Equipartition theorem of energy

The basic formula connecting system temperature and its average energy

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stay classic statistical mechanics Medium, Equipartition theorem of energy (Equipartition Theorem) is a contact system temperature And its average energy Basic formula of. The equipartition theorem of energy is also called Law of equipartition of energy 、 Principle of equipartition of energy 、 Energy equipartition , or just Equipartition 。 The initial concept of energy sharing is heat balance Time energy is equally divided into various forms of motion; For example, a molecule Translational motion Average of hours kinetic energy Should be equal to what it does Rotary motion Average kinetic energy at.

Chinese name: Equipartition theorem of energy
Foreign name: Equipartition Theorem

Content: classic statistical mechanics
be applied to: forecast fixed star Nature of
Solid: specific heat

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▪ Failure caused by quantum effect
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The equipartition theorem of energy can Quantitative prediction 。 be similar to Mean work theorem For a system with a given temperature, the total average kinetic energy and potential energy of the system can be calculated by using the equipartition theorem, and the heat capacity 。 The equipartition theorem can also give the average value of each component of energy, such as the kinetic energy of a specific particle or a Spring Potential energy. For example, it predicts that ideal gas The average kinetic energy of each particle in is (3/2)k B T , where k B by Boltzmann constant and T Is the temperature. More generally, no matter how complex, it can be applied to any heat balance Of Classical system Medium. The equipartition theorem of energy can be used to derive classic Ideal gas law , and solids specific heat Of Duron Petty law 。 It can also be applied to forecasting fixed star Because even considering relativity The theorem is still valid.

Although the equipartition theorem can provide a very accurate prediction of physical phenomena under certain conditions, when Quantum effect When it becomes significant (e.g. at sufficiently low temperatures), the prediction based on this theorem becomes inaccurate. Specifically, when heat energy k B T More specific freedom When the quantum energy level spacing is small, the average energy and heat capacity under this degree of freedom are smaller than the values predicted by the equipartition theorem. When the thermal energy is much smaller than the energy level spacing, such a degree of freedom is said to be "frozen". For example, many kinds of motions are frozen at low temperatures, so the heat capacity of solids at low temperatures will decline, instead of being kept constant as originally measured by the equipartition theorem. For physicists in the 19th century, this decline in heat capacity was the first sign that classical physics was no longer correct and needed new physics. Equipartition theorem in prediction electromagnetic wave The failure of (called“ ultraviolet catastrophe ”）Cause Plonk It is proposed that light itself is quantized into photon , and this revolutionary theory quantum mechanics and Quantum field theory The development of has played an important role.^[1]

Basic concepts and simple examples

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See: kinetic energy and ideal gas

Using the Boltzmann statistical method, we can get that when the gas is in equilibrium, the average energy of any degree of freedom of the molecule is equal, which is kT/2. This is the equipartition theorem of energy according to the degree of freedom, referred to as the equipartition theorem of energy. "Equal share" in the name means "share or similar share". The original concept of the equipartition theorem of energy is that when the system reaches thermal equilibrium kinetic energy Divided equally by each independent component. The equipartition theorem also makes quantitative predictions for these energies. For example, it predicts inert gas Each atom of T When the thermal balance is reached, there will be a translational average kinetic energy (3/2)K B T , where K B by Boltzmann constant 。 What follows is that at isothermal temperature xenon The velocity of heavy atoms of helium Its lighter atoms are lower. Figure 2 shows the atomic velocities of four inert gases Maxwell Boltzmann distribution 。

In this case, the key point is that kinetic energy is a quadratic homogeneous function of velocity. The equipartition theorem shows that anything that occurs only twice in energy freedom (for example, a component of a particle's position or velocity) is equal to ½K B T The average energy of the heat capacity ½ K provided B 。 This result has many applications.

Specific heat capacity of solid

An important application of the equipartition theorem is the specific heat capacity of crystalline solids. In this way, each atom of the solid can oscillate in three independent directions, so the solid can be regarded as having its own independent 3N individual Simple harmonic oscillator System, where N Is the number of atoms in the lattice. Since each harmonic oscillator has an average energy k B T , so the average total energy of the solid is 3Nk B T , while the specific heat capacity is 3Nk B 。 If selected N by Avogadro constant N A , and use R = N A k B This connection gas constant R And Boltzmann constant k B The solid can be obtained by Molar specific heat capacity Of Duron Petty law The law states that the atomic heat capacity per mole in the lattice is 3R ≈ 6cal/( mol · K )。

However, due to the quantum effect, this law is not accurate at low temperatures; This is also inconsistent with the experimental results The third law of thermodynamics The third law states that Molar specific heat capacity It must be zero at absolute zero. Albert Einstein (1907) and Peter Debye (1911) added quantum effect to the foundation and developed a more accurate theory.

The vibration of each atom in the solid is not independent, and can be used in groups Coupling oscillator As a model. So the model of oscillator can be decomposed into Simple mode , this Piano string Vibration mode and Organ The resonance modes of are similar. On the other hand, the equipartition theorem generally fails when applied to such systems, because there is no energy exchange between normal modes. In an exceptional case, the modes are independent and their energy is independently conserved. This shows that there is some kind of energy mixing, officially called Ergodicity It is very important for the establishment of the equipartition theorem.^[1]

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The concept of equipartition of kinetic energy was first introduced in 1843, or more accurately in 1845, by John James Whataston Proposed. In 1859, James Clerk Maxwell It claimed that the dynamic heat energy of gas was divided into linear and rotating energy in 1876, Ludwig Boltzmann The principle is further extended because it shows that the average energy is divided equally by each independent component of a system. Boltzmann also applied the equipartition theorem to the solid Specific heat capacity Of Duron Petty law A theoretical explanation is proposed.

The history of equipartition theorem of energy and mole Specific heat capacity Both were studied in the 19th century. In 1819, French physicist Pierre Louis Duron And Alexis Therese Petit It is found that the specific heat capacity of solids at all room temperatures is almost the same, about 6 cal /( mol · K )。 Their laws have been used as measures for many years Atomic mass However, later James Dewar and Heinrich Friedrich Weber Research shows that Duron Petty law Only at high temperature; At low temperatures or like diamond This exceptionally hard solid has a lower specific heat.

The experimental observation of gas specific heat also raises doubts about the validity of the equipartition theorem. Theorem predicts that the molar specific heat capacity of simple monatomic gas should be about 3 cal /( mol · K ）And diatomic gas is about 7 cal /( mol · K )。 The former is verified experimentally, but it is found that the typical molar specific heat capacity of diatomic gas is about 5 cal /( mol · K ）And dropped to about 3 at low temperature cal /( mol · K )。 James Clerk Maxwell In 1875, he pointed out that the incompatibility between the experiment and the equipartition theorem was much worse than those implied by these figures; Since atoms have internal parts, the heat energy should move towards these internal parts, so that the predicted specific heat of single atoms and double atoms is 3 cal /( mol · K )7 cal /( mol · K ）Much higher.

The third relevant discrepancy is the specific heat of the metal. According to the classic Drude model Metal electrons behave like almost ideal gases, so they should N e k B Heat capacity of, where N e Is the number of electrons. However, experiments show that the supply of electrons to heat capacity is not much: the molar specific heat capacity of many metals is almost the same as that of insulators.

Several explanations for the failure of averaging have been proposed. Boltzmann He defended that his deduction of the equipartition theorem was correct, but he proposed that the gas might be ether Interacting without being in thermal equilibrium. Because it is inconsistent with the experiment, Lord Kelvin The derivation of the equipartition theorem must be uncertain, but nothing is wrong. instead rayleigh Lord put forward a more thorough view, that is, the equipartition theorem and the assumption that the system is in thermal equilibrium during the experiment all correct; In order to make the two coincide, he pointed out that a new principle was needed to provide the equipartition theorem with a way to escape from destructive simplicity. Albert Einstein In 1907, he showed that these specific heat anomalies are caused by quantum effects, especially the quantization of elastic modal energy of solids. Einstein used the failure of the equipartition theorem as the argument for the need for a new quantum theory of matter. Walther Nernst In 1910, the measurement of specific heat at low temperature supported Einstein's theory and caused physicists to Quantum Theory Wide recognition.^[2]

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Ergodic requirements

See: Ergodicity 、 chaos theory and Kolmogorov Arnold Moser theorem

The equipartition law is only applicable to ergodic The system is effective, which means that the possibility of state migration of the same energy is necessarily the same. Therefore, the system must be able to exchange all its various forms of energy, or Canonical ensemble Middle heel one Thermal reservoir Together. The number of systems that have been proved to traverse is small; Yakov Sinai Of Hard ball system It is a famous example. Let the isolation system ensure that Ergodicity ——Therefore, the demand of the equipartition theorem has been studied, and the research has also promoted dynamic system chaos theory Development of. A chaos hamilton system It is not necessarily an ergodic system, although it is usually accurate enough to assume that it is.

Sometimes energy and no Divided by its various forms, and at this time, the equipartition theorem Microcanonical ensemble no Yes, the coupled harmonic oscillator system is an example often cited in this case. If the system is isolated from the outside world Normal mode The energy of is constant; Energy is not transferred from one mode to another. Therefore, the equipartition theorem is invalid in such a system; The amount of each modal energy is fixed by its initial value. If energy When there is enough nonlinear quantity in the function, the energy may be transferred in the normal mode, making the system ergodic and the equipartition law effective. However, Kolmogorov Arnold Moser theorem It is clearly pointed out that unless the disturbance is strong enough, the energy will not be exchanged; If the disturbance is small, the minimum energy will continue to be trapped in some modes.

Failure caused by quantum effect

See: ultraviolet catastrophe 、 History of Quantum Mechanics and Identical particle

When heat energy k B T When the difference between energy levels is much smaller, the equipartition rule will fail. The equipartition is no longer true at this time, because the energy level composition is smooth Continuous energy spectrum This assumption is not similar to the actual situation, and this assumption It is useful in the derivation of the equipartition theorem above 。 Historically, the classical equipartition theorem is used to explain specific heat and Blackbody radiation The failure of quantum mechanics and Quantum field theory ）Played a key role.^[3]

Equipartition theorem of energy

catalog

brief introduction

Basic concepts and simple examples

Specific heat capacity of solid

history

limit

Ergodic requirements

Failure caused by quantum effect

See also