of supplier (dealers); 2) subsystem of demander (shops, pharmacies, etc).
(Fig. 25)
The processes inside of this complex supplier-demander system are stochastical. The stochastical processes are
such random processes which evaluate in time and are variable with time. [34].
Let, the quantity of products is N0st of one article of the first subsystem of dealers at the initial moment of time t
=0 . This value of products of one article N0st is well planned statistically due to of the long period of activity of
the system and due to of statistical processes and analyses of the data. This is like a stationary value of the
products of one article.
The processes of receiving of the stocks by the shops evaluate with time and during the time the quantity of
products is increased. Which type of functional law of the amounts of products as a function of time takes
place? The result of the amount of transferred products for the respective interval of time is influenced by a lot
of factors: the price of product, socio-economical status of the customers (patients), geographical place, stock
reserves that are supplied at this respective moment of time, weather conditions, so all the factors that are
described by marketing mix policies. For example, if ten thousand products are sold during ten months, then it
means one hundred products averagely within three days and the linear functional law takes place:
tN 
3
100
0 ; t - days ( 1-st day, 2 -nd day,.....).

Fig.25. Schematical representation of the interaction between demander and supplier with the result of
transfer of stocks from first subsystem to the second subsystem
Is this linear law is valid for all rating marketing groups? Or, has this linear law some limits of application ?
The research papers [35], [36], [37], [38], [39], [40], [41], [42] explain these processes and after the processing
of big amounts of statistical data and analyses the following exponential expression with asymptotical
increasing of inventory level of one article )(tNo is suggested:
)1()( 0
bt
sto eNtN 
 (46)
stN0 is saturation value of inventory level of one article; t - interval of time; b - exponential rate constant that
depends on a lot of factors regarding marketing mix policies. The meaning of this exponential rate constant b is
the inverse interval of time during which the quantity of products of the first subsystem is decreased e times. (e
≈2.71). The measurement unit of b is [b]=day-1
, month-1
, year-1
, etc.
The respective graphic (Fig. 26) of the expression (46) with asymptotical increasing is:
Fig. 26 The graphic of )(tNo with asymptotical increasing of the inventory model
It can be observed from Fig. 26 that for the small values of time the shape has the linear segment and for the
bigger values of time then it is increased till the saturation value of stN0 . The exponential function has linear
approximation for the small values of b∙t : [43]
bte bt

1 tbNeNtN stst o
bt
oo  
)1()(
(47)
The question about which segments of time is valid for such linear approximation can be answered when the
comparison of linear and exponential graphs are plotted on the same plane Fig. 27.
It can be seen from the Fig. 27 that if the value of b is increased then the segment of linear approximation is
decreased and also if the value of stN0 is increased, the segment of linear approximation is decreased too.
Referring to Fig. 25, we have that the supplier subsystem at initial moment of time (t=0) contains the amount
of products stN0 and this subsystem after the interaction with the subsystem of demanders evaluate with time
like as the dependence that is represented on the Fig. 28:

Fig. 27. The numerical simulation of the expression with asymptotical increasing in comparison with the
linear approximation as the dependence of time for two values of exponential rate constants b
Fig. 28. The evaluation with time of the amount 0N of products of supplier
The process of the purchasing from the supplier to the demander of some amount of products ΔN during the
interval of time t is represented schematically on the Fig 29 :
Fig. 29 The evaluation with time of the amount of products of the supplier and the demander
So, according to the Fig. 29 the amount of the products of the supplier at the moment of time t is
NNost  and the respective amount of the products that are transferred from the supplier to the demander is
N . The total sum of the amounts of the first subsystem and the second subsystem is constant with time:
 stst o
subsystemIIsubsystemI
o NconstNNN 



(48)
The amount of products of the first subsystem at the moment of time t is :

)()( tNNtN ooo st
 ; but the value )(tNo is:
bt
oo eNtN st

)( (49)
Referring to Exp. 48, then : ststst oo
bt
oooo NtNeNNtNtN  
)(;)()(
(50)
and the final expression of )(tN for the second subsystem at this moment of time t is:
)1()( bt
o eNtN st

  (51)
Referring to the marketing rating groups A, B, C, X, Z with the stocks articles and the amount of stock articles
is Noj of one article, then the Exp. 51 can be written analogically as:
AjZeNtN
tb
ooj
j
jst


);1()( (52)
The values of ΔNoj are increased like as the shapes of the dependences that are represented on the Fig. 27 and
asymptotically reaches the stationary value
jstoN for the interval of time t . Such form of asymptotically
reaching of the dependence is explained by the fact that some products have seasonal characters and finite
product life cycles of some products of the system of stocks [44], [45]. If the demand is continuously and
permanently and the product exist on the market permanently for a very long time then the increasing of the
amounts takes place by linear function. [44], [45].
The numerical simulation of the Exp. 52 for various values of the stationary amounts for one stock
article
jstoN and various values of exponential decay constants b is represented on the Fig. 30.
Fig. 30. The numerical simulation of the behavior with time of sold amounts ΔNoj at various values of
exponential decay constants b
The results of numerical simulation that is represented on the Fig. 30 can take place in general for all
rating marketing groups. It is observed from the graphic that if the value b is increased the saturation till the
value N0st is reached more quickly with the interval of time shorter than for the small values of b. For the value
b=0.1 we have that the stationary value N0st = 2000 is reached during greater interval of time in comparison with
the value N0st = 500.
Rating coefficients of the stock of the rating groups that show the capacities of turnovers from one stock article
also reach stationary states coinciding with the series of Fibonacci numbers.
The expression (52) AjZeNtN
tb
ooj
j
jst


);1()( allows to represent the dependence of Kj(t)
as:    tb
jj
tb
p
j
p
jtb
ooj
j
st
jstj
jst
eKtKe
P
K
P
tK
eNtN





 1)(1
)(
)1()(
(53)
Schematically, this dependence on time for the various values of exponential decay constants b is represented in
the (Fig.31).
Also, it is observed that if the value of the exponential decay constant b is increased the reaching of the
saturation takes place at more shorter interval of time (Fig. 31).

Fig. 31. The simulation of the rating coefficients of the stocks as the function of time for different values
of exponential decays constants b
Regarding the items (stock articles) the similar dependence in the form of asymptotical increasing takes place
that is represented on the Fig.32.
Fig. 32. The behavior with time of the items of the rating groups with asymptotical decreasing
The expression about the amount of stock articles as the function of time can be written analogically as the
expression (52) like: AjZeNtN
tB
jartart
j
stj


);1()( (54)
the rate exponential constant of the stock articles in general can be different of exponential decay constant b and
is signed as B. The measurement unit of B is the same as for b.
It is necessary to mention that the scheme of the transferring of products from the dealers (supplier) to the
demander is valid also for the case of the interaction between the shops and the customers. In this situation the
shop plays the role of supplier but the customers play the role of demanders.
The rating marketing group A has bigger exponential rate constant B and the level of saturation is situated
higher than of B, C, X, Z (Fig.32). The respective interval of time is shorter for the bigger value of B in
comparison with the smaller one.
In order to understand the processes that takes place as the result of the selling, then the Fig. 30 that shows
numerical simulation can be applied for some examples of products. For example, one OTC pharmaceutical
product is researched with continuous permanently demand with big fluctuations within the values from 15 till
55 products each month. (Fig. 33, a)
These sold products of the respective month are signed by the value ΔNom (meaning momentary amounts of
sold products of the respective month). In order to use these model of asymptotical increasing it is necessary to
sum previous values of ΔNom till the respective last moment of time and then the amount of sold products for the
respective stock article is  omo NtN )( .
The graphic of the function )(0 tfN  in general could be linear or with asymptotical saturation. For this
OTC product the following linear graphic is obtained that is represented on Fig. 33, b. Nevertheless that the

demand values ΔNom are uncertainty with big instant fluctuations all time, the linear dependence of cumulative
amounts )(0 tfN  is obtained and the coefficient of correlation R=0.998 indicates the strong belonging to
the linear function. The proportional coefficient of this linear function shows the average amount sold during
one month (≈32 products per month), (Fig. 33, b). The graphic of the function )(ln tf
NN
N
ost
st







of
course for this case is nonlinear (fig. 33, c) due to of the fact that the dependence ΔNo(t) is not with the
asymptotical saturation.
Fig. 33. The application of the model with asymptotical increasing to the real example of OTC product.
In general we can remark that the dependence )(ln tf
NN
N
ost
st







gives the answer about the processes
that are developed with time. If such dependence )(ln tf
NN
N
ost
st







is not linear, then we can conclude
that the activity is expected to be instant and stable with the stable demand just if the big fluctuations exist and
the respective interval of time of these fluctuations is stable in time (Fig. 33,a). Only the case of linear form of
the dependence )(ln tf
NN
N
ost
st







suggests the seasonal character of the process or in some case could
be just finite life cycle of the products.
Another example is about well known product Panthenol spray. This product has seasonal character and the
values Nom have the peaks that are represented on the Fig. 34, a. The peaks represent the great demand at the
respective moment of time (7-th - 8-th months of the year). If we take only the interval of time one year then an
asymptotical saturation is observed on the Fig. 34, b. The graphic )(ln tf
NN
N
ost
st







that is represented
on Fig. 34, c contains two linear segments: first till 7-th month and another till 12-th month. As two linear
segments exist then the conclusion is that this interval of time of one year contains one peak at seventh month
that means the great demand at this moment of time (Fig. 34,c). The Fig. 34, d contains six peaks corresponding
to the peaks of demand. The graphic of Fig. 34, e gives more detailed information. Beside that, it gives the
information about six peaks during the entire period of time and also the answer about of the linearity of the
graphic )(ln tf
NN
N
ost
st







is given. One important moment we can remark, that if
)(ln tf
NN
N
ost
st







is linear then the criterion of the saturation takes place (seasonal character or finite
life cycle).

Fig. 34. The application of the model of asymptotical increasing for the real example of Panthenol spray
products
Regarding the product with the finite life cycle, we have that the demand of this product is decreased gradually
with time with some exceptions that sometimes the fluctuations from minimal till maximal values take place but
finally the demand reaches zero (Fig. 35, a).
Fig. 35. The application of the model of asymptotical increasing to the real example of the product with
the final life cycle
The cumulative value ΔNo(t) reaches asymptotically the saturation (Fig. 35, b). The total answer about the
degree of asymptotical saturation gives the graphic of the dependence )(ln tf
NN
N
ost
st







, (Fig. 35, c).
The approximated linear dependence of )(ln tf
NN
N
ost
st







that is represented on the Fig. 35, c shows
the character of the finite life cycle of this product. The exponential rate constant b can be found from this linear
dependence by the slope to the axis x.
This value b is b=0.099(months-1
), (Fig. 35, c). This constant b can also be found from Fig. 35, b taking into
consideration only the linear segment corresponding to the small values of the interval of time 27 months.
 
tbNbtNeNtN stost
bt
oo st 0)11()1()( tbNtN stoo  )(
(56)

The values 770 oN ; 1040stoN ; 27t (months) from the Fig. 35, b, and:
)(027.0
1040.27
770 1
 monthsb . The value b from Fig. 35, c that corresponds strictly to the linear
dependence gives: )(03.0
26
8.0 1
 monthsb
The correlation calculations for the entire linear segment of the Fig. 35, c gives b=0.099(months-1
).
The full segment till 70 months almost is good described by linear dependence. The value of correlation
coefficient R2
states about the strong connection to the linear dependence. The question is which values of b are
valid? Both values are valid. Just if we have planned stock reserve, the interval of time for total selling of this
stock reserve is calculated as:
)(33
03.0
11
months
bbN
N
t
ost
o


 or )(10
099.0
11
months
bbN
N
t
ost
o



(57)
It means that the full stock reserve with the quantity Nost=1040 can be sold minimum during ten months till
maximum thirty three months. The respective amounts during 10 months
is: )(3121003.01040 productsNo 
So, the question is again, why all this information is necessary? Each activity is based on experience and
practice. In order to have more performed the activity it is necessary to have a large information about previous
activity till the recent moments. We can forecast the activity for the future if we have the value of the rate
exponential constant b. Just if the forecasted amounts of products deviate from real ones then the remained
reserve will be used forward with the condition if the expiration date is far. And therefore it is necessary to
consider as long as is the interval of time the probability is bigger to have small deviations from real amounts.
So, the values of exponential rate constant b serve as the criterion of levelling of the forecasted amounts and if
the interval of time is bigger then the more real results are obtained.
The next example is about the subsystem of two products: the Panthenol spray and one of OTC product. Both
product with their momentary amounts Nom are represented on Fig. 36, a. The panthenol spray has seasonal
character but OTC one has all time the demand with big fluctuations. The cumulative value in this case is
calculated as the average of two products:
2
)()(
)(
 
 panthenol OTC
omom
o
tNtN
tN (58)
The respective graphic ΔNo(t) is represented on Fig.36, b. In general this graphic is linear and flexible points are
observably corresponding to their six peaks of the product panthenol that are similar to the Fig. 34, d. The
peaks are attenuated by the fluctuations of OTC product but the peaks are bigger than the fluctuations and
therefore the thresholds are visible on Fig. 36, b that corresponds to six seasons of the entire interval of time.
The points on Fig. 36, b are almost arranged on the straight line and the coefficient a of the linear function
shows the average amount of both product per month.
Fig. 36. The application of the model of asymptotical increasing to the subsystem of two products:
Panthenol and one of OTC product
The following question is about if there is no a big error for the forecasting if the value a is considered for all
items. The found value a is more real for Panthenol than for the second product of OTC due to of the fact that
the second product has the big fluctuations. Just if this value a is applied for the forecasting of the amounts of

the OTC product then maybe the eventually surplus of stock is used forward. In order do not have the big errors
during the process of the forecasting it is necessary to have small interval of time for the forward like several
days till several weeks. Another important moment it is necessary to mention that promotional packages also
influences on the planning of necessary amounts. So, in the conditions of high hyper competition the forecasting
politics based on the found value of exponential decay plays an important role. In general it would be avoidable
to use the value of exponential rate constant for the full rating group and this value will serve for the forecasting
of each item for small interval of time taking into consideration promotional packages.
Generally speaking, each item has a lot of peculiarities in the real conditions of competition and a series of
factors act on the demand of products like seasonal character, finite life cycle of product due to of the fact that a
lot of generics substitute the previous ones. The prices of generics are usually significantly lower than the
corresponding originals. The appearance of generic products on the market often leads to a decrease in the price
of the original as well as of the individual competing products. Generally, a generic product is offered at a price
that is many times lower than the original price of the original. There are about ≈25% in Bulgaria of generic
products, and ≈75% original pharmaceutical products [46].
It was mentioned before that the exponential rate constant b has the property of the leveling of the amounts for
each item that serves as the criteria of the forecasting. For example, the subsystem of eight items that is
represented on the Fig. 37, a shows the items which have different behaviors, some of them are seasonal, some
with finite life cycle, and some with fluctuations of the demand all time.
Fig. 37. The application of the model of asymptotical increasing to the subsystem of eight products
The average cumulative amount ΔNo(t) of the item is calculated as:
art
Nart
i
om
o
N
tN
tN
i

 1
)(
)( (59)
All these peculiarities in time of the items lead to such asymptotical increasing represented on Fig. 27,b. The
respective dependence )(ln tf
NN
N
ost
st







is almost linear for this subgroup of products meaning that the
model of asymptotical increasing is valid and applicable.
Regarding the stock articles the situation has the specific peculiarities. The amount of the stock articles can be
various as the dependence of the specific interval of time and could have the seasonal character. Sometimes the
set of articles is fixed and non changeable but sometimes it could be in the state of the decreasing of the set or in
the state if increasing. The inventory in this case could contains initially the quantity of articles
jstartN and this
inventory is evaluating in time depending on the value )(tN jart .
Referring to the Exp. 54, it is possibly to calculate the value of exponential rate constant B if the quantity of the
stock articles at the stationary state
jstartN and the quantity of the items jartN that are sold during the interval
of time t (expressed in quantities of months) for the respective rating group j is known. So, the expression of the
rate constant B of the articles is written as:
t
tNN
N
B
jartart
art
jst
jst










)(
ln
(60)

Mechanics Phenomenological Econophysics For The Description Of Microeconomical Systems Of Stocks

Mechanics Phenomenological Econophysics For The Description Of Microeconomical Systems Of Stocks

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