The science of negative temperatures. temperature

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

Where N 1 – number of particles with energy 𝜀 1 , N 2 – number of particles with energy 𝜀 2 .

Taking logarithms of this formula, we get

In the equilibrium state of the system N 2 is always less N 1 if 𝜀 2 > 𝜀 1 . 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 only achievable in systems that have a finite maximum energy value, or in systems that have 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.

Let's consider a system with 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 you increase the temperature of the system 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 there are the same number of particles at both levels. Applying formula (6.27) to this state, we obtain that T = at N 1 = N 2, i.e. a uniform energy distribution of particles in a system corresponds to an infinitely high temperature. If in some way additional energy is imparted to the system, then the transition of particles from the lower level to the upper one will continue, and N 2 will be greater than N 1 . Obviously, in this case the temperature, in accordance with formula (6.27), will take a negative value. The more energy is supplied to the system, the more particles will be at the upper level and the more negative the temperature will be. In the limiting case, one can imagine a state in which all particles are collected at the top level; wherein N 1 = 0. Therefore, this state will correspond to a temperature of 0 K or, as they say, the temperature of negative absolute zero. However, the energy of the system in this case will be infinitely large.

As for 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.

Unlike 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 section shown by the dotted line corresponds to negative values ​​of absolute temperature.

To more clearly explain this behavior of entropy, let us turn again to the example of a two-level system discussed above. At absolute zero temperature (+0K), when N 2 = 0, i.e. all particles are at the lower level, the system is maximally ordered and its entropy is 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 across energy levels. Since this state of the system can be represented in the greatest number of ways, it will correspond to the maximum entropy value. The further transition of particles to the upper level leads to some ordering of the system in comparison with what took place with an uneven energy distribution of particles. Consequently, despite the increase in the energy of the system, its entropy will begin to decrease. At N 1 = 0, when all the particles gather at the upper level, the system will again have maximum order and therefore its entropy will become 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 furthest from the usual absolute zero (+0K). This is due to the fact that on the temperature scale the region of negative absolute temperatures is located above the 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 like this:

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 the states with temperatures +0K and –0K have the same entropy, equal to zero, and correspond to the maximum order of the system, they are two completely different states. At +0K the system has a maximum energy value and if it could be achieved, it would be a state of stable equilibrium of the system. An isolated system could not come out of such a state on its own. At –0K the system has a maximum energy value and if it could be achieved, it would be a metastable state, i.e. a state of unstable equilibrium. It could only be maintained with a continuous supply of energy to the system, since otherwise the system, left to itself, would immediately come out of this state. All states with negative temperatures are equally unstable.

If a body with a negative temperature is brought into contact with a body with a positive temperature, then energy will transfer from the first body to the second, and not vice versa (as with bodies with ordinary 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 (second particular formulation)

can be written in the form

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

Obviously, this inequality can be satisfied if 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 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 domestic physicists N.G. Basov and A.M. Prokhorov. Negative temperatures can be obtained using a gas discharge in semiconductors exposed to a pulsed electric field, and in a number of other cases.

It is interesting to note that since systems with negative temperatures 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 appear, 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, scientific reports on the experimental implementation of systems with negative absolute temperature have become increasingly common. Although each time scientists understood exactly what they were talking about, it remained unclear how widely this term is allowed to be used in thermodynamics - after all, it is known that strict thermodynamics does not accept negative temperatures. A methodological article recently published in the magazine Nature Physics, puts things in their places.

The essence of the work

At school they learn that absolute temperature - the one that is measured 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 often encounters articles about exotic systems characterized by negative absolute temperature. A standard example is a collection of atoms, each of which can exist in only two energy states. If you make sure that the number of atoms in the upper energy state is greater than in the lower one, then you get a negative temperature (Fig. 1). At the same time, it must be 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 first done back 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 some kind of curious effective description of unusual situations, but to normal thermodynamic systems in which heat is associated with spatial movement, it doesn't apply.

The situation has begun to change in recent years. Several years ago, systems with a negative temperature associated with the movement of particles were predicted (see the news A gas with a negative kinetic temperature is predicted, “Elements”, 08/29/2005), and literally this year an experimental implementation of a similar situation appeared (for details, see, for example, in the note In the experiment, it was possible to obtain a stable temperature below absolute zero, “Compulenta”, 01/09/2013). Moreover, scientists not only obtained such systems, but also began to seriously talk about real thermodynamics with negative temperatures (heat 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, methodological, not research, but several important things were clearly formulated in it:

• 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, differ radically. Thus, the usual determination of temperature can give a negative result, but another determination is always positive.
• Rigorous thermodynamics requires that the thermodynamic temperature be 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 to it.

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

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

Detailed Explanation

We are accustomed to the fact that heat - and therefore temperature as a numerical measure of heat - is something so tangible and understandable. It would seem that if there are problems with temperature in physics, then they may concern the measurement of temperature in some complex cases, but not its determination. However, a new article says that there are two temperatures and one of them is in some sense “wrong.” 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 neat formulation. Thermodynamics is the science of thermal processes, everything is true, but 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, a certain 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 must also have certain mathematical properties that can be carefully 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, entropy is the quantity from which all conventional thermodynamics follows. In particular, temperature (more precisely, 1/T) is defined as the rate of change of 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 in this strict mathematical construction of thermodynamics there is not a word about what entropy is equal to, how exactly it depends on internal energy. This mathematical formulation is a kind of “universal container” 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.

Another science is doing this - statistical physics. This is also a very serious and respected discipline, based on the quantum mechanics of multi-particle systems and neat mathematics. In particular, you can count not only the energy of a collective of several particles in a given configuration, but also, conversely, find the number of states - how many different configurations there can be with a given total energy. 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, not an experimental step: we need decide, how to calculate entropy from the number of states. Of course, this imposes the requirement 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 statistical physics, two slightly different methods were proposed: Boltzmann entropy, S B, and Gibbs entropy, S G. Boltzmann entropy characterizes the concentration of energy states near a given energy, Gibbs entropy characterizes the total number of states with energy less than a given energy; see explanations in fig. 2. Accordingly, the temperatures in these two pictures were different: Boltzmann temperature, T B, and Gibbs temperature, T G. It turns out, it is possible to construct two different thermodynamics for the same system.

For all real situations, these two thermodynamics are so close that it is simply impossible to distinguish 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 basis. But if the appropriate temperature T B is used in some exotic situations, then it can indeed take on 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 how justified the application of thermodynamic concepts to such systems is at all.

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

The authors of the article summarize, which is very typical for many controversial situations in physics: you can use any definition, but you must always remember 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 they propose using the Gibbs definition of temperature as a more reliable basis for difficult situations. It is also not forbidden to try to expand the boundaries of thermodynamics by coming up with 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.

negative absolute temperature, a quantity introduced to describe nonequilibrium 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 E i - system energy levels, k- Boltzmann constant, T- absolute temperature characterizing the average energy of the equilibrium system U = Σ (W n E n), From (1) it is clear that when T> 0 lower energy levels are more populated by particles than upper ones. If a system, under the influence of external influences, goes into a nonequilibrium state, characterized by a greater 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 determined through the reciprocal value 1/ T, equal to the derivative of entropy (See Entropy) S based on the average energy of the system with other parameters constant X:

From (2) it follows that O. t. means a decrease in entropy with increasing average energy. However, 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, very weakly interacting with thermal vibrations of the crystal lattice (See Vibrations of the crystal lattice), that is, practically isolated from thermal motion. The time required to establish thermal equilibrium between the spins and the 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 passed under external influence.

In a narrower sense, OPT is a characteristic of the degree of population inversion of two selected 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 - absolute temperature of a substance. From (3) it follows that N 2 < N 1. If you disturb the equilibrium of the system, for example, influence the system with monochromatic electromagnetic radiation, the frequency of which is close to the frequency of transition between levels: ω 21 = ( E 2 - E 1)/ħ and differs from the frequencies of other transitions, then it is possible to obtain a state in which the population of the upper level is higher than the lower one 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. using the formula:

Thermodynamic systems in which the probability of finding the system in a microstate with a higher energy is higher than in a microstate with a lower one.

In quantum statistics, this means that it is more likely to find a system at one energy level with a higher energy than at one level with a lower energy. An n-fold degenerate level is considered to be n levels.

In classical statistics, this corresponds to a higher probability density for points in phase space with higher energy compared to points with lower energy. At a positive temperature, the ratio of probabilities or their densities is the opposite.

For the existence of equilibrium states with negative temperatures, the statistical sum must converge at this temperature. Sufficient conditions for this are: in quantum statistics - the finiteness of the number of energy levels of the system, in classical statistical physics - that the phase space accessible to the system has a limited volume, and all points in this accessible space correspond to energies from a certain finite interval.

In these cases, there is the possibility that the energy of the system will be higher than the energy of the same system in an equilibrium distribution with any positive or infinite temperature. An infinite temperature will correspond to a uniform distribution and a final energy below the maximum possible. If such a system has an energy higher than the energy at infinite temperature, then the equilibrium state at such an energy can only be described using a negative absolute temperature.

The negative temperature of the system remains long enough if this system is sufficiently well insulated from bodies with a positive temperature. In practice, negative temperature can be realized, for example, in a system of nuclear spins.

At negative temperatures, equilibrium processes are possible. When there is thermal contact between two systems with different temperature signs, the system with a positive temperature begins to heat up, and the one with a negative temperature begins to cool. For the temperatures to become equal, one of the systems must pass through an infinite temperature (in a particular case, the equilibrium temperature of the combined system will remain infinite).

Absolute temperature + ∞ (\displaystyle +\infty ) And − ∞ (\displaystyle -\infty )- this is the same temperature (corresponding to a uniform distribution), but the temperatures T=+0 and T=-0 differ. Thus, a quantum system with a finite number of levels will be concentrated at the lowest level at T=+0, and at the highest at T=-0. Passing through a series of equilibrium states, the system can enter a temperature region with a different sign only through infinite temperature.

In a level system with population inversion, the absolute temperature is negative if it is determined, that is, if the system is sufficiently close to equilibrium.

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Negative temperature

negative absolute temperature, a quantity introduced to describe nonequilibrium 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 E i - system energy levels, k- Boltzmann constant, T- absolute temperature characterizing the average energy of the equilibrium system U = Σ (W n E n), From (1) it is clear that when T> 0 lower energy levels are more populated by particles than upper ones. If a system, under the influence of external influences, goes into a nonequilibrium state, characterized by a greater population of the upper levels compared to the lower ones, then formally we can use formula (1), putting in it T

In thermodynamics, absolute temperature T determined through the reciprocal value 1/ T, equal to the derivative of entropy (See Entropy) S based on the average energy of the system with other parameters constant X:

From (2) it follows that O. t. means a decrease in entropy with increasing average energy. However, 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, very weakly interacting with thermal vibrations of the crystal lattice (See Vibrations of the crystal lattice), that is, practically isolated from thermal motion. The time required to establish thermal equilibrium between the spins and the 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 passed under external influence.

In a narrower sense, OPT is a characteristic of the degree of population inversion of two selected 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 - absolute temperature of a substance. From (3) it follows that N 2 N 1. If you disturb the equilibrium of the system, for example, influence the system with monochromatic electromagnetic radiation, the frequency of which is close to the frequency of transition between levels: ω 21 = ( E 2 - E 1)/ħ and differs from the frequencies of other transitions, then it is possible to obtain a state in which the population of the upper level is higher than the lower one 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. using the formula:

Despite the formal nature of this definition, it turns out to be convenient in a number of cases; for example, it allows one to describe fluctuations in equilibrium and nonequilibrium systems with O. t. by similar formulas. The concept of optical energy is used in quantum electronics (see Quantum electronics) for the convenience of describing the processes of amplification and generation in media with population inversion.

D. N. Zubarev.

Great Soviet Encyclopedia. - M.: Soviet Encyclopedia. 1969-1978 .

See what “Negative temperature” is in other dictionaries:

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negative temperature- Characteristics of the inverse state, meaning the transition temperature... Polytechnic terminological explanatory dictionary

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Dimension Θ Units of measurement SI K ... Wikipedia

A negative quantity having the dimension of temperature, characterizing the degree of population inversion of energy levels of systems (atoms, ions, molecules) ... Big Encyclopedic Dictionary