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statistical physics
- Aizenshitz R. Statistical theory of irreversible processes. M.: Ed. Foreign lit., 1963 (djvu)
- Anselm A.I. Fundamentals of statistical physics and thermodynamics. Moscow: Nauka, 1973 (djvu)
- Akhiezer A.I., Peletminsky S.V. Methods of statistical physics. Moscow: Nauka, 1977 (djvu)
- Bazarov I.P. Methodological problems of statistical physics and thermodynamics. M.: Publishing House of Moscow State University, 1979 (djvu)
- Bogolyubov N.N. Selected writings in statistical physics. M.: Publishing House of Moscow State University, 1979 (djvu)
- Bogolyubov N.N. (Jr.), Sadovnikov B.I. Some questions of statistical mechanics. M.: Higher. school, 1975 (djvu)
- Bonch-Bruevich V.L., Tyablikov S.V. Green's function method in statistical mechanics. Moscow: Fizmatlit, 1961 (djvu, 2.61Mb)
- Vasiliev A.M. Introduction to statistical physics. M.: Higher. school, 1980 (djvu)
- Vlasov A.A. Nonlocal statistical mechanics. Moscow: Nauka, 1978 (djvu)
- Gibbs JW Basic principles of statistical mechanics (expounded with a special application to the rational justification of thermodynamics). M.-L.: OGIZ, 1946 (djvu)
- Gurov K.P. Foundations of the kinetic theory. Method N.N. Bogolyubov. Moscow: Nauka, 1966 (djvu)
- Zaslavsky G.M. Statistical irreversibility in nonlinear systems. Moscow: Nauka, 1970 (djvu)
- Zakharov A.Yu. Lattice models of statistical physics. Veliky Novgorod: NovGU, 2006 (pdf)
- Zakharov A.Yu. Functional methods in classical statistical physics. Veliky Novgorod: NovGU, 2006 (pdf)
- Ios G. Course of theoretical physics. Part 2. Thermodynamics. Statistical physics. Quantum theory. Nuclear physics. M.: Enlightenment, 1964 (djvu)
- Ishihara A. Statistical Physics. M.: Mir, 1973 (djvu)
- Kadanov L., Beim G. Quantum statistical mechanics. Methods of Green's functions in the theory of equilibrium and nonequilibrium processes. M.: Mir, 1964 (djvu)
- Katz M. Probability and related issues in physics. M.: Mir, 1965 (djvu)
- Katz M. Several probabilistic problems of physics and mathematics. Moscow: Nauka, 1967 (djvu)
- Kittel Ch. Elementary statistical physics. M.: IL, 1960 (djvu)
- Kittel Ch. Statistical thermodynamics. M: Science, 1977 (djvu)
- Kozlov V.V. Thermal equilibrium according to Gibbs and Poincare. Moscow-Izhevsk: Institute for Computer Research, 2002 (djvu)
- Kompaneets A.S. Laws of physical statistics. shock waves. Super dense substance. M.: Nauka, 1976 (djvu)
- Kompaneets A.S. Course of theoretical physics. Volume 2. Statistical laws. M.: Enlightenment, 1975 (djvu)
- Kotkin G.L. Lectures on Statistical Physics, NSU (pdf)
- Krylov N.S. Works on substantiation of statistical physics. M.-L.: From the Academy of Sciences of the USSR, 1950 (djvu)
- Kubo R. Statistical mechanics. M.: Mir, 1967 (djvu)
- Landsberg P. (ed.) Problems in thermodynamics and statistical physics. M.: Mir, 1974 (djvu)
- Levich V.G. Introduction to Statistical Physics (2nd ed.) M.: GITTL, 1954 (djvu)
- Libov R. Introduction to the theory of kinetic equations. M.: Mir, 1974 (djvu)
- Mayer J., Geppert-Mayer M. Statistical mechanics. M.: Mir, 1980 (djvu)
- Minlos R.A. (ed.) Mathematics. New in foreign science-11. Gibbs states in statistical physics. Digest of articles. M.: Mir, 1978 (djvu)
- Nozdrev V.F., Senkevich A.A. Course of statistical physics. M.: Higher. school, 1965 (djvu)
- Prigogine I. Nonequilibrium Statistical Mechanics. M.: Mir, 1964 (djvu)
- Radushkevich L.V. Course of statistical physics (2nd ed.) M.: Prosveshchenie, 1966 (djvu)
- Reif F. Berkeley Physics Course. Volume 5. Statistical physics. M.: Nauka, 1972 (djvu)
- Rumer Yu.B., Ryvkin M.Sh. Thermodynamics, statistical physics and kinetics. M.: Nauka, 1972 (djvu)
- Rumer Yu.B., Ryvkin M.Sh. Thermodynamics Statistical Physics and Kinetics (2nd ed.). Moscow: Nauka, 1977 (djvu)
- Ruel D. Statistical mechanics. M.: Mir, 1971 (djvu)
- Savukov V.V. Refinement of the axiomatic principles of statistical physics. SPb.: Balt. state tech. univ. "Voenmekh", 2006
10. Basic postulates of statistical thermodynamics
When describing systems consisting of a large number of particles, two approaches can be used: microscopic and macroscopic. In the first approach, based on classical or quantum mechanics, the microstate of the system is characterized in detail, for example, the coordinates and momenta of each particle at each moment of time. A microscopic description requires the solution of classical or quantum equations of motion for a huge number of variables. Thus, each microstate of an ideal gas in classical mechanics is described by 6 N variables ( N- number of particles): 3 N coordinates and 3 N impulse projections.
The macroscopic approach, which is used by classical thermodynamics, characterizes only the macrostates of the system and uses a small number of variables for this, for example, three: temperature, volume, and number of particles. If the system is in equilibrium, then its macroscopic parameters are constant, while the microscopic parameters change with time. This means that each macrostate corresponds to several (actually, infinitely many) microstates.
Statistical thermodynamics establishes a connection between these two approaches. The basic idea is as follows: if each macrostate corresponds to many microstates, then each of them contributes to the macrostate. Then the properties of a macrostate can be calculated as an average over all microstates, i.e. summing their contributions, taking into account the statistical weight.
Averaging over microstates is carried out using the concept of a statistical ensemble. Ensemble is an infinite set of identical systems that are in all possible microstates corresponding to one macrostate. Each ensemble system is one microstate. The whole ensemble is described by some distribution function in coordinates and momenta ( p, q, t), which is defined as follows:
(p, q, t) dp dq is the probability that the ensemble system is in the volume element dp dq near the point ( p, q) at time t.
The meaning of the distribution function is that it determines the statistical weight of each microstate in the macrostate.
The definition implies the elementary properties of the distribution function:
1. Normalization
. (10.1)
2. Positive definiteness
(p, q, t) i 0 (10.2)
Many macroscopic properties of a system can be defined as average value coordinate and momentum functions f(p, q) by ensemble:
For example, the internal energy is the mean value of the Hamilton function H(p,q):
The existence of a distribution function is the essence basic postulate of classical statistical mechanics:
The macroscopic state of the system is completely specified by some distribution function that satisfies conditions (10.1) and (10.2).
For equilibrium systems and equilibrium ensembles, the distribution function does not explicitly depend on time: = ( p,q). The explicit form of the distribution function depends on the type of ensemble. There are three main types of ensembles:
1) microcanonical the ensemble describes isolated systems and is characterized by variables: E(energy), V(volume), N(number of particles). In an isolated system, all microstates are equally probable ( postulate of equal prior probability):
2) Canonical Ensemble describes systems that are in thermal equilibrium with their environment. Thermal equilibrium is characterized by temperature T. Therefore, the distribution function also depends on temperature:
(10.6)
(k\u003d 1.38 10 -23 J / K - Boltzmann's constant). The value of the constant in (10.6) is determined by the normalization condition (see (11.2)).
A special case of the canonical distribution (10.6) is Maxwell distribution in terms of velocities v, which is valid for gases:
(10.7)
(m is the mass of a gas molecule). Expression (v) d v describes the probability that a molecule has an absolute velocity value between v and v + d v. The maximum of function (10.7) gives the most probable velocity of molecules, and the integral
The average speed of the molecules.
If the system has discrete energy levels and is described quantum mechanically, then instead of the Hamilton function H(p,q) use the Hamilton operator H, and instead of the distribution function, the density matrix operator:
(10.9)
The diagonal elements of the density matrix give the probability that the system is in i-th energy state and has energy E i:
(10.10)
The value of the constant is determined by the normalization condition: S i = 1:
(10.11)
The denominator of this expression is called the sum over states (see Chapter 11). It is of key importance for the statistical evaluation of the thermodynamic properties of the system From (10.10) and (10.11) one can find the number of particles N i having energy E i:
(10.12)
(N is the total number of particles). The distribution of particles (10.12) over energy levels is called Boltzmann distribution, and the numerator of this distribution is the Boltzmann factor (multiplier). Sometimes this distribution is written in a different form: if there are several levels with the same energy E i, then they are combined into one group by summing the Boltzmann factors:
(10.13)
(gi- number of energy levels E i, or statistical weight).
Many macroscopic parameters of a thermodynamic system can be calculated using the Boltzmann distribution. For example, the average energy is defined as the average of the energy levels given their statistical weights:
, (10.14)
3) Grand canonical ensemble describes open systems in thermal equilibrium and capable of exchanging matter with the environment. Thermal equilibrium is characterized by temperature T, and the equilibrium in the number of particles - chemical potential. Therefore, the distribution function depends on temperature and chemical potential. We will not use an explicit expression for the distribution function of the grand canonical ensemble here.
In statistical theory, it is proved that for systems with a large number particles (~ 10 23) all three types of ensembles are equivalent to each other. The use of any ensemble leads to the same thermodynamic properties, so the choice of one or another ensemble for describing a thermodynamic system is dictated only by the convenience of mathematical processing of distribution functions.
EXAMPLES
Example 10-1. The molecule can be at two levels with energies 0 and 300 cm -1 . What is the probability that the molecule will be at the top level at 250°C?
Solution. It is necessary to apply the Boltzmann distribution, and to convert the spectroscopic unit of energy cm -1 into joules, use the factor hc (h\u003d 6.63 10 -34 J. s, c\u003d 3 10 10 cm / s): 300 cm -1 \u003d 300 6.63 10 -34 3 10 10 \u003d 5.97 10 -21 J.
Answer. 0.304.
Example 10-2. A molecule can be at a level with energy 0 or at one of three levels with energy E. At what temperature will a) all molecules be at the lower level, b) the number of molecules at the lower level will be equal to the number of molecules at the upper levels, c) the number of molecules at the lower level will be three times less than the number of molecules at the upper levels?
Solution. We use the Boltzmann distribution (10.13):
A) N 0 / N= 1; exp(- E/kT) = 0; T= 0. As the temperature decreases, the molecules accumulate at the lower levels.
b) N 0 / N= 1/2; exp(- E/kT) = 1/3; T = E / [k log(3)].
V) N 0 / N= 1/4; exp(- E/kT) = 1; T= . At high temperatures, the molecules are evenly distributed over energy levels, because all Boltzmann factors are almost the same and equal to 1.
Answer. A) T= 0; b) T = E / [k log(3)]; V) T = .
Example 10-3. When any thermodynamic system is heated, the population of some levels increases, while others decrease. Using the Boltzmann distribution law, determine what the level energy must be in order for its population to increase with increasing temperature.
Solution. Population - the proportion of molecules that are at a certain energy level. By condition, the derivative of this quantity with respect to temperature must be positive:
In the second line we used the definition of average energy (10.14). Thus, the population increases with increasing temperature for all levels exceeding the average energy of the system.
Answer. .
TASKS
10-1. The molecule can be at two levels with energies 0 and 100 cm -1 . What is the probability that the molecule will be lowest level at 25 o C?
10-2. The molecule can be at two levels with energies 0 and 600 cm -1 . At what temperature will there be half as many molecules at the top level as at the bottom?
10-3. A molecule can be at a level with energy 0 or at one of three levels with energy E. Find the average energy of the molecules: a) at very low temperatures, b) at very high temperatures.
10-4. When cooling any thermodynamic system, the population of some levels increases, while others decrease. Using the Boltzmann distribution law, determine what the level energy must be in order for its population to increase with decreasing temperature.
10-5. Calculate the most probable velocity of carbon dioxide molecules at a temperature of 300 K.
10-6. Calculate average speed helium atoms under normal conditions.
10-7. Calculate the most probable velocity of ozone molecules at -30°C.
10-8. At what temperature is the average velocity of oxygen molecules equal to 500 m/s?
10-9. Under certain conditions, the average speed of oxygen molecules is 400 m/s. What is the average speed of hydrogen molecules under the same conditions?
10-10. What is the proportion of molecules with a mass m having a speed above the average at a temperature T? Does this fraction depend on the mass of molecules and temperature?
10-11. Using the Maxwell distribution, calculate the average kinetic energy of the movement of molecules with a mass m at a temperature T. Is this energy equal kinetic energy at medium speed?
Let there be two identical vessels connected to each other in such a way that gas from one vessel can flow into another, and let at the initial moment all gas molecules are in one vessel. After some time, there will be a redistribution of molecules, leading to the emergence of an equilibrium state, characterized by an equal probability of finding molecules in both vessels. A spontaneous transition to the initial nonequilibrium state, in which all molecules are concentrated in one of the vessels, is practically impossible. The process of transition from an equilibrium to a nonequilibrium state is very unlikely, since the size of the relative fluctuations of the parameters at large quantities particles in the vessels is very small.
This conclusion corresponds to the second law of thermodynamics, which states that a thermodynamic system spontaneously passes from a nonequilibrium state to an equilibrium state, while reverse process possible only under external influences on the system.
Entropy and Probability
Entropy is a thermodynamic quantity that characterizes the direction of spontaneous thermodynamic processes. The most probable equilibrium state corresponds to the entropy maximum.
Let there be a vessel with a volume V 0 containing one molecule. The probability that a particle will be found inside a certain volume V< V 0 , isolated inside the vessel, is equal to . If the vessel contains not one, but two particles, then the probability of their simultaneous detection in the specified volume is determined as the product of the probabilities of finding each of the particles in this volume:
.
For N particles the probability of their simultaneous detection in the volume V will be
.
If two volumes are distinguished in this vessel V 1 And V 2 then we can write the ratios of the probabilities that all molecules are in the indicated volumes:
.
Let us determine the increment of entropy in the isothermal process
expansion of an ideal gas from V 1 before V 2 :
Using the ratio, the probabilities are:
.
The resulting expression does not determine the absolute value of the entropy in any state, but only makes it possible to find the difference in entropies in two different states.
For an unambiguous definition of entropy, use statistical weight G , whose value is expressed as an integer positive number and in proportion to the probability: G ~ P .
The statistical weight of the macrostate is called a quantity, numerically equal to the number equilibrium microstates, with the help of which the considered macrostate can be realized.
Passing to the statistical weight allows us to write the relation for the entropy in the form Boltzmann formulas for statistical entropy :
Lecture 15
Transfer phenomena
Thermodynamic flows
Thermodynamic flows , associated with the transfer of matter, energy or momentum from one part of the medium to another, arise if the values of certain physical parameters differ in the volume of the medium.
by diffusion called the process of spontaneous alignment of the concentrations of substances in mixtures. The diffusion rate is highly dependent on state of aggregation substances. Diffusion occurs faster in gases and very slowly in solids Oh.
thermal conductivity called the phenomenon that leads to the equalization of temperature at different points in the environment. The high thermal conductivity of metals is due to the fact that heat transfer in them is carried out not due to the chaotic movement of atoms and molecules, as, for example, in gases or liquids, but by free electrons, which have much higher thermal motion velocities.
Viscosity or internal friction call the process of the emergence of a resistance force when a body moves in a liquid or gas and the attenuation of sound waves when they pass through various media.
For a quantitative description of the thermodynamic flow, the quantity is introduced
, Where
Thermodynamics. The works of Mayer, Joule, Helmholtz made it possible to develop the so-called. “the law of conservation of forces” (the concepts of “force” and “energy” were not strictly distinguished at that time). However, the first clear formulation of this law was obtained by physicists R. Clausius and W. Thomson (Lord Kelvin) on the basis of an analysis of the study of the operation of a heat engine, which was carried out by S. Carnot. Considering the transformation of heat and work in macroscopic systems, S. Carnot actually laid the foundation for new science, which Thomson later called thermodynamics. Thermodynamics is limited to the study of the features of the transformation of a thermal form of motion into others, without being interested in questions of the microscopic motion of the particles that make up matter.
Thermodynamics, therefore, considers systems between which energy exchange is possible, without taking into account the microscopic structure of the bodies that make up the system, and the characteristics of individual particles. Distinguish between the thermodynamics of equilibrium systems or systems passing to equilibrium (classical or equilibrium thermodynamics) and the thermodynamics of non-equilibrium systems (non-equilibrium thermodynamics). Classical thermodynamics is most often called simply thermodynamics, and it is precisely this that forms the basis of the so-called Thermodynamic Picture of the World (TCM), which was formed by the middle of the 19th century. Non-equilibrium thermodynamics was developed in the second half of the 20th century and plays a special role in the consideration of biological systems and the phenomenon of life in general.
Thus, in the study of thermal phenomena, two scientific directions:
1. Thermodynamics, which studies thermal processes without taking into account the molecular structure of matter;
2. Molecular-kinetic theory (development of the kinetic theory of matter as opposed to the theory of caloric);
Molecular-kinetic theory. In contrast to thermodynamics, molecular-kinetic theory is characterized by the consideration of various macroscopic manifestations of systems as the results of the total action of a huge set of randomly moving molecules. Molecular kinetic theory uses a statistical method, being interested not in the movement of individual molecules, but only in the average values that characterize the movement of a huge collection of particles. Hence the second name of the molecular-kinetic theory is statistical physics.
First law of thermodynamics. Based on the work of Joule and Mayer, Klausnus was the first to express an idea that was later formed in the first law of thermodynamics. He concluded that every body has an internal energy U. Clausius called it the heat contained in the body, in contrast to the "heat Q communicated to the body." Internal energy can be increased in two equivalent ways: by performing mechanical work -A on the body, or by imparting to it the amount of heat Q.
In 1860, W. Thomson, having finally replaced the obsolete term “force” with the term “energy”, writes down the first law of thermodynamics in the following formulation:
The amount of heat imparted to the gas is used to increase the internal energy of the gas and to perform external work by the gas (Fig. 1).
For infinitesimal changes we have
The first law of thermodynamics, or the law of conservation of energy, states the balance of energy and work. His role can be compared with the role of a kind of "accountant" in the mutual transformation various kinds energy into each other.
If the process is cyclic, the system returns to its original state and U1 = U2 , and dU = 0. In this case, all summed heat is coming to do outside work. If, in addition, Q = 0, then A = 0, i.e. a process is impossible, the only result of which is the production of work without any changes in other bodies, i.e. work of "perpetuum mobile" (perpetuum mobile).
Mayer in his work compiled a table of all the "forces" (energies) of nature considered by him and cited 25 cases of their transformations (heat ® mechanical work ® electricity, chemical "force" of matter ® heat, electricity). Mayer extended the concept of the conservation and transformation of energy to living organisms (food intake ® chemical processes ® thermal and mechanical effects). These examples were subsequently reinforced by the works of Hess (1840), who studied the conversion of chemical energy into heat, as well as Faraday, Lenz and Joule, as a result of which the Joule-Lenz law (1845) was formulated on the relationship between electrical and thermal energy Q = J2Rt.
Thus, gradually, over more than four decades, one of the greatest principles of modern science, which led to the unification of the most diverse natural phenomena. This principle is as follows: There is a certain quantity called energy, which does not change under any transformations that occur in nature. There are no exceptions to the law of conservation of energy.
1. Why did the study of thermal phenomena and phase transitions reveal the failure of Laplacian determinism?
2. What are microparameters, macroparameters in the study of thermal phenomena?
3. What was the study of thermal phenomena connected with and when did it begin?
4. Name the scientists whose works formed the basis of the physics of thermal phenomena.
5. What are conservative forces? dissipative forces? Give examples.
6. For which systems is the law of conservation of mechanical energy valid?
7. What is potential energy? Is it only to mechanical systems What is the concept of potential energy? Explain.
8. Explain briefly the theory of caloric.
9. What experiments refuting the theory of caloric were carried out by Rumfoord?
10. Why are the heat capacities of gas in processes at constant pressure (Ср) and at constant volume (Сv) different? Which scientist first discovered this fact?
11. What is thermodynamics? What is she studying?
12. What does molecular kinetic theory study?
13. What is statistical physics? Where does this name come from?
14. Formulate the first law of thermodynamics.
15. With what (whom) can figuratively compare the first law of thermodynamics?
Literature
1. Diaghilev F.M. Concepts modern natural science. – M.: Ed. IMPE, 1998.
2. Concepts of modern natural science./ ed. prof. S.A. Samygin, 2nd ed. - Rostov n / a: "Phoenix", 1999.
3. Dubnishcheva T.Ya. Concepts of modern natural science. Novosibirsk: YuKEA Publishing House, 1997.
4. Remizov A.N. Medical and biological physics. – M.: graduate School, 1999.
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Books
- Statistical physics, Klimontovich Yu.L. This course differs from the existing ones both in content and in the nature of presentation. All material is presented on the basis of a single method - the theory of non-equilibrium state serves as the core ...
- Statistical Physics, L. D. Landau, E. M. Lifshits. 1964 edition. The safety is good. The book is clear general principles statics and, if possible, a more complete exposition of their many applications. The second edition adds...
