negative absolute temperature, a quantity introduced to describe non-equilibrium states of a quantum system in which higher energy levels are more populated than lower ones. At equilibrium, the probability of having energy E n is determined by the formula:

Here Ei- system energy levels, k- Boltzmann constant, T- absolute temperature characterizing the average energy of the equilibrium system U = Σ (W n E n), It can be seen from (1) that at T> 0, the lower energy levels are more populated by particles than the upper ones. If the system, under the influence of external influences, passes into a non-equilibrium state, characterized by a higher population of the upper levels compared to the lower ones, then formally we can use formula (1), putting in it T < 0. Однако понятие О. т. применимо только к квантовым системам, обладающим конечным числом уровней, так как для создания О. т. для пары уровней необходимо затратить определённую энергию.

In thermodynamics, absolute temperature T is defined in terms of the reciprocal of 1/ T, equal to the derivative of entropy (See Entropy) S by the average energy of the system with the remaining parameters constant X:

From (2) it follows that O. t. means a decrease in entropy with an increase in the average energy. However, the theory of thermodynamics is introduced to describe nonequilibrium states, to which the application of the laws of equilibrium thermodynamics is conditional.

An example of a system with O. t. is a system of nuclear Spins in a crystal located in a magnetic field, interacting very weakly with thermal vibrations of the crystal lattice (see. Crystal lattice vibrations), that is, practically isolated from thermal motion. The time for establishing thermal equilibrium of spins with a lattice is measured in tens of minutes. During this time, the system of nuclear spins can be in a state with O. t., into which it has passed under external influence.

In a narrower sense, O. t. is a characteristic of the degree of population inversion of two chosen energy levels of a quantum system. In the case of thermodynamic equilibrium of the population N 1 And N 2 levels E 1 And E 2 (E 1 < E 2), i.e., the average numbers of particles in these states are related by the Boltzmann formula:

where T - the absolute temperature of a substance. From (3) it follows that N 2 < N 1. If the balance of the system is disturbed, for example, the system is affected by monochromatic electromagnetic radiation, the frequency of which is close to the frequency of the transition between levels: ω 21 = ( E 2 - E 1)/ħ and differs from the frequencies of other transitions, then you can get a state in which the population of the upper level is higher than the lower N 2 > N 1. If we conditionally apply the Boltzmann formula to the case of such a nonequilibrium state, then with respect to a pair of energy levels E 1 And E 2 you can enter O. t. according to the formula:

The absolute temperature in the molecular kinetic theory is defined as a quantity proportional to the average kinetic energy of the particles (see Section 2.3). Since kinetic energy is always positive, the absolute temperature cannot be negative either. The situation will be different if we use a more general definition of absolute temperature, as a quantity characterizing the equilibrium distribution of the particles of the system according to the energy values ​​(see Section 3.2). Then, using the Boltzmann formula (3.9), we have

where N 1 is the number of particles with energy 𝜀 1 , N 2 is the number of particles with energy 𝜀 2 .

Taking the logarithm of this formula, we get

In the equilibrium state of the system N 2 is always smaller N 1 if 𝜀 2 > 𝜀 one . This means that the number of particles with a higher energy value is less than the number of particles with a lower energy value. In this case, always T > 0.

If we apply this formula to such a nonequilibrium state, when N 2 > N 1 at 𝜀 2 > 𝜀 1 , then T < 0, т.е. состоянию с таким соотношением числа частиц можно формально по аналогии с предыдущим случаем приписать определенную отрицательную абсолютную температуру. Поскольку при этом формула Больцмана применена к неравновесному распределению частиц системы по энергии, то отрицательная температура является величиной, характеризующей неравновесные системы. Поэтому отрицательная температура имеет иной физический смысл, чем понятие обычной температуры, определение которой неразрывно связано с равновесием.

Negative temperature is achievable only in systems with a finite maximum energy value, or in systems with a finite number of discrete energy values ​​that particles can accept, i.e. with a finite number of energy levels. Since the existence of such systems is associated with the quantization of energy states, in this sense the possibility of the existence of systems with a negative absolute temperature is a quantum effect.

Consider a system with a negative absolute temperature, which has, for example, only two energy levels (Fig. 6.5). At absolute zero temperature, all particles are at the lowest energy level, and N 2 = 0. If the temperature of the system is increased by supplying energy to it, then the particles will begin to move from the lower level to the upper one. In the limiting case, one can imagine a state in which the number of particles is the same at both levels. Applying formula (6.27) to this state, we obtain that T = at N 1 = N 2 , i.e. an infinitely high temperature corresponds to a uniform distribution of the particles of the system in terms of energy. If additional energy is added to the system in some way, then the transition of particles from the lower level to the upper one will continue, and N 2 becomes more than N one . Obviously, in this case, the temperature, in accordance with formula (6.27), will take a negative value. The more energy will be supplied to the system, the more particles will be at the upper level and the greater will be the negative temperature value. In the limiting case, one can imagine a state in which all the particles are collected at the upper level; wherein N 1 = 0. Therefore, this state will correspond to the temperature - 0K or, as they say, the temperature of negative absolute zero. However, the energy of the system in this case will already be infinitely large.

As for the entropy, which, as is known, is a measure of the disorder of a system, depending on the energy in ordinary systems, it will increase monotonically (curve 1, Fig. 6.6), so

as in conventional systems, there is no upper limit to the energy value.

In contrast to conventional systems, in systems with a finite number of energy levels, the dependence of entropy on energy has the form shown by curve 2. The area shown by the dotted line corresponds to negative values ​​of the absolute temperature.

For a clearer explanation of this behavior of entropy, let us turn again to the above example of a two-level system. At absolute zero temperature (+0K), when N 2 = 0, i.e. all particles are at the lower level, the maximum ordering of the system takes place and its entropy is equal to zero. As the temperature rises, the particles will begin to move to the upper level, causing a corresponding increase in entropy. At N 1 = N 2 particles will be evenly distributed in energy levels. Since such a state of the system can be represented in the largest number of ways, it will correspond to the maximum value of entropy. A further transition of particles to the upper level already leads to a certain ordering of the system in comparison with what took place in the case of a non-uniform distribution of particles over energies. Therefore, despite the increase in the energy of the system, its entropy will begin to decrease. At N 1 = 0, when all particles are collected at the upper level, the maximum ordering of the system will again be, and therefore its entropy will become equal to zero. The temperature at which this happens will be the temperature of negative absolute zero (-0K).

Thus, it turns out that the point T= - 0K corresponds to the state most remote from the usual absolute zero (+0K). This is due to the fact that on the temperature scale the region of negative absolute temperatures is above an infinitely large positive temperature. Moreover, the point corresponding to an infinitely large positive temperature coincides with the point corresponding to an infinitely large negative temperature. In other words, the sequence of temperatures in ascending order (from left to right) should be:

0, +1, +2, … , +

It should be noted that a negative temperature state cannot be achieved by heating a conventional system in a positive temperature state.

The state of negative absolute zero is unattainable for the same reason that the state of positive absolute zero temperature is also unattainable.

Despite the fact that states with a temperature of +0K and -0K have the same entropy equal to zero and correspond to the maximum ordering of the system, they are two completely different states. At +0K, the system has a maximum energy value, and if it could be achieved, then it would be a state of stable equilibrium of the system. An isolated system could not get out of such a state on its own. At –0K, the system has a maximum energy value, and if it could be achieved, then it would be a metastable state, i.e. state of unstable equilibrium. It could be preserved only with a continuous supply of energy to the system, since otherwise the system, left to itself, would immediately leave such a state. All states with a negative temperature are just as unstable.

If a body with a negative temperature is brought into contact with a body with a positive temperature, then the energy will transfer from the first body to the second, and not vice versa (as in bodies with the usual positive absolute temperature). Therefore, we can assume that a body with any finite negative temperature is “warmer” than a body with any positive temperature. In this case, the inequality expressing the second law of thermodynamics (the second particular formulation)

can be written in the form

where is the value by which the heat of a body with a positive temperature changes over a short period of time, is the value by which the amount of heat of a body with a negative temperature changes over the same time.

Obviously, this inequality can be satisfied for and only if the value = – is negative.

Since the states of a system with a negative temperature are unstable, in real cases it is possible to obtain such states only if the system is well isolated from the surrounding bodies with a positive temperature and provided that such states are maintained by external influences. One of the first methods for obtaining negative temperatures was the method of sorting ammonia molecules in a molecular generator created by Russian physicists N.G. Basov and A.M. Prokhorov. Negative temperatures can be obtained using a gas discharge in semiconductors under the influence of a pulsed electric field, and in a number of other cases.

It is interesting to note that since systems with a negative temperature are unstable, when radiation of a certain frequency passes through them, as a result of the transition of particles to lower energy levels, additional radiation will arise, and the intensity of the radiation passing through them will increase, i.e. systems have negative absorption. This effect is used in the operation of quantum generators and quantum amplifiers (in masers and lasers).

Note that the difference between the usual absolute zero temperature and negative is that we approach the first from the side of negative temperatures, and the second - from the side of positive ones.

In recent years, there have been more and more scientific reports on the experimental implementation of systems with a negative absolute temperature. Although every time it was clear to scientists what exactly they were talking about, it remained unclear how widely this term was allowed to be used in thermodynamics - after all, it is known that strict thermodynamics does not accept negative temperatures. Methodical article published recently in the journal nature physics puts things in their place.

The essence of the work

At school, they pass that the absolute temperature - the one that is counted from absolute zero and measured in kelvins, and not in degrees Celsius - must be positive. However, in modern physics, and after it in popular materials, one can often find articles about exotic systems characterized by a negative absolute temperature. A standard example is a collection of atoms, each of which can be in only two energy states. If we make it so that the number of atoms in the upper energy state is greater than in the lower one, then, as it were, a negative temperature is obtained (Fig. 1). At the same time, it is necessarily emphasized that negative temperatures are not very cold temperatures, below absolute zero, but, on the contrary, extremely hot, hotter than any positive temperature.

Such situations can even be obtained experimentally; this was done for the first time in 1951. But since these situations themselves were unusual, for the time being, the attitude of scientists to this topic was moderately calm: this is a kind of curious effective description of unusual situations, but for normal thermodynamic systems in which heat is associated with spatial movement, it does not apply.

The situation has begun to change in recent years. Several years ago, systems with a negative temperature associated with the motion of particles were predicted (see the news A gas with a negative kinetic temperature is predicted, Elements, 29.08.2005), and literally this year appeared with an experimental realization of a similar situation (for details, see, for example, in the note In the experiment, it was possible to obtain a stable temperature below absolute zero, Compulent, 01/09/2013). Moreover, scientists not only obtained such systems, but also began to seriously talk about real thermodynamics with negative temperatures (thermal engines with efficiency above 100%) and even about its possible role in the mystery of dark energy. Thus, at least for some physicists, negative temperatures ceased to seem like a mathematical trick, but became something quite real.

The other day in a magazine nature physics came out, which raised the question of the physicality of the term "negative temperature" in real thermodynamics. This article was, in essence, methodical, not research, but it clearly articulates several important things:

• The concept of temperature can be defined in many ways, and all talk about negative temperature refers to only one specific definition. For the vast majority of systems, these different temperatures are virtually indistinguishable, so it doesn't matter which definition you use.
• For unusual systems, these temperatures can differ, and moreover, they can differ dramatically. So, the usual definition of temperature can give a negative result, and another definition is always positive.
• Rigorous thermodynamics requires that the thermodynamic temperature is always positive. Therefore, the definition that leads to negative values ​​is fake temperature. It can be used, no one forbids it, but it cannot be substituted into real thermodynamic formulas or given an excessively physical meaning.

In other words, this article calls for moderating the excitement generated by recent experimental advances.

For an inexperienced reader, all this may seem strange: how so - several temperatures? what is strict thermodynamics? Therefore, we give below a slightly more detailed, but also more technical description of the situation.

Detailed explanation

We are used to the fact that heat - and hence temperature as a numerical measure of heat - is something so tangible, understandable. It would seem that if there are problems with temperature in physics, then they can relate to temperature measurement in some difficult cases, but not its definition. However, the new article says that there are two temperatures and one of them is “wrong” in some sense. What does it mean?

To explain the situation, we need to step back a little, move away from the applied aspects of thermodynamics and look into its essence, into its accurate formulation. Thermodynamics is the science of thermal processes, that's right, but only the concept of "temperature" does not appear in it at all at the first stage. Thermodynamics starts with mathematics, with the introduction of certain abstract quantities and the establishment of their mathematical properties. It is believed that the system has a volume, an amount of matter, some internal energy - these are still mechanical characteristics - as well as a new characteristic called entropy. It is with the introduction of entropy that thermodynamics begins, but what entropy is is not discussed at this stage. Entropy, too, must have certain mathematical properties that can be neatly formulated as real axioms. Those wishing to briefly get acquainted with this real mathematical side of the issue can recommend the article A Guide to Entropy and the Second Law of Thermodynamics, published in a mathematical (!) Journal. In principle, all this was more or less known a century ago, but in such a neat mathematical form it was formulated only in recent decades.

So, it is entropy that is the quantity from which all the usual thermodynamics follows. In particular, temperature (more precisely, 1/T) is defined as the rate of change in entropy with increasing internal energy. And if you follow all the axioms of thermodynamics, then this real thermodynamic temperature must be positive.

Everything would be fine, but only in this strict mathematical construction of thermodynamics there is not a word about what entropy equals, how exactly it depends on internal energy. This mathematical formulation is a kind of "universal receptacle" for a variety of real-life situations, but it does not say exactly how it should be applied to specific systems. The problem arises of how to fit real systems consisting of a large number of atoms and molecules into thermodynamics.

This is already another science - statistical physics. This is also a very serious and respected discipline, based on the quantum mechanics of systems of several particles and on accurate mathematics. In particular, in it you can count not only the energy of a collective of several particles in a given configuration, but also, on the contrary, find the number of states - how many different configurations with a given total energy can be. This is all good too, but there is no entropy in this picture yet.

There is only one step left - the transition from statistical physics to thermodynamics. This is also a theoretical step, not an experimental one: we need to decide, how to calculate the entropy from the number of states. Of course, here the requirement is imposed that the entropy calculated in this way must have the correct properties - at least for all life situations. And here the ambiguity appears: it turns out that this can be done in different ways.

Back in the era of building statistical physics, two slightly different methods were proposed: entropy according to Boltzmann, S B , and Gibbs entropy, S G. Entropy according to Boltzmann characterizes the concentration of energy states near a given energy, entropy according to Gibbs - the total number of states with an energy less than a given energy; see explanations in fig. 2. Accordingly, the temperatures in these two pictures were different: temperature according to Boltzmann, T B , and Gibbs temperature, T G. It turns out, one can construct two different thermodynamics for the same system.

For all real situations, these two thermodynamics are so close that it is simply unrealistic to distinguish between them. Therefore, in most textbooks on statistical physics and thermodynamics, this distinction is not made at all, and thermodynamics according to Boltzmann is chosen as a support. But if the appropriate temperature T B is used in some exotic situations, then it can indeed take a negative value. The simplest examples given in the article are the standard situation (many particles at two energy levels) and a single quantum particle in a one-dimensional rectangular potential. In both cases, it is not clear to what extent the application of thermodynamic concepts to such systems is generally justified.

But the definition of temperature according to Gibbs, T G , always remains meaningful, even in those exotic situations where the applicability of thermodynamics is debatable. With an increase in the average energy, the temperature gradually increases, but never becomes infinite and then does not jump into negative values. Therefore, if we undertake to build thermodynamics for such systems, then we must identify the real temperature precisely with T G , not c T B; the thermodynamics constructed in this way will satisfy all the axioms of the theory.

The authors of the article sum up, which is very typical for many disputable situations in physics: any definition can be used, but one should always keep in mind the assumptions made and the resulting limitations of applicability. The standard definition of temperature suffers from the fact that in exotic situations it ceases to meet the mathematical requirements of thermodynamic theory, and is also not an adequate measure of heat. Therefore, the authors urge physicists not to attach too much importance to negative temperatures, and as a more reliable support for difficult situations, they suggest using the definition of temperature according to Gibbs. It is also not forbidden to try to expand the boundaries of thermodynamics by inventing some generalizations of this theory - but one must always remember that this will no longer be real thermodynamics and that in these situations not all real thermodynamic results work.

First, we note that the idea of ​​states with a negative absolute temperature does not contradict Nerst's theorem on the impossibility of reaching absolute zero.

Consider a system with a negative absolute temperature that has only two energy levels. At absolute zero temperatures, all particles are at the lower level. As the temperature rises, some of the particles begin to move from the lower level to the upper one. The ratio between the number of particles at the first and second levels at different temperatures will satisfy the energy distribution in the form:

As the temperature rises, the number of particles at the second level will approach the number of particles at the first level. In the limiting case of infinitely high temperatures, there will be the same number of particles at both levels.

Thus, for any ratio of the number of particles in the interval

our system can be assigned a certain statistical temperature in the interval defined by equality (12.44). However, under special conditions it is possible to ensure that in the system under consideration the number of particles at the second level is greater than the number of particles at the first level. By analogy with the first considered case, a state with such a ratio of the number of particles can also be assigned a certain statistical temperature or distribution modulus. But, as follows from (12.44), this modulus of the statistical distribution must be negative. Thus, the considered state can be assigned a negative absolute temperature.

From the example considered, it is clear that the negative absolute temperature introduced in this way is by no means a temperature below absolute zero. Indeed, if at absolute zero the system has a minimum internal energy, then with increasing temperature the internal energy of the system increases. However, if we consider a system of particles with only two energy levels, then its internal energy will change as follows. When all particles are at the lower level with energy, therefore, the internal energy At an infinitely high temperature, the particles are evenly distributed between the levels (Fig. 71) and the internal energy:

i.e., has a finite value.

If we now calculate the energy of the system in the state to which we have attributed a negative temperature, it turns out that the internal energy in this state will be greater than the energy in the case of an infinitely large positive temperature. Really,

Thus, negative temperatures correspond to higher internal energies than positive ones. During thermal contact of bodies with negative and positive temperatures, energy will transfer from bodies with negative absolute temperatures to bodies with positive temperatures. Therefore, bodies at negative temperatures can be considered "hotter" than at positive ones.

Rice. 71. To the explanation of the concept of negative absolute temperatures

The above considerations about the internal energy with a negative distribution modulus allow us to consider the negative absolute temperature, as it were, higher than the infinitely large positive temperature. It turns out that on the temperature scale the region of negative absolute temperatures is not "below absolute zero", but "above the infinite temperature". In this case, an infinitely large positive temperature "is next to" an infinitely large negative temperature, i.e.

A decrease in the absolute value of the negative temperature will lead to a further increase in the internal energy of the system. When the energy of the system will be maximum, since all particles will be collected at the second level:

The entropy of the system turns out to be symmetrical with respect to the sign of the absolute temperature in equilibrium states.

The physical meaning of the negative absolute temperature is reduced to the concept of the negative modulus of the statistical distribution.

Whenever the state of a system is described by a statistical distribution with a negative modulus, the concept of a negative temperature can be introduced.

It turns out that such states can be realized for some systems under different physical conditions. The simplest of them is the finiteness of the energy of the system with weak interaction with the surrounding systems with positive temperatures and the ability to maintain this state by external forces.

Indeed, if you create a state with a negative temperature, i.e., do more, then thanks to spontaneous transitions, particles will be able to move from a state with a state with a lower energy. Thus, the state with a negative temperature will be unstable. To maintain it for a long time, it is necessary to replenish the number of particles at the level by reducing the number of particles at the level

It turned out that the systems of nuclear magnetic moments satisfy the requirement that the energy be finite. Indeed, spin magnetic moments have a certain number of orientations and, therefore, energy levels in a magnetic field. On the other hand; in a system of nuclear spins, with the help of nuclear magnetic resonance, it is possible to transfer most of the spins to the state with the highest energy, i.e., to the highest level. For the reverse transition to the lower level, the nuclear spins will have to exchange energy with the crystal lattice, which will take quite a long time. During time intervals shorter than the spin-lattice relaxation time, the system can be in states with a negative temperature.

The considered example is not the only way to obtain systems with a negative temperature.

Systems with a negative temperature have one interesting feature. If radiation with a frequency corresponding to the energy level difference is passed through such a system, then the transmitted radiation

will stimulate transitions of particles to the lower level, accompanied by additional radiation. This effect is used in the operation of quantum generators and quantum amplifiers (masers and lasers).

If we proceed from the definition of temperature that was given at the beginning of this book, i.e., that the temperature is proportional to the average kinetic energy of the particles, then the title of this paragraph seems to be devoid of a / syll: after all, the kinetic energy cannot be negative! And for those atomic systems, in which the energy contains only the kinetic energy of particle motion, the negative temperature really has no physical meaning.

But let us recall that in addition to the molecular-kinetic definition of temperature, we in Chap. I also noted the role of temperature as a quantity that determines the energy distribution of particles (see p. 55). If we use this more general concept of temperature, then we will come to the possibility of the existence (at least in principle) of negative temperatures.

It is easy to see that the Boltzmann formula (9.2)

formally "allows" the temperature to take not only positive, but also negative values.

Indeed, in this formula, this is the fraction of particles that are in a state with energy, and this is the number of particles in a state with some initial energy, from which the energy is counted. It can be seen from the formula that the higher the smaller the fraction of particles with this energy. So, for example, at times less than the base of natural logarithms). And a much smaller fraction of particles already has energy: in this case, times less. It is clear that in an equilibrium state, to which, as we know, Boltzmann's law applies, it is always less than

Taking the logarithm of equality (9.2), we obtain: whence

From this expression for it is clear that if then

If, however, it turned out that there is such an atomic system in which there can be more than that, this would mean that the temperature can also take negative values, since at becomes negative.

It will be easier for us to understand under what circumstances this is possible if we consider not a classical system (in which a negative temperature cannot be realized), but a quantum one, and we use, in addition, the concept of entropy, which,

as we have just seen, is a quantity that determines the degree of disorder in the system.

Let the system be represented by a diagram of its energy levels (see, for example, Fig. 1, p. 17). At absolute zero temperature, all particles of our system are at their lowest energy levels, and all other levels are empty. The system under such conditions is maximally ordered and its entropy is equal to zero (its heat capacity is also equal to zero).

If we now raise the temperature of the system by supplying energy to it, then the particles will also pass to higher energy levels, which, thus, also turn out to be partially populated, and the higher the temperature, the greater the “population” of higher energy levels. The distribution of particles over energy levels is determined by the Boltzmann formula. This means that it will be such that there will be fewer particles at higher levels than at lower ones. The "settlement" of particles over many levels, of course, increases the disorder in the system and its entropy increases with increasing temperature. The greatest disorder, and hence the maximum entropy, would be achieved with such an energy distribution of particles in which they are uniformly distributed over all energy levels. Such a distribution would mean that in the formula means, Therefore, a uniform distribution of particles in energy corresponds to an infinitely high temperature and maximum entropy.

However, in the quantum system in question, such a distribution is impossible, because the number of levels is infinitely large, and the number of particles is finite. Therefore, the entropy in such a system does not pass through a maximum, but grows monotonically with temperature. At an infinitely high temperature, the entropy will also be infinitely high.

Let us imagine now such a system (quantum), which has an upper limit of its internal energy, and the number of energy levels is finite. This, of course, is only possible in a system in which the energy does not include the kinetic energy of particle motion.

In such a system, at absolute zero temperature, particles will also occupy only the lowest energy levels, and the entropy will be zero. As the temperature rises, the particles "settle" at higher levels, causing a corresponding increase in entropy. On fig. 99a shows a system with two energy levels. But, since the number of energy levels of the system, as well as the number of particles in it, is now finite, then in the end a state can be reached in which the particles are evenly distributed over the energy levels. As we have just seen, this state corresponds to an infinitely high temperature and maximum entropy.

In this case, the energy of the system will also be some maximum, but not infinitely large, so that our old definition of temperature, as the average energy of particles, becomes inapplicable.

If now, in some way, additional energy is imparted to the system, already at an infinitely high temperature, then the particles will continue to move to a higher energy level, and this will lead to the fact that the "population" of this high energy level will become greater than that of the lower one. (Fig. 99, b). It is clear that such a predominant accumulation of particles at high levels means already some ordering in comparison with the complete disorder that existed at, i.e., with a uniform distribution of particles over energies. The entropy that has reached its maximum at begins, therefore, to decrease with further energy supply. But if the entropy does not increase with increasing energy, but falls, then this means that the temperature is not positive, but negative.

The more energy will be supplied to the system, the more particles will be at the highest energy levels. In the limit, one can imagine a state in which all particles will be collected at the highest levels. Such a state, obviously, is also quite ordered. It is in no way "worse" than the state when all particles occupy the lowest levels: in both cases, the system is dominated by complete order, and the entropy is zero. We can therefore designate the temperature at which this second well-ordered state is established by -0, in contrast to the "ordinary" absolute zero. The difference between these two "zeros" is that we arrive at the first of them from the side of to the second - from the side of positive temperatures.

Thus, the conceivable temperatures of the system are not limited to the interval from absolute zero to infinity, but extend from through to , and coincide with each other. On fig. 100 shows the curve of dependence of entropy on the energy of the system. The part of the curve to the left of the maximum corresponds to positive temperatures, to the right of it - negative. At the maximum point, the temperature value is

From the point of view of ordering, and hence entropy, the following three extreme states are possible:

1. Complete ordering - particles are concentrated at the lowest energy levels. This state corresponds to the "ordinary" absolute zero

2. Complete disorder - particles are evenly distributed over all energy levels. This state corresponds to the temperature

3. Complete ordering again - particles occupy only the highest energy levels. The temperature corresponding to this state is assigned the value -0.

We are dealing here, therefore, with a paradoxical situation: in order to reach negative temperatures, we had to not cool the system below absolute zero, which is impossible, but, on the contrary, increase its energy; the negative temperature is higher than the infinitely high temperature!

There is a very important difference between the two well-ordered states we have just mentioned, the states with temperatures.

The state of "ordinary" absolute zero, if it could be created in the system, would be preserved in it for an arbitrarily long time, provided that it is reliably isolated from the environment, isolated in the sense that energy is not supplied to the system from this environment. This state is a state of stable equilibrium, from which the system itself, without outside intervention, cannot exit. This is due to the fact that the energy of the system in this state has a minimum value.

On the other hand, the state of negative absolute zero is an extremely non-equilibrium state, since. maximum energy of the system. If it were possible to bring the system to this state, and then leave it to itself, then it would immediately come out of this non-equilibrium, unstable state. It could be preserved only with a continuous supply of energy to the system. Without this, particles that are at higher energy levels will certainly "fall" to lower levels.

A common property of both "zeros" is their unattainability: to reach them, an infinite amount of energy is required.

However, unstable, non-equilibrium is not only the state corresponding to a temperature of -0, but also all states with negative temperatures. All of them correspond to the values ​​and for equilibrium, the inverse relation is necessary

We have already noted that negative temperatures are higher temperatures than positive ones. Therefore, if you bring

a body heated (one cannot say: cooled) to negative temperatures, in contact with a body whose temperature is positive, then energy will transfer from the first to the second, and not vice versa, which means that its temperature is higher, although it is negative. When two bodies with a negative temperature come into contact, energy will transfer from a body with a lower absolute value of temperature to a body with a higher numerical temperature value.

Being in an extremely non-equilibrium state, a body heated to a negative temperature very willingly gives up energy. Therefore, in order for such a state to be created, the system must be reliably isolated from other bodies (in any case, from systems that are not similar to it, i.e., do not have a finite number of energy levels).

However, a state with a negative temperature is so non-equilibrium that even if a system in this state is isolated and there is no one to transfer energy to it, it can still give off energy in the form of radiation until it passes into a state (equilibrium) with a positive temperature .

It remains to add that atomic systems with a limited set of energy levels, in which, as we have seen, it is possible to realize a state with a negative temperature, is not only a conceivable theoretical construction. Such systems really exist, and negative temperatures can indeed be obtained in them. The radiation that occurs during the transition from a state with a negative to a state with an ordinary temperature is practically used in special devices: molecular generators and amplifiers - masers and lasers. But we cannot dwell on this issue in more detail here.