Statistical physics occupies a prominent place in modern science and deserves special consideration. It describes the formation of macrosystem parameters from the movements of particles. For example, thermodynamic parameters such as temperature and pressure are reduced to the pulse-energy characteristics of molecules. She does this by specifying some probability distribution. The adjective "statistical" comes from the Latin word status(Russian - state). This word alone is not enough to express the specifics of statistical physics. Indeed, any physical science studies the states of physical processes and bodies. Statistical physics deals with an ensemble of states. The ensemble in the case under consideration presupposes a plurality of states, but not any, but correlating with the same aggregate state, which has integrative characteristics. Thus, statistical physics involves a hierarchy of two levels, often called microscopic and macroscopic. Accordingly, it examines the relationship between micro- and macrostates. The integrative features mentioned above are constituted only if the number of microstates is sufficiently large. For specific states it has a lower and an upper limit, the determination of which is a special task.
As already noted, a characteristic feature of the statistical approach is the need to refer to the concept of probability. Using distribution functions, statistical average values (mathematical expectations) of certain characteristics that are inherent, by definition, at both the micro and macro levels are calculated. The connection between the two levels becomes particularly clear. The probabilistic measure of macrostates is entropy ( S). According to the Boltzmann formula, it is directly proportional to the statistical weight, i.e. number of ways to realize a given macroscopic state ( R):
Entropy is greatest in the state of equilibrium of the statistical system.
The statistical project was developed within the framework of classical physics. It seemed that it was not applicable in quantum physics. In reality, the situation turned out to be fundamentally different: in the quantum field, statistical physics is not limited to classical concepts and acquires a more universal character. But the very content of the statistical method is significantly clarified.
The character of the wave function is of decisive importance for the fate of the statistical method in quantum physics. It determines not the values of physical parameters, but the probabilistic law of their distribution. L this means that the main condition of statistical physics is satisfied, i.e. assignment of probability distribution. Its presence is a necessary and, apparently, sufficient condition for the successful extension of the statistical approach to the entire field of quantum physics.
In the field of classical physics, it seemed that the statistical approach was not necessary, and if it was used, it was only due to the temporary absence of methods truly adequate to the nature of physical processes. Dynamic laws, through which unambiguous predictability is achieved, are more relevant than statistical laws.
Future physics, they say, will make it possible to explain statistical laws using dynamic ones. But the development of quantum physics presented scientists with a clear surprise.
In fact, the primacy of not dynamic, but statistical laws became clear. It was statistical patterns that made it possible to explain dynamic laws. The so-called unambiguous description is simply a recording of events that are most likely to occur. It is not unambiguous Laplacean determinism that is relevant, but probabilistic determinism (see paradox 4 from paragraph 2.8).
Quantum physics, by its very essence, is a statistical theory. This circumstance testifies to the enduring importance of statistical physics. In classical physics, the statistical approach does not require solving the equations of motion. Therefore, it seems that it is essentially not dynamic, but phenomenological. The theory answers the question “How do processes occur?”, but not the question “Why do they happen this way and not differently?” Quantum physics gives the statistical approach a dynamic character, phenomenology acquires a secondary character.
Statistical physics and thermodynamics
Statistical and thermodynamic research methods . Molecular physics and thermodynamics are branches of physics in which they study macroscopic processes in bodies, associated with the huge number of atoms and molecules contained in the bodies. To study these processes, two qualitatively different and mutually complementary methods are used: statistical (molecular kinetic) And thermodynamic. The first underlies molecular physics, the second - thermodynamics.
Molecular physics - a branch of physics that studies the structure and properties of matter based on molecular kinetic concepts, based on the fact that all bodies consist of molecules in continuous chaotic motion.
The idea of the atomic structure of matter was expressed by the ancient Greek philosopher Democritus (460-370 BC). Atomism was revived again only in the 17th century. and develops in works whose views on the structure of matter and thermal phenomena were close to modern ones. The rigorous development of molecular theory dates back to the middle of the 19th century. and is associated with the works of the German physicist R. Clausius (1822-1888), J. Maxwell and L. Boltzmann.
The processes studied by molecular physics are the result of the combined action of a huge number of molecules. The laws of behavior of a huge number of molecules, being statistical laws, are studied using statistical method. This method is based on the fact that the properties of a macroscopic system are ultimately determined by the properties of the particles of the system, the features of their movement and averaged values of the dynamic characteristics of these particles (speed, energy, etc.). For example, the temperature of a body is determined by the speed of the chaotic movement of its molecules, but since at any moment of time different molecules have different speeds, it can only be expressed through the average value of the speed of movement of the molecules. You can't talk about the temperature of one molecule. Thus, the macroscopic characteristics of bodies have a physical meaning only in the case of a large number of molecules.
Thermodynamics- a branch of physics that studies the general properties of macroscopic systems in a state of thermodynamic equilibrium and the processes of transition between these states. Thermodynamics does not consider the microprocesses that underlie these transformations. This thermodynamic method different from statistical. Thermodynamics is based on two principles - fundamental laws established as a result of generalization of experimental data.
The scope of application of thermodynamics is much wider than that of molecular kinetic theory, since there are no areas of physics and chemistry in which the thermodynamic method cannot be used. However, on the other hand, the thermodynamic method is somewhat limited: thermodynamics does not say anything about the microscopic structure of matter, about the mechanism of phenomena, but only establishes connections between the macroscopic properties of matter. Molecular kinetic theory and thermodynamics complement each other, forming a single whole, but differing in various research methods.
Basic postulates of molecular kinetic theory (MKT)
1. All bodies in nature consist of a huge number of tiny particles (atoms and molecules).
2. These particles are in continuous chaotic(disorderly) movement.
3. The movement of particles is related to body temperature, which is why it is called thermal movement.
4. Particles interact with each other.
Evidence of the validity of MCT: diffusion of substances, Brownian motion, thermal conductivity.
Physical quantities used to describe processes in molecular physics are divided into two classes:
microparameters– quantities that describe the behavior of individual particles (mass of an atom (molecule), speed, momentum, kinetic energy of individual particles);
macro parameters– quantities that cannot be reduced to individual particles, but characterize the properties of the substance as a whole. The values of macroparameters are determined by the result of the simultaneous action of a huge number of particles. Macro parameters are temperature, pressure, concentration, etc.
Temperature is one of the basic concepts that plays an important role not only in thermodynamics, but also in physics in general. Temperature- a physical quantity characterizing the state of thermodynamic equilibrium of a macroscopic system. In accordance with the decision of the XI General Conference on Weights and Measures (1960), only two temperature scales can currently be used - thermodynamic And International practical, graduated respectively in kelvins (K) and degrees Celsius (°C).
On the thermodynamic scale, the freezing point of water is 273.15 K (at the same
pressure as in the International Practical Scale), therefore, by definition, thermodynamic temperature and International Practical Temperature
scale are related by the ratio
T= 273,15 + t.
Temperature T = 0 K is called zero kelvin. Analysis of various processes shows that 0 K is unattainable, although approaching it as close as desired is possible. 0 K is the temperature at which theoretically all thermal movement of particles of a substance should cease.
In molecular physics, a relationship is derived between macroparameters and microparameters. For example, the pressure of an ideal gas can be expressed by the formula:
position:relative; top:5.0pt">- mass of one molecule, - concentration, font-size: 10.0pt">From the basic MKT equation you can obtain an equation convenient for practical use:
font-size: 10.0pt">An ideal gas is an idealized gas model in which it is believed that:
1. the intrinsic volume of gas molecules is negligible compared to the volume of the container;
2. there are no interaction forces between molecules (attraction and repulsion at a distance;
3. collisions of molecules with each other and with the walls of the vessel are absolutely elastic.
An ideal gas is a simplified theoretical model of a gas. But, the state of many gases under certain conditions can be described by this equation.
To describe the state of real gases, corrections must be introduced into the equation of state. The presence of repulsive forces that counteract the penetration of other molecules into the volume occupied by a molecule means that the actual free volume in which molecules of a real gas can move will be smaller. Whereb - the molar volume occupied by the molecules themselves.
The action of attractive gas forces leads to the appearance of additional pressure on the gas, called internal pressure. According to van der Waals calculations, internal pressure is inversely proportional to the square of the molar volume, i.e. where A - van der Waals constant, characterizing the forces of intermolecular attraction,V m - molar volume.
In the end we will get equation of state of real gas or van der Waals equation:
font-size:10.0pt;font-family:" times new roman> Physical meaning of temperature: temperature is a measure of the intensity of thermal motion of particles of substances. The concept of temperature is not applicable to an individual molecule. Only for a sufficiently large number of molecules creating a certain amount of substance, It makes sense to include the term temperature.
For an ideal monatomic gas, we can write the equation:
font-size:10.0pt;font-family:" times new roman>The first experimental determination of molecular speeds was carried out by the German physicist O. Stern (1888-1970). His experiments also made it possible to estimate the speed distribution of molecules.
The “confrontation” between the potential binding energies of molecules and the energies of thermal motion of molecules (kinetic molecules) leads to the existence of various aggregate states of matter.
Thermodynamics
By counting the number of molecules in a given system and estimating their average kinetic and potential energies, we can estimate the internal energy of a given system U.
font-size:10.0pt;font-family:" times new roman>For an ideal monatomic gas.
The internal energy of a system can change as a result of various processes, for example, performing work on the system or imparting heat to it. So, by pushing a piston into a cylinder in which there is a gas, we compress this gas, as a result of which its temperature increases, i.e., thereby changing (increasing) the internal energy of the gas. On the other hand, the temperature of a gas and its internal energy can be increased by imparting a certain amount of heat to it - energy transferred to the system by external bodies through heat exchange (the process of exchanging internal energies when bodies come into contact with different temperatures).
Thus, we can talk about two forms of energy transfer from one body to another: work and heat. The energy of mechanical motion can be converted into the energy of thermal motion, and vice versa. During these transformations, the law of conservation and transformation of energy is observed; in relation to thermodynamic processes this law is first law of thermodynamics, established as a result of generalization of centuries-old experimental data:
In a closed loop, therefore font-size:10.0pt;font-family:" times new roman>Heat engine efficiency: 
.
From the first law of thermodynamics it follows that the efficiency of a heat engine cannot be more than 100%.
Postulating the existence of various forms of energy and the connection between them, the first principle of TD says nothing about the direction of processes in nature. In full accordance with the first principle, one can mentally construct an engine in which useful work would be performed by reducing the internal energy of the substance. For example, instead of fuel, a heat engine would use water, and by cooling the water and turning it into ice, work would be done. But such spontaneous processes do not occur in nature.
All processes in nature can be divided into reversible and irreversible.
For a long time, one of the main problems in classical natural science remained the problem of explaining the physical nature of the irreversibility of real processes. The essence of the problem is that the motion of a material point, described by Newton’s II law (F = ma), is reversible, while a large number of material points behave irreversibly.
If the number of particles under study is small (for example, two particles in figure a)), then we will not be able to determine whether the time axis is directed from left to right or from right to left, since any sequence of frames is equally possible. That's what it is reversible phenomenon. The situation changes significantly if the number of particles is very large (Fig. b)). In this case, the direction of time is determined unambiguously: from left to right, since it is impossible to imagine that evenly distributed particles by themselves, without any external influences, will gather in the corner of the “box”. This behavior, when the state of the system can only change in a certain sequence, is called irreversible. All real processes are irreversible.
Examples of irreversible processes: diffusion, thermal conductivity, viscous flow. Almost all real processes in nature are irreversible: this is the damping of a pendulum, the evolution of a star, and human life. The irreversibility of processes in nature, as it were, sets the direction on the time axis from the past to the future. The English physicist and astronomer A. Eddington figuratively called this property of time “the arrow of time.”
Why, despite the reversibility of the behavior of one particle, does an ensemble of a large number of such particles behave irreversibly? What is the nature of irreversibility? How to justify the irreversibility of real processes based on Newton's laws of mechanics? These and other similar questions worried the minds of the most outstanding scientists of the 18th–19th centuries.
Second law of thermodynamics sets the direction laziness of all processes in isolated systems. Although the total amount of energy in an isolated system is conserved, its qualitative composition changes irreversibly.
1. In Kelvin's formulation, the second law is: “There is no process possible whose sole result would be the absorption of heat from a heater and the complete conversion of this heat into work.”
2. In another formulation: “Heat can spontaneously transfer only from a more heated body to a less heated one.”
3. The third formulation: “Entropy in a closed system can only increase.”
Second law of thermodynamics prohibits the existence perpetual motion machine of the second kind , i.e., a machine capable of doing work by transferring heat from a cold body to a hot one. The second law of thermodynamics indicates the existence of two different forms of energy - heat as a measure of the chaotic movement of particles and work associated with ordered movement. Work can always be converted into its equivalent heat, but heat cannot be completely converted into work. Thus, a disordered form of energy cannot be transformed into an ordered one without any additional actions.
We complete the transformation of mechanical work into heat every time we press the brake pedal in a car. But without any additional actions in a closed cycle of engine operation, it is impossible to transfer all the heat into work. Part of the thermal energy is inevitably spent on heating the engine, plus the moving piston constantly does work against friction forces (this also consumes a supply of mechanical energy).
But the meaning of the second law of thermodynamics turned out to be even deeper.
Another formulation of the second law of thermodynamics is the following statement: the entropy of a closed system is a non-decreasing function, that is, during any real process it either increases or remains unchanged.
The concept of entropy, introduced into thermodynamics by R. Clausius, was initially artificial. The outstanding French scientist A. Poincaré wrote about this: “Entropy seems somewhat mysterious in the sense that this quantity is inaccessible to any of our senses, although it has the real property of physical quantities, since, at least in principle, it is completely measurable "
According to Clausius's definition, entropy is a physical quantity whose increment is equal to the amount of heat , received by the system, divided by the absolute temperature:
font-size:10.0pt;font-family:" times new roman>In accordance with the second law of thermodynamics, in isolated systems, i.e. systems that do not exchange energy with the environment, a disordered state (chaos) cannot independently transform into order Thus, in isolated systems, entropy can only increase. This pattern is called principle of increasing entropy. According to this principle, any system strives for a state of thermodynamic equilibrium, which is identified with chaos. Since an increase in entropy characterizes changes over time in closed systems, entropy acts as a kind of arrows of time.
We called the state with maximum entropy disordered, and the state with low entropy ordered. A statistical system, if left to itself, goes from an ordered to a disordered state with maximum entropy corresponding to given external and internal parameters (pressure, volume, temperature, number of particles, etc.).
Ludwig Boltzmann connected the concept of entropy with the concept of thermodynamic probability: font-size:10.0pt;font-family:" times new roman> Thus, any isolated system, left to its own devices, over time passes from a state of order to a state of maximum disorder (chaos).
From this principle follows a pessimistic hypothesis about heat death of the Universe, formulated by R. Clausius and W. Kelvin, according to which:
· the energy of the Universe is always constant;
· The entropy of the Universe is always increasing.
Thus, all processes in the Universe are directed towards achieving a state of thermodynamic equilibrium, corresponding to the state of greatest chaos and disorganization. All types of energy degrade, turning into heat, and the stars will end their existence, releasing energy into the surrounding space. A constant temperature will be established only a few degrees above absolute zero. Lifeless, cooled planets and stars will be scattered in this space. There will be nothing - no energy sources, no life.
This grim prospect was predicted by physics until the 1960s, although the conclusions of thermodynamics contradicted the results of research in biology and social sciences. Thus, Darwin's evolutionary theory testified that living nature develops primarily in the direction of improvement and complexity of new species of plants and animals. History, sociology, economics, and other social and human sciences have also shown that in society, despite individual zigzags of development, progress is generally observed.
Experience and practical activity have shown that the concept of a closed or isolated system is a rather crude abstraction that simplifies reality, since in nature it is difficult to find systems that do not interact with the environment. The contradiction began to be resolved when in thermodynamics, instead of the concept of a closed isolated system, the fundamental concept of an open system was introduced, that is, a system exchanging matter, energy and information with the environment.
Thermodynamics and statistical physics
Guidelines and test assignments for distance learning students
Shelkunova Z.V., Saneev E.L.
Methodological instructions and test assignments for distance learning students of engineering, technical and technological specialties. Contains sections of the programs “Statistical Physics”, “Thermodynamics”, examples of solving typical problems and variants of test tasks.
Key words: Internal energy, heat, work; isoprocesses, entropy: distribution functions: Maxwell, Boltzmann, Bose – Einstein; Fermi – Dirac; Fermi energy, heat capacity, characteristic temperature of Einstein and Debye.
Editor T.Yu.Artyunina
Prepared for printing. Format 6080 1/16
Conditional p.l. ; ed.l. 3.0; Circulation ____ copies. Order no.
___________________________________________________
RIO VSTU, Ulan-Ude, Klyuchevskaya, 40a
Printed on the rotaprint of VSTU, Ulan-Ude,
Klyuchevskaya, 42.
Federal Agency for Education
East Siberian State
University of Technology
PHYSICS No. 4
(Thermodynamics and statistical physics)
Guidelines and control tasks
for distance learning students
Compiled by: Shelkunova Z.V.
Saneev E.L.
Publishing House VSTU
Ulan-Ude, 2009
Statistical physics and thermodynamics
Topic 1
Dynamic and statistical patterns in physics. Thermodynamic and statistical methods. Elements of molecular kinetic theory. Macroscopic condition. Physical quantities and states of physical systems. Macroscopic parameters as average values. Thermal equilibrium. Ideal gas model. Equation of state of an ideal gas. The concept of temperature.
Topic 2
Transference phenomena. Diffusion. Thermal conductivity. Diffusion coefficient. Coefficient of thermal conductivity. Thermal diffusivity. Diffusion in gases, liquids and solids. Viscosity. Viscosity coefficient of gases and liquids.
Topic 3
Elements of thermodynamics. The first law of thermodynamics. Internal energy. Intensive and extensive parameters.
Topic 4
Reversible and irreversible processes. Entropy. Second law of thermodynamics. Thermodynamic potentials and equilibrium conditions. Chemical potential. Conditions of chemical equilibrium. Carnot cycle.
Topic 5
Distribution functions. Microscopic parameters. Probability and fluctuations. Maxwell distribution. Average kinetic energy of a particle. Boltzmann distribution. Heat capacity of polyatomic gases. Limitations of the classical theory of heat capacity.
Topic 6
Gibbs distribution. Model of the system in the thermostat. Canonical Gibbs distribution. Statistical meaning of thermodynamic potentials and temperature. The role of free energy.
Topic 7
Gibbs distribution for a system with a variable number of particles. Entropy and probability. Determination of the entropy of an equilibrium system through the statistical weight of a microstate.
Topic 8
Bose and Fermi distribution functions. Planck's formula for weighted thermal radiation. Order and disorder in nature. Entropy as a quantitative measure of chaos. The principle of increasing entropy. The transition from order to disorder about the state of thermal equilibrium.
Topic 9
Experimental methods for studying the vibrational spectrum of crystals. The concept of phonons. Dispersion laws for acoustic and optical phonons. Heat capacity of crystals at low and high temperatures. Electronic heat capacity and thermal conductivity.
Topic 10
Electrons in crystals. Approximation of strong and weak coupling. Free electron model. Fermi level. Elements of band theory of crystals. Bloch function. Band structure of the electron energy spectrum.
Topic 11
Fermi surface. The number and density of the number of electronic states in the zone. Zone fillings: metals, dielectrics and semiconductors. Electrical conductivity of semiconductors. The concept of hole conductivity. Intrinsic and impurity semiconductors. The concept of p-n junction. Transistor.
Topic 12
Electrical conductivity of metals. Current carriers in metals. Insufficiency of classical electronic theory. Electron Fermi gas in metal. Current carriers as quasiparticles. The phenomenon of superconductivity. Cooper pairing of electrons. Tunnel contact. Josephson effect and its application. Magnetic flux capture and quantization. The concept of high temperature conductivity.
STATISTICAL PHYSICS. THERMODYNAMICS
Basic formulas
1. Amount of substance of a homogeneous gas (in moles):
Where N- number of gas molecules; N A- Avogadro's number; m- mass of gas; -molar mass of gas.
If the system is a mixture of several gases, then the amount of substance in the system
,
,
Where i , N i , m i , i - respectively, the amount of substance, the number of molecules, mass, molar mass i- components of the mixture.
2. Clapeyron-Mendeleev equation (equation of state of an ideal gas):
Where m- mass of gas; - molar mass; R- universal gas constant; = m/ - amount of substance; T-thermodynamic temperature Kelvin.
3. Experimental gas laws, which are special cases of the Clapeyron-Mendeleev equation for isoprocesses:
Boyle-Mariotte law
(isothermal process - T=const; m=const):
or for two gas states:
Where p 1 and V 1 - pressure and volume of gas in the initial state; p 2 and V 2
Gay-Lussac's law (isobaric process - p=const, m=const):
or for two states:
Where V 1 And T 1 - volume and temperature of the gas in the initial state; V 2 And T 2 - the same values in the final state;
Charles' law (isochoric process - V=const, m=const):
or for two states:
Where R 1 And T 1 - pressure and temperature of the gas in the initial state; R 2 And T 2 - the same values in the final state;
combined gas law ( m=const):
Where R 1 , V 1 , T 1 - pressure, volume and temperature of the gas in the initial state; R 2 , V 2 , T 2 - the same values in the final state.
4. Dalton’s law, which determines the pressure of a gas mixture:
p = p 1 + p 2 + ... +r n
Where p i- partial pressures of the mixture components; n- number of mixture components.
5. Molar mass of a mixture of gases:
Where m i- weight i-th component of the mixture; i = m i / i- amount of substance i-th component of the mixture; n- number of mixture components.
6. Mass fraction i i th component of the gas mixture (in fractions of a unit or percent):
Where m- mass of the mixture.
7. Concentration of molecules (number of molecules per unit volume):
Where N-the number of molecules contained in a given system; is the density of the substance. The formula is valid not only for gases, but also for any state of aggregation of a substance.
8. Basic equation of the kinetic theory of gases:
,
Where<>- average kinetic energy of translational motion of a molecule.
9. Average kinetic energy of translational motion of a molecule:
,
Where k- Boltzmann constant.
10. Average total kinetic energy of a molecule:
Where i- the number of degrees of freedom of the molecule.
11. Dependence of gas pressure on the concentration of molecules and temperature:
p = nkT.
12. Molecular speeds:
mean square 
;
arithmetic mean 
;
most likely 
,
Definition 1
Statistical thermodynamics is a broad branch of statistical physics that formulates laws that connect all the molecular properties of physical substances with quantities measured during experiments.
Figure 1. Statistical thermodynamics of flexible molecules. Author24 - online exchange of student works
The statistical study of material bodies is devoted to the substantiation of the postulates and methods of thermodynamics of equilibrium concepts and the calculation of important functions using molecular constants. The basis of this scientific direction is made up of hypotheses and assumptions confirmed by experiments.
Unlike classical mechanics, in statistical thermodynamics only average readings of coordinates and internal momenta are studied, as well as the possibility of the emergence of new values. Thermodynamic properties of a macroscopic medium are considered as general parameters of random characteristics or quantities.
Today, scientists distinguish between classical (Boltzmann, Maxwell) and quantum (Dirac, Fermi, Einstein) thermodynamics. The basic theory of statistical research: there is an unambiguous and stable relationship between the molecular features of the particles that make up a particular system.
Definition 2
An ensemble in thermodynamics is an almost infinite number of thermodynamic concepts that are in different, equally probable microstates.
The average parameters of a physically observed element over a long period of time begin to equate to the overall value for the ensemble.
Basic idea of statistical thermodynamics
Figure 2. Statistical formulation of the 2nd law of thermodynamics. Author24 - online exchange of student works
Statistical thermodynamics establishes and implements the interaction of microscopic and macroscopic systems. In the first scientific approach, based on classical or quantum mechanics, the internal states of the medium are described in detail in the form of the coordinates and momentum of each individual particle at a certain moment in time. The microscopic formulation requires solving complex equations of motion for many variables.
The macroscopic method used by classical thermodynamics characterizes exclusively the external state of the system and uses a small number of variables for this:
- physical body temperature;
- volume of interacting elements;
- number of elementary particles.
If all substances are in an equilibrium state, then their macroscopic indicators will be constant, and their microscopic coefficients will gradually change. This means that each state in statistical thermodynamics corresponds to several microstates.
Note 1
The main idea of the branch of physics being studied is the following: if each position of physical bodies corresponds to many microstates, then each of them as a result makes a significant contribution to the overall macrostate.
From this definition we should highlight the elementary properties of the statistical distribution function:
- normalization;
- positive certainty;
- the average value of the Hamilton function.
Averaging over existing microstates is carried out using the concept of a statistical ensemble located in any microstates corresponding to one macrostate. The meaning of this distribution function is that it generally determines the statistical weight of each state of the concept.
Basic concepts in statistical thermodynamics
To statistically and competently describe macroscopic systems, scientists use ensemble and phase space data, which allows them to solve classical and quantum problems using the probability theory method. The microcanonical Gibbs ensemble is often used to study isolated systems with a constant volume and number of identically charged particles. This method is used to carefully describe systems of stable volume that are in thermal equilibrium with the environment at a constant index of elementary particles. The state parameters of a large ensemble make it possible to determine the chemical potential of material substances. The Gibbs isobaric-isothermal system is used to explain the interaction of bodies that are in thermal and mechanical equilibrium in a certain space at constant pressure.
Phase space in statistical thermodynamics characterizes a mechanical-multidimensional space, the axes of which are all generalized coordinates and the associated internal impulses of a system with constant degrees of freedom. For a system consisting of atoms, the indicators of which correspond to the Cartesian coordinate, the set of parameters and thermal energy will be designated according to the initial state. The action of each concept is represented by a point in phase space, and the change in a macrostate in time is represented by the movement of a point along the trajectory of a specific line. To statistically describe the properties of the environment, the concepts of distribution function and phase volume are introduced, characterizing the probability density of finding a new point depicting the real state of the system, as well as in matter near a line with certain coordinates.
Note 2
In quantum mechanics, instead of a phase volume, the concept of a discrete energy spectrum of a system of finite volume is used, since this process is determined not by coordinates and momentum, but by a wave function, which in a dynamic state corresponds to the entire spectrum of quantum states.
The distribution function of the classical system will determine the possibility of implementing a specific microstate in one element of the volume of the phase medium. The probability of finding particles in an infinitesimal space can be compared with the integration of elements over the coordinates and momenta of the system. The state of thermodynamic equilibrium should be considered as a limiting indicator of all substances, where solutions to the equation of motion of the particles that make up the concept arise for the distribution function. The type of such a functional, which is the same for quantum and classical systems, was first established by the theoretical physicist J. Gibbs.
Calculation of statistical functions in thermodynamics
To correctly calculate the thermodynamic function, it is necessary to apply any physical distribution: all elements in the system are equivalent to each other and correspond to different external conditions. The microcanonical Gibbs distribution is used mainly in theoretical studies. To solve specific and more complex problems, ensembles are considered that have energy with the environment and can exchange particles and energy. This method is very convenient for studying phase and chemical equilibria.
Partition functions allow scientists to accurately determine the energy and thermodynamic properties of a system, obtained by differentiating indicators according to relevant parameters. All these quantities acquire statistical meaning. Thus, the internal potential of a material body is identified with the average energy of the concept, which allows us to study the first law of thermodynamics, as the basic law of conservation of energy during the unstable movement of the elements that make up the system. Free energy is directly related to the partition function of the system, and entropy is directly related to the number of microstates in a particular macrostate, therefore, to its probability.
The meaning of entropy, as a measure of the emergence of a new state, is preserved in connection with an arbitrary parameter. In a state of complete equilibrium, the entropy of an isolated system has a maximum value under initially correctly specified external conditions, that is, the equilibrium general state is a probable result with maximum statistical weight. Therefore, a smooth transition from a nonequilibrium position to an equilibrium one is a process of change to a more real state.
This is the statistical meaning of the law of increasing internal entropy, according to which the parameters of a closed system increase. At absolute zero, any concept is in a stable state. This scientific statement represents the third law of thermodynamics. It is worth noting that for an unambiguous formulation of entropy it is necessary to use only a quantum description, since in classical statistics this coefficient is defined with maximum accuracy up to an arbitrary term.