Editorial Feature

Irreversible Thermodynamic Equilibrium in 1D Crystal Lattices

Crystal dynamics were developed to account for crystal structural characteristics, interatomic potential models, crystal defects, nonlinear effects, and so on. A system of noninteracting oscillators is the same as a system of interacting particles. In crystal dynamics, there are hardly any works on the development of the retardation effect of interaction. A paper in the journal Quantum Reports hopes to change this.

crystals, crystal, oscillators, crystal defects, nonlinear effects

Image Credit: Kolbakova Olga/Shutterstock.com

There is an irreversible flow of energy from particles to the field through which particles interact in all nonlinear oscillator models. The phenomenon of irreversibility is driven by field retardation of interactions, which expresses itself in a transition in the spectrum of elementary excitations in crystals.

Equations of particle dynamics and field dynamics describe the laws of thermodynamics. These equations are invariant concerning time reversal, but they feature a character that differentiates them apart from Newtonian laws of classical mechanics. The study of the method for establishing thermodynamic equilibrium in a system of particles is concerned with the dynamics of systems of particles having retarded interactions between them.

New research aims to find a microscopic dynamic substantiation of the process for attaining thermodynamic equilibrium in a system of particles by studying the effect of interatomic interaction retardation on the dynamics of a one-dimensional crystal lattice.


The investigations revealed that delaying particle interactions induces a significant restructuring of the dynamics of a one-dimensional harmonic chain. Stationary free oscillations in the chain are difficult due to the retardation of interactions.

Because the existence of free oscillations of increasing amplitudes leads to the chain’s annihilation, a criterion for the absence of increasing oscillations in the system has been established. This criterion is a requisite for the chain’s stability.

The researchers also noted that when a stable chain of particles with retarded interactions is immersed in an alternating external field, the system reaches a stationary state that is dependent on both the system’s properties and the external field’s characteristics. A dynamic equilibrium between a chain and an external field was described as this stationary condition.

As a result, the following phenomenon—the presence of thermodynamic equilibrium and the phenomena of irreversibility—occurred within the dynamics of a one-dimensional crystal lattice with retarded particle interactions.

Two basic physical laws may be used to validate, explain, and define the zeroth law of thermodynamics: the field nature of particle interaction and the principle of causality. As part of the theory of quantum mechanics, both of these phenomena are postulated in phenomenological thermodynamics and statistical mechanics.


The essence of the potential energy of particle interactions remained a type of “thing in itself” within the framework of pre-relativistic physics. It was established that the potential energy of interacting bodies is equal to the kinetic energy of hidden particles.

An instantaneous interaction of particles separated by a particle distance from one another is impossible in relativistic physics. As a result, the dynamics of a system of atoms can only be described using the Hamiltonian of a system of particles in the non-relativistic approximation. The field operates as a mediator in particle interactions.

A proper explanation of the dynamics of a system of particles interacting via the field must include the field’s dynamics.

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The article constructs a conventional relativistic dynamic theory of a system of point charges, which interacts through the electromagnetic field they create. The dynamics of such a system are characterized by microscopic (non-averaged) distribution functions. Because the advanced electromagnetic fields were excluded according to the concept of causality, this system of equations is not invariant when time is reversed.

Energy dissipation in classical electrodynamics is qualitatively comparable to the process of lattice oscillation damping with retarded interactions (dipole radiation). The physical mechanics underlying these phenomena, however, are different.

A nonlocal effect occurs in crystal dynamics because each lattice particle interacts with the field formed by all other particles, taking into consideration the retardation. In dipole radiation, on the other hand, a local effect is created by the accelerated movement of charges in their own electromagnetic field, which is known as self-action.


The behavior of the system under investigation in relativistic physics was different from that of the equivalent system in pre-relativistic physics, where an isolated chain of atoms oscillated continuously. Both irreversibility and a state of dynamic equilibrium occur in relativistic physics. In pre-relativistic physics, there is no such phenomenon as one or the other.

Journal Reference:

Zakharov, A. Y., & Zakharov, M. A. (2021) Microscopic Dynamic Mechanism of Irreversible Thermodynamic Equilibration of Crystals. Quantum Reports, 3(4), pp.724–730. Available Online: https://www.mdpi.com/2624-960X/3/4/45/htm

References and Further Reading

  1. Born, M & von Karman, T (1912) Über Schwingungen im Raumgittern. Physikalische Zeitschrift, 13, pp. 297–309.
  2. Born, M & Huang, K (1954) Dynamical Theory of Crystal Lattices; Clarendon Press: Oxford, UK.
  3. Brillouin, L (1953) Wave Propagation in Periodic Structures: Electric Filters and Crystal Lattices; Dover Publications: Mineola, NY, USA.
  4. Maradudin, A. A., et al. (1971) Theory of Lattice Dynamics in the Harmonic Approximation; Academic Press: New York, NY, USA.
  5. Kosevich, A M (2005) The Crystal Lattice: Phonons, Solitons, Dislocations, Superlattices; Wiley–VCH Verlag: Weinheim, Germany.
  6. Lamb, H (1900) On a Peculiarity of the Wave-System due to the Free Vibrations of a Nucleus in an Extended Medium. Proceedings of the London Mathematical Society, s1-32, pp. 208–211. doi.org/10.1112/plms/s1-32.1.208.
  7. Love, A E H (1905) Some Illustrations of Modes of Decay of Vibratory Motions. Proceedings of the London Mathematical Society, s2-2, pp. 88–113. doi.org/10.1112/plms/s2-2.1.88.
  8. Jahan, A (2021) The Lamb Problem with a Nonuniform String. Physics Letters A, 392, p. 127133. doi.org/10.1016/j.physleta.2020.127133.
  9. Jahan, A (2021) The Lamb Problem with a Nonuniform String II: Perturbative Solutions. Physics Letters A, 400, p. 127320. doi.org/10.1016/j.physleta.2021.127320.
  10. Synge, J L (1940) The Electromagnetic Two-Body Problem. Proceedings of the Royal Society A, 177, pp. 118–139. doi.org/10.1098/rspa.1940.0114.
  11. Driver, R D (1963) A Two-Body Problem of Classical Electrodynamics: The One-Dimensional Case. Annals of Physics, 21, pp. 122–142. doi.org/10.1016/0003-4916(63)90227-6.
  12. Hsing, D K (1977) Existence and Uniqueness Theorem for the One-Dimensional Backwards Two-Body Problem of Electrodynamics. Physical Review D, 16, pp. 974–982. https://doi.org/10.1103/PhysRevD.16.974.
  13. Hoag, J T & Driver, R D (1990) A Delayed-Advanced Model for the Electrodynamics Two-Body Problem. Nonlinear Analysis: Theory, Methods & Applications, 15, pp. 165–184. doi.org/10.1016/0362-546X(90)90120-6.
  14. Zakharov, AY (2019) On Physical Principles and Mathematical Mechanisms of the Phenomenon of Irreversibility. Physica A: Statistical Mechanics and its Applications, 525, pp. 1289–1295. doi.org/10.1016/j.physa.2019.04.047.
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  26. Zakharov, A Y (2019) On the Probability-Free Mechanism of Macroscopic Irreversibility and Microscopic Foundation of Thermodynamics. Physics of the Solid State, 61, pp. 2442–2445. doi.org/10.1134/S1063783419120606.
Skyla Baily

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Skyla Baily

Skyla graduated from the University of Manchester with a BSocSc Hons in Social Anthropology. During her studies, Skyla worked as a research assistant, collaborating with a team of academics, and won a social engagement prize for her dissertation. With prior experience in writing and editing, Skyla joined the editorial team at AZoNetwork in the year after her graduation. Outside of work, Skyla’s interests include snowboarding, in which she used to compete internationally, and spending time discovering the bars, restaurants and activities Manchester has to offer!


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