System state, standard state. The concept of the state of the system of the System as well as the state

A system of bodies, or simply a system, we will call the totality of the bodies under consideration. An example of a system is a liquid and vapor in equilibrium with it. In particular, the system can consist of one body.

Any system can be in various states, differing in temperature, pressure, volume, etc. Such values characterizing the state of the system are called state parameters.

Not always any parameter has a specific meaning. If, for example, the temperature at different points of the body is not the same, then the body cannot be assigned a certain value of the parameter T. In this case, the state is called nonequilibrium. If such a body is isolated from other bodies and left to itself, then the temperature will equalize and take the same value T for all points - the body will go into an equilibrium state. This value of T does not change until the body is brought out of equilibrium by external action.

The same can be the case for other parameters, for example for pressure. If we take a gas enclosed in a cylindrical vessel, closed by a tightly fitted piston, and begin to quickly push the piston in, then a gas cushion will form under it, the pressure in which will be greater than in the rest of the gas volume. Consequently, the gas in this case cannot be characterized by a certain pressure value, and its state will be nonequilibrium. However, if you stop moving the piston, then the pressure at different points in the volume will equalize and the gas will go into an equilibrium state.

The process of transition of a system from a nonequilibrium state to an equilibrium state is called the relaxation process or simply relaxation. The time taken for such a transition is called the relaxation time. The relaxation time is taken as the time during which the initial deviation of any value from the equilibrium value decreases by a factor. Each parameter of the system has its own relaxation time. The longest of these times plays the role of the relaxation time of the system.

So, the equilibrium state of a system is a state in which all the parameters of the system have certain values that remain constant under unchanged external conditions for an arbitrarily long time.

If the values of any two parameters are plotted along the coordinate axes, then any equilibrium state of the system can be depicted by a point on the coordinate plane (see, for example, point 1 in Fig. 81.1). The nonequilibrium state cannot be depicted in this way, because at least one of the parameters will not have a certain value in the nonequilibrium state.

Any process, that is, the transition of a system from one state to another, is associated with a violation of the equilibrium of the system. Consequently, when a process occurs in the system, it passes through a sequence of nonequilibrium states. Turning to the already considered process of gas compression in a vessel closed by a piston, we can conclude that the imbalance when the piston is pushed in is the more significant, the faster the gas is compressed. If the piston is pushed in very slowly, then the equilibrium is disturbed insignificantly and the pressure at different points differs little from a certain average value. In the limit, if the gas is compressed infinitely slowly, the gas at each moment of time will be characterized by a certain pressure value. Consequently, in this case, the state of the gas at each moment of time is equilibrium, and the infinitely slow process will consist of a sequence of equilibrium states.

A process consisting of a continuous sequence of equilibrium states is called equilibrium or quasi-static. It follows from what has been said that only an infinitely slow process can be in equilibrium.

With a sufficiently slow course, real processes can approach the equilibrium one arbitrarily close.

The equilibrium process can be carried out in the opposite direction, and the system will go through the same states as in the forward course, but in the reverse sequence. Therefore, equilibrium processes are also called reversible.

A reversible (i.e., equilibrium) process can be depicted on the coordinate plane of the corresponding curve (see Fig. 81.1). We will conventionally depict irreversible (i.e., nonequilibrium) processes with dashed curves.

The process in which the system, after a series of changes, returns to its original state is called a circular process or cycle. The cycle is graphically depicted as a closed curve.

The concepts of an equilibrium state and a reversible process play an important role in thermodynamics. All quantitative conclusions of thermodynamics are strictly applicable only to equilibrium states and reversible processes.

Molecular kinetic approach. Molecular physics is based on two main principles:

any body - solid, liquid or gaseous - consists of separate particles, which we call molecules (atoms, ions, etc.);

particles of any substance are in random chaotic motion, which, in the absence of external forces of force, does not have any preferential direction. This movement is called thermal, since its intensity determines the temperature of the substance.

In the first paragraph, apart from electrically neutral atoms and molecules, electrically charged particles - ions - are mentioned as particles of which a substance can be composed. First of all, this is a very important case of the plasma state of matter. According to available estimates, approximately 95% of the visible matter in the Universe is in the plasma state. In addition, in solutions - for example, table salt in water - the dissolved substance exists in the form of ions and, further, metals are a set of positive ions vibrating about equilibrium positions (crystal lattice sites) and free electrons, which form an electron gas. In what follows, the main attention will be paid to the "ordinary" "state of matter, when its constituent particles are electrically neutral. Plasma, as a special state of matter, solutions and metals will be considered separately. influences do not have any preferential direction. "Note the following in this regard: in anisotropic crystals there are preferred directions due to the interaction of the particles that make up the crystal, and not associated with external force fields. Consideration of such situations is beyond the scope of this chapter.

Molecular kinetic theory sets itself the goal of interpreting those properties of a substance that are directly observed experimentally (viscosity, thermal conductivity, etc.) as the total result of the action of molecules. At the same time, she uses a statistical method, being interested not in the movement of each individual molecule, but only in such average values that characterize the movement and interaction of the entire set of molecules. Molecular kinetic theory operates with the basic laws of physics acting on microscopic level - the laws of classical mechanics, electrodynamics, etc. Therefore, she is able to predict the values of many physical parameters of the system on the basis of, as they say, first principles. In this chapter, we will focus on the derivation of well-known laws for ideal gases based on molecular kinetic theory.

State of the system. In any branch of physics, the study of phenomena begins with the isolation of a set of bodies, which is called system.

Imagine, for example, a gas (system) in a closed cylinder under a piston (medium), Fig. 1.1.

Rice. 1.1. Gas in a closed cylinder under the piston

Changing the position of the piston or the temperature of the cylinder walls changes the state of the system.

The state of such simple systems as gas is characterized by the following macroscopic parameters: volume, pressure, temperature . Naturally, we also need parameters that determine the system - its mass m,relative molecular weight M(or moth mass m).

In total, four quantities: volume , pressure , temperature, weight . Or, with a known mole mass of a substance in the system, the number of moles. If the system is a mixture of different substances, then it is necessary to add the relative concentrations of the components of the mixture: , here is the mass of the substance. Obviously, in the latter case, there are not four parameters, but more.

Recall that

Another - equivalent - definition of a mole reads:

Note that the modern definition of the Avogadro number says that the Avogadro number is equal to the number of atoms of the isotope 12 C contained in 0.012 kilograms of carbon-12. Thus, a mole can be defined as follows:

When solving problems, the values of the relative molecular weight M elements are taken from the periodic table. Molar mass is easy to calculate:

For example, for gold

For complex substances, it is necessary to perform simple arithmetic operations, for example, for carbon dioxide:

Generally speaking, such parameters of the system as pressure, temperature, density of matter can have different meanings at different points. In this case, the system as a whole cannot be assigned certain values of these parameters, the system is in non-equilibrium state. Experience shows, however, that if external conditions are unchanged, then the system over time comes to equilibrium state: the pressures and temperatures of its individual parts are equalized, so that the parameters of the system take on certain values that remain constant for an arbitrarily long time. In this case, the external conditions should be such that there is no transfer of matter, energy, momentum, etc. in the system.

Let us consider, for simplicity, a system, the total mass of which is unchanged, its composition and the relative concentrations of its constituent substances are unchanged. This is the case, for example, when there are no chemical reactions in the system. With a more general approach: in the system there are no processes of birth and destruction of its constituent particles. For example, the reaction of the formation of water molecules from oxygen and hydrogen molecules

can be considered as the process of destruction of particles and and the creation of particles. In a number of cases, for example, in a gas of photons (thermal radiation), the presence of production processes in the destruction of particles is fundamentally important.

Additional Information

http://www.femto.com.ua/articles/part_2/4471.html - Physical encyclopedia. Chemical potential: a physical quantity required to describe the properties of thermodynamic systems with a variable number of particles;

http://www.femto.com.ua/articles/part_1/0017.html - Physical encyclopedia. Avogadro's law;

http://marklv.narod.ru/mkt/mkt.htm - School lesson with pictures on the molecular kinetic hypothesis;

As will be seen in what follows, for a complete description of the equilibrium state of such a system, only three parameters are sufficient:. Moreover, if the state is in equilibrium, then there is a relationship between these three parameters: the given two parameters of the system (for example, its temperature and volume) uniquely determine the third one (in this case, pressure). Mathematically, this connection can be characterized the equation of state of the system

where is the specific kind of function F depends on the properties of the system. An example is equations Clapeyron - Mendeleev for perfect or Van der Waals for imperfect gases (these equations will be considered below).

Thus, at equilibrium system with constant mass, composition and relative concentrations its constituent substances - in the future we will not stipulate this every time - there are only two independent parameters and its equilibrium state can be depicted graphically by a point on a plane (Fig. 1.2), where any two of the three parameters are plotted along the axes -, or:

Rice. 1.2. Equilibrium states of the system on the diagrams (p, V), (p, T) and (V, T)

Process is any transition of the system from one state to another.

The process is always associated with a violation of the thermodynamic equilibrium of the state of the system. At the moment, under a thermodynamically equilibrium state, it is sufficient to understand a state in which all possible energy exchange processes are absent: 1) none of the subsystems of the system does work on other subsystems; 2) none of the subsystems of the system exchanges heat with other subsystems of the system; 3) none of the subsystems of the system exchanges particles with other subsystems of the system. As will be seen later, there are no other types of energy exchange in ordinary ones (in which there are no processes of creation and destruction of particles). From this, ultimately, it follows that it is sufficient to specify only three independent parameters (for example: the number of particles, volume and internal energy) to describe the thermodynamic equilibrium state of a one-component system.

If the state of the system changes over time, then some process takes place in the system. The converse, generally speaking, is not true: the state of the system may not change, although there is a process in it - a stationary, but nonequilibrium state of the system. For example, in a stationary process of heat transfer, the state of the system is nonequilibrium, although it remains unchanged in the sense that the distributions of temperature, pressure, density, etc., over the volume of the system do not change.

With an infinitely slow course of the process, it can be assumed that at any given moment in time, the state of the system is in equilibrium. Physically, this means that the time characteristic of the process is much longer than the time for establishing equilibrium in the system, which is also called the relaxation time . This process is called equilibrium process.

It is obvious that the equilibrium process is another idealization. In order for the process to be considered, with some finite accuracy, to be in equilibrium, it is necessary that the inequality

and the better it is performed, the closer the process is to equilibrium.

An equilibrium process can be thought of as a sequence of equilibrium states. In the future, only equilibrium processes will be studied (unless otherwise specifically stated).

Since the state of the system is depicted by a point on the diagram, and the process is a sequence of equilibrium states, such a process is depicted on the diagram by a line. Each point on the line is a conditionally equilibrium intermediate state of the system. An equilibrium process is a process reversible, that is, it can flow in the opposite direction, going through the same intermediate states in the reverse order, and no changes will remain in the surrounding bodies.

Naturally, then no forces similar to friction forces should act in the system. Below we will get acquainted with diagrams describing some characteristic processes in thermodynamic systems.

Knowing the state of the system, we can find various state functions - physical characteristics that depend only on the state of the system, that is, they take the same values every time the system is in a given state, regardless of its prehistory.

Temperature. Any system has a certain margin internal energy, not related to the position or movement of the system as a whole relative to the external environment. We will talk about internal energy in more detail, but now we have enough intuitive understanding that, having thrown an egg at some speed, we will not cook it, although the kinetic energy of the egg will increase. To cook a soft-boiled egg, you do not need to throw it, but warm it up.

To quantitatively characterize the internal energy, the concept is introduced temperature. Temperature occupies a special place among physical quantities. Experience shows that it characterizes the state of thermal equilibrium of bodies. If two bodies with different temperatures are brought into contact, then as a result of the interaction between the molecules, these bodies will exchange energy. After a while, the temperatures will equalize and the transfer of heat will stop, a state of thermal equilibrium will come. The state of thermal equilibrium is the state into which any isolated system passes over time.

Conventional methods for determining temperature are based on the dependence of a number of properties of bodies (volume, pressure, etc.) on it. In this case, a thermometric body and a graduation of the temperature scale are selected. The most common is the centigrade scale (Celsius scale, Fig. 1.3).

Rice. 1.3. Centigrade Celsius

The section of this scale between the freezing points (water crystallization or, what is the same, ice melting) and water boiling at normal atmospheric pressure is divided into 100 equal parts. This part is called degrees Celsius(denoted by t ° C). Thus, the crystallization point of water corresponds to 0 ° C, and the boiling point is 100 ° C... We emphasize that both at a normal pressure of 760 mm Hg. Art. In the United States, the Fahrenheit scale is also used (denoted t ° F). For the zero of his Fahrenheit scale, he chose the lowest temperature that he could reproduce in his laboratory - the melting point of a mixture of salt and ice. The freezing point of water on this scale corresponds to the temperature 32 ° F, and the boiling point is 212 ° F. This interval is divided not into one hundred, but into 180 parts (similar to angular degrees). Therefore, Fahrenheit is less than Celsius (factor 100/180 = 5/9 ). The relationship of temperatures in these two scales is given by the formulas

Rice. 1.4. Correspondence between scales

Physics uses a thermodynamic (old name: absolute) temperature scale (Kelvin scale), which does not depend on the thermometric body, but is established on the basis of the laws of thermodynamics.

Currently, one kelvin is defined as follows: kelvin is a unit of thermodynamic temperature equal to a fraction of the thermodynamic temperature of the triple point of water. The triple point of water was chosen instead of its boiling point because the temperature of the triple point is independent of pressure and is more accurately determined. On the Celsius scale, the triple point of water corresponds to temperature. The value of one kelvin (denoted by K) coincides with the value of the degree Celsius. Taking into account the indicated difference of 0.01 kelvin, for the relationship of temperatures on the thermodynamic scale and centigrade Celsius, we obtain

Examples of typical temperatures in nature are shown in Fig. 1.5.

Rice. 1.5. Temperature of various physical processes

Additional Information

http://kvant.mirror1.mccme.ru/1990/08/temperatura_teplota_termometr.htm - Kvant Journal, 1990, No. 8, pp. 10 - 19, A. Kikoin, Temperature, heat, thermometer;

Do you know physics? Library "Kvant", issue 82, Science, 1992. P. 130, questions 113, 115 on the uniformity of temperature scales (answer on pages 136-138);

http://ilib.mirror1.mccme.ru/djvu/bib-kvant/kvant_82.htm - Perelman Ya.I. - Do you know physics? Library "Kvant", issue 82, Science, 1992. P. 130, question 112: origin of the temperature scale Fahrenheit (see answer on pages 135-136);

http://www.femto.com.ua/articles/part_2/4070.html - Physical encyclopedia. Instruments for measuring temperatures from the highest to the lowest are described.

Rice. 1.6. Thermogram of a cup of hot tea

Biomedical relevance of the topic

Thermodynamics is a branch of physical chemistry that studies any macroscopic systems, changes in state of which are associated with the transfer of energy in the form of heat and work.

Chemical thermodynamics is the theoretical basis of bioenergy - the science of energy transformations in living organisms and the specific features of the transformation of some types of energy into others in the process of life. In a living organism, there is a close relationship between the processes of metabolism and energy. Metabolism is the source of energy for all life processes. The implementation of any physiological functions (movement, maintaining a constant body temperature, secretion of digestive juices, synthesis in the body of various complex substances from simpler ones, etc.) requires energy. The source of all types of energy in the body are nutrients (proteins, fats, carbohydrates), the potential chemical energy of which in the process of metabolism is converted into other types of energy. The main way of releasing chemical energy necessary to maintain the body's vital activity and the implementation of physiological functions is oxidative processes.

Chemical thermodynamics makes it possible to establish a relationship between the energy costs of a person performing a certain work and the caloric content of nutrients, makes it possible to understand the energetic essence of biosynthetic processes occurring due to the energy released during the oxidation of nutrients.

Knowledge of the standard thermodynamic values of a relatively small number of compounds allows thermochemical calculations to be performed for the energy characterization of various biochemical processes.

The use of thermodynamic methods makes it possible to quantify the energetics of structural transformations of proteins, nucleic acids, lipids and biological membranes.

In the practice of a doctor, thermodynamic methods are most widely used to determine the intensity of the basal metabolism in various physiological and pathological conditions of the body, as well as to determine the calorie content of food.

Chemical thermodynamics problems

1. Determination of the energy effects of chemical and physicochemical processes.

2. Establishment of criteria for the spontaneous course of chemical and physicochemical processes.

3. Establishment of criteria for the equilibrium state of thermodynamic systems.

Basic concepts and definitions

Thermodynamic system

A body or a group of bodies separated from the environment by a real or imaginary interface is called a thermodynamic system.

Depending on the ability of the system to exchange energy and matter with the environment, isolated, closed and open systems are distinguished.

Isolated A system is a system that does not exchange either matter or energy with the environment.

A system that exchanges energy with the environment and does not exchange matter is called closed.

An open system is a system that exchanges both matter and energy with the environment.

System state, standard state

The state of the system is determined by the totality of its physical and chemical properties. Each state of the system is characterized by certain values of these properties. If these properties change, then the state of the system also changes, if the properties of the system do not change over time, then the system is in a state of equilibrium.

To compare the properties of thermodynamic systems, it is necessary to accurately indicate their state. For this purpose, the concept is introduced - a standard state, for which for an individual liquid or solid body is taken such a physical state in which they exist at a pressure of 1 atm (101315 Pa) and a given temperature.

For gases and vapors, the standard state corresponds to a hypothetical state in which a gas at a pressure of 1 atm obeys the laws of ideal gases at a given temperature.

The values referring to the standard state are written with the "o" subscript and the subscript indicates the temperature, most often it is 298K.

Equation of state

The equation that establishes the functional relationship between the values of the properties that determine the state of the system is called the equation of state.

If the equation of state of a system is known, then to describe its state it is not necessary to know the numerical values of all properties of the system. For example, the Clapeyron – Mendeleev equation is the equation of state for an ideal gas:

where P is the pressure, V is the volume, n is the number of moles of an ideal gas, T is its absolute temperature, and R is the universal gas constant.

It follows from the equation that to determine the state of an ideal gas, it is enough to know the numerical values of any three of the four quantities P, V, n, T.

State functions

Properties, the values of which during the transition of the system from one state to another depend only on the initial and final state of the system and do not depend on the path of the transition, are called state functions. These include, for example, pressure, volume, system temperature.

Processes

The transition of a system from one state to another is called a process. Depending on the flow conditions, the following types of processes are distinguished.

Circular or cyclical- a process, as a result of which, the system returns to its original state. At the end of the circular process, changes in any function of the state of the system are equal to zero.

Isothermal- a process that takes place at a constant temperature.

Isobaric- a process that takes place at constant pressure.

Isochoric- a process in which the volume of the system remains constant.

Adiabatic- a process that takes place without heat exchange with the environment.

Equilibrium- a process considered as a continuous series of equilibrium states of the system.

Unbalanced- the process by which the system passes through non-equilibrium states.

Reversible thermodynamic process- the process after which the system and the systems interacting with it (environment) can return to the initial state.

Irreversible thermodynamic process- a process after which the system and the systems interacting with it (environment) cannot return to their initial state.

The latter concepts are discussed in more detail in the section "Thermodynamics of Chemical Equilibrium".

STATE OF THE SYSTEM

in physics - is determined by a set of values characteristic of a given system of physical. quantities, called. state parameters. For example, mechanical condition system at each moment of time is characterized by the values of coordinates and impulses of all material points, forming this system. State electromagnetic field characterized by the values of the electric tension. and magnetic fields at all points of the field at each moment of time.

Big Encyclopedic Polytechnic Dictionary. 2004 .

See what "SYSTEM STATE" is in other dictionaries:

State of the system- characteristics of the system at the moment of its functioning. Since the system is described by a certain set of essential variables and parameters, then in order to express the SS, it is necessary to determine the values taken ... ... Economics and Mathematics Dictionary

state of the system- 3.2 system state: A specific combination of element states. Note Several system states can be combined into one state. Source: GOST R 51901.15 2005: Risk management. Application of Markov methods ... ...

state of the system- State of System System state Characteristic of the system at the moment of its functioning. Since the system is described by a certain set of essential variables and parameters, in order to express the state of the system, you need ... ... Explanatory English-Russian Dictionary of Nanotechnology. - M.

state of the system- sistemos būsena statusas T sritis automatika atitikmenys: angl. state of system vok. Systemzustand, m rus. system state, n pranc. état du système, m ... Automatikos terminų žodynas

state of the system- sistemos būsena statusas T sritis chemija apibrėžtis Makroskopiniais parametrais apibūdinama sistemos būsena. atitikmenys: angl. state of system rus. state of the system … Chemijos terminų aiškinamasis žodynas

state of the system- sistemos būsena statusas T sritis fizika atitikmenys: angl. state of system vok. Systemzustand, m rus. system state, n pranc. état du système, m ... Fizikos terminų žodynas

Aircraft system failure state- 14 Source: GOST 27332 87: Flight conditions of aircraft. Terms and definitions original document ... Dictionary-reference book of terms of normative and technical documentation

Aircraft system status- 10. State of the aircraft system State of the system Situation of the system Operation parameters of the aircraft system, determined by the nature of its activation and its operable or failure state, the presence of faults when ... ... Dictionary-reference book of terms of normative and technical documentation

aircraft system failure- failure state of the system Inoperative state of the aircraft system, characterized by the considered violation of the function of the system as a whole, regardless of the causes that caused it. [GOST 27332 87] Topics of flight conditions for flying ... ... Technical translator's guide

Failure state of the aircraft system- 14. Failure state of the aircraft system Failure state of the system Failure situation (title = Amendment, IMS 8 88). Inoperative state of the aircraft system, characterized by the considered violation of the system function ... Dictionary-reference book of terms of normative and technical documentation

Books

Radio control systems. Book 1. State and development trends of radio control systems. The authors of the collective monograph are well-known scientists, leading developers and specialists in the field of radio control systems. The book examines the state and development trends of radioelectronic ... Category: Electronics Series: Scientific and technical series Publisher: Radio Engineering, Manufacturer: Radiotechnics,
Radio control systems. Issue 1. State and development trends of radio control systems, VS Verba , The authors of the collective monograph are renowned scientists, leading developers and specialists in the field of radio control systems. The book examines the state and development trends of radioelectronic ... Category: Radio. Radio engineering Series: Publisher:

The state of any real system, at any given moment in time, can be described using a set that characterizes the system of quantities - parameter.

The number of parameters, even for a relatively simple system, can be very large, and therefore, in practice, for describing systems, only the most essential, characteristic parameters, corresponding to the specific goals of studying objects, are used. So, to study the state of human health from the point of view of the need to release him from work, the values of such parameters as temperature and blood pressure are taken into account first of all.

The state of a certain economic system is characterized by such parameters as the quantity and quality of products, labor productivity, return fund, etc.

Methods such as verbal description, tabular or matrix descriptions, mathematical expressions and graphic images can be used to describe the state and movement of the system.

Verbal description is reduced to a sequential listing and characteristics of the parameters of the system, trends in their change, the sequence of changes in the state of the system. The verbal description is very approximate and gives only general ideas about the system, in addition, it is largely subjective, since displays not only the true characteristics of the system, but also the attitude of the person describing them to them.

Tables and matrices received the most widespread use for the quantitative characteristics of the system, expressed by the values of their parameters at a certain fixed point in time. According to the data of a table or a set of tables corresponding to various points in time, diagrams and graphs can be built, giving a visual representation of the dynamics of the system.

To describe the movement of the system and change its elements, use mathematical expressions, which in turn are interpreted by graphs that reflect the course of certain processes in the system.

However, the most profound and adequate is formalized geometric interpretation states and movements of the system in the so-called state space or phase space.

System state space

System state space a space is called, at each point of which a certain state of the considered dynamic system uniquely corresponds, and each process of changing the state of the system corresponds to a certain trajectory of movement of the representing point in space.

To describe the movements of dynamic systems, a method is widely used based on the so-called phase space(n dimensional Euclidean space), along the axes of which the values of all n generalized coordinates of the considered dynamical system are plotted. In this case, a one-to-one correspondence between the state of the system and the points of the phase space is achieved by choosing the number of dimensions equal to the number of generalized coordinates of the dynamical system under consideration.

Let us denote by the parameters of some system by the symbols z1, z2… zn, which can be considered as the coordinates of the vector z, n of the dimensional space. Such a vector is a collection of real numbers z = (z1, z2..zn). The parameters z1, z2… zn will be called the phase coordinates of the system, and the states (phase of the system) will be represented by the point z in the phase space. The dimension of this space is determined by the number of phase coordinates, that is, by the number of selected by us to describe the system, its essential parameters.

In the case when the states of the system can be characterized by only one parameter z1 (for example, the distance from the point of departure of a train moving along a given route), then the phase space will be one-dimensional and displayed as a portion of the z-axis.

If the state of the system is characterized by two parameters z1 and z2 (for example, the movement of the car, expressed by the angle relative to a given direction and the speed of its movement), then the phase space will be two-dimensional.

In cases where the state of the system is described by three parameters (for example, speed and acceleration control), it will be represented by a point in three-dimensional space, and the trajectory of the system will be spatially curved in this space.

In the general case, when the number of parameters characterizing the system is arbitrary and, as in most complex economic systems, is much more than 3, the geometric interpretation loses clarity. However, the geometric terminology in these cases remains convenient for describing the state and motion of systems, in the so-called n-dimensional or multidimensional phase space (hyper space).

The number of independent parameters of the system is called number of degrees of freedom or the variance of the systems.

In real conditions, the system and its parameters (phase coordinates), as a rule, can change only in some limited chapters. So the speed of the car is limited by side-altars from 0 to 200 km per hour, the temperature of a person is from 35 degrees to 42, etc.

The region of the phase space beyond the limits of which the representative point cannot go is called the area of admissible states of the system... When researching and designing systems, it is always assumed that the system is within the range of its permissible states.

If the depicting point goes beyond this area, then this threatens the destruction of the integrity of the system, the possibility of its disintegration into elements, the violation of existing connections, that is, the complete cessation of its functioning as a given system.

The area of admissible states, which can be called the field of the system, includes all kinds of phase trajectories, that is, the lines of behavior of systems. The set of phase trajectories is called phase portrait of the considered dynamical system. In all cases, when the parameters of the system can take any values in a certain interval, that is, the smoothly depicting point changes, which can be located at any point within the region of permissible states, while we are dealing with the so-called continuous state space. However, there are a large number of technical, biological and economic systems in which a number of parameters - coordinates can only take discrete values.

Only discretely it is possible to measure the number of machines in a workshop, the number of certain organs and cells in a living organism, etc.

The state space of such systems should be considered as discrete, therefore, their point representing the state of such a system cannot be located anywhere, in the region of admissible states, but only in certain fixed points of this region. A change in the state of such systems, that is, their movement, will be interpreted by jumps of the representing point from one state to another, to a third, etc. Accordingly, the trajectory of motion of the representing point will have a discrete, discontinuous character.