Statistical physics and thermodynamics. Dynamic and statistical laws. The principle of increasing entropy. Statistical thermodynamics Basic concepts in statistical thermodynamics

Definition 1

Statistical thermodynamics is an extensive branch of statistical physics that formulates laws that relate all the molecular properties of physical substances to the quantities measured during experiments.

Figure 1. Statistical thermodynamics of flexible molecules. Author24 - online exchange of student papers

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 by molecular constants. The basis of this scientific direction is hypotheses and assumptions confirmed by experiments.

Unlike classical mechanics, in statistical thermodynamics only the average readings of coordinates and internal momenta are studied, as well as the possibility of new values. The 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. Basic theory of statistical research: there is an unambiguous and stable relationship of 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 observable element over a long period of time begin to equate to the total value for the ensemble.

Basic idea of statistical thermodynamics

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Figure 2. Statistical formulation of the 2nd law of thermodynamics. Author24 - online exchange of student papers

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 coordinates and momentum of each individual particle at a certain point in time. The microscopic formulation requires the solution of 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:

• the temperature of the physical body;
• volume of interacting elements;
• the number of elementary particles.

If all substances are in an equilibrium state, then their macroscopic indicators will be constant, and the microscopic coefficients will gradually change. This means that each state in statistical thermodynamics corresponds to several microstates.

Remark 1

The main idea of the branch of physics under study is as follows: if many microstates correspond to each position of physical bodies, then each of them, as a result, makes a significant contribution to the general macrostate.

From this definition, one should single out the elementary properties of the statistical distribution function:

• normalization;
• positive definiteness;
• the mean 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

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For a statistical and competent description of macroscopic systems, scientists use the data of the ensemble and phase space, which makes it possible to solve classical and quantum problems using the method of probability theory. The microcanonical Gibbs ensemble is often used in the study of isolated systems that have a constant volume and the number of equally charged particles. This method is used to carefully describe systems of stable volume, which 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 in thermal and mechanical equilibrium in a certain space at constant pressure.

The phase space in statistical thermodynamics characterizes a mechanical multidimensional space, the axes of which are all generalized coordinates and their conjugate internal momenta 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 the phase space, and the change in the macrostate in time is represented by the movement of the point along the trajectory of a specific line. For a statistical description of the properties of the environment, the concepts of distribution function and phase volume are introduced, which characterize the probability density of finding a new point representing the real state of the system, as well as in a substance near a line with certain coordinates.

Remark 2

In quantum mechanics, instead of the 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 realizing a specific microstate in one volume element of the phase medium. The probability of finding particles in an infinitely small 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 the limiting indicator of all substances, where solutions of the equation of motion of the particles that make up the concept arise for the distribution function. The form of such a functional, which is the same for quantum and classical systems, was first established by the theoretical physicist J. Gibbs.

Statistical function calculations in thermodynamics

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For the correct calculation of 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 medium and can exchange particles and energy. This method is very convenient in the study of phase and chemical equilibria.

Statistical sums allow scientists to accurately determine the energy and thermodynamic properties of a system, obtained by differentiating the indicators by the appropriate parameters. All these quantities acquire a statistical meaning. Thus, the internal potential of a material body is identified with the average energy of the concept, which makes it possible to study the first law of thermodynamics as the basic law of conservation of energy in the case of unstable motion 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, hence, 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 the maximum statistical weight. Therefore, a smooth transition from a non-equilibrium position to an equilibrium one is a process of change into 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 a temperature of 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.

Lecture 2

Thermodynamics, statistical physics, informational entropy

1. Information from thermodynamics and statistical physics. distribution function. Liouville's theorem. Microcanonical distribution. First law of thermodynamics. adiabatic processes. Entropy. statistical weight. Boltzmann formula. The second law of thermodynamics. Reversible and irreversible processes.

2. Shannon information entropy. Bits, nats, trits, etc. Relationship between entropy and information.

This part belongs to Lecture 1. It is better considered in section V (“The concept of entanglement of quantum states”).

LE CNOT is depicted as:

We store the value of (ku) bit a, while (ku) bit b changes according to the XOR law:

bit b(target = target) changes its state if and only if the state of the control bit a corresponds to 1; in this case, the state of the control bit does not change.

The logical XOR operation (CNOT) illustrates why classical data can be cloned but quantum data cannot. Note that, in the general case, by quantum data we mean superpositions of the form

, (1)

where and are complex numbers or state amplitudes, moreover, .

According to the truth table, if XOR is applied to boolean data in which the second bit is in the “0” state (b), and the first bit is in the “X” state (a), then the first bit is not changed, and the second becomes its copy:

U XOR (X, 0) = (X, X), where X = “0” or “1”.

In the quantum case, the superposition (1) should be considered as the data denoted by the symbol “X”:

.

Physically, the data can be encoded, for example, in the polarization basis |V> = 1, |H> = 0 (H,V)= (0,1):

and

It can be seen that the copying of the state actually takes place. The no-cloning theorem states that copying is impossible. arbitrary quantum state. In the considered example, copying took place because the operation was performed in its own basis (|0>, |1>), i.e. in private case of a quantum state.

It would seem that the XOR operation can also be used to copy superpositions of two Boolean states, such as |45 0 > ? |V> + |H>:

But it's not! The unitarity of quantum evolution requires that the superposition of input states be transformed into the corresponding superposition of output states:

(2)

This is the so-called. an entangled state (Ф+), in which each of the two output qubits does not have a specific value (in this case, polarization). This example shows that logical operations performed on quantum objects follow different rules than in classical computational processes.

The following question arises: Seems to be state in output mode a can again be represented as a superposition like the state of the art b. How to show that this is not so, i.e., that it makes no sense at all to talk about the states of the mode (bit) a and mods (bit) b?

Let's use the polarization analogy when

(3).

There are two ways. Path 1 is long but more consistent. It is necessary to calculate the average values of the Stokes parameters for both output modes. The averages are taken over the wave function (2). If all but turn out to be equal to zero, then this state is unpolarized, i.e. mixed and superposition (3) has no meaning. We work in the Heisenberg representation, when the operators are transformed, but the wave function is not.

So, we find in fashion a.

is the total beam intensity a,

- proportion of vertical polarization,

- share +45 0th polarization,

- proportion of right-circular polarization.

The wave function over which the averaging is performed is taken in the form (2):

where are the birth and death operators in mods a and b operate according to the rules:

(Make calculations in section V (see notebook). In the same place, calculate the probability of registering coincidences or a correlator of the form }

Path II - more visual, but less “honest”!

Let us find the dependence of the light intensity in the mode a on the angle of rotation of the polaroid placed in this mode. This is the standard quantum-optical way to check the state (2) - the intensity should not depend on the rotation. At the same time, a similar dependence of the number of coincidences has the form

. For the first time, such dependences were obtained by E. Fry (1976) and A. Aspek (1985) and are often interpreted as proof of the nonlocality of quantum mechanics.

So, the experimental situation is shown in the figure:

By definition

where is the annihilation operator in mode a. It is known that the transformation of the operators of two orthogonally polarized modes x and y when light passes through a polaroid oriented at an angle has the form:

.

(only the first, fourth, fifth and eighth terms are non-zero) =

(only the first and eighth terms are different from zero) = - does not depend on the angle?!

Physically, this happens because the wave function (2) is not factorized and there is no point in talking about states in modes a and b separately. Thus, it cannot be argued that the mode a is in the superposition state (3)!

Comment. The calculations done (Path II) do not at all prove that the state is in fashion. a unpolarized. For example, in the presence of circularly polarized light in this mode, the result would be the same. Rigorous proof - for example, via Stokes parameters (in Section V).

Note that, proceeding in the same way, we can prove that the state in the a mode before the CNOT element is polarized.

Here, averaging should be carried out over the wave function of the initial state (3). The result is like this:

those. the maximum readings are reached at = 45 0 .

Information and entropy.

Without introducing yet the “operational” term “information”, we will argue using “everyday” language. Those. information is some knowledge about an object.

The following example speaks for the fact that the concepts of information and entropy are closely related. Consider an ideal gas in thermodynamic equilibrium. A gas consists of a huge number of molecules that move in volume V. The state parameters are pressure and temperature. The number of states of such a system is enormous. The gas entropy at TD equilibrium is maximum and, as follows from the Boltzmann formula, it is determined by the number of microstates of the system. At the same time, we do not know anything about what specific state the system has at a given time, we do not have - the information is minimal. Let us suppose that somehow we managed, with the help of a very fast instrument, to “peep into the state of the system at a given moment in time. So we got some information about her. One can even imagine that we photographed not only the coordinates of the molecules, but also their speeds (for example, by taking several photographs one after another). At the same time, at every moment of time when information about the state of the system is available to us, the entropy tends to zero, since the system is in only one specific state out of all their huge variety, and this state is highly non-equilibrium. This example shows that indeed information and entropy are somehow connected, and the nature of the connection is already emerging: the more information, the less entropy.

Information from thermodynamics and statistical physics.

Physical quantities characterizing the macroscopic states of bodies (many molecules) are called thermodynamic (including energy, volume). There are, however, quantities that appear as a result of the action of purely statistical regularities and that make sense when applied only to macroscopic systems. Such, for example, are entropy and temperature.

classical statistics

*Liouville's theorem. The distribution function is constant along the phase trajectories of the subsystem (we are talking about quasi-closed subsystems, so the theorem is valid only for not very long time intervals during which the subsystem behaves like a closed one).

Here - - distribution function or probability density. It is introduced through the probability w detect subsystem in phase space element at this point in time: dw = ( p 1 ,..., ps , q 1 ,..., qs ) dpdq , and

Finding the statistical distribution for any subsystem is the main task of statistics. If the statistical distribution is known, then it is possible to calculate the probabilities of various values of any physical quantities depending on the states of this subsystem (i.e., on the values of coordinates and momenta):

.

*Microcanonical distribution.

The distribution for the totality of two subsystems (they are assumed to be closed, i.e., weakly interacting) is equal. That's why - logarithm of the distribution function - magnitude additive. It follows from Liouville's theorem that the distribution function must be expressed in terms of such combinations of variables p and q, which must remain constant during the motion of the subsystem as a closed one (such quantities are called integrals of motion). This means that the distribution function itself is an integral of motion. Moreover, its logarithm is also an integral of motion, and additive. In total, there are seven integrals of motion in mechanics - energy, three components of momentum and three components of angular momentum - (for subsystem a: E a (p, q), P a (p, q), M a (p, q)). The only additive combination of these quantities is

moreover, the coefficients (there are seven of them) must remain the same for all subsystems of a given closed system, and is chosen from the normalization conditions (4).

For the normalization condition (4) to be satisfied, it is necessary that the function (p, q) addressed in points E 0, R 0 , M 0 to infinity. A more precise formulation gives the expression

Microcanonical distribution.

The presence of - functions ensures vanishing for all points of the phase space at which at least one of the quantities E, R, M is not equal to its specified (average) value E 0, R 0 , M 0 .

From six integrals P and M can be disposed of by enclosing the system in a solid box in which it rests.

.

Physical entropy

Again we use the concept of an ideal gas.

Let a monatomic ideal gas with density n and temperature T occupies volume V. We will measure the temperature in energy units - the Boltzmann constant will not appear. Each gas atom has an average kinetic energy of thermal motion equal to 3T/2. Therefore, the total thermal energy of the gas is

It is known that the gas pressure is p = nt. If the gas can exchange heat with the environment, then the gas energy conservation law looks like this:

. (5)

Thus, a change in the internal energy of a gas can occur both due to the work done by it, and due to the receipt of a certain amount of heat dQ from the outside. This equation expresses the first law of thermodynamics, i.e. law of energy conservation. It is assumed that the gas is in equilibrium, i.e. p = const throughout the volume.

If we assume that the gas is also in a state of TD equilibrium, T =const, then relation (5) can be considered as an elementary process of variation of the gas parameters during their very slow change, when the TD equilibrium is not disturbed. It is for such processes that the concept of entropy S is introduced using the relation

Thus, it is argued that, in addition to internal energy, an equilibrium gas has one more internal characteristic associated with the thermal motion of atoms. According to (5, 6) at a constant volume dV= 0, the change in energy is proportional to the change in temperature, and in the general case

Because where N = nV = const is the total number of gas atoms, then the last relation can be written as

After integration, we get

The expression in square brackets is the entropy per particle.

Thus, if both temperature and volume change in such a way that VT 3/2 remains constant, then the entropy S does not change. According to (6), this means that the gas does not exchange heat with the environment, i.e. the gas is separated from it by heat-insulating walls. Such a process is called adiabatic.

Because the

where = 5/3 is called the adiabatic exponent. Thus, in an adiabatic process, temperature and pressure change with density according to the law

Boltzmann formula

As follows from Liouville's theorem, the distribution function? has a sharp maximum at E = E 0 (average value) and is nonzero only in the vicinity of this point. If you enter the width E of the curve (E), defining it as the width of a rectangle whose height is equal to the value of the function (E) at the maximum point, and the area is equal to one (with proper normalization). One can pass from the range of energy values to the number of states Γ with energies belonging to E (this is, in fact, the average fluctuation of the energy of the system). Then the quantity Γ characterizes the extent to which the macroscopic state of the system is spread over its microscopic states. In other words, for classical systems, Г is the size of that region of the phase space in which the given subsystem spends almost all the time. phase space has a cell with volume , where s is the number of degrees of freedom

The value of Г is called the statistical weight of the macroscopic state, it can be written as:

The logarithm of the statistical weight is called the entropy:

where - statistical weight = the number of microstates covered by the considered macrostate of the system.

.

In quantum statistics it is shown that = 1. Then

Where by is understood the statistical matrix (density). In view of the linearity of the logarithm of the energy distribution function (*) , where the averaging is carried out over the distribution function .

Since the number of states is in any case not less than one, the entropy cannot be negative. S determines the density of levels of the energy spectrum of a macroscopic system. In view of the additivity of entropy, we can say that the average distances between the levels of a macroscopic body decrease exponentially with an increase in its size (ie, the number of particles in it). The highest value of entropy corresponds to complete statistical equilibrium.

Characterizing each macroscopic state of the system by the distribution of energy between different subsystems, we can say that a number of states sequentially passed by the system corresponds to an increasingly probable distribution of energy. This increase in probability is large due to its exponential nature. eS- in the exponent there is an additive value - entropy. That. processes occurring in a non-equilibrium closed system proceed in such a way that the system continuously passes from states with lower entropy to states with higher entropy. As a result, the entropy reaches the highest possible value, corresponding to complete statistical equilibrium.

Thus, if a closed system at some point in time is in a nonequilibrium macroscopic state, then the most likely consequence at subsequent points in time will be a monotonic increase in the entropy of the system. It - second law of thermodynamics (R. Clausius, 1865). Its statistical justification was given by L. Boltzmann in 1870. Other definition:

if at some point in time the entropy of a closed system is different from the maximum, then at subsequent moments the entropy does not decrease. It increases or, in the limiting case, remains constant. According to these two possibilities, all processes occurring with macroscopic bodies are usually divided into irreversible and reversible . irreversible - those processes that are accompanied by an increase in the entropy of the entire closed system (processes that would be their repetitions in reverse order cannot occur, since in this case the entropy would have to decrease). Note that the decrease in entropy can be caused by fluctuations. reversible processes are called in which the entropy of a closed system remains constant and which, therefore, can also take place in the opposite direction. A strictly reversible process is an ideal limiting case.

During adiabatic processes, the system does not absorb or release heat. ? Q = 0 .

Comment: (essential). The assertion that a closed system must for a sufficiently long time (longer than the relaxation time) go into a state of equilibrium applies only to a system that is in stationary external conditions. An example is the behavior of a large region of the Universe accessible to our observation (the properties of nature have nothing to do with the properties of an equilibrium system).

Information.

Consider a tape divided into cells - a classic register. If only one of the two characters can be placed in each cell, then the cell is said to contain a bit of information. It is obvious (see lecture 1) that in the register containing N cells contains N bit of information and it can be written 2 N messages. So, information entropy is measured in bits:

(7)

Here QN = 2 N is the total number of distinct messages. From (7) it is clear that information entropy is simply equal to the minimum number of binary cells that can be used to record some information.

Definition (7) can be rewritten in a different way. Let us have many QN various messages. Find the probability that the message we need will match the one randomly selected from the total number QN various messages. It is obviously equal P N = 1/ QN. Then definition (7) can be written as:

(8)

The more cells N, the less likely P N and the greater the information entropy H B contained in this particular message.

Example . The number of letters of the alphabet is 32 (without the letter ё). The number 32 is the fifth power of two 32 = 2 5 . To match each letter with a certain combination of binary numbers, you must have 5 cells. By adding capital letters to lowercase letters, we double the number of characters we want to encode - there will be 64 = 2 6 - i.e. extra bit of information added H B= 6. Here H B- the amount of information per letter (lowercase or uppercase). However, such a direct calculation of informational entropy is not entirely accurate, since there are letters in the alphabet that occur less often or more often. Those letters that are less common can be given a larger number of cells, and for frequently occurring letters, save and give them those register states that occupy a smaller number of cells. The exact definition of information entropy was given by Shannon:

(9)

Formally, the derivation of this relation can be substantiated as follows.

We showed above that

due to the additivity of the logarithm of the distribution function and its linearity in energy.

Let p- distribution function of some discrete value f i (for example, the letter “o” in this text). If using the function p build a probability distribution function of various values of the quantity f = f 1 , f 2 ,... f N, then this function will have a maximum at , where and (normalization). Then p()= 1 and (generally speaking, this is true for the class of functions satisfying the condition (*))

The summation is carried out over all symbols (letters of the alphabet), and pi means the probability of occurrence of a symbol with a number i. As you can see, this expression covers both frequently used letters and letters that are unlikely to appear in a given message.

Since the natural logarithm is used in expression (9), the corresponding unit of information is called “nat”.

Expression (9) can be rewritten as

where brackets mean the usual classical averaging using the distribution function p i .

Comment . In the following lectures it will be shown that for quantum states

where is the density matrix. Formally, expressions (10) and (11) coincide, but there is also a significant difference. Classical averaging is performed over orthogonal (eigen) states of the system, while for the quantum case states can also be non-orthogonal (superpositions). Therefore, always H quant H class !

Formulas (8) and (9) use logarithms in different bases. In (8) - by base 2, and in (9) - by base e. The information entropies corresponding to these formulas can be easily expressed through each other. We use the relation in which M is an arbitrary number

.

Then, considering that and we get

- the number of bits is almost one and a half times the number of nats!

Arguing similarly, one can obtain the ratio between the entropies expressed in trits and bits:

In computer technology, information is used in binary base (in bits). For reasoning in physics, it is more convenient to use information according to Shannon (in nats), which can characterize any discrete information. You can always find the number of matching bits.

RELATIONSHIP OF ENTROPY AND INFORMATION. Maxwell's Demon

This paradox was first considered by Maxwell in 1871 (see Fig. 1). Let some "supernatural" force open and close the damper in a vessel partitioned into two parts and containing gas. The shutter is controlled according to the rule: it opens if fast molecules moving from right to left touch it or if slow molecules hit it moving in the opposite direction. Thus, the demon introduces the temperature difference between the two volumes without doing work, which violates the second law of thermodynamics.

Demon Maxwell. The demon sets the pressure difference by opening the damper when the number of gas molecules hitting it from the left exceeds the number of hits from the right. This can be done in a completely reversible way, as long as the random results of its observations of molecules are stored in the demon's memory. Therefore, the demon's memory (or its head) heats up. The irreversible step is not that information is accumulated, but that information is lost when the daemon then clears the memory. Above: Filling the daemon's memory with bits of information is a random process. To the right of the dotted line is an empty memory area (all cells are in state 0, to the left are random bits). Below is a demon.

A number of attempts have been made to resolve the paradox or exorcise the demon. For example, it was assumed that the demon could not extract information without doing work or without perturbing (ie heating) the gas - but it turned out that this was not the case! Other attempts boiled down to the fact that the second principle can be violated under the influence of some "reasonable" or "thinking" forces (creatures). In 1929 Leo Szilard significantly "advanceed" the solution of the problem, reducing it to a minimal formulation and highlighting the essential components. The main thing that the Demon needs to do is to establish whether a single molecule is to the right or to the left of the sliding shutter, which would allow heat to be extracted. Such a device was called the Szilard engine. However, Szilard did not resolve the paradox, because his analysis did not take into account how the measurement by which the demon knows whether the molecule is on the right or on the left affects the increase in entropy (see figure Szilard_demon.pdf). The engine runs on a six-step cycle. The engine is a cylinder, at the ends of which pistons are placed. A damper is inserted in the middle. The work of pushing in the septum can be reduced to zero (this was shown by Szilard). There is also a memory device (UP). It can be in one of three states. "Empty", "Molecule on the right" and "Molecule on the left". Initial state: BP = “Empty”, pistons are depressed, the partition is extended, the molecule has an average speed, which is determined by the temperature of the thermostat (slide 1).

1. The septum is inserted, leaving the molecule on the right or left (slide 2).

2. The memory device determines where the molecule is and goes into the "right" or "left" state.

3. Compression. Depending on the state of the UE, the piston is pushed in from the side where there is no molecule. This step does not require any work to be done. Because the vacuum is compressed (slide 3).

4. The partition is removed. The molecule begins to exert pressure on the piston (slide 4).

5. Working stroke. The molecule hits the piston, causing it to move in the opposite direction. The energy of the molecule is transferred to the piston. As the piston moves, its average speed should decrease. However, this does not happen, since the walls of the vessel are at a constant temperature. Therefore, the heat from the thermostat is transferred to the molecule, keeping its speed constant. Thus, during the working stroke, the thermal energy supplied from the thermostat is converted into mechanical work performed by the piston (slide 6).

6. Purification of the UE, returning it to the "Empty" state (slide 7). The cycle is complete (slide 8 = slide 1).

Surprisingly, this paradox was not resolved until the 1980s. During this time, it was found that, in principle, any process can be made reversible, i.e. without "payment" by entropy. Finally, Bennett in 1982 established the final connection between this statement and Maxwell's paradox. He suggested that the demon could actually find out where the molecule is in the Szilard engine without doing work or increasing the entropy of the environment (thermostat) and thus doing useful work in one engine cycle. However, information about the position of the molecule must remain in the demon's memory (psi.1). As more cycles are performed, more and more information accumulates in memory. To complete the thermodynamic cycle, the daemon must erase the information stored in memory. It is this operation of erasing information that has to be classified as a process of increasing the entropy of the environment, as required by the second principle. This completes the fundamentally physical part of the device of the Maxwell demon.

Some development of these ideas was received in the works of D.D. Kadomtsev.

Consider an ideal gas consisting of only one particle (Kadomtsev, "dynamics and information"). This is not absurd. If one particle is enclosed in a vessel of volume V with walls at a temperature T, then sooner or later it will come into equilibrium with these walls. At each moment of time, it is at a well-defined point in space and at a well-defined speed. We will carry out all the processes so slowly that the particle will have time, on average, to fill the entire volume and repeatedly change the magnitude and direction of the velocity during inelastic collisions with the walls of the vessel. Thus, the particle exerts an average pressure on the walls, has a temperature T and its velocity distribution is Maxwellian with temperature T. This system of one particle can be compressed adiabatically, its temperature can be changed, giving it the opportunity to come into equilibrium with the walls of the vessel.

Average pressure on the wall at N = 1 , equals p= T/V, and the average density n = 1/ V. Consider the case of an isothermal process, when T =const. From the first start T =const. and p= T/V we get

, because the

From this we find that the change in entropy does not depend on temperature, so that

Here the constant of integration is introduced: “particle size”<

Operation in an isothermal process

work is determined by the difference of entropies.

Suppose we have ideal partitions that can be used to divide the vessel into parts without expending energy. We divide our vessel into two equal parts with a volume V/2 each. In this case, the particle will be in one of the halves - but we do not know which one. Suppose we have a device that allows us to determine in which part the particle is located, for example, a precision balance. Then from a symmetrical probability distribution of 50% to 50% of being in two halves, we get 100% probability for one of the halves - there is a “collapse” of the probability distribution. Accordingly, the new entropy will be less than the original entropy by the value

By reducing entropy, work can be done. To do this, it is enough to move the partition towards the empty volume until it disappears. The work will be equal If nothing changed in the external world, then by repeating these cycles, it is possible to build a perpetual motion machine of the second kind. This is Maxwell's demon in Szilard's version. But the second law of thermodynamics forbids the receipt of work only due to heat. So something must be happening in the outside world. What is this? Detection of a particle in one of the halves changes information about the particle - of the two possible halves, only one is indicated, in which the particle is located. This knowledge corresponds to one bit of information. The process of measurement reduces the entropy of the particle (transfer to a nonequilibrium state) and exactly the same increases the information about the system (particle). If you make repeated divisions in half of the previously obtained halves, quarters, eighths, etc., then the entropy will consistently decrease, and the information will increase! In other words

The more known about a physical system, the lower its entropy. If everything is known about the system, this means that we have transferred it to a highly non-equilibrium state, when its parameters are as far as possible from equilibrium values. If in our model the particle can be placed in the unit cell of the volume V 0 , then at the same time S = 0 , and the information reaches its maximum value because the probability pmin find a particle in a given cell is equal to V 0 / V. If at subsequent moments of time the particle begins to fill a larger volume, then the information will be lost, and the entropy will increase. We emphasize that information must be paid for (according to the second law) by an increase in entropy Se external system, and Indeed, if for one bit of information the device (external system) increased its entropy by less than one bit, then we could reverse the heat engine. Namely, by expanding the volume occupied by a particle, we would increase its entropy by ln2 getting a job Tln2 , and the total entropy of the particle plus device system would decrease. But this is impossible according to the second law. Formally, , therefore, the decrease in the entropy of the system (particle) is accompanied by an increase in the entropy of the device .

So the information entropy is a measure of the lack (or degree of uncertainty) of information about the actual state of the physical system.

Shannon information entropy:

, where (this applies to two-level systems, such as bits: “0” and “1”. If the dimension is n, then H = log n. Yes, for n = 3, H =log 3 moreover, = 3.)

Amount of information I(or simply information) about the state of a classical system, obtained as a result of measurements by an external device connected to the system under consideration by some communication channel, is defined as the difference in the information entropy corresponding to the initial uncertainty of the system state H 0 , and the information entropy of the final state of the system after the measurement H. In this way,

I + H = H 0 = const .

In the ideal case, when there is no noise and interference created by external sources in the communication channel, the final probability distribution after the measurement is reduced to one specific value p n= 1, i.e. H = 0, and the maximum value of the information obtained during the measurement will be determined by: Imax = H 0 . Thus, Shannon's informational entropy of a system has the meaning of the maximum information contained in the system; it can be determined under ideal conditions for measuring the state of the system in the absence of noise and interference, when the entropy of the final state is zero:

Consider a classical logical element that can be in one of two equiprobable logical states “0” and “1”. Such an element, together with the environment - a thermostat and a signal generated by an external heat-insulated object, forms a single non-equilibrium closed system. The transition of an element to one of the states, for example, to the state “0”, corresponds to a decrease in stat. the weight of its state compared to the initial state by 2 times (for three-level systems - by 3 times). Let's find the decrease information entropy Shannon, which corresponds to an increase in the amount of information about the element by one, which is called bit:

Therefore, the information entropy determines the number of bits that are required to encode information in the system or message in question.

LITERATURE

1. D. Landau, I. Lifshits. Statistical physics. Part 1. Science, M 1976.

2. M.A. Leontovich. Introduction to thermodynamics. Statistical physics. Moscow, Nauka, 1983. - 416s.

3. B.B. Kadomtsev. Dynamics and information. UFN, 164, No. 5, 449 (1994).

STATISTICAL THERMODYNAMICS, statistic section. physics, dedicated to the substantiation of the laws of thermodynamics on the basis of the laws of interaction. and motion of the particles that make up the system. For systems in equilibrium, statistical thermodynamics makes it possible to calculate thermodynamic potentials, write equations of state, phase and chemical conditions. equilibria. Non-equilibrium statistical thermodynamics provides a rationale for the relationships (equations of energy transfer, momentum, mass and their boundary conditions) and allows you to calculate the kinetic transfer equations included in the equation. coefficients. Statistical thermodynamics sets quantities. connection between micro- and macro-properties of physical. and chem. systems. Calculation methods of statistical thermodynamics are used in all directions of modern. theoretical chemistry.

Basic concepts. For statistical macroscopic descriptions. systems J. Gibbs (1901) proposed to use the concept of statistical. ensemble and phase space, which makes it possible to apply methods of probability theory to problem solving. Statistical ensemble - a set of a very large number of identical systems pl. particles (i.e., "copies" of the system under consideration) that are in the same macrostate, which is determined by the state parameters ; The microstates of the system may differ in this case. Main statistical ensembles-microcanonical, canonical, grand canonical. and isobaric-isothermal.

Microcanonical The Gibbs ensemble is used when considering isolated systems (not exchanging energy E with the environment) that have a constant volume V and the number of identical particles N (E, V and N are the state parameters of the system). Canonical the Gibbs ensemble is used to describe systems of constant volume that are in thermal equilibrium with the environment (abs. temperature T) at a constant number of particles N (state parameters V, T, N). Grand Canon. the Gibbs ensemble is used to describe open systems that are in thermal equilibrium with the environment (t-ra T) and material equilibrium with a reservoir of particles (particles of all sorts are exchanged through the "walls" surrounding the system with volume V). The state parameters of such a system are V, T and m - the chemical potential of the particles. Isobaric-isothermal The Gibbs ensemble is used to describe systems found in thermal and fur. equilibrium with the environment at constant pressure P (state parameters T, P, N).

Phase space in statistical. a mechanics-multidimensional space, the axes of which are all generalized coordinates q i and their conjugate momenta p i (i =1,2,..., M) of a system with M degrees of freedom. For a system consisting of N atoms, q i and p i correspond to the Cartesian coordinate and momentum component (a = x, y, z) of some atom j and M = 3N. The set of coordinates and momenta are denoted by q and p, respectively. The state of the system is represented by a point in the phase space of dimension 2M, and the change in the state of the system in time is represented by the movement of the point along the line, called. phase trajectory. For statistical description of the state of the system, the concepts of phase volume (an element of the volume of the phase space) and the distribution function f(p, q) are introduced, which characterizes the probability density of finding a point representing the state of the system in an element of the phase space near the point with coordinates p, q. In quantum mechanics, instead of the phase volume, the concept of discrete energy is used. spectrum of a system of finite volume, since the state of an individual particle is determined not by the momentum and coordinates, but by the wave function, which is in a stationary dynamic. the state of the system corresponds to the energetic. spectrum of quantum states.

distribution function classical system f(p, q) characterizes the probability density of realization of a given microstates (р, q) in the volume element dГ of the phase space. The probability of N particles being in an infinitely small volume of the phase space is equal to:

where dГ N is the element of the phase volume of the system in units of h 3N , h is Planck's constant; divisor N! takes into account the fact that the permutation of identities. particles does not change the state of the system. The distribution function satisfies the normalization condition t f(p, q)dГ N = 1, because the system is reliably located in the c.-l. condition. For quantum systems, the distribution function determines the probability w i , N of finding a system of N particles in a quantum state given by a set of quantum numbers i , with energy E i,N under the condition of normalization

The average value at time t (i.e. overinfinitely small time interval from t to t + dt) any physical. the quantity A(p, q), which is a function of the coordinates and momenta of all particles of the system, using the distribution function is calculated according to the rule (including for non-equilibrium processes):

Integration over coordinates is carried out over the entire volume of the system, and integration over momenta from - , to +, . thermodynamic state. equilibrium of the system should be considered as a limit m: , . For equilibrium states, distribution functions are determined without solving the equations of motion of the particles that make up the system. The form of these functions (the same for classical and quantum systems) was established by J. Gibbs (1901).

In the microcanonical Gibbs ensemble all microstates with a given energy E are equiprobable and the distribution function for the classical. systems looks like:

f(p,q) = A d,

where d-delta-f-tion of Dirac, H (p, q)-f-tion of Hamilton, which is the sum of the kinetic. and potent. energies of all particles; the constant A is determined from the condition of normalization of the function f(p, q). For quantum systems, with an accuracy of specifying a quantum state equal to D E, in accordance with the uncertainty relation between energy and time (between momentum and particle coordinate), function w (E k) = -1, if ЕE k E + D E, and w (E k) = 0 if E k< Е и E k >E + D E. The quantity g(E, N, V)-i.e. called statistical weight equal to the number of quantum states in energetic. layer D E. An important relation of statistical thermodynamics is the relationship of the entropy of the system with the statistical. weight :

S(E, N, V) = klng(E, N, V), where k-Boltzmann is a constant.

In canonical In the Gibbs ensemble, the probability of the system being in a microstate determined by the coordinates and momenta of all N particles or by the values of E i,N has the form: f(p, q) = exp (/kT); w i,N = exp[(F - E i,N)/kT],where F-free. energy (Helmholtz energy) depending on the values of V, T, N:

F = -kTlnZN ,

where Z N -statistic. sum (in the case of a quantum system) or statistical. integral (in the case of a classical system), determined from the condition of normalization of functions w i,N or f(p, q):

Z N = m exp[-H(p, q)/kT]dpdq/(N!h 3N)

(the sum over r is taken over all quantum states of the system, and the integration is carried out over the entire phase space).

In the grand canonical Gibbs ensemble f-tion distribution f (p, q) and statistical. the sum X determined from the normalization condition has the form:

where W - thermodynamic potential depending on the variables V, T, m (the summation is over all positive integers N). In the isobaric-isothermal Gibbs ensemble f-tion of distribution and statistical. the sum Q determined from the normalization condition has the form:

where G is the Gibbs energy of the system (isobaric-isothermal potential, free enthalpy).

To calculate the thermodynamic f-tion can use any distribution: they are equivalent to each other and correspond to different physical. conditions. Microcanonical the Gibbs distribution applies Ch. arr. in the theoretical research. To solve specific problems, ensembles are considered in which there is an exchange of energy with the medium (canonical and isobaric-isothermal) or an exchange of energy and particles (grand canonical ensemble). The latter is especially convenient for studying the phase and chem. equilibria. Statistical the sums Z N and Q make it possible to determine the Helmholtz energy F, the Gibbs energy G, and also the thermodynamic. St. Islands of the system obtained by differentiation of statistical. sums for the relevant parameters (per 1 mol in-va): ext. energy U = RT 2 (9 lnZ N /9 T) V , enthalpy H = RT 2 (9 lnQ/9 T) P , entropy S = RlnZ N + RT(9 lnZ N /9 T) V = R ln Q + RT(9 ln Q/9 T) P , heat capacity at constant volume С V = 2RT(9 lnZ N /9 T) V + RT 2 (9 2 lnZ N /9 T 2) V , heat capacity at constant pressure С Р = 2RT (9 lnZ N /9 T) P + + RT 2 (9 2 lnZ N /9 T 2) P etc. Resp. all these quantities acquire and statistical. meaning. Thus, the internal energy is identified with the average energy of the system, which allows us to consider the first law of thermodynamics as the law of conservation of energy during the motion of the particles that make up the system; free energy is related to statistic. the sum of the system, entropy, with the number of microstates g in a given macrostate, or statistical. the weight of the macrostate, and, consequently, with its probability. The meaning of entropy as a measure of the probability of a state is preserved with respect to arbitrary (non-equilibrium) states. In a state of equilibrium, entropy is isolated. system has the maximum possible value for given ext. conditions (E, V, N), i.e., the equilibrium state is the most. probable state (with max. statistical weight). Therefore, the transition from a non-equilibrium state to an equilibrium state is a process of transition from less probable states to more probable ones. This is the statistical the meaning of the entropy increase law, according to which the entropy of a closed system can only increase (see the Second Law of Thermodynamics). At t-re abs. zero, any system is in the core. state, in which w 0 = 1 and S = 0. This statement is the third law of thermodynamics (see Thermal theorem). It is essential that for an unambiguous definition of entropy it is necessary to use the quantum description, since in the classic statistics entropy m. b. is defined only up to an arbitrary term.

ideal systems. Calculation of statistical sums of most systems is a difficult task. It is greatly simplified in the case of gases, if the contribution of the potential. energy into the total energy of the system can be neglected. In this case, the total distribution function f(p, q) for N particles of an ideal system is expressed through the product of one-particle distribution functions f 1 (p, q):

The distribution of particles over microstates depends on their kinetics. energy and from quantum sv-in the system, due toparticle identity. In quantum mechanics, all particles are divided into two classes: fermions and bosons. The type of statistics that particles obey is uniquely related to their spin.

The Fermi-Dirac statistics describes the distribution in a system of identities. particles with half-integer spin 1 / 2 , 3 / 2 ,... in units ђ = h/2p . A particle (or quasi-particle) subject to the specified statistics, called. fermion. Fermions include electrons in atoms, metals and semiconductors, atomic nuclei with an odd atomic number, atoms with an odd difference between the atomic number and the number of electrons, quasiparticles (for example, electrons and holes in solids), etc. This statistics was proposed by E. Fermi in 1926; in the same year, P. Dirac found out its quantum mechanics. meaning. The wave function of the fermion system is antisymmetric, i.e. changes its sign upon permutation of the coordinates and spins of any pair of identities. particles. Each quantum state can contain no more than one particle (see the Pauli principle). The average number of particles n i of an ideal fermion gas in a state with energy E i is determined by the Fermi-Dirac distribution function:

n i =(1+exp[(E i - m )/kT]) -1 ,

where i is a set of quantum numbers characterizing the state of the particle.

Bose-Einstein statistics describes systems of identities. particles with zero or integer spin (0, ђ, 2ђ, ...). A particle or quasi-particle that obeys the specified statistics, called. boson. This statistics was proposed by S. Bose (1924) for photons and developed by A. Einstein (1924) in relation to ideal gas molecules, considered as composite particles from an even number of fermions, for example. atomic nuclei with an even total number of protons and neutrons (deuteron, 4 He nucleus, etc.). Bosons also include phonons in solids and liquid 4 He, excitons in semiconductors and dielectrics. The wave function of the system is symmetrical with respect to the permutation of any pair of identities. particles. The occupation numbers of quantum states are not limited by anything, i.e. any number of particles can be in one state. The average number of particles n i of an ideal gas of bosons that are in a state with energy E i is described by the Bose-Einstein distribution function:

n i =(exp[(E i - m )/kT]-1) -1 .

Boltzmann statistics is a special case of quantum statistics, when quantum effects can be neglected (high temperatures). It considers the distribution of particles of an ideal gas in terms of momenta and coordinates in the phase space of one particle, and not in the phase space of all particles, as in the Gibbs distributions. As a minimum volume units of the phase space, which has six dimensions (three coordinates and three projections of the particle's momentum), in accordance with quantum mechanics. uncertainty relation , you can not choose a volume smaller than h 3 . The average number of particles n i of an ideal gas in a state with energy E i is described by the Boltzmann distribution function:

n i =exp[( m -E i)/kT].

For particles, to-rye move according to the laws of the classical. mechanics in ext. potent. field U(r), the statistically equilibrium distribution function f 1 (p, r) in momentum p and coordinates r of particles of an ideal gas has the form:f 1 (p, r) = A exp( - [р 2 /2m + U(r)]/kT). Here p 2 /2m-kinetic. the energy of molecules with mass w, the constant A is determined from the normalization condition. This expression is often called the Maxwell-Boltzmann distribution, and the Boltzmann distribution is called. function

n(r) = n 0 exp[-U(r)]/kT],

where n(r) = t f 1 (p, r)dp is the density of the number of particles at the point r (n 0 is the density of the number of particles in the absence of an external field). The Boltzmann distribution describes the distribution of molecool in the gravitational field (barometric f-la), molecules and highly dispersed particles in the field of centrifugal forces, electrons in non-degenerate semiconductors, and is also used to calculate the distribution of ions in a dilute. solutions of electrolytes (in the volume and at the boundary with the electrode), etc. At U(r) = 0, the Maxwell-Boltzmann distribution follows from the Maxwell-Boltzmann distribution, which describes the velocity distribution of particles in the statistic. equilibrium (J. Maxwell, 1859). According to this distribution, the probable number of molecules per unit volume of the velocity components, which lie in the intervals from u i to u i + du i (i = x, y, z), is determined by the function:

The Maxwell distribution does not depend on the interaction. between Particles and is true not only for gases, but also for liquids (if a classical description is possible for them), as well as for Brownian particles suspended in liquids and gases. It is used to count the number of collisions of gas molecules with each other during chem. p-tion and with atoms pov-sti.

The sum over the states of the molecule. Statistical the sum of an ideal gas in the canonical Gibbs ensemble is expressed in terms of the sum over the states of one molecule Q 1:

where E i is the energy of the i-th quantum level of the molecule (i = O corresponds to the zero level of the molecule), g i -statistic. weight of the i-th level. In the general case, the individual types of motion of electrons, atoms and groups of atoms in a molecule, as well as the motion of the molecule as a whole, are interconnected, but approximately they can be considered as independent. Then the sum over the states of the molecule can be presented as a product of individual components associated with step-by-step. movement (Q post) and vnutrimol. movements (Q ext):

Q 1 \u003d Q post Q ext, Q post \u003d l (V / N),

where l \u003d (2p mkT / h 2) 3/2. For atoms Q ext is the sum over the electronic and nuclear states of the atom ; for molecules Q ext - the sum of the electronic, nuclear, vibrational. and rotate. states. In the region of t-r from 10 to 10 3 K, an approximate description is usually used, in which each of the indicated types of motion is considered independently: Q ext \u003d Q el Q poison Q rotation Q count /g, where g is the symmetry number , equal to the identity number. configurations arising from the rotation of a molecule consisting of identical atoms or groups of atoms.

The sum of the states of electronic motion Q el is equal to statistical. weight P t main. the electronic state of the molecule. In many cases of main the level is non-degenerate and separated from the nearest excited level means. energy: (P t \u003d 1). However, in some cases, eg. for the O 2 molecule, P t \u003d h, in the main. state, the moment of the number of motion of the molecule is nonzero and there is a degeneracy of the energy levels, and the energies of the excited states can be. low enough. The sum over the states Q kernel, due to the degeneracy of nuclear spins, is equal to:

where s i is the spin of the nucleus of atom i, the product is taken over all atoms of the molecule. The sum of the states oscillate. movementsmolecules where v i -frequencies of holessmall fluctuations, n is the number of atoms in a molecule. Sum by states rotate. motions of a polyatomic molecule with large moments of inertia can be considered classically [high-temperature approximation, T/q i 1, where q i = h 2 /8p 2 kI i (i = x, y, z), I t is the main moment of inertia of rotation around the i axis] : Q vr \u003d (p T 3 / q x q y q z) 1/2. For linear molecules with moment of inertia I statistic. sum Q vr \u003d T / q, where q \u003d h 2 / 8p 2 * kI.

When calculating at temperatures above 10 3 K, it is necessary to take into account the anharmonicity of atomic vibrations, the effects of interaction. oscillating and rotate. degrees of freedom (see. Nonrigid molecules), as well as the multiplicity of electronic states, the population of excited levels, etc. At low temperatures (below 10 K), quantum effects must be taken into account (especially for diatomic molecules). Yes, they rotate. the movement of a hetero-nuclear molecule AB is described by the f-le:

l-rotation number states, and for homonuclear molecules A 2 (especially for hydrogen molecules H 2, deuterium D 2, tritium T 2) nuclear and rotation. degree of freedom interaction. friendwith a friend: Q poison. rot. Q poison Q rotation

Knowing the sum over the states of the molecule allows you to calculate the thermodynamic. Holy Islands of an ideal gas and a mixture of ideal gases, incl. chemical constants. equilibrium, equilibrium degree of ionization, etc. Importance in the theory of abs. speeds p-tions has the ability to calculate the equilibrium constant of the process of education aktivir. complex (transition state), a cut is presented as a modified. particle, one of the vibrations. degrees of freedom to-roy replaced by a degree of freedom act. movement.

Imperfect systems. In real gases, molecules interact. together. In this case, the sum over the states of the ensemble does not reduce to the product of the sums over the states of individual molecules. If we consider that the intermol. interaction do not affect the internal states of molecules, statistical. the sum of the system in the classical approximation for a gas consisting of N identities. particles has the form:

where

Here<2 N-config. integral, taking into account the interaction. molecules. Naib, often potent. the energy of molecules U is considered as the sum of pair potentials: U = = where U(r ij) is the potential center. forces depending ondistances r ij between molecules i and j. Many-particle contributions to the potential are also taken into account. energy, molecular orientation effects, etc. The need to calculate the configuration. integral arises when considering any condenser. phases and phase boundaries. The exact solution of the problem pl. bodies is almost impossible, therefore, to calculate the statistical. sum and all thermodynamic. sv-in, obtained from the statistical. sums by differentiation with respect to the relevant parameters, use decomp. approximate methods.

According to the so-called. group decomposition method, the state of the system is considered as a set of complexes (groups) consisting of a different number of molecules , and configurations. the integral splits into a set of group integrals. This approach allows us to represent any thermodynamic f-tion of a real gas in the form of a series in degrees of density. Naib. An important relation of this kind is the virial equation of the state.

For the theoretical descriptions of St. in dense gases, liquids and solids, solutions of non-electrolytes and electrolytes and interfaces in these systems are more convenient than direct calculation of statistical. sums, is the method of n-particle distribution functions. In it, instead of counting statistical weights of each state with fixed. energy use the relationship between distribution functions f n , which characterize the probability of finding particles simultaneously at points in space with coordinates r 1 ,..., r n ; for n = N f N = b t f(p, r)dp (here and below q i = r i). Single-particle function f 1 (r 1) (n = 1) characterizes the density distribution in the islands. For a solid body, this is periodic. f-tion with maxima at the nodes of the crystal. structures; for gases or liquids in the absence of ext. field is a constant value equal to macroscopic. density in-va river. The two-particle distribution function (n = 2) characterizes the probability of findingtwo particles at points 1 and 2, it determines the so-called. correlation function g(|r 1 - r 2 |) = f 2 (r 1 , r 2)/r 2 characterizing the mutual correlation in the distribution of particles. Corresponding experimental information is provided by X-ray structural analysis.

Distribution functions of dimensions n and n + 1 are connected by an infinite system of linked integro-differentials. equations of Bogolyubov-Born-Green-Kirkwood-Yvon, the solution to which is extremely difficult, so the effects of correlation between particles are taken into account by the introduction of decomp. approximations, which determine how the function f n is expressed in terms of functions of lower dimension. Resp. developed by several approximate methods for calculating functions f n , and through them, all thermodynamic. characteristics of the system under consideration. Naib. Approximations Perkus-Ievik and hyperchain.

Lattice models of condensers. states are widely used in thermodynamic. consideration of almost all fiz.-chem. tasks. The entire volume of the system is divided into local regions with a characteristic size of the order of the size of the molecule u 0 . In the general case, in different models, the size of the local area can be both greater and less than u 0 ; in most cases they match. The transition to a discrete distribution of molecules in space greatly simplifies the calculation of decomp. molecular configurations. Lattice models take into account the interaction. molecules with each other; interaction energy. described energetically. parameters. In a number of cases, lattice models allow exact solutions, which makes it possible to estimate the nature of the approximations used. With their help it is possible to consider many-particle and specific. interaction, orientation effects, etc. Lattice models are the main ones in the study and implementation of applied calculations for solutions of non-electrolytes and polymers, phase transitions, critical phenomena, and highly inhomogeneous systems.

Numerical methods for determining thermodynamic. sv-in become increasingly important as the development of compute. technology. In the Monte Carlo method, a direct calculation of multidimensional integrals is carried out, which allows you to directly obtain the statistical. mean of observedvalues A(r1.....r N) according to any of the statistical. ensembles(eg, A is the energy of the system). So, in the canonical thermodynamic ensemble. the average looks like:

This method is applicable to almost all systems; the average values obtained with its help for limited volumes (N = 10 2 -10 5) serve as a good approximation for describing the macroscopic. objects and can be considered as exact results.

In the method they say dynamics, the evolution of the state of the system is considered with the help of numerical integration of Newton's equations for the motion of each particle (N = 10 2 -10 5) at given potentials of interparticle interaction. The equilibrium characteristics of the system are obtained by averaging over phase trajectories (over velocities and coordinates) at large times, after the establishment of the Maxwellian distribution of particles over velocities (the so-called thermalization period).

Limitations in the use of numerical methods in the main. determined by the capabilities of the computer. Specialist. calculate. techniques allow you to bypass the difficulties associated with the fact that we are considering not a real system, but a small volume; this is especially important when taking into account long-range interaction potentials, analyzing phase transitions, and so on.

Physical kinetics - a section of statistical. physics, to-ry gives a justification for the relations of thermodynamics of irreversible processes, describing the transfer of energy, momentum and mass, as well as the influence of external processes on these processes. fields. Kinetic macroscopic coefficients. characteristics of a continuous medium that determine the dependence of the flows of physical. quantities (heat, momentum, mass of components, etc.) fromcausing these flows of gradients t-ry, concentration, hydrodynamic. speeds, etc. It is necessary to distinguish between the Onsager coefficients included in the equations that connect flows with thermodynamic. forces (thermodynamic equations of motion), and transfer coefficients (diffusion, thermal conductivity, viscosity, etc.) included in the transfer equations. The first m. expressed in terms of the second using the relationship between macroscopic. characteristics of the system, therefore, in the future, only the coefficient will be considered. transfer.

To calculate the macroscopic coefficient transfer, it is necessary to average over the probabilities of realizations of elementary transfer acts using a non-equilibrium distribution function. The main difficulty lies in the fact that the analyte. the form of the distribution function f(p, q, t) (t-time) is unknown (in contrast to the equilibrium state of the system, which is described using the Gibbs distribution functions obtained at t : , ). Consider n-particle distribution functions f n (r, q, t), which are obtained from f-tions f (p, q, t) by averaging over the coordinates and momenta of the remaining (N - n) particles:

For them, m. a system of equations was compiled, which makes it possible to describe arbitrary non-equilibrium states. The solution of this system of equations is very difficult. As a rule, in the kinetic theory of gases and gaseous quasi-particles in a solid (fermions and bosons) uses only the equation for a one-particle distribution function f 1 . Under the assumption that there is no correlation between the states of any particles (the hypothesis of mol. chaos), the so-called. kinetic ur-tion of Boltzmann (L. Boltzmann, 1872). This ur-tion takes into account the change in the function of the distribution of particles under the influence of external. force F(r, m) and pair collisions between particles:

where f 1 (u, r, t) and particle distribution functions up tocollisions, f " 1 (u", r, t) and distribution functionsafter the collision; u and -velocities of particles before collision, u" and -velocities of the same particles after collision, u = |u -|-modulus of relative velocity of colliding particles, q - angle between relative velocity of u - colliding particles and the line connecting their centers , s (u,q )dW is the differential effective cross section of particle scattering at the solid angle dW in the laboratory coordinate system, which depends on the law of interaction particles For the model of molecules in the form of elastic rigid spheres with radius R, we take s = 4R 2 cosq In the framework of classical mechanics, the differential cross section is expressed in terms of the collision parameters b and e (respectively, the impact distance and the azimuth angle of the line of centers): s dW = bdbde , and the molecules are considered as centers of forces with a potential depending on the distance. the expression for the differential effective cross section is obtained on the basis of quantum mechanics, taking into account the influence of symmetry effects on the collision probability.

If the system is in statistical equilibrium , the collision integral Stf is equal to zero and the solution of the kinetic. the Boltzmann equation will be the Maxwell distribution. For non-equilibrium states of the solution kinetic. Boltzmann's equations are usually sought in the form of an expansion into a series of functions f 1 (u, r, m) in small parameters with respect to the function of Maxwell's distribution. In the simplest (relaxation) approximation, the collision integral is approximated as St f gases with internal degrees of freedom of symmetry the thermal conductivity of the liquid, you can use a locally equilibrium single-particle distribution function with t-swarm, chem. potentials and hydrodynamic. speed, to-rye correspond to the considered small volume of liquid . To it, you can find a correction proportional to the gradients of the t-ry, hydrodynamic. speed and chem. potentials of the components, and calculate the fluxes of momentum, energy and in-va, as well as justify the Navier-Stokes equations, thermal conductivity and diffusion. In this case, the coefficient transfer are proportional to space-time correlations. f-tions of energy flows, momentum and in-va of each component.

The lattice model of a condenser is widely used to describe the processes of matter transfer in solids and at interfaces with a solid. phases. The evolution of the state of the system is described in the main. kinetic equation (master equation) with respect to the distribution function P(q, t):

where P(q,t)= t f (p, q, t) du - distribution function, averaged over the momenta (velocities) of all N particles, describing the distribution of particles over the nodes of the lattice structure (their number is N y , N< N y), q- номер узла или его координата. В модели "решеточного газа " частица может находиться в узле (узел занят) или отсутствовать (узел свободен); W(q : q") is the probability of the transition of the system per unit time from state q, described by a complete set of particle coordinates, to another state q". The first sum describes the contribution of all processes in which the transition to a given state q is carried out, the second sum is the exit from this state. In the case of an equilibrium distribution of particles (t : , ) P(q) = exp[-H(q)/kT]/Q, where Q is a statistic. sum, H(q)-energy of the system in state q. The transition probabilities satisfy the detailed equilibrium principle: W(q" : q)exp[-H(q")/kT] = W(q : q")exp[-H(q)/kT]. On the basis of the equations for the functions P(q, t), a kinetic is built. ur-tion for n-particle distribution functions, to-rye obtained by averaging over the locations of all other (N - n) particles. For small h kinetic. ur-tion m. b. solved analytically or numerically and with their help m. coefficients received. diffusion, self-diffusion, shear viscosity, mobility, etc. This approach is applicable to the processes of transfer of matter in monatomic crystals of relaxation of the system to an equilibrium state allows us to consider decomp. transient processes in the study of the kinetics of phase transformations, crystal growth, the kinetics of surface p-tions, etc. and determine their dynamics. characteristics, including and coefficient. transfer.

To calculate the coefficient transfer in gaseous, liquid and solid phases, as well as at the phase boundaries, various variants of the pier method are actively used. dynamics, which allows you to follow in detail the evolution of the system from times ~10 -15 s to ~10 -10 s Newtonian terms containing a stochastic term on the right-hand side).

For systems with chem. p-tions on the nature of the distribution of particles is greatly influenced by the ratio between the characteristic transfer times of the reagents and their chemical. transformations. If the speed of the chem. the transformation is small, the distribution of particles does not differ much from the case when p-tion is absent. If the speed of the p-tion is large, its influence on the nature of the distribution of particles is great and it is impossible to use the average concentrations of particles (i.e., distribution functions with n = 1), as is done when using the law of mass action. It is necessary to describe in more detail the distribution of reagents using distribution functions f n with n > 1. Importance in describing the reactions. particle flows on the surface and rates of diffusion-controlled reactions have boundary conditions (see Macrokinetics)., 2nd ed., M., 1982; Berkeley course of physics, trans. from English, 3rd ed., vol. 5-Reif F., Statistical Physics, M., 1986; Tovbin Yu.K., Theory of physical and chemical processes at the gas-solid boundary, M., 1990. Yu.K. Tovbin.

Statistical physics and thermodynamics

Statistical and thermodynamic research methods . Molecular physics and thermodynamics are branches of physics that study macroscopic processes in bodies, associated with a 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 that are in continuous chaotic motion.

The idea of the atomic structure of matter was expressed by the ancient Greek philosopher Democritus (460-370 BC). Atomistics 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 work 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 regularities, 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 motion and averaged the 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 in time different molecules have different speeds, it can only be expressed through the average value of the speed of the molecules. It is impossible to 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 statistics. Thermodynamics is based on two principles - fundamental laws established as a result of generalization of experimental data.

The field of application of thermodynamics is much wider than that of molecular-kinetic theory, because there are no areas of physics and chemistry in which it would be impossible to use the thermodynamic method. However, on the other hand, the thermodynamic method is somewhat limited: thermodynamics says nothing about the microscopic structure of a substance, about the mechanism of phenomena, but only establishes connections between the macroscopic properties of a substance. Molecular-kinetic theory and thermodynamics mutually complement each other, forming a single whole, but differing in different 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(random) movement.

3. The movement of particles is related to the temperature of the body, which is why it is called thermal motion.

4. Particles interact with each other.

Evidence for the validity of the MKT: diffusion of substances, Brownian motion, thermal conductivity.

Physical quantities used to describe processes in molecular physics are divided into two classes:

microparameters– quantities describing the behavior of individual particles (mass of an atom (molecule), velocity, momentum, kinetic energy of individual particles);
macro parameters- quantities that are not reducible to individual particles, but characterize the properties of the substance as a whole. The values of macro parameters 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 play an important role not only in thermodynamics, but also in physics in general. Temperature- 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, the thermodynamic temperature and the temperature according to the International Practical

scale are related by the ratio

T= 273,15 + t.

Temperature T = 0 K is called zero kelvin. An analysis of various processes shows that 0 K is unattainable, although it is possible to approach it as close as desired. 0 K is the temperature at which, theoretically, any thermal motion of particles of matter should stop.

In molecular physics, a relationship is derived between macro-parameters and micro-parameters. 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 get an equation convenient for practical use:

font-size: 10.0pt"> An ideal gas is an idealized model of a gas, in which it is considered that:

1. the intrinsic volume of gas molecules is negligible compared to the volume of the vessel;

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 oppose the penetration of other molecules into the volume occupied by the molecule comes down to the fact that the actual free volume in which real gas molecules can move will be less. whereb - the molar volume occupied by the molecules themselves.

The action of the forces of attraction of the gas leads to the appearance of additional pressure on the gas, called internal pressure. According to Van der Waals calculations, the 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.

As a result, we will get real gas equation of state or van der Waals equation:

font-size:10.0pt;font-family:" times new roman> The physical meaning of temperature: temperature is a measure of the intensity of the thermal motion of particles of substances. The concept of temperature is not applicable to a single molecule. Only for a sufficiently large number of molecules that create a certain amount of matter, it makes sense to refer the term temperature.

For an ideal monatomic gas, the equation can be written:

font-size:10.0pt;font-family:" times new roman>The first experimental determination of the velocities of molecules was made by the German physicist O. Stern (1888-1970). His experiments also made it possible to estimate the distribution of molecules over velocities.

The "confrontation" between the potential energies of the binding of molecules and the energies of the 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, one 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, such as doing work on the system or imparting heat to it. So, by moving the piston into the cylinder in which the gas is located, we compress this gas, as a result of which its temperature rises, i.e., thereby changing (increasing) the internal energy of the gas. On the other hand, the temperature of the gas and its internal energy can be increased by imparting to it a certain amount of heat - the energy transferred to the system by external bodies through heat transfer (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; as applied 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, so 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 relationship between them, the first principle of TD does not say anything about the direction of processes in nature. In full accordance with the first law, one can mentally design an engine in which useful work would be done by reducing the internal energy of the substance. For example, instead of fuel in a heat engine, water would be used, 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.

One of the main problems in classical natural science has long been the problem of explaining the physical nature of the irreversibility of real processes. The essence of the problem lies in the fact 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 where 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, heat conduction, viscous flow. Almost all real processes in nature are irreversible: 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 a single 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 the laws of Newtonian mechanics? These and other similar questions agitated the minds of the most prominent scientists of the 18th-19th centuries.

Second law of thermodynamics sets the direction the 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 whose only result would be the absorption of heat from the heater and the complete conversion of this heat into work."

2. In another formulation: "Heat can spontaneously transfer only from a hotter body to a less hot one."

3. The third formulation is: "The entropy in a closed system can only increase."

Second law of thermodynamics forbids 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 motion of particles and the work associated with ordered motion. 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 form without any additional actions.

We do a complete conversion 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 is constantly doing 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, in 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 a real property of physical quantities, since, at least in principle, it is quite measurable ".

According to Clausius, entropy is such a physical quantity, the increment of which is equal to the amount of heat obtained 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 turn into order Thus, in isolated systems, entropy can only increase.This regularity is called principle of increasing entropy. According to this principle, any system tends to a state of thermodynamic equilibrium, which is identified with chaos. Since an increase in entropy characterizes changes in time of 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, passes from an ordered to a disordered state with a 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 itself, over time moves from a state of order to a state of maximum disorder (chaos).

From this principle follows the 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 the state of thermodynamic equilibrium, corresponding to the state of the greatest chaos and disorganization. All types of energy degrade, turning into heat, and the stars will end their existence, giving energy to 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 sources of energy, no life.

Such a gloomy 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 mainly in the direction of improving and complicating new types of plants and animals. History, sociology, economics, and other social and human sciences have also shown that in society, despite some zigzags of development, progress is generally being made.

Experience and practical activity testified that the concept of a closed or isolated system is a rather crude abstraction that simplifies reality, since it is difficult to find systems in nature 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.

From FFWiki.

Subject		Thermodynamics and statistical physics		Semester		7-8		Type of		lecture, seminar		Reporting		exam		Chair		Department of Quantum Statistics and Field Theory

About the subject

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Thermodynamics and statistical physics. The first question when you see this subject on the schedule is: how so? Indeed, in the 1st year they already talked about molecular physics, where there were all 3 principles of thermodynamics, and potentials, and the Maxwell distribution. It would seem, what else can be new in nature?

It turns out that what was on the 1st course is baby talk compared to real thermodynamics and statistical physics. The one with which Landau calculated liquid helium and received the Nobel Prize.

It is important not to get into a mess, thinking that once in 1 lecture they tell what you knew at school, then it will continue to be so. Already from mid-September, you will witness amazing tricks-fitting with partial derivatives, and by the end of the autumn semester, very furious topics in statistical physics will go:

• Calculation of statistical sums and Gibbs distributions
• Quantum gases - Fermi and Bose gases with different conditions
• Phase transitions and their properties
• Non-ideal gases - Bogolyubov chains, plasma and electrolyte models

Although the author of these words was able to prepare for excellent 4 days before the exams, he is very repentant of this and does not advise anyone to repeat such violence against his brain :) The tasks and questions for the exam have been known since the beginning of the year and it is very useful to prepare part of the material in advance.

In the spring semester, there are both easy and difficult topics. For example, the theory for Brownian motion is written out quite easily. But at the end of the course there are various kinetic equations, which are much more difficult to deal with.

Exam

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The exam in the fall goes pretty well, they don’t give you much to write off. For the most part, teachers do not bring down, but no special freebies have been noticed either. You need to know the theory. The diploma includes an assessment for the exam in the spring. The spring exam is more difficult than the autumn exam, but they are usually accepted more loyally. However, the theory should also be well known.

In the ticket both in autumn and in spring there are 2 theoretical questions and one task.

Be careful on the stats, several people (the number varies from 2 to 10!) Regularly graduate by failing this exam. And it's not just anyone, but hardened fourth-year students.

materials

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Fall semester

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• Leaflet 7 semester (pdf) - useful leaflet
• Answers to questions on thermodynamics (pdf) - 1 question per ticket
• Answers to questions on statistical physics (pdf) - 2 question in the ticket
• https://vk.com/doc231234703_450962027?hash=b2eb6c2220f95a7795&dl=cf7b5295f12e0fd4e0 assignments for the first colloquium
• https://vk.com/doc181113102_453848269?hash=6d6f68abc3358e2adf&dl=1606060abdd44d226e to the second

Spring semester

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• Answers to questions for the exam, theory (pdf) - answers to the theoretical questions of the exam, neatly typed on computers.
• - problem solving
• Problem solving for the exam (pdf) - more problem solving

Literature

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problem books

• Assignments in thermodynamics and statistical physics for 4th year students of the Faculty of Physics, Moscow State University (autumn semester - theory of equilibrium systems) (pdf)